Monograph on the occasion of 100th birthday anniversary of Zygmunt Zahorski 978-83-7880-206-8, 837880206X

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Monograph on the occasion of 100th birthday anniversary of Zygmunt Zahorski
 978-83-7880-206-8, 837880206X

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Monografie Wydział Matematyki Stosowanej Politechniki Śląskiej

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Monografie

Politechnika Śląska Wydział Matematyki Stosowanej Instytut Matematyki

Monograph on the Occasion of 100

th

Birthday

Anniversary of Zygmunt Zahorski Edited by Roman WITUŁ A Damian SŁOTA Waldemar HOŁUBOWSKI WYDAWNICTWO POLITECHNIKI ŚLĄSKIEJ GLIWICE 2015 ISBN 978-83-7880-206-8

„Lubię

tylko

trudne

tematy,

łatwe

zostawię

początkującym (którzy zresztą też nieraz zrobili coś trudnego). W każdym razie brać się za trudniejsze od tych, które zrobiłem. Jest to jednak przyjemność (walki z trudnościami) połączona z nieprzyjemnością, gdy się jest całkiem bezradnym.” „I like only difficult topics, the easy ones I leave for the beginners (who has done as well sometime something difficult anyway). In any case, to deal with problems more difficult than the ones that I have done. It is yet a

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pleasure (of struggle with difficulties) joined with an unpleasure when you are completely helpless.” Zygmunt Zahorski

Preface p. 7 PDF

Part I Zygmunt ZAHORSKI Zarys biografii pp. 13-26 PDF Zygmunt ZAHORSKI The biographical sketch pp. 27-40 PDF Zygmunt ZAHORSKI List of papers and PhD students pp. 41-44 PDF Edyta HETMANIOK, Mariusz PLESZCZYŃ

SKI and Roman WITUŁ A

Selected scientific achievements of Professor Zygmunt Zahorski pp. 45-50 PDF Bronisław SZLĘ

K

Professor Zygmunt Zahorski - a memoir pp. 51-52 PDF Henryk FAST On passing away of Zygmunt Zahorski pp. 53-54 PDF Jerzy MIODUSZEWSKI My memories of Professor Zygmunt Zahorski pp. 55-56 PDF Edyta HETMANIOK, Mariusz PLESZCZYŃSKI and Roman WITUŁ A Gliwice PhD students of Professor Zahorski and scientific backgrounds of their thesis pp. 57-62 PDF

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Monografie Wydział Matematyki Stosowanej Politechniki Śląskiej

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Edyta HETMANIOK, Damian SŁ OTA and Roman WITUŁ A Pál Erdös and Zygmunt Zahorski pp. 63-66 PDF Zygmunt ZAHORSKI and Roman WITUŁA - komentarze i uwagi Profesora Zygmunta Zahorskiego „wykład” o pochodnych - przygotowany do publikacji przez sprawcę zamieszania Romana Witułę pp. 67-80 PDF Zygmunt ZAHORSKI and Roman WITUŁA - comments and remarks Professor Zygmunt Zahorski's "lecture" on derivatives - prepared for publication by Roman Wituła, the turmoil maker pp. 81-96 PDF

Part II Władysław WILCZYŃ

SKI

Zygmunt Zahorski and contemporary real analysis pp. 99-108 PDF Ewa KORCZAK-KUBIAK, Anna LORANTY and Ryszard J. PAWLAK On the topological entropy of discontinuous functions. Strong entropy points and Zahorski classes pp. 109-123 PDF Małgorzata FILIPCZAK and Małgorzata TEREPETA Similarity and topologies generated by iterations of functions pp. 125-140 PDF Jacek HEJDUK, Anna LORANTY and Renata WIERTELAK On density points on the real line with respect to sequences tending to zero pp. 141-154 PDF Ján BORSÍK Maximal classes for some families of Darboux-like and quasicontinuous-like functions pp. 155-168 PDF Roman WITUŁ A Permutations preserving the convergence or the sum of series - a survey pp. 169-190 PDF

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Jerzy MIODUSZEWSKI Urysohn Lemma or Luzin-Menshov Theorem? pp. 191-199 PDF

Part III Andrzej KASPERSKI Approximation in Musielak-Orlicz sequence vector spaces of multifunctions pp. 203-218 PDF Jakub Jan LUDEW Nemytskij operator, Rådström embedding and set-valued functions pp. 219-237 PDF Ewelina MAINKA-NIEMCZYK and Jakub Jan LUDEW Functional equations and Nemytskij operator pp. 239-257 PDF Andrzej STAROSOLSKI Fun with cascades pp. 259-271 PDF Marcin ADAM A fixed point approach to the stability of some functional equation connected with additive and quadratic mappings pp. 273-289 PDF Viktor KULYK and Dariusz PĄCZKO Selected issues in the theory of nonlinear oscillations pp. 291-304 PDF Olga MACEDOŃSKA and Piotr SŁ ANINA On identities satisfied by cancellative semigroups and their groups of fractions pp. 305-312 PDF Piotr SŁANINA On some Möbius transformations generating free semigroup pp. 313-323 PDF Edyta HETMANIOK, Piotr LORENC, Damian SŁ OTA and Roman WITUŁ A Periodic orbits of boundary logistic map and new kind of modified Chebyshev polynomials

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pp. 325-343 PDF Ł ukasz PRZONTKA and Alicja SAMULEWICZ Continua and dimension pp. 345-354 PDF Bożena PIĄTEK A survey of the fixed point property in CAT(κ) spaces pp. 355-364 PDF

Appendix Zygmunt ZAHORSKI 3

Przykład łuku prostego w R z indykatrysą nigdzie niespójną (zerowymiarową) pp. 367-379 PDF

Created by Ewelina Domalik

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Preface

In the year 2014, the 100th anniversary of birthday of a remarkable mathematician, Professor Zygmunt Zahorski, is passing. A hierarchy of the Zahorski classes of sets and functions or proof of the Kolmogorov hypothesis have found a permament place in the history of the greatest mathematical achivements. This extraordinary occasion for memories could not be wasted, especially by us, His former students, fascinated till today by His knowledge and charisma. Such facts and feelings turned into this monograph dedicated to the memory of Professor Zahorski. We have divided the monograph into three parts. The first part contains the biography, memoirs, unique pieces of information and, which is the most important, Professor’s letter on derivatives, not published till now. The second part was created on the basis of papers of our respectable Guests. Subjects of these works refer to Professor’s interests. Finally, in the third part there are included the papers of scientists presently employed in the Institute of Mathematics at Silesian University of Technology – the Institute, in creation of which Professor contributed and the unquestionable mainstay of which He always was. The monograph is ended by the manuscript of Professor’s paper, not published till now. For the convenience of potential readers rewritten and edited by us, but according to Professor’s remarks. In the course of elaborating this monograph we have discovered a number of new interesting facts about Professor. We are very glad that the remembrance of Professor is still vivid and His works constantly inspire mathematicians for research – we have been convinced of that by participating in the annual International Summer Conference on Real Functions Theory organized for many years by the Slovak Academy of Sciences, University of L´ od´z, L´ od´z University of Technology and Pomeranian University in Slupsk. ´ We declare our sincere gratitude to Professor Janina Sladkowska-Zahorska for her kindness towards our plans of publishing this monograph as well as for giving us the access to the materials. We would like to express our thanks to Professor Wladyslaw Wilczy´ nski for sharing with us the photographs and the copy of Professor Zahorski’s manuscript, as well as for many valuable pieces of information. Furthermore, we are grateful to Anna Grajner, Edyta Hetmaniok, Ewelina Mainka-Niemczyk, Iwona Nowak, Mariusz Pleszczy´ nski and Alicja Wr´ obel for their help in preparing this monograph. We thank very much all the Authors for their creative contribution in preparing this elaboration. We also express our gratitude to the Reviewers for their hard work and critical, but constructive remarks, which were very helpful in maintaining the high scientific level of the monograph. We hope that the monograph will become a source of valuable information and inspiration in the scientific work for the respected Readers.

Roman Witula Damian Slota Waldemar Holubowski

Zarys biografii Zygmunt Zahorski

Urodziłem się 30 kwietnia 1914 r. we wsi Szubina (w pobliżu Kutna i Krośniewic). Ojciec – Stanisław Zahorski, który był tam nauczycielem szkoły podstawowej, zmarł 18 stycznia 1921 r. Ponieważ mieszkanie służbowe należało przekazać następcy, rodzina (matka i młodsza siostra) została bez domu i środków do życia. Starania o rentę wdowią rozbijały się o przeszkody biurokratyczne (a na adwokatów nie było pieniędzy) i trwały siedem lat. Matka była za słaba do pracy fizycznej, a do pracy urzędniczej nie miała wykształcenia. Reszta oszczędności zniknęła w ówczesnej błyskawicznej dewaluacji. W zaistniałej sytuacji matka do roku 1928 korzystała z mieszkania i pomocy kolejno trzech ze swych czterech braci. W lecie 1923 r. i w latach 1924–1926 pracowała jako służąca w majątku „Zameczek” koło Chodcza. Do szkoły podstawowej chodziłem w latach 1924–1929 w Chodczu, Krośniewicach i Pułtusku. Od początku nauki szkolnej zauważyłem, że mam zamiłowanie i zdolności do matematyki, której uczyłem się jako samouk ponad program, a nawet stawiałem już różne problemy, niektóre z nich udało mi się rozwiązać. Były to jednak „odkrycia” dawno znane, pierwszy wynik naprawdę nowy uzyskałem dopiero w wieku lat 23 na ostatnim roku studiów matematycznych. We wrześniu 1928 r. zdałem egzamin i zostałem przyjęty do piątej klasy męskiego gimnazjum humanistycznego im. ks. Piotra Skargi w Pułtusku. W tym okresie mieszkałem u brata matki w leśniczówce na Popławach, 2 km od Pułtuska, następnie przeniosłem się na stancję w Pułtusku w roku 1931. Utrzymywałem się tam z korepetycji, których mi nie brakowało ze względu na to, że cieszyłem się opinią najlepszego matematyka w gimnazjum. W roku 1932 skończyłem gimnazjum i we wrześniu tego roku, bez przygotowania, zdałem z odznaczeniem wstępny egzamin konkursowy na Wydział Mechaniczny Politechniki Warszawskiej. Do studiowania zamierzałem obrać – po zaliczeniu dwóch lat studiów – najbardziej zmatematyzowaną sekcję, którą w tym czasie była sekcja lotnicza. Odznaczonych po egzaminie wstępnym było czterech na 190 przyjętych i na ok. 360 zdających. Obierając studia techniczne nie kierowałem się względami zarobkowymi w przyszłości, lecz niechęcią do zawodu nauczyciela szkoły średniej. Jednakże Jest to przedruk obszernych fragmentów wspomnień profesora Zygmunta Zahorskiego, napisanych przez niego z okazji Jubileuszu 70-lecia urodzin i pierwotnie wydrukowanych w Zeszytach Naukowych Politechniki Śląskiej, ser. Matematyka-Fizyka 48 (1986), 7–25.



R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 13–26. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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właśnie w roku 1932 znacznie podniesiono czesne na wyższych uczelniach, a studenci pierwszego roku nie mieli prawa do odroczeń czesnego ani do stypendiów państwowych. W ten sposób, a nie przez podniesienie wymagań egzaminacyjnych, ówczesne władze przeciwdziałały „nadprodukcji inteligencji”. Czesne na pierwszym roku politechniki wynosiło 320 zł i płatne było w dwóch ratach. Do tego dochodziła opłata za egzamin wstępny w kwocie 40 zł. Była to dla mnie zawrotna kwota. Renta mojej matki w owym czasie wynosiła z początku 80 zł miesięcznie, a później została zmniejszona, w miarę przekraczania 16 lat przez dzieci i w ramach ogólnych obniżek kryzysowooszczędnościowych płac i emerytur. Pod koniec, w roku 1938, kiedy matka zmarła, wynosiła już tylko 30 zł miesięcznie. Sytuacja na rynku korepetycji była wówczas dla mnie dużo gorsza niż w czasach szkoły średniej w Pułtusku. W Warszawie nikt mnie nie znał, a studentów poszukujących korepetycji – jako zarobku – było tysiące, dlatego na pierwszym i drugim roku studiów korepetycji w ogóle nie udzielałem. Z odłożonych pieniędzy, zaledwie 300 zł, starczyło na egzamin wstępny, pierwszą ratę czesnego oraz zapłacenie za pierwszy miesiąc 75 zł za mieszkanie z wyżywieniem w bursie ZNP. Stypendium ZNP, w kwocie 75 zł miesięcznie za pobyt w tej bursie (tj. zwolnienie od płacenia za pobyt), nie zostało mi przyznane. Moje podanie o to stypendium opiniował kierownik bursy, niechętny mi i zawistny nauczyciel podstawówki. Oświadczyłem więc owemu kierownikowi, że będę mieszkał nie płacąc, gdyż moim zdaniem stypendium mi się należało. Pogróżka wyrzucenia siłą była nierealna, bo woźny musiałby się mocować z moimi 32 kolegami, a zakaz wstępu do stołówki nie działał, gdyż kelnerki dawały moją porcję kolegom, a ci przynosili mi ją do pokoju. Nie udało się też kierownikowi wciągnąć mnie w awanturę, w której kierownik miałby świadka, a student nie. Tego typu incydent byłby dla kierownika pretekstem do usunięcia studenta, tym razem na mocy wyroku sądowego. Po dramatycznych staraniach o przesunięcie choćby o miesiąc terminu płatności drugiej raty czesnego, a następnie po rozpaczliwym szukaniu pieniędzy na zapłacenie tej wierzytelności, zdobyłem wymagane 160 zł z jednorazowego zasiłku starostwa w Kutnie, który był właśnie w tej wysokości. Ta resztówka budżetowa starostwa uchroniła odznaczonego – na egzaminach wstępnych – studenta od wyrzucenia ze studiów za nieuiszczenie czesnego. Mieszkać, i to ze darmo, w bursie ZNP nie można było jednak przez całe studia. Na wakacje wyprowadzali się bowiem z niej wszyscy, w tym i ci koledzy, którzy chronili niezamożnego studenta, gdyż w tym okresie czasu przeprowadzano remonty i po wakacjach składano od nowa podania o przyjęcie do bursy, które, zwłaszcza dla mieszkańców z ubiegłego roku, opiniował kierownik. Oczywiście szkoda było czasu na pisanie takiego podania, niewykonalne też było wprowadzenie się na zajęte miejsce. Drugi rok studiów rozpocząłem z 20 zł w kieszeni. Moja sytuacja materialna się wprawdzie nieco poprawiła, gdyż mając dużo dobrych i bardzo dobrych ocen z zimowej i czerwcowej sesji egzaminacyjnej dostałem całkowite odroczenie czesnego, tj. kwoty 290 zł, na „po studiach”, ale w dalszym ciągu nie miałem środków do życia – nie dostałem nawet połówkowego stypendium w wysokości 60 zł miesięcznie. W owym czasie otrzymanie obu tych pomocy na Politechnice Warszawskiej było praktycznie niemożliwe, choć być może jakieś protekcyjne wyjątki się zdarzały. Nie tylko urzędnicy kwestury, ale i władze uczelni traktowały studentów jak „natrętne muchy”, od których należało się opędzać, bo, o zgrozo, zachciało im się studiować i to bez pieniędzy. Oczywiście lepiej było dostać połowę stypendium – bez odroczenia czesnego, które wtedy przy oszczędnym życiu można było zapłacić. Złożyłem też podanie o sty-

Zarys biografii

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pendium miejskie miasta Warszawy, 75 zł/mies., ale data rozpatrywania tych podań przez władze miejskie nie była zgodna z datą początku roku studiów. Posiadane na jesienną sesję egzaminacyjną 20 zł rozdzieliłem tak: 10 zł na mieszkanie, tzw. kątem w kuchni u pewnego fryzjera, co dawało mi też prawo do herbaty trzy razy dziennie; drugie 10 zł przeznaczyłem na suchy chleb w ciągu miesiąca. Po miesiącu właścicielka mieszkania orzekła, że na ogół inni biorę 15 zł za kąt, ale w owym czasie nie miałem już nawet 10 zł. Wtedy postanowiłem zamieszkać w domu noclegowym dla bezdomnych przy ul. Jagiellońskiej na Pradze. W domu tym można było przebywać od godz. 21 do 6 rano, w dzień dom ten musiał być pusty. Bezdomni spali na gołych drewnianych pryczach, na gazetach położonych na podłodze czy też na posadzce – gdy prycze były zajęte. Często spano nawet na schodach – z parteru na pierwsze piętro – gdy zimno wypędziło nocujących pod mostami i w parkach. To „mieszkanie” w brudzie i przenikliwym zimnie trwało przez około 6 tygodni (w listopadzie i grudniu 1933 r.). W początku grudnia 1933 r. dostałem od Bratniaka Politechniki, ogólnostudenckiej organizacji samopomocowej, kilkunastozłotowe stypendium mieszkaniowe w małym domu akademickim przy ul. Ceglanej 1 (duży dom na pl. Narutowicza Ministerstwo Oświaty Bratniakom odebrało) i zamieszkałem wreszcie w ludzkich warunkach. Nieco wcześniej dostałem również od Bratniaka stypendium obiadowe, 75-groszowe bloczki na obiady w studenckiej stołówce przy ul. Koszykowej, w tym czasie pomogli mi też byli współlokatorzy z bursy i dr J. Stawiński. W lutym 1934 r. magistrat Warszawy zawiadomił mnie o przyznaniu mi stypendium miejskiego, i to płatnego od września 1933 r. wstecz. Wtedy zrzekłem się stypendiów obiadowego i mieszkaniowego, mogłem też znaczną część pieniędzy posłać matce mieszkającej w Krośniewicach. Stypendium miejskie miało jednak tę wadę, że było jednoroczne. W następnym roku ta pozycja budżetu miejskiego już nie istniała, więc nie dostał go nikt ze studentów. Przy okazji należy sprostować mylną informacje o mnie w książce wspomnieniowej „Czas przed burzą”, Nasza Księgarnia, Warszawa 1973, strony 118, 163, 164, 170, 215, napisanej przez mego nieco starszego kolegę z Politechniki Warszawskiej. Autorem jej jest dr inż. S. Minorski. W książce tej, miejscami humorystycznej, niektóre fakty są prawdziwe, ale wydarzenie najważniejsze, to o domu noclegowym, zostało przez autora opuszczone. Pisał ją bowiem wiele lat po wojnie, więc widocznie zawiodła go pamięć. Dodatkowo miejscami myli mnie z innymi osobami, a znaczna część tych informacji jest błędna. Rzeczą poglądów subiektywnych jej autora jest jego nieco negatywny stosunek do wielkiego filozofa i wzorowego człowieka, prof. T. Kotarbińskiego, ale informacja o stypendiach na matematyce w Uniwersytecie Warszawskim, więc na terenie nieznanym bezpośrednio autorowi, natomiast znanym mnie, jest całkowicie fałszywa. Autor tej książki zmarł w czerwcu 1981 r. i nie wiadomo, czy będą dalsze jej wydania, wobec czego przytoczyłem wyżej częściowe sprostowanie informacji podanych w tej książce, z braku innej możliwości polemizowania z opisem autora. Przedmioty czysto techniczne, takie jak części maszyn czy odlewnictwo, nie interesowały mnie, natomiast wykłady z matematyki, zwłaszcza niektórych jej działów, które budziły moje zainteresowanie, na Politechnice Warszawskiej były za skromne – bardzo mało algebry, zupełny brak teorii liczb i teorii mnogości, przedmiotów istotnie wówczas inżynierom niepotrzebnych.

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Kierując się własnymi zainteresowaniami, w jesieni 1934 r., kiedy zmiana ustawy pozwoliła na studia na dwóch uczelniach, za zgodą rektorów zapisałem się na matematykę na Uniwersytecie Warszawskim. Będąc studentem Uniwersytetu nie opuszczałem jeszcze Wydziału Mechanicznego Politechniki Warszawskiej, na którym miałem zaliczone niemal całe pierwsze dwa lata studiów, łącznie z częścią trzymiesięcznej praktyki fabrycznej. Na Uniwersytecie trafiłem na zupełnie inny stosunek wykładających do studentów. Profesorowie Sierpiński, Mazurkiewicz, Kuratowski byli to wielcy uczeni, którzy popierali ludzi uzdolnionych matematycznie. Dziekan Mazurkiewicz, gdy został zaczepiony na dziedzińcu Uniwersytetu po skończonym urzędowaniu, bez ważniactwa, jak przystało na prawdziwego uczonego, nie dał wymijającej odpowiedzi, że przyjmuje tylko w godzinach urzędowych w kolejności zapisanych u sekretarki, tylko z miejsca obiecał mi obejście przepisów o studentach pierwszego roku z racji studiów na Politechnice i przyrzekł pełne odroczenie czesnego oraz pół stypendium, tj. 60 zł/mies., na pierwszym roku. Po bardzo dobrze zdanych egzaminach sesji zimowej i letniej 1935 r. do końca studiów uniwersyteckich korzystałem już z pełnego stypendium w wysokości 120 zł/mies. i pełnego odroczenia czesnego. Były to dla jednego człowieka warunki wręcz luksusowe, lecz zostały one nieco pogorszone, gdyż około połowy uzyskanych w ten sposób pieniędzy posyłałem matce. Pełnych stypendiów było dwa na wszystkich czterech latach na całym Wydziale (matematyka, fizyka, astronomia, chemia, biologia i geografia), natomiast połówkowych było dość dużo. Drugie stypendium miał też student matematyki, ale nie ten odpowiadający opisowi S. Minorskiego, gdyż był on nie z ONR-ru, lecz z Legionu Młodych, organizacji sanacyjnej, co oczywiście było pomagającą okolicznością, jednak „pozanaukową”, aczkolwiek nie jedyną. Był to istotnie zdolny matematyk, lecz nie tej klasy co ja. Najzdolniejszy matematyk, studiujący w owym czasie na Wydziale, stypendium nie potrzebował, gdyż był zamożny i nie spieszył się z zaliczeniem lat studiów. Pozwoliło mu to już na I lub II roku studiów pisać prace, w których uzyskiwał nowe wyniki. Studia matematyczne trwały formalnie 4 lata, faktycznie na ogół dłużej. Ja skończyłem je w ciągu 3 i pół roku w 1938 r. Na Politechnice w tym czasie studiowałem w zwolnionym tempie, zaliczając jednak – jak wspomniałem – całe pierwsza dwa lata, łącznie z 3-miesięczną praktyką fabryczną. Zapisałem się na sekcję lotniczą, zaliczając niektóre przedmioty trzeciego roku studiów. Było to nieco więcej niż tzw. umownie półdyplom, urzędowo – pierwszy egzamin dyplomowy, choć faktycznie żadnego osobnego egzaminu nie było. Przy tym nie miałem już zamiaru kończyć Politechniki, na której studia były mi potrzebne tylko do uzyskania dłuższego niż z Uniwersytetu odroczenia ze służby wojskowej, tj. do 26 roku życia, nie do 22 jak z innych uczelni. Nie unikając w zasadzie służby wojskowej nie chciałem jednak, aby przerwała mi ona studia matematyczne. W tym czasie zmieniłem znacznie kierunek specjalności: nie aerodynamika, więc i nie równania różniczkowe o pochodnych cząstkowych, lecz teoria funkcji rzeczywistych, trochę teoria funkcji analitycznych (której fragmenty były też w programie sekcji lotniczej) i szeregi trygonometryczne stały się domeną moich zainteresowań. W roku 1937 na życzenie prof. Mazurkiewicza wydrukowałem pierwszą krótką pracę, bez wyniku nowego, ale z nową metodą konstrukcji pewnego rodzaju funkcji. Robił to już m.in. Mazurkiewicz 22 lata wcześniej, ale w sposób dużo bardziej skomplikowany. W tymże roku uzyskałem pierwszy wynik nowy, trudny i znacznej wagi, który został uzupełniony w roku 1938. Miał on być pracą doktorską wykonywaną u Mazurkiewicza, a zakończenie przewodu było zaplanowana na wrzesień 1939 r.

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Oczywiście sfinalizowaniu tych planów przeszkodziła wojna. Po przekładzie tej pracy na język francuski i wysłaniu jej do druku w Bulletin de la Societ´e Math´ematique de France w lecie 1939 r. Francuskie Towarzystwo Matematyczne (Soc. Math. de France) zaproponowało mi, abym się do niego zapisał, co też uczyniłem wpłacając niskie zresztą wpisowe (11 zł). Zapewne wkrótce zostałem skreślony z listy członków wobec niemożności przesyłania – w czasie wojny – równie niskiej składki. Nie wiadomo, czy gdzieś pozostał ślad tego zapisu do Soc. Math. de France. Dzieje rękopisu omawianej pracy były pechowe. Choć okupacja we Francji była dużo łagodniejsza niż w Polsce i wychodziły tam czasopisma naukowe, rękopis przeleżał całą wojnę – poza redakcją – i został wydrukowany w Bull. de la Soc. Math. de France dopiero w roku 1946. Jednakże wcześniej, w 1941 r., praca ta została wydrukowana w Moskwie, po rosyjsku, i to w poszerzonej nieco formie. W 1937 r. zacząłem pracować jako asystent dra inż. S. Neumarka, dra inż. J. Bondera i dra P. Szymańskiego w Szkole Podchorążych Lotnictwa (grupa Techniczna) w Warszawie przy mechanice teoretycznej, wytrzymałości materiałów i matematyce, z programem nieco niższym niż na politechnikach. Posadę tę zawdzięczałem doktorowi A. Wundheilerowi, wybitnemu twórcy w geometrii różniczkowej, którego dr Neumark chciał zatrudnić jako asystenta. Wundheiler prowadził na Uniwersytecie Warszawskim ćwiczenia z mechaniki u prof. Przeborskiego, nie miał jednak wtedy szans na zatrudnienie w szkole wojskowej z powodów „personalnych”, które wprawdzie dotyczyły i dra Neumarka, ale on pracował już od dawna w Instytucie Lotnictwa i w wojsku oraz w Instytucie Aerodynamicznym Politechniki. Jednakże Neumark jak i Bonder, choć „syjoniści”, byli nie do zastąpienia jako inżynierowie, a zarazem i matematycy. Wobec tego Wundheiler polecił Neumarkowi mnie, jako swego najlepszego studenta znanego mu z ćwiczeń z mechaniki teoretycznej. Pracowałem w tej szkole aż do czasu jej ewakuacji w nieznanym kierunku we wrześniu 1939 r. Jednocześnie w marcu 1939 r. skończyły się wszystkie odroczenia i dostałem wezwania do stawienia się 2 października 1939 r. w Szkole Podchorążych Piechoty w Zambrowie, w celu odbycia służby wojskowej. Dnia 7 września 1939 r., gdy Niemcy zbliżali się do Warszawy, a pułk. Umiastowski przez radio wezwał mężczyzn mających kategorię wojskową A do wyjścia z Warszawy na wschód, gdzie będą wzięci do wojska, wyszedłem na wschód z Esterą Steinbok, absolwentką matematyki. W Siedlcach dowódca miejscowej jednostki wojskowej, po okazaniu wezwania do Podchorążówki, odmówił przyjęcia mnie do wojska jako nieprzeszkolonego. Wówczas jako uchodźcy udaliśmy się do Brześcia, gdzie po silnym nalocie i zbliżeniu się Niemców do Wysokiego Mazowieckiego zrozumieliśmy, że Polska wojnę na razie przegrała. Postanowiliśmy więc uciec jak najbliżej granicy radzieckiej, aby, gdy podejdą Niemcy i odejdą jednostki graniczne KOP, przejść przez granicę i schronić się przed Niemcami w ZSRR. Dnia 14 września byliśmy w Łunińcu, 60 km od granicy, ale jej przekraczanie nie było potrzebne, gdyż 18 września armia radziecka weszła do Łunińca. Wkrótce udałem się do Lwowa, gdzie czynne były wyższe uczelnie. Estera Steinbok chwilowo została w Łunińcu. We Lwowie zostałem asystentem S. Banacha na Uniwersytecie Lwowskim, a od marca 1941 r. aspirantem (wraz z nieco młodszym od siebie mgr A. Alexiewiczem, później profesorem Uniwersytetu im. A. Mickiewicza w Poznaniu). Z aspiranturą wiązało się też stypendium doktoranckie. Banach zgodził się przyjąć jako doktorską tę

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pracę, którą przyjął jako doktorską Mazurkiewicz, ale wtedy jeszcze nie była rozstrzygnięta przez ministerstwo w Kijowie kwestia, czy Uniwersytet Lwowski będzie miał prawo nadawania stopni doktorskich (w tamtejszej terminologii – kandydackich). We Lwowie pracowało na Uniwersytecie w owym czasie wielu uchodźców, wymienię tutaj Saksa, Knastera, Marczewskiego1, Wojdysławskiego oraz mnie, także prof. Orlicza z Poznania, Boy-Żeleńskiego. W listopadzie 1940 r. ożeniłem się z Esterą Steinbok. W tym okresie wciągnąłem się w wir pracy naukowej, byłem bardzo zajęty, redagowałem swoje prace już zrobione i przystępowałem do pisania nowych, aby ocalały w przypadku śmierci na wojnie. Zredagowane prace wysyłałem do Moskwy, Japonii, czasopism Lwowskich – te ostatnie prace później zaginęły, ale po ponownym ich napisaniu, po wojnie, zostały wydrukowane w Indiach i USA. Nowe problemy pochodziły częściowo od Banacha i Mazura (jednak nie odnosiły się one do analizy funkcjonalnej), częściowo z własnej inicjatywy. Moje obawy, że pożoga wojenna w taki czy inny sposób dotrze do Lwowa, okazały się słuszne. 30 czerwca 1941 r. Niemcy wkroczyli do Lwowa. Dalsza ucieczka na wschód była technicznie niemożliwa od pierwszego dnia wojny, tj. od 22 czerwca. Powstał też dodatkowy kłopot – należało zmienić mieszkanie, gdyż w książce meldunkowej moja żona figurowała jako Żydówka, a na razie korzystała z bardzo kruchego fałszywego dokumentu. Uczelnie oczywiście zamknięto (nawet dla Ukraińców), zresztą nie tylko nie pracowałbym na hitlerowskiej uczelni, ale nawet nie mógłbym się przyznać do wyższego wykształcenia wiedząc, że daje to pierwszeństwo w wywozie do obozu koncentracyjnego. Warunki odżywiania znacznie się pogorszyły, a dorywcze zarobki, głównie handlowe, były bardzo małe. Pogorszyło się więc po raz pierwszy moje zdrowie (gruźlica), a syn urodzony 25 sierpnia 1941 r. zmarł 27 września tegoż roku z braku należytego odżywiania. W marcu 1942 r. wróciłem do Warszawy, a w maju 1942 r. dr Stawiński skierował mnie do znajomego dyrektora technicznego fabryki Philipsa, z zatajeniem wyższego wykształcenia. Fabryka była skonfiskowana przez Niemców i dawała tzw. ausweis, chroniący częściowo przed wywózką do Niemiec. Dawała też zupę fabryczną rano i na obiad, dość podłą, ale czasem z jakimiś ochłapami mięsa jakości gorszej niż „nur fur Deutsche”. Praca tam dawała też możliwość słuchania, bez żadnego ryzyka, zabronionych pod karą śmierci (m.in. według afiszów wiszących w halach fabryki) audycji z Londynu, Moskwy, a czasem przypadkowo i z krótkofalówek partyzanckich. Przy ówczesnych środkach technicznych nie było możliwe stwierdzić, czego się słucha w słuchawkach, a skalujący aparat według falomierza służbowo musiał mieć słuchawki na uszach. Należało tylko, gdy jakiś Niemiec wchodził na salę, spokojnie zejść z podejrzanej fali, odłączyć zwykły drut o charakterze wewnętrznej anteny, a podłączyć leżący o 5 cm obok przewód falomierza. Wyrobiłem żonie Esterze bardzo mocną aryjską kennkartę na prawdziwej metryce mojej siostry, Heleny Barbary Zahorskiej, która wtedy mieszkała na terenie tzw. Rzeszy. Siostra po wojnie uznała to za bardzo ryzykowne, mogące ściągnąć wówczas śmierć na nią, jak i na całą rodzinę, u której mieszkała. Jednakowoż, wobec trudności podróży Polaków z Rzeszy do Generalnej Guberni, znalezienie dwóch bliźniaczek Zahorskich o identycznych obu imionach było mało prawdopodobne. W tym okresie pomagałem też innym Żydom, czasem nieznajomym dzieciom, ale to ostatnie nie było ryzykowne. Nie udało się natomiast wykupić od granatowej policji żony wybitnego matematyka lwowskie1

Edward Szpilrajn ukrywał się z początku pod nazwiskiem Zawadzki, później Marczewski.

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go, J. Schaudera, zastrzelonego na ulicy we Lwowie2 . Pani Schauderowa przyjechała wtedy ukrywać się w Warszawie, ale, jak i jej mąż, nie mogła wytrzymać stałego pobytu w mieszkaniu. Na ulicy rozpoznały ją dzieci i krzyczały „Żydówka”, nie wiedząc zapewne co robią, a wtedy zatrzymał ją granatowy policjant. Druga Żydówka ukrywana przez tę samą gospodynię, znająca Esterę, zaalarmowała nas. Gospodyni ta miała dojście do jakiegoś z policjantów, a ja byłem jednym z ogniw pośrednich między tą gospodynią a panią Tarską, zbierającą pieniądze na wykup. Bezpośredniego kontaktu ze sferami policyjnymi, nawet polskimi, wolałem unikać. Policjanci jednak dwukrotnie, po zebraniu żądanej kwoty, podbijali cenę, za trzecim razem pani Schauderowej już nie było. Gestapowcy zabrali wszystkich zatrzymanych Żydów z okazji kolejnej wizytacji komisariatu, a do gestapowców ani pani Tarska ani owa gospodyni dojścia już nie miały. Ponieważ w fabrykach niemieckich należało pracować ślamazarnie, w znacznej części godzin pracy fabrycznej pisałem dalsze prace. Chociaż prace te później zginęły w gruzach powstania, niemniej jednak ich streszczenia (w kilku egzemplarzach, dane różnym ludziom, a jedno trzymane stale przy sobie) ocalały, co pozwoliło je po wojnie odtworzyć. W latach 1942 i 1943 przebywałem po 2–3 miesiące w sanatorium – w Rudce pod Mińskiem Mazowieckim, postawionym na dobrym poziomie jak na czasy okupacji. W sanatorium tym zostało ufundowanych 12 miejsc dla pacjentów z fabryki przez jej polski personel (składkowo). Miałem znajomych partyzantów zarówno z AK (kolegę ze swej klasy maturalnej), jak i z AL (Michała Tetmajera, nieznanego mi osobiście przed wojną, spotkanego tylko raz na zebraniu towarzyskim) i zamierzałem się do nich przyłączyć. Jednak na dłuższy pobyt w warunkach partyzanckich zdrowie już mi nie pozwalało. Nastąpiło dalsze pogorszenie zdrowia, którego stan był alarmujący pod koniec 1943 r. W styczniu 1944 r. znalazłem się w szpitalu, a po operacji, w lutym, stan mój się pogorszył, a w maju był całkowicie beznadziejny. Jako pesymista „dla asekuracji” liczyłem wtedy najwyżej na miesiąc życia. Okazało się, że nieobiektywnie i że za mało pesymistycznie: lekarze bowiem liczyli wtedy, że pożyję tylko kilka dni. Ale mylili się, nastąpiła powolna poprawa, tak iż w czerwcu mogłem już chodzić bez prowadzenia. Niewątpliwie pomogło mi dożywianie przesyłane przez kuzynkę Jankę Wł. z Siedlec, ale głównie jakaś niespodziewana odporność, bo inni pacjenci „poczekalni do trupiarni”, zwłaszcza ci, którzy dostawali ze wsi i od rodzin znacznie lepsze dożywianie, pomarli wszyscy – ja pozostałem przy życiu . Przez czas okupacji salę tę opuściło dwóch żywych – jednym byłem ja, a drugim (wcześniej) był jakiś marynarz. Przejściowe pogorszenie stanu zdrowia nastąpiło później, w czasie Powstania Warszawskiego. Jednak w tym szpitalu (przy ul. Nowogrodzkiej, Szpital Dzieciątka Jezus), prawie od początku zajętym przez Niemców, warunki były znacznie lepsze niż w innych dzielnicach. Niemcy nie mordowali chorych ani nawet personelu, jak w Szpitalu Wolskim czy na Lesznie, gdzie chodzących chorych rozstrzeliwano na miejscu, a niechodzących spalono w budynku; wśród tych ostatnich był teść prof. Borsuka. Trzy tygodnie po powstaniu Rada Główna Opiekuńcza szpital ten oraz chorych, rannych z powstania i starców z innych szpitali i przytułków, ewakuowała do prowizorycznego szpitala w Domu Medyków przy ul. Grzegórzeckiej w Krakowie. Z tego nowego miejsca wypisać się wolno było tylko po wyrażeniu zgody przez gestapo. Jednak w szpitalu warty nie postawili. Skorzystałem z tej „swobody”, odszukałem matematyków krakowskich i na tajnym zebraniu oddziału Polskiego Towarzystwa Matematycznego zreferowałem im swoje prace napisane w czasie wojny. Wkrótce zostałem powiadomiony, że tajna 2

Czy w Drohobyczu? Dokładnie nie wiem. Zadenuncjował go podobno student Ukrainiec.

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Z. Zahorski

Polska Akademia Umiejętności zorganizuje mi pomoc żywnościową. Mięso dostarczył prof. S. Turski, były więzień obozu koncentracyjnego w Sachsenhausen, wtedy pracownik rzeźni, po wojnie organizator i pierwszy rektor Politechniki Gdańskiej, później – Uniwersytetu Warszawskiego, przez pewien czas Dyrektor Departamentu w Ministerstwie. W styczniu 1945 r. Niemcy zamierzali pacjentów Szpitala z Grzegórzeckiej wywieźć do obozu Oświęcim-Brzezinka, ale nie zdążyli tego uczynić. Pacjentów tego szpitala, i wielu innych ludzi, uratowała błyskawiczna ofensywa radziecka, zaczęta 12 stycznia od Sandomierza. Po tej dacie zdążyli rozstrzelać tylko tych ludzi, którzy znajdowali się w więzieniu, po czym szybko uciekali przed okrążeniem. Już 18–19 stycznia Kraków był zdobyty prawie bez walk, po dalekim okrążeniu. Poprawiły się warunki odżywiania i szybko poprawiło się moje zdrowie. W lutym 1945 r. stanąłem przed komisją poborową. Kapitan lekarz sprawdził, czy bandaż na żebrach nie jest fikcyjny, po czym bez rozmowy wypisał zwolnienie. Od chwili otwarcia Uniwersytetu Jagiellońskiego pracowałem na nim jako asystent. W czerwcu 1945 r. dostałem małe mieszkanie przy ul. Podwale. Dnia 11 lutego 1946 r. zakończyłem przewód doktorski z innej pracy niż przedwojenna, a ta ostatnia, znacznie uzupełniona, była podstawą habilitacji, planowanej i przygotowanej na 15 października 1947 r. Brak jednak czasu u jednego z recenzentów, prof. Mazura, spowodował opóźnienie i przesunięcie kolokwium habilitacyjnego na grudzień 1947 r. Jednocześnie Uniwersytet Jagielloński wystąpił z wnioskiem o powołanie na profesora nadzwyczajnego i zatrudnił mnie jako zastępcę profesora. W październiku 1948 r. dostałem nominację na profesora nadzwyczajnego i przeniesienie na Uniwersytet Łódzki. Dnia 1 marca 1949 r. Helena (Estera) Zahorska urodziła córkę Elżbietę (zachowała też pierwsze imię z okupacyjnej kennkarty). W roku 1949 dostałem nagrodę Polskiego Towarzystwo Matematycznego3, jako dwunasty z matematyków polskich. Te nagrody, wtedy od roku 1946 po trzy rocznie na całą Polskę, były najwyżej cenione, nie co do kwoty, lecz co do znaczenia. Tylko dwóch matematyków dostało ją dwa razy: H. Steinhaus (za życia) i M. Biernacki (drugą pośmiertnie, ale wniosek mógł być wystawiony tylko za życia). Nagrody Ministra dostałem trzy razy, w 1948 r. (nagroda młodych, tj. do 40 lat), w 1962 r. (nagroda II stopnia za pracę naukową) i w roku 1984 za całokształt działalności, w tym w szczególności za badania naukowe i rozwój młodej kadry. Pomijam tutaj kilka nagród rektorskich w Łodzi (i jedną w Gliwicach) za pracę naukową lub dydaktyczną. W roku 1954 Rada Wydziału Matematyki, Fizyki i Chemii Uniwersytetu Łódzkiego wystąpiła jednomyślnie o awansowanie mnie na profesora zwyczajnego, co udaremniłem, nie składając niezbędnej ankiety, gdyż uważałem, że jeszcze mi się ten awans nie należy. Zgodziłem się na awans dopiero w 1960 r., po rozwiązaniu problemu Kołmogorowa (dowód pewnej hipotezy z roku 1926, którą Kołmogorow opublikował w roku 1927 bez dowodu i bez żadnych szczegółów). Był to istotnie wynik na skalę światową. Od roku 1927 na całym świecie wielu wybitnych specjalistów z teorii szeregów trygonometrycznych próbowało bezskutecznie rozwiązać ten problem, udało mi się to w maju 1960 r., po trzech tygodniach pracy. Pewne ułatwienie w uzyskaniu tego wyniku stanowiła moja bezskuteczna praca od roku 1940, a nawet trochę wcześniej, bo od 1936 r., zaś intensywnie od 1942 r., nad inną hipotezą Łuzina. Niemal wyłącznym, a na pewno głównym, celem mego życia były badania nad hipotezą Łuzina postawioną w roku 1912. Hipoteza ta jest uogólnieniem problemu du Bois Reymonda z roku 1876, a po3

Nagroda im. Stanisława Zaremby (red.).

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dobno nawet Riemanna, a więc sprzed roku 1866. Pracowałem nad nią do reanimacji na jesieni 1980 r. Hipoteza Łuzina, też z teorii szeregów trygonometrycznych, ale jeszcze trudniejsza i starsza, miała pewne aspekty, które powodowały, że pomysły dla niej nieskuteczne okazały się skuteczne w hipotezie Kołmogorowa. W roku 1961 pospiesznie, przed zredagowaniem czystopisu, anonsowałem w C.R. Akademii Paryskiej rozwiązanie hipotezy Łuzina, a w trzy tygodnia później, redagując czystopis i referaty na seminarium prof. Mazura w Instytucie Matematycznym PAN, zauważyłem błąd. Ponieważ anons był już wydrukowany, sprostowałem natychmiast błąd w liście do prof. A. Zygmunda, recenzenta tej noty (z C.R.) w światowym czasopiśmie bibliograficznym Mathematical Reviews, ale naprawić tego błędu nie potrafiłem do roku 1980 ani później. Błąd ten psychologicznie stanowił mocne uderzenie w wiarę we własne siły i stanowił odtąd główne moje nieszczęście, przesłaniające, czy raczej wchłaniające, nawet nieszczęścia wojenne. Obiektywnie nie jest to takie złe, sam Lebesgue ogłosił w 1905 r. pracę z fałszywym wynikiem, a największy chyba matematyk XX wieku, D. Hilbert, w roku 1925 „udowodnił” hipotezę continuum – błąd zauważyli zaraz Kuratowski i von Neumann. Natomiast hipoteza Łuzina okazała się prawdziwą, co udowodnił poprawnie w 1966 r. szwedzki matematyk Lennart Carleson (młodszy ode mnie), później przez parę lat prezydent światowej Unii Matematycznej. Błąd mój tkwił więc tylko w dowodzie, jednak prawdziwość tej hipotezy nie jest moją zasługą. Bo, nie wchodząc w szczegóły, problem miał dwie możliwe odpowiedzi – tak lub nie. W roku 1961 zostałem wytypowany przez dzielnicowy Front Jedności Narodu na radnego do Dzielnicowej Rady Narodowej (DRN) Łódź-Górna. Propozycję tę przyjąłem mając nadzieję, że będę mógł coś pomóc ludziom, bo, mimo małej ilości czasu wolnego (czas pochłaniała praca naukowa), ówczesny stan zdrowia i energii pozwalały mi na dodatkową działalność. Będąc przeciwnikiem pracy czysto frazeologicznej i uważając, że najlepszym przykładem jest konkretna praca (w tym i naukowa), uważałem pracę radnego za coś konkretnego. Okazało się to słuszne, ale w małym stopniu. W DRN Łódź-Górna byłem radnym i członkiem Komisji Oświaty w latach 1961– 1970, a członkiem Prezydium DRN od 1965 do 1970 r., tj. do czasu przeprowadzki do Gliwic, razem przez dwie kadencje. W latach 1951–1953 i 1959–1961 byłem prezesem Oddziału Łódzkiego Polskiego Towarzystwa Matematycznego, w latach 1975–1977 członkiem Zarządu Głównego PTM, a niemal stale delegatem na Walne Zgromadzenia PTM. W Łodzi na Uniwersytecie zostałem odznaczony Medalem X-lecia PRL i Złotym Krzyżem Zasługi w latach 1953–1954, a w roku 1968 22 lipca (po sprostowaniu błędu w nazwisku, 17 kwietnia 1969) Krzyżem Kawalerskim Orderu Odrodzenia Polski. W przeniesieniu do Gliwic pomogli mi nieżyjący już ludzie: prof. M. Mochnacki i doc. dr hab. W. Sobieszek, były mój student z Uniwersytetu Łódzkiego pracujący w Gliwicach. Mimo wyleczenia, całkowicie dopiero w roku 1964, skutków wojny zdrowiu zagroziła, i to mocniej, choroba serca. Od roku 1955 lekka, głównie nerwicowa, później i organiczna, od 1972 poważna, od 1976 ciężka, a 24 października 1980 z reanimacją. W latach 1977–1978 czyniłem starania o przeniesienie się do Białegostoku, ze względów zdrowotnych i klimatycznych. Starania te nie kończą się sukcesem. Pozostaję nadal w Gliwicach. W roku 1966 moja córka Elżbieta skończyła III Liceum w Łodzi i wybrała kierunek studiów, którego w Łodzi nie było, a w następnym roku zdała, za moją namową, na

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fizykę na Uniwersytecie Warszawskim, choć wolałaby matematykę. Uważałem fizykę za ważniejszą i ciekawszą od matematyki, nie tylko ze względów na zastosowania techniczne, choć sam jestem matematykiem. Fizyka dawałaby też większy wybór zatrudnienia po studiach. Próba studiowania jednocześnie fizyki i matematyki była zbyt czasochłonna, wreszcie córka po dwóch latach fizyki, mimo na ogół dobrych wyników, przeniosła się na matematykę, stwierdzając, że nie ma zdolności doświadczalnych, i w roku 1979 skończyła studia. W roku 1970 rozwiodłem się, po czym ożeniłem się z Janiną Śladkowską, docentem Politechniki Śląskiej, pracującą twórczo w mało mi znanej dziedzinie funkcji analitycznych. Również ze względów rodzinnych i mieszkaniowych przeniosłem się do Gliwic, do pracy w Politechnice Śląskiej, mimo większego na Śląsku zadymienia i zapylenia atmosfery, szkodliwego klimatu. Janina Śladkowska-Zahorska urodziła syna. Syn Jaś gruntownie nie lubi matematyki, być może obrzydziła mu ją szkoła, a zmuszać go nikt nie chce, nigdy przymus nie spowodował zainteresowania. Ma wprawdzie dobre oceny z matematyki, ale interesuje się chemią i elektroniką. Córka często odwiedzała mnie w Gliwicach, a po skończeniu studiów przez osiem miesięcy mieszkała u nas w Gliwicach. Następnie została zatrudniona w WSI w Radomiu. Ja miałem również propozycję przeniesienia się do Radomia ze względów klimatycznych, ale nie chciałem podjąć pospiesznej decyzji, a po zorientowaniu się w tamtejszych warunkach w 1981 r. zrezygnowałem z tej przeprowadzki. W czerwcu 1982 r., w czasie przymusowej miesięcznej nieobecności żony w Gliwicach (ja byłem wtedy w sanatorium kardiologicznym w Nałęczowie), córka zaopiekowała się Jasiem wywożąc go do znajomej gospodyni pod Ciechocinkiem. Mam publikacje w czasopismach naukowych w Polsce, Francji, Japonii, Indiach, USA, ZSRR i Czechosłowacji, jeden skrypt dla początkowych lat studiów i jeden artykuł popularny dla czytelników z wykształceniem od niepełnego podstawowego wzwyż. Artykuł popularny wypadł nie najlepiej, bo nie miałem możności zrobienia korekty przed publikacją, korektę zrobiła redakcja „Przekroju”, zmieniając miejscami na . . . gorzej. Odliczając z publikacji naukowych bibliografię, recenzje, problemy postawione, ale nie rozwiązane przez autora i dublety (np. anons, streszczenie, a później pełny tekst) i przekłady, prac naukowych jest w tym 17, z nich jedna błędna, jedna istotnie dobrej wartości, osiem średniej jakości, pozostałe przyczynkowe. Doktorantów mam ośmiu w Łodzi i dwóch w Gliwicach. Wybitnymi mymi uczniami są prof. zw. dr hab. Jan Lipiński, przez pewien czas dyrektor Instytutu Matematyki Uniwersytetu Gdańskiego i prof. dr hab. Tadeusz Świątkowski, który wprawdzie doktoryzował się w innej katedrze, ale najlepsze prace ma w mojej problematyce. Szkoła stworzona przez ludzi uprawiających tę problematykę wykracza poza granice Polski. Pracują w niej m.in. matematycy czescy, amerykańscy, rumuńscy, radzieccy i inni. Z polskich można wymienić docentów Filipczaka i Wilczyńskiego oraz kilku uczniów prof. Lipińskiego czy prof. Świątkowskiego. Brałem udział w kilku międzynarodowych kongresach matematycznych, skupiających od 4 do 6 tys. matematyków różnych specjalności, organizowanych co 4 lata w różnych krajach przez matematyków kraju organizującego (wybieranego na poprzednim kongresie) i Międzynarodową Unię Matematyczną. W Sztokholmie w roku 1962 i w Moskwie w roku 1966 byłem na koszt własny, w Moskwie organizatorzy obrali mnie przewodniczącym na referatach sekcji funkcji rzeczywistych. Na Kongres w Nicei w roku 1970 byłem wytypowany z Uniwersytetu Łódzkiego przez Ministerstwo, lecz

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z uwagi na „cięcia” finansowe nie mogłem w nim wziąć udziału. W kongresach 1978 r. w Helsinkach i w 1983 r. w Warszawie byłem na koszt Politechniki. W mniejszych konferencjach zagranicznych brałem udział z wygłoszeniem referatów – w Pradze w roku 1949, w Berlinie w 1960 r., w Palermo na Sycylii w 1976 r. i w Warnie (nie referat, tylko problem w czasie dyskusji) w 1967 r.

Podsumowanie i zakończenie tej biografii Oczywiście przypadek mojej fascynacji matematyką nie jest typu ambicjonalnego, bo to mogło odgrywać rolę najwyżej w początkach nauki szkolnej, kiedy zależy na ocenie – a i nawet wtedy nie wyłącznie. Ale rodzi się pytanie: czy ta fascynacja ma stanowić końskie okulary, z którymi można dobrze i nawet przyjemnie – mimo przeszkód – przeżyć życie baz ogólniejszej refleksji? Na pytania, w jakim stosunku jest matematyka do ogólnych celów ludzkości, odpowiadam w skrócie: z ludzkiego punktu widzenia najważniejszymi i najbardziej zadziwiającymi tworami przyrody są organizmy żywe, zwłaszcza „mające duszę”, tj. widzące, słyszące, czujące ból i przyjemność, decydujące o swych ruchach, czyli zwierzęta. A z nich zwierzęta myślące, czyli ludzie. Rozwój myśli ludzkiej, nawet jeśli genetycznie jest ona jednym ze środków zdobywania warunków do życia, od chwili jej usamodzielnienia jest już ważny nie tylko w kierunkach utylitarnych, ale i bezinteresownie poznawczych. Matematyka ma w tym drugorzędną rolę, bo poznaje najbardziej zewnętrzną, formalną i banalną stronę rzeczywistości, ilościową. Fizyka z chemią, czyli właściwie też fizyka ultramikroskopowa i biologia, zwłaszcza ta z mikroskopów elektronowych, są znacznie ważniejsze, choć według chyba słusznej tezy Kanta, i one nie poznają „istoty rzeczy w sobie”. Zarówno materialiści, jak i większość filozofów wierzących, uważają, że matematyka i każda nauka, nawet logika, powstała wskutek obserwacji świata i w jakimś stopniu go poznaje. Ale gdyby nawet uznać, że matematyka jest absolutną abstrakcją, „czystym tworem wolnego ducha ludzkiego”, to i wtedy byłaby jakoś związana z rzeczywistością, bo umysł ludzki jest częścią rzeczywistości. Lecz matematyka to jałowa część rzeczywistości, choć wcale nie „zimna” czy „sucha”, jak mówią ludzie ślepi na urok matematyki. Wobec tego cenię matematykę głównie od strony artystycznej, jako sztukę dla sztuki. Jest to hasło mylnie sformułowane, bo każda sztuka jest dla ludzi. Muzyka poważna (chyba bez treści umoralniającej czy społecznej) jest dla małej ilości ludzi, ale są tacy. Odbiorców sztuki matematycznej jest chyba więcej, a ilość ich będzie rosła w miarę wzrostu oświaty. Żądanie, aby sztuka była zrozumiała dla mas, jest w przypadku matematyki nonsensem, trzeba by wtedy zatrzymać matematykę na poziomie V klasy podstawówki. Sztuka nie powinna zniżać się do poziomu mas, lecz podnosić masy do swego poziomu. Dla jakich mas ma być zrozumiała – ciemnych? Ciemne masy to nazwa z przeszłości, w przyszłości ludzi ciemnych nie będzie. „Chodzi o to, aby podnieść ludzi na wyżyny filozofii” – zdanie chyba Marksa. Uważam, że nauki podstawowe dla samego poznania świata, będącego jednym z głównych celów ludzkości (wbrew zdaniu S. Staszica, uważającego naukę za luksus), nie są celem wyłącznym. Równie ważne są zastosowania, ale nie do produkcji igieł czy samochodów, choć i to potrzebne. Są to badania:

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1. w medycynie, przedłużenie przeciętnego wieku ludzi choćby tylko do 150 lat z najwyżej 5 latami starości, a w przyszłości może nieograniczenie; 2. zwiększenie produkcji środków żywności dla już przeludnionego świata; 3. wynalezienie silników bezspalinowych, odtrucie wody, gleby i atmosfery, nowe źródła energii, bo i uranu na świecie mało; 4. stworzenie na innych planetach warunków atmosferycznych i termicznych do kolonizacji miliardami ludzi, którzy zamieszkają tam stale, poza tym, co łatwiejsze, wynalezienie skutecznych, nieszkodliwych, wygodnych i tanich w masowej produkcji środków antykoncepcyjnych. Inaczej nieuchronnie stoczymy się do automatycznej regulacji przeludnienia poprzez głód, zatrucie spalinami i ściekami lub wojnę nuklearną. Czyli dalsze istnienie ludzkości zależy od nauki stosowanej. Co do zastosowań przeciw ludziom – to nie jest winą uczonych, że ich odkrycia zostały wykorzystane do zbrodniczych celów. Mimo to bomba atomowa miała na celu rzecz tak słuszną, jak walka z ludobójcami hitlerowskimi. W tej sprawie mogę na szczęście oburącz podpisać się pod słowami wielkiego matematyka angielskiego, G.H. Hardy’ego: „Cieszę się, że żadne z moich odkryć nie przydaje się do produkcji maszyn do zabijania ludzi i ujarzmiania narodów”. Bo w ogóle nie mam prac z matematyki stosowanej, choćby do celów pokojowych. Pośrednio może znajdują zastosowanie w pracy inżynierów wiadomości z moich elementarnych wykładów z analizy, natomiast wykłady monograficzne nie były stosowane. Nie jestem specjalistą żadnego z działów zastosowań matematyki. Uważam, że takich specjalistów w ogóle poza Wrocławiem i Warszawą jest w Polsce mało. Pracę dydaktyczną traktuję drugorzędnie, głównie jako zarobkową. Wprawdzie są instytuty pracy czysto naukowej, jednak mogą one ulec redukcji, np. ze względów oszczędnościowych, a co ważniejsze, choć instytuty takie nie narzucają tematów, jednak wymagają osiągnięć. Jednak podzielam zdanie prof. A. Alexiewicza: „gdyby nawet wolno było wcale nie wykładać, to bym wziął choć dwie godziny tygodniowo wykładu dla kontaktu z ludźmi i młodzieżą w szczególności”. Podzielam zdanie większości fachowców, że bez pracy naukowej nie można dobrze wykładać matematyki wyższej. Nie pomoże tu ani tzw. wrodzony talent dydaktyczny – choć się zdarza – ani wieloletnia rutyna, ani stopnie naukowe z dydaktyki ogólnej. Nienaukowiec łatwo popada w błędy, nawet starannie przygotowując wykłady, i wtedy jego talent dydaktyczny czy wiedza z dydaktycznej habilitacji działają na szkodę, gdyż tym skuteczniej i bardziej przekonująco psychologicznie nauczy błędów. W rezultacie, drugorzędnie traktowany (ale jednak solidnie) wykład naukowca jest dużo lepszy od rutynowego wykładu różnego rodzaju wykładowców powtarzających utarte myśli. Do większości wykładów nie przygotowywałem się – i to właśnie dobrze. Wykład bez pamiętania szczegółów, ale przy znajomości metody – improwizowany – pokazuje, jak się robi matematykę, a nie tylko opowiada, co w niej jest. Naukowiec z własnego doświadczenia rozumie metodę i myśl przewodnią, pokazuje, jak do niej dorobić szczegóły. Tego nigdy nie zrobi rutyniarz. Do trudniejszych wykładów jednak przygotować się trzeba, zawierają one bowiem też sporo materiału pamięciowego, a tego nie można szybko zrekonstruować na wykładzie, można bowiem zastanawiać się przed tablicą pięć sekund, natomiast nic nie powiedzieć przez pół godziny byłoby stratą czasu, zwłaszcza przy błędnej opinii początkujących studentów (za co winę ponosi szkoła), że nauczyciel rzekomo wie wszystko. Praca na uczelniach zawiera też cześć administracyjną, której zawsze nie znosiłem. Dziwne kwestionariusze z często niezrozumiałymi pytaniami o rzeczy,

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których nie ma, wypełnione siłą rzeczy „z sufitu”, planowania, sprawozdania, a między nimi – krótki czas na produkcję naukową, kłopotliwe i czasochłonne stanowiska kierownicze, których unikałem. Moim zdaniem naukowiec nie powinien mieć nawet śladu żądzy władzy, jednej z najgorszych i najszkodliwszych cech ludzkich. Niektórzy uważają, że ma ją każdy: „kto nie rządzi w domu, ten rządzi w miejscu pracy, a kto nie rządzi w miejscu pracy, jest tyranem domowym”. Jest to nieprawda, bo despota jest nieznośny i w domu i w pracy, a naukowiec despotą być nie potrzebuje, nie musi on bowiem rządzić ani w domu, ani w pracy, wystarcza mu aż nadto władza nad problemami, które rozwiązuje, a ten typ władzy jest dla ludzi nieszkodliwy, i nad umysłami przyszłych czytelników jego prac. Moim zdaniem matematyka powinna mieć, i na pewnym poziomie rzeczywiście ma, implikacje etyczne. Człowiek umiejący myśleć powinien też umieć wybrać dobro. Niezależnie od nazw rzekomo je dyskwalifikujących – czy nazwać to współczuciem, czy instynktem stadnym. Jasne, że słuszna nie jest głoszona przez Nietzschego etyka silnych – w gruncie rzeczy pochwała ludobójstwa, lecz właśnie etyka słabych. Wolnomyślicielski humanitaryzm, chrześcijańska miłość bliźniego czy też humanitaryzm socjalistyczny, a nie nienawiść. Wreszcie całkowicie potwierdzam – przeczytane w jakiejś pracy prof. A. Wakulicza – słowa wielkiego algebraisty niemieckiego, Kroneckera: „matematyka uczy skromności”. O tym wie każdy twórczy matematyk, który zmagał się godzinami i latami z kolosalnie trudnymi problemami, przerastającymi często mniej czy bardziej utalentowany umysł. Wobec trudności jesteś niczym. Tego można się dowiedzieć walcząc tam, „gdzie twardym murem trudny problem stał”. Moje osiągnięcia naukowe zostały uzyskane przy pokonywaniu wielu przeciwności losu, zarówno bezosobowych, jak nędza, choroby, wojna, jak i spowodowanych przez konkretnych ludzi. Przeciwności te – dodatkowe, poza trudnością samych problemów – ujawniam chętnie, nie dla zyskania współczucia, którego nie chcę i nie potrzebuję, lecz ze słusznej dumy: trzy metry w skoku o tyczce to daleko do rekordu przekraczającego 5,5 metra, ale trzy metry z plecakiem zawierającym 25 kg cegieł, to super rekord. Choć w ogóle traktuję te sprawy nie w kategoriach pychy czy skromności, stosując m.in. zasadę „lepiej się nie doceniać niż przeceniać”, tzw. pesymizm asekuracyjny (nie asekurancki). Po reanimacji w roku 1980 odłożyłem na czas bliżej nieoznaczony szukanie poprawnego, i prostszego, dowodu twierdzenia Carlesona. Zająłem się natomiast dla rozrywki dużo starszym problemem, w myśl zasady: próbować każdemu wolno. Stwierdzam natomiast, że nadzieja rozwiązania tego problemu jest prawie żadna – w tym przypadku byłoby to grube przecenianie swoich możliwości, nawet przy uwzględnieniu maksymy „mierz siły na zamiary”. Zdolności maleją przecież z wiekiem, a biorąc pod uwagę stan zdrowia i ilość lat czy tygodni stojących do dyspozycji, szans nie ma wcale. Toteż nie ujawniam tego problemu. Ujawnię, jeśli cudem, jak ślepej kurze ziarno, uda mi się zagadnienie rozwiązać.

26

Z. Zahorski

Posłowie (od redakcji) W roku 1987 Uniwersytet Łódzki nadał Profesorowi Zygmuntowi Zahorskiemu tytuł doktora honoris causa. Walne Zgromadzenie Polskiego Towarzystwa Matematycznego uchwałą z dnia 3 IX 1993 r. nadało profesorowi Zygmuntowi Zahorskiemu godność członka honorowego za wybitne osiągnięcia wzbogacające matematykę. Do emerytury, na którą przeszedł w 1984 roku, Profesor pracował w Instytucie Matematyki Politechniki Śląskiej. Profesor zmarł, po ciężkiej chorobie, 8 maja 1998 roku w Gliwicach, gdzie został pochowany.

Janina Śladkowska-Zahorska and Zygmunt Zahorski

The biographical sketch Zygmunt Zahorski

I was born on April 30 1914 in the Szubina village (near Kutno and Kro´sniewice). My father – Stanislaw Zahorski, who had been a teacher in a local primary school, died on January 18 1921. Since the official flat had to be handed over to the successor, my family (mother and younger sister) was left with no home nor livelihood. Because of bureaucratic problems (there was no money to hire a lawyer) it took seven years to procure a widow’s pension. My mother wasn’t strong enough for a manual labor and didn’t have any education for an clerical one. All savings disappeared in the then rapid devaluation. In these circumstances, till 1928 my mother used help of three out of four of her brothers. In the summer of 1923 and in years 1924–1926 she worked as a maid in the “Zameczek” estate near Chodcz. I went to primary school in the years 1924–1929 in Chodcz, Kro´sniewice and Pultusk. Since the beginning of schooling I noticed my passion and capacity for mathematics, which I learned by myself beyond the course of study. I even set some problems already, some of which I managed to solve. However, my “discoveries” were, in fact, long known facts, I obtained the first truly new result at the age of 23 on the last year of mathematical studies. In September 1928 I passed the exam and was accepted to the fifth grade of a male humanistic junior high school in the name of Piotr Skarga in Pultusk. At the time I lived with my mother’s brother in his forester lodge in Poplawy, 2 km away from Pultusk. Then I moved to a lodgings in Pultusk in 1931. I earned my keep by giving private lessons, since I was considered the best mathematician in the school. In 1932 I finished the junior high school and in September of this year I passed with honors an entrance exam for the Mechanics Faculty of the Warsaw University of Technology. I intended to choose – after completing two years of studies – the most mathematicised, aerial section. There were only four people out of 190 accepted and 360 examinees, who passed the exam with honors. I chose the technical studies not because of the future salary, but because of my aversion towards being a high school teacher. However, exactly in 1932 the tuition on universities was significantly raised and the first year students did not have the right to get a payment deferment or to ∗ It is a reprint of comprehensive parts of professor Zygmunt Zahorski’s memoirs, written by him ´ askiej, on the occasion his of 70’s birthday and originally printed in Zeszyty Naukowe Politechniki Sl¸ Matematyka-Fizyka 48 (1986), 7–25.

R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 27–40. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

28

Z. Zahorski

get a state scholarship. This way, not by raising the entrance exam requirements, the then authorities countered the “overproduction of intellectuals”. The tuition on the first year on the technical university was 300 zl in two payments. Additionally the entrance exam fee was 40 zl. This amount of money was staggering for me. My mother’s pension in that time was 80 zl at first, then was reduced, as children reached the age of 16, according to general crisis-economical reduction of salaries and pensions. At the end, in 1938 when my mother died, it was only 30 zl monthly. In that time my situation on the tutoring market was much worse than in junior high school in Pultusk. I wasn’t known in Warsaw and in the same time there were thousands of students intending to earn as tutors. Therefore during first and second year of studies I wasn’t tutoring at all. My savings, 300 zl, was enough only for: the entrance exam fee, the first installment of the tuition and one month accommodation fee – 75 zl – in the ZNP1 dormitory. I did not get the ZNP scholarship, 75 zl monthly, for staying in the dormitory (i.e. exemption from accommodation fee). My application for this scholarship was evaluated by the dormitory manager, jealous and adverse teacher of mine from primary school. Hence I pledged to this manager that I was going to stay without paying, since in my opinion the scholarship was due to me. The threat of throwing me out was not realistic, since the janitor would have to struggle with 32 friends of mine. Prohibited entry to the canteen wasn’t a problem for me either, since waitresses gave extra food for me to my friends, who brought it to my room. The manager also didn’t succeed in arranging an argument with me, for which he would have a witness, while the student – me – didn’t. Such an incident would be an excuse for turning me out, by a judgment of a court of law. After a month of dramatic efforts to postpone the payment date of the second installment of the tuition and then a desperate search of the money required, I acquired the needed 160 zl from a one-off allowance from Kutno district authority office. This budget leavings of the district authority office saved me – a student who passed with honors the entrance exam – from being thrown out from studies because of not acquitting the tuition. But to live – especially for free – in the ZNP dormitory during whole studies was not possible. Everyone had to move out for the summer break, including my protective friends, since during this time renovations were made and after vacation everyone had to apply for a room in the dormitory again. Opinion on applications, especially of the last year residents, was given by the manager. Therefore it was obviously useless to apply for a room there, moving again in my old room was not feasible. I started second year of studies with 20 zl in my pocket. Admittedly, my financial situation improved a little, since because of getting a lot of good and very good exam grades I was given a deferment of the tuition payment, i.e. 290 zl, date for “after studies”, but still I had no livelihood. I didn’t even get the partial scholarship: 60 zl monthly. In that time getting both these aids on Warsaw University of Technology was practically impossible, maybe there were some favored exceptions. Not only bursary clerks, but also the university authorities treated students as “intrusive flies”, which had to be repelled, because, dreadfully, they had a whim to study despite having no money. Obviously, it was better to get half of scholarship than a tuition deferment, which a student would be able to pay, providing he was thrifty. I also applied for the city of Warsaw authority office scholarship, 75 zl monthly, but the date of consideration of these applications was not accordant with the beginning of an academic year. 20 zl, 1

azek Nauczycielstwa Polskiego (eds.). Polish Teachers’ Union, Polish: Zwi¸

The biographical sketch

29

which I had for the autumn exam session, I spent as follows: 10 zl for the rent, lodgings in a kitchen at some barber’s flat, where the tea three times a day was included; the remaining 10 zl I destined for a plain bread to eat during the month. After one month the flat owner stated that in most cases the rent is 15 zl, but I didn’t have even 10 zl. I decided I would live in a shelter for the homeless, on the Jagiello´ nska street in the Praga district. It was allowed to stay there from 9 pm. till 6 am., during the day the building had to stay empty. The homeless slept on raw wooden bunks, on some newspapers placed on the floor or directly on the floor – when all bunks were occupied; often even on the stairs leading from the ground floor to the first floor – when the cold drove them away from under the bridges or parks. This “living” in the dirt and perishing cold lasted about 6 weeks (in November and December 1933). At the beginning of December 1933 Bratniak – a students’ society self-help organization, granted me an accommodation scholarship in a small student house on the Ceglana street (the big student house on the Narutowicz square was taken away from Bratniak by the Ministry of Education) and eventually I started to live in bearable conditions. A little earlier I also got, from Bratniak too, a boarding scholarship – vouchers for lunches in a student canteen on the Koszykowa street. In that time I also got help from my ex-roommates from the dormitory and from Dr. J. Stawi´ nski. In February 1934 the Warsaw authority office informed me that I was granted a scholarship since September 1933, paid aback. I resigned then the accommodation and boarding scholarships. I could send a significant part of these money to my mother, who lived in Kro´sniewice. This scholarship had this particular disadvantage of being one-year; in the next year it was not anticipated in the city’s budget and no student got it. On the occasion, a piece of information about me, which appeared in the book “Czas przed burz¸a” (“A time before a storm”), Nasza Ksi¸egarnia, Warsaw 1973, pages 118, 163, 164, 170, 215, written by my older colleague from the Warsaw University of Technology, should be rectified. It’s author is Dr. eng. S. Minorski. In this, partly humorist, book some facts are real, but the most important one, about the homeless shelter, was skipped by the author. He wrote it many years after the war, so apparently his memory failed him. Also, occasionally he mistakes me with other people, and significant part of the information is errant. The author’s negative attitude to the great philosopher and exemplary man, Prof. T. Kotarbi´ nski, is due to his subjective opinion, but the information about scholarships on the University of Warsaw, the unknown ground for the author, while known well for me – is totally false. The author of this book died in June 1981 and it is unknown whether there will be further editions of this book, therefore I brought up the partial rectification mentioned above, since I don’t have any other possibility to engage in polemics with the author’s description. I wasn’t much interested in strictly technical subjects, such as machinery or molding, and there was too little of mathematics lectures, which I put interest in. Very little of algebra, complete lack of numbers theory and set theory, subjects indeed redundant for engineers. Following my interests, in the autumn of 1934, when the new law let studying on two fields of study and with an approval from rectors, I signed for mathematics on the University of Warsaw. Already being a student of the University of Warsaw, I didn’t leave the Mechanics faculty on the Warsaw University of Technology yet,

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Z. Zahorski

where I completed almost whole two years of studies, including three months of factory apprenticeship. On the University of Warsaw I came across completely different attitude of lecturers towards students. Professors Sierpi´ nski, Mazurkiewicz, Kuratowski were great scientists, who supported people mathematically talented. Dean Mazurkiewicz, when I accosted him on the university yard, after his office hours, without any beadledom, as befits a true scholar, didn’t send me away saying, that he receives students only during office hours and after making an appointment through his secretary, but on the spot he promised me, against first year students regulations, a full tuition deferment and partial scholarship, i.e. 60 zl monthly during the first year. After finishing the winter and summer 1935 exam sessions with very good grades, till the end of studies I got a full scholarship, 120 zl, and tuition deferment. As for one person, these conditions were downright luxurious, but they got worsen a little, because I was sending a half of the money to my mother. There were only two full scholarships on all four years on the whole faculty (mathematics, physics, astronomy, chemistry, biology and geography), in contrast, there was a lot of partial ones. The second scholarship was given to another student of mathematics, but not the one described by S. Minorski, because he was not a member of ONR2 , but of the Legion of the Young, a remedial organization, which was certainly a favorable circumstance, although not the only one. He was a capable mathematician, but out of my league. The most talented mathematician studying on that time on the university didn’t need the scholarship, he was wealthy and didn’t hasten with completing years of studies, what let him write papers with original results during first and second year. Formally, the mathematical studies lasted four years, but in reality often longer. I finished it after 3 years and a half in 1938. At that time I studied on the Warsaw University of Technology in slow mode, but still completed – as I mentioned before – two first years, including the factory apprenticeship. I signed for the aerial section, passing some of the subjects on the third year of studies. It was a little more than so-called contractually half-diploma, officially – the first diploma exam, although there was no a separate exam. I did not intend to complete studies on the University of Technology, which I needed only to get a postponement of military service till 26 years of age, not only till 22, as on other universities. I did not avoid the military service, but I did not want it to interrupt my mathematical studies. In that time I changed my field of interest: not aerodynamics, hence also not partial differential equations, but real functions theory, a little of analytic functions theory (which excerpts were included in the aerial section program also) and trigonometric series became my main interest. In 1937, on Prof. Mazurkiewicz request, I published my first short paper, without original result, but with a new construction method of some function. Mazurkiewicz did it already, 22 years earlier, but his method was much more complicated. In the same year I got a new result, difficult and of a big importance, which was complemented in 1938. It was supposed to be my doctoral thesis supervised by Mazurkiewicz, and the completion of my PhD was planned for September 1939. Obviously, the war prevented finalizing these plans. After translating this paper to French and sending it to Bulletin de la Societ´e Math´ematique de France in the summer of 1939, the French Mathematical Society (Soc. Math. de France) offered me to sign up, which I did, paying quite a low registration fee (11 zl). Undoubtedly I was removed from the list of members due to inability to send – during the war – the equally low membership 2

oz Narodowo Radykalny (eds.). The National Radical Camp, Polish: Ob´

The biographical sketch

31

fee. It is unknown if there is any trace of that signing up to Soc. Math. de France. The history of the mentioned manuscript was unlucky. Although the occupation in France was much milder than in Poland and scientific journals kept publishing, the manuscript was published in Bull. de la Soc. Math. de France only after the war in 1946. However earlier, in 1941, the paper was printed in Moscow, in Russian, even in an extended form. In 1937 I started to work as an assistant of Dr. eng. S. Neumark, Dr. eng. J. Bonder and Dr. P. Szyma´ nski in the Aviation Cadet School (Technical group) in Warsaw, to theoretical mechanics, durability of materials and mathematics, with a little narrower program than on technical universities. I owed this job to Dr. Al. Wundheiler, an outstanding creator in differential geometry, whom Dr. Neumark wanted to employ as an assistant. Wundheiler conducted classes in mechanics on the University of Warsaw to Prof. Przeborowski’s lecture, but he had no chances to get employed in an officer school due to “personal” reasons – which, in fact, concerned also Dr. Neumark, but he had worked for a long time already in the Institute of Aviation and in the army, also in the Institute of Aerodynamics in the University of Technology. However, both Neumark and Bonder, although “Zionists”, were irreplaceable both as engineers and mathematicians. Wundheiler recommended me to Neumark, as his best student, whom he knew from theoretical mechanics classes. I worked in this school until its’ evacuation to an unknown destination in 1939. At the same time, in March 1939 all the postponements ended and I got a summons to appear on October 2nd, 1939, at the Officer Cadet School in Zambr´ow, to serve in the army. On September 7 1939, when the Germans approached Warsaw and colonel Umiastowski on the radio appealed to men with military category A to leave Warsaw and go east, where they would be conscripted, I went east with Estera Steinbok, mathematics graduate. Commander of a local military unit in Siedlce, after I showed him the summons to the cadet school, refused to take me to the army, because I was untrained. Then, as refugees, we went to Brze´s´c, where, seeing the heavy air-strike and the Germans approaching Wysokie Mazowieckie, we understood that, as for the moment, Poland had lost the war. We decided then to run away as close to the Soviet border as possible, so when the Germans would come and border units of KOP3 would leave, we would cross the border and take refuge from the Germans in USSR. On September 14 we were in Luniniec, 60 km from the border, but crossing it was unnecessary, since on September 18 the Soviet army came in Luniniec. Soon I went to Lviv where universities were open, Estera Steinbok temporarily stayed in Luniniec. In Lviv I became an assistant of St. Banach, on the Lviv University, and since the beginning of March 1941 an aspirant (together with my younger colleague MSc A. Alexiewicz, later a professor on the Adam Mickiewicz University in Pozna´ n). There was a doctoral scholarship associated with the postgraduate studies. Banach agreed to accept as doctoral thesis the same treatise as Mazurkiewicz had, but there was still an unsettled by the Ministry in Kiev matter whether the Lviv University would have the right to confer the degree of doctor (candidate in the local terminology). There was a lot of refugees working on the Lviv University back then, to name some of

3

The Border Protection Corps, Polish: Korpus Ochrony Pogranicza (eds.).

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Z. Zahorski

them: Saks, Knaster, Marczewski4, Wojdyslawski and me, Prof. Orlicz from Pozna´ n, ˙ nski. Boy-Zele´ In November 1940 I married Estera Steinbok. During this period of time I dragged myself into research, I was very busy, I edited the finished papers and started writing new ones, so that they survive in the case of my death in war. The edited ones I sent to Moscow, Japan and Lviv – the last ones got lost, but after I rewrote them – after war – they got published in India and USA. New problems came partly from Banach and Mazur (but they did not concern functional analysis) and partly from myself. My concerns that the conflagration of war would reach Lviv one way or another turned out to be correct on June 30, 1941, when the Germans marched into Lviv. Further escape east was technically impossible since the first day of war, i.e. June 22. Also, another problem appeared: we had to move, because my wife figured in the registration book as a Jew, and for that time she used a fragile fake ID. Obviously, universities got closed (even for the Ukrainians), anyway not only I would not work at the Nazi university, but I could not even admit to having higher education, knowing that it prioritizes for a concentration camp. The nutrition conditions got worse, casual earnings, mostly from trade, were very small. In that situation my health worsened for the first time (tuberculosis). My son, born on August 25, 1941, died on September 27, the same year, because of inadequate nutrition. In March 1942 I came back to Warsaw and in May 1942 Dr. Stawi´ nski referred me to a familiar technical director in the Philips factory, under the condition of concealment of higher education. The factory was confiscated by the Germans and it gave the so-called ausweis, which partly protected from transportation to Germany. It also provided a soup in the morning and for the lunch, which was quite awful, but sometimes with meat offals of worse quality than “nur fur Deutsche”. Working there gave the opportunity to listen, without any risk, to the radio broadcasts from London and Moscow, which were prohibited under death penalty (according to, among others, posters hanging in factory halls), and sometimes by chance even from partisans short-wave radios. With the technical possibilities of those times it was impossible to ascertain what a person was listening to in headphones, and it was obligatory for a worker calibrating a device with a wavemeter to have headphones on. A person should only, when some German entered the hall, calmly leave a suspicious wave, disconnect an ordinary cable used as an internal antenna and connect a wavemeter cable laid 5 cm away. I established for my wife Estera a solid Aryan kennkarte using a real birth certificate of my sister, Helena Barbara Zahorska, who lived at that time inside of the so-called Reich. After the war my sister deemed it very dangerous, putting her and all the family, with whom she lived, in a risk of death. However, considering difficulties with Poles traveling from Reich to General Government, finding two Zahorski twins with both names identical was very much unlikely. In this time I also helped other Jews, sometimes unfamiliar children, the last not being risky. I didn’t manage to ransom from the navy-blue police the wife of a remarkable Lviv mathematician, J. Schauder, who was shot on a street in Lviv5 . Mrs Schauder came to hide in Warsaw then, but, like her husband, she could not stand staying at home all the time. Some children recognized her on the street and started shouting “Jew”, most likely without awareness what that could cause, and then a navy-blue policeman arrested her. Another Jew who was hidden by the 4

Edward Szpilrajn used to hide first under the name Zawadzki, and Marczewski later on.

5

I do not know exactly if it was maybe in Drohobycz. Supposedly he was denounced by a Ukrainian.

The biographical sketch

33

same woman, familiar with Estera, alarmed us. This hostess had some connection with one of the policemen, I was a link in a chain connecting her to Mrs Tarska, who was collecting money for the ransom. I preferred to avoid a direct contact with police circles, even Polish ones. But after collecting a required amount of money, the policemen raised it twice, and for the third time Mrs Schauder wasn’t there anymore. During the next inspection of the police station the Gestapo agents took all taken Jews. Neither Mrs Tarska, nor the hostess, had any access to a gestapo agent. Since one should work very sluggishly in German factories, I had lots of time to write more articles. Although these papers got lost in the debris after the rebellion, nevertheless theirs abstracts (in a few copies, some given to other people, one I kept always with me) survived, what let me recreate them after the war. In 1942 and in 1943 I spend 2–3 months in a sanatorium in Rudka near Mi´ nsk Mazowiecki, which was put up in a good standard as for the occupation time. There were 12 places funded for patients from the factory by its workers (by contribution). I had some acquaintances among partisans from AK6 (a colleague from a high school class), and from AL7 (Michal Tetmajer, whom I didn’t know personally before the war, I met him only once on some social gathering) and I intended to join them. But my health condition wasn’t good enough to spend long time in partisans terms: there was a setback of my health, which state at the end of 1943 was alarming. In January 1941 I was in a hospital, after a surgery in February my condition got worse, and in May it was completely hopeless. For the ”pessimistic security” I counted for one month of life at most. It turned out, it was judgmental and not pessimistic enough: doctors gave me at most a few days. But they were wrong, there was a slow improvement, so in June I could even walk unassisted. Undoubtedly the food sent me by my cousin Janka Wl. from Siedlce was of much help, but mainly – it was some unexpected endurance, because other patients from “the waiting room to a mortuary”, especially the ones getting much better food from the countryside and from families, all died. I stayed alive. During the occupation there were only two people who left this room alive: one of them was me, the other one (earlier) some sailor. A temporary health setback appeared later, during The Warsaw Uprising. However, in this hospital (on Nowogrodzka Street, Baby Jesus Hospital) almost from the beginning occupied by the Germans, the conditions were much better than in other districts. The Germans didn’t murder the patients, nor the medical personnel, like in The Wolski Hospital, or in Leszno, where the ill who could walk were shot at the spot, and the bedridden were burned inside the building. Among the last ones was the father-in-law of Prof. Borsuk. Three weeks after the uprising the Social Council evacuated this hospital, as well as the ill, the wounded from the uprising and the elder from other hospitals and shelters, to a provisional hospital in the Medics House on Grzeg´ orzecka street in Cracow. This place could be left only after getting an approval from gestapo. However, there was no guard duty in the hospital, so I made use of this “freedom”. I found the Cracow mathematicians and during a secret gathering of a branch of Polish Mathematical Society I offered them my papers written during the war. Soon I got informed that the secret Polish Academy of Arts and Sciences would organize a food aid for me. The meat was provided by Prof. S. Turski, a former prisoner of the concentration camp of Sachsenhausen, an employee of a butchery at that time, after the war an organizer and the first rector 6

The Home Army, Polish: Armia Krajowa (eds.).

7

The People’s Army, Polish: Armia Ludowa (eds.).

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Z. Zahorski

of the Gda´ nsk University of Technology, later of the University of Wroclaw, for some time a Director of a Department in a Ministry. In January 1945 the Germans intended to transport patients from the hospital on Grzeg´orzecka to the Auschwitz-Birkenau concentration camp, but they didn’t manage to do it on time. Patients of this hospital, as well as many other people, were saved by an instant Soviet offensive, which started on January 12 in Sandomierz. After that the Germans only managed to shoot people held in prisons, and then they ran away not to get surrounded. On 18–19 January Cracow was conquered already, almost without any fight, after a far envelopment. My dietary intake improved and health got better, so in February 1945 I stood in front of a recruitment board. The doctor verified that a bandage on my ribs is not fake and then, without any question, he wrote me a dismissal. Since the opening of the Jagiellonian University I worked there as an assistant. In June 1945 I got a small apartment on Podwale street. On February 11, 1946, I finished my doctorate. I used a different paper than the one from before the war, and that one, significantly complemented, was a base for my postdoctoral degree, planned and prepared for October 15, 1947. A lack of time of one of the reviewers, Prof. Mazur, caused a delay and the postdoctoral colloquium was postponed to December 1947. Simultaneously, the Jagiellonian University applied for an appointment to an associate professor and employed me on a post of a deputy professor. In October 1948 I got an appointment of an assistant professor and a transfer to University of L´ od´z. On March 3, 1949, Helena (Estera) Zahorska gave birth to a daughter El˙zbieta. (She kept first name from the occupation kennkarte). In 1949 I got a Polish Mathematical Society award8, as the twelfth of Polish mathematicians. These awards, three a year were given since 1946, were most appreciated, not because of the money, but the meaning of them. Only two mathematicians got it twice: H. Steinhous (during the lifetime) and M.Biernacki (the second one posthumously, but an application could be issued only during his lifetime). I got three Ministry awards, in 1948 (an award for the young, i.e. until 40 years of age), in 1962 (a II degree award for the research) and in 1984 for the overall work, in particular for my research and development of the young cadre. I omit here a few Rector awards in L´od´z (and one in Gliwice) for research or didactic work. In 1954 the Council of Mathematics, Physics and Chemistry Faculty on University of L´ od´z unanimously came forward to promote me to a full professor, which I foiled by not filling a required form, because I did not feel worthy of this promotion yet. I accepted the promotion only in 1960, after solving a Kolmogorov problem (a proof of some hypothesis from 1926, which Kolmogorov published in 1927 without a proof or any details). This was in fact a worldwide result, since 1927 on the whole world many outstanding specialists of trigonometric series theory tried to solve it, unsuccessfully. I succeeded in May 1960, after three weeks of work. My earlier inconclusive efforts, since 1940 and even earlier, because since 1936, and intensively since 1942, on the Luzin hypothesis, were of some help with solving it. Almost exclusive and for sure the main goal of my life was the work on Luzin hypothesis, put in 1912. This hypothesis is a generalization of du Bois Reymond problem from 1876, and supposedly even of the Riemann problem, hence from before 1866. I worked on it until the reanimation in autumn 1980. Luzin hypothesis, also concerning trigonometric series theory, but even more difficult and older, had some aspects causing my ideas to be ineffectual. The same ideas turned out to be effective with the 8

The Stanislaw Zaremba Prize (eds.).

The biographical sketch

35

Kolmogorov problem. In 1961 in a rush, before I drafted a final version, I announced the solution of the Luzin hypothesis in C.R. Paris Academy, and three weeks later, while writing a clean copy and a presentation for the seminar of Prof. Mazur in the Institute of Mathematics of the Polish Academy of Sciences, I noticed a mistake. Since the announcement had already been published, I rectified my mistake immediately – in a letter to Prof. A. Zygmund, a reviewer of this note (from C.R.) in a worldwide bibliographic journal Mathematical Reviews, but I couldn’t fix the mistake until 1980, neither later. This mistake, psychologically, had a huge impact on my faith in my own capability and from then on it was my greatest misfortune, obscuring, or rather absorbing even the calamities of war. Objectively, it is not that bad, Lebesgue himself announced in 1905 a false result, and the supposedly greatest mathematician of XX century, D. Hilbert, in 1925 “proved” the continuum hypothesis; the mistake was noticed right away by Kuratowski and von Neumann. By contrast, the Luzin hypothesis turned out to be true, what was proved correctly in 1966 by a Swedish mathematician Lennart Carleson (younger than me), who was later the president of the world Mathematical Union for a few years. Hence my mistake was only in the proof, but, whether it was true or not, was not down to me. Because, not going into details, the problem had two possible answers – yes or no. In 1961 I was named by the district Front of National Unity for a district councillor in a District National Council (DRN), L´ od´z-G´ orna. I accepted this proposition hoping that I would be able to help people, because despite very little spare time (research took a lot of time), the then health and energy condition allowed some additional activity. I was against purely phraseological work and I thought that the best example is a specific work (including research). Working as a councillor was such a specific work for me. It turned out to be true, but only by a little bit. I was a councillor and a member of Education Council in DRN L´od´z-G´orna in years 1961–1970, and a member of the DRN Presidium from 1965 to 1970, i.e. until relocation to Gliwice, two terms of office in total. In 1951–1953 and 1959–1961 I was a president of L´od´z Branch of Polish Mathematical Society, in 1975–1977 a member of The Main Board of Polish Mathematical Society, and almost constantly a delegate to The General Assembly of Polish Mathematical Society. On the University of L´ od´z I got The Tenth Anniversary of PRL9 Medal and The Gold Cross of Merit in 1953–1954, and on July 22, 1968 (April 17, 1969, after name rectification) Knight’s Cross Order of Polonia Restituta. I got help with relocation to Gliwice from people now already deceased: prof. M. Mochnacki and doc. dr hab. W. Sobieszek, my former student from University of L´od´z, working in Gliwice. Despite curing, completely only in 1964, the effects of war, my health was put in, even bigger, danger by a heart disease. Since 1955 light, mostly neurotic, later also physical, since 1976 serious, and since October 24, 1980 with reanimation. In 1977–1978 I tried to move to Bialystok, for health and climate reasons. My efforts stay unsuccessful, I still live in Gliwice. In 1966 my daughter El˙zbieta finished III High School in L´od´z and chose a field of study, which was not available in L´ od´z, and in the next year she passed, on my advice, the entrance exam for physics at the University of Warsaw, although she would 9 The Polish People’s Republic, Polish: Polska Rzeczpospolita Ludowa – the official name of Poland from 1952 to 1989 (eds.).

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prefer mathematics. I considered physics to be more important and interesting than mathematics, not only because of its technical applications, although I am a mathematician myself. Physics would also give a bigger variety of possible employment after studies. An attempt to study physics and mathematics at the same time was too timeconsuming, so eventually, after completing two years of physics, despite mostly good grades, my daughter transferred to mathematics claiming that she had no ability for experiments and she finished studies in 1979. ´ In 1970 I got divorced and then married Janina Sladkowska, a docent on Silesian University of Technology, doing research in a very little known by me field of analytic functions. For the family and housing reasons I moved to Gliwice to work on the Silesian University of Technology – despite the atmosphere in Silesia is much more ´ smoky and the climate is harmful to health. Janina Sladkowska-Zahorska gave birth to a son. Our son Ja´s sincerely hates mathematics, maybe the revulsion was caused by his school. No one wants to force him into mathematics, constraint never results in interest. Admittedly, he has good grades in mathematics, but he is more interested in chemistry and electronics. My daughter used to visit me in Gliwice quite often, and after finishing studies she lived at our place in Gliwice for eight months. Then she got a job on WSI in Radom. I also got an offer to transfer to Radom on grounds of the climate, but I did not want to make a rushed decision, and after figuring out the conditions there, in 1981, I decided to give up the move. In June 1982, during a one-month compulsory absence of my wife in Gliwice (I was in a cardiac sanatorium in Nalecz´ow at that time), my daughter took care of Ja´s, taking him to an acquaintance hostess near Ciechocinek. I have published papers in scientific journals in Poland, France, Japan, India, USA, USSR and Czechoslovakia, one script for initial years of study and one popular science article for readers with incomplete basic education and higher. The popular article came out not the best, because I did not have a chance to correct it before publishing – the revision was made by the editorial board of “Przekr´oj”, who changed some elements ... for worse. Excluding from the set of papers the bibliography, reviews, problems set but not solved by the author, doublets (eg. announcement, abstract and then full paper) and translations, there are 17 research papers, one of them errant, one of really good quality, eight of average quality, others contributory. There are eight PhD students of mine in L´od´z and two in Gliwice. I have two distinguished students: prof. zw. dr hab. Jan Lipi´ nski, for some time director of Institute ´ of Mathematics in University of Gda´ nsk and prof. dr hab. Tadeusz Swiatkowski, who admittedly made his doctorate in another department, but his best papers are from my field. The group of people working with this subject matter spreads wide outside Poland: there are, among others, the Czech, American, Romanian, Soviet mathematicians and others. From Poland one can name docents Filipczak and Walczy´ nski and ´ several students of prof. Lipi´ nski, or prof. Swiatkowski. I participated in several international mathematical congresses, gathering from 4 to 6 thousands of mathematicians with different specializations, organized once in 4 years in various countries by mathematicians from the organizing country (chosen on the previous congress) and the International Mathematical Union. I went to Stockholm in 1962 and to Moscow in 1966 to the prime cost, in Moscow I got chosen by organizers for a chairman on the real functions session. I was marked out from the University of L´od´z by the Ministry for a congress in Nice in 1970, but due to financial cuts

The biographical sketch

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I couldn’t take part in it. To congresses in 1978 in Helsinki and in 1983 in Warsaw I went at the expense of the Silesian University of Technology. I also took part and gave presentations on smaller foreign conferences in Prague in 1949, Berlin in 1960, Palermo on Sicily in 1976 and Varna (without a lecture, but with a problem during a discussion) in 1967.

Summary and conclusion of this biography Obviously, my fascination for mathematics is not ambitious, this aspect could only be significant during school times, when one cares about grades – and even then not only. But a question arises: should this fascination be like a horse’s blinders, which let you live good and pleasant – despite obstacles – life without more general reflection? For questions about the attitude of mathematics to general aims of humanity I answer shortly: from a human point of view the most important and astonishing creations of nature are the living organisms, especially the ones that “have a soul”, i.e. seeing, feeling pain and pleasure and self aware, so animals. And among them the thinking animals – humans. Development of a human thought, even if genetically it is one of the means of obtaining livelihood conditions, since the moment when it became independent, it is important not only in utilitarian fields, but also selfless cognitive. Mathematics plays a secondary role in it, because it lets to learn about the most exterior, formal and trivial feature of reality – quantitative. Physics with chemistry, so as a matter of fact also with ultra-microscopic physics and biology, especially the one from electron microscope, are much more important, although, according to probably correct Kant’s thesis, even they don’t recognize “the essence of things”. Both materialists and majority of philosophers think that mathematics and any science, logics even, arose as a consequence of observing the world and recognizes it to some extent. But, even assuming that mathematics is an absolute abstraction, “a pure creation of a free human spirit”, it would be still connected to reality somehow, because human mind is a part of reality. But mathematics is a jejune part of reality, although it is not “cold” or “arid”, as people blind for the beauty of mathematics say. In that case I value mathematics mainly from its artistic side, as the art for the sake of art. This phrase is erroneously formulated, since every art is for the people. The classical music (maybe without its moralizing and social content) is for a small amount of people, but there are some. There are probably more recipients of the art of mathematics, and there will be more of them as education improves. Demand for the art to be understandable for the masses is, in the case of mathematics, a nonsense. Mathematics should be then stopped at the level of fifth grade of primary school. The art shouldn’t lower itself to match the masses, but make the masses rise to match the art. Which masses should it be understandable for, thick-heads? Thickheads is a name from the past, in the future there will be no thick-heads. “It is all about rising people to the heights of philosophy” – a sentence probably by Marks. I think that basic sciences for the recognition of the world, one of the main goals of humanity (against St. Staszic opinion, who considered science to be a luxury), aren’t the exclusive goal. Applications are equally important, but not to produce needles or cars, although there are important too. These are:

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1. in medicine, to extend people’s lifespan to at least 150 years, with 5 years of senility, and in the future maybe without any limits; 2. tp incrise food production for the already over-populated world; 3. to invent engines with no exhaust fumes, depoison the water, earth and atmosphere, new energy sources, because there is not much of uranium in the world; 4. to create atmospheric and thermal conditions on other planets to colonize them with billions of people, who should live there permanently, additionally, what’s easier, to invent effective, harmless, comfortable and cheap for mass production contraceptives. Otherwise we will inevitably come to an automatic regulation of population by: hunger, fumes and sewage poisoning or nuclear war. Hence, the further existence of humanity depends on applied science. For applications against people, it is not the scientists fault that their discoveries were used for flagitious purposes. Despite that, the atomic bomb was for purpose as rightful, as fighting the Nazi genocides. In this matter, I can sign with both my hands the sentence by great English mathematician G.H. Hardy: “I’m happy, that none of my discoveries is useful to produce killing machines or subduing nations”. It is because I don’t have any results on applied mathematics, even for peaceful purposes. Maybe my basic lectures on analysis find some application in the work of engineers, whereas my monographic lectures were not applied. I’m not a specialist of any branch of applied mathematics. In my opinion there are very little such specialists in Poland, outside Wroclaw and Warsaw. Didactic work is of secondary meaning for me, mostly for earnings. There are, admittedly, institutes with only research work, however they might be reduced, eg. for savings reasons, and, what is more important, although such institutes don’t impose subjects, they require results. I agree with prof. Alexiewicz: “even if it was allowed not to give lectures at all, I would take at least two hours of lectures a week to have contact with other people, especially the youth”. I share an opinion with the majority of professionals, that without doing research one can not give good lectures on higher mathematics. Of no help would be the so-called natural didactic talent – although it occurs – or long term routine, or degrees in general didactics. A non-scientist makes mistakes easily, even preparing lectures conscientiously, and then his didactic talent or knowledge from post doctoral didactic degree causes even more harm, because he would more effectively and convincingly teach those mistakes. As the result, even treated secondarily (but still soundly), lecture of a scientist is better than a routine lecture of lecturers of various kinds who just repeat commonplace thoughts. I did not prepare for the majority of my lectures – and that was good. A lecture without remembering all the details, but knowing the method – improvised – shows how mathematics is done, not only tells what’s in it. A scientist, thanks to his experience, understands the method and the keynote and shows how to add the missing details. A routinist would never do that. But there are still more difficult lectures, which must be prepared in advance, they include also a lot of information to be memorized, which can’t be reconstructed during a lecture: for you can spend five seconds at the blackboard cogitating, but to not say anything for half an hour would be a waste of time, especially considering an errant opinion of first years students (for what school is to blame) that a teacher knows everything. Work on universities includes also an administrative part, which I always hated: strange questionnaires often with incomprehensive questions about non-existing things, perforce taken out of blue, planning, reports, an in between –

The biographical sketch

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short periods of time for research; troublesome and time-consuming management positions, which I avoided. In my opinion a scientist should not have any trace of lust for power, one of the worst and most harmful human features. Some people think that everyone has it: “who does not rule at home, rules at work, and who does not rule at work, is a home tyrant”. It is not true, because a despot is aggravating both at home, and work, while a scientist doesn’t have to be a despot. He doesn’t have to rule neither at home, nor at work. It is sufficient for him to have power over the problems he solves, and this type of power is harmless for people, and over the minds of future readers of his papers. In my opinion mathematics should have, and at some level in fact it does, ethical implications. A person that can think, should also be able to choose good. Independently from names – ostensibly disqualifying it – whether called compassion, or the herd instinct. It is clear, that not correct is the ethics of the strong proclaimed by Nietzsche – essentially approval of genocide – but the ethics of the poor. Freethinking humanitarianism, Christian love to one’s neighbor, or socialist humanitarianism, not hatred. Eventually I confirm absolutely – read in some paper of prof. A. Wakulicz – words of Kronecker, a great German algebraist: “mathematics teaches humbleness”. It is known by every creative mathematician, who struggled for hours and years with enormously difficult problems, often surpassing a more or less talented mind. Facing difficulties you’re nothing. This is what you can learn when fighting “where a difficult problem stood as a hard wall”. My scientific accomplishments were obtained through overcoming adversities: both impersonal, like poverty, illnesses, war, and caused by specific people. These adversities – additional, besides difficulty of the problems themselves – I divulge not to get compassion, which I don’t want or need, but because of the rightful pride: three meters for a pole vault is far from the record, which is over 5,5 meter, but three meters with a backpack containing 25 kg of bricks is a super-record. Although I treat these matters in general not in terms of pride or modesty, I comply, among others, with a rule: “it is better to underrate, than to overrate yourself”, the so-called security measures pessimism (not a non-committal one). After the reanimation in 1980 I put aside for a indefinite period of time searching for a correct and simpler proof of Carleson theorem. Instead, for an entertainment, I gave attention to much older problem, according to the rule: everyone is allowed to try. But, in my opinion, there is almost no hope for solving it. In this case it would be a big overrating of my capabilities, even considering the sentence: “bite off more than one can chew”. After all, the capabilities decrease with aging, and considering additionally my health condition and the number of years or weeks available, there are no chances. Therefore I do not reveal this problem. I will if, miraculously, I will solve it.

Afterword (of Editorial Board) In 1987 the University of L´ od´z honored Professor Zygmunt Zahorski with the honoris causa doctorate. General Assembly of the Polish Mathematical Society, by the

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decision on November 3rd, 1993, gave to Professor Zygmunt Zahorski the dignity of honorable member in appreciation for his outstanding achievements enriching mathematics. Till the retirement, into which Professor went in 1984, he worked in the Institute of Mathematics at the Silesian University of Technology. Professor died, after serious illness, on 8 May 1998 in Gliwice where he is buried.

Zygmunt Zahorski

List of papers ¨ 1. Uber die Konstruktion einer differentierbaren, monotone, nicht konstanten Funktion, mit u ¨berall dichter Menge von Konstanzintervallen. Sprawozd. Tow. Nauk. Warszawskiego, Wydz. III 30 (1937), 202–206. ¨ 2. Uber die Menge der Punkte in welchen die ableitung unendlich ist. Tˆohoku Math. J. 48 (1941), 321–330 (https://www.jstage.jst.go.jp/article/tmj1911/48/0/48 0 321 / pdf). 3. On the set of points of nondifferentiability of a continuous function. Rec. Math. (Mat. Sbornik, N. Ser.) 9(51) (1941), 487–510 (in Russian, http://www.mathnet.ru /php/archive.phtml?wshow =paper&jrnid=sm&paperid=6102&option lang=eng). 4. Sur les int´egrales singuli`eres. C. R. Acad. Sci. Paris 223 (1946), 399–401 (http://gallica.bnf.fr/ark:/12148/bpt6k31759/f399.image.langEN). 5. Sur les d´eriv´ees des fonctions partout d´erivables. C. R. Acad. Sci. Paris 223 (1946), 415–417 (http://gallica.bnf.fr/ark:/12148/bpt6k31759/f415.image.langEN). 6. Probl`emes de la th´eorie des ensembles et des fonctions. C. R. Acad. Sci. Paris 223 (1946), 449–451 (http://gallica.bnf.fr/ark:/12148/bpt6k31759 /f449.image.langEN). 7. Un probl`eme de la th´eorie des ensembles et des fonctions. C. R. Acad. Sci. Paris 223 (1946), 465–467 (http://gallica.bnf.fr/ark:/12148/bpt6k31759 /f465.image.langEN). 8. Sur l’ensemble des points de non-d´erivabilit´e d’une fonction continue. Bull. Soc. Math. France 74 (1946), 147-178 (http://www.numdam.org/item?id= BSMF 1946 74 147 0). 9. Sur les ensembles des points de divergence de certaines int´egrales singuli`eres. Ann. Soc. Polon. Math. 19 (1946), 66–105. 10. Sur l’ensemble de points singuliers d’une fonction d’une variable r´eelle admettant les deriv´ees de tous les ordres. Fund. Math. 34 (1947), 183–245 (http://matwbn.icm.edu.pl/ksiazki/fm/fm34/fm34124.pdf). 11. On a problem of M.F. Leja. Ann. Soc. Polon. Math. 20 (1947), 215–222. 12. On zeros of quasi-analytic (B) functions. Bull. Calcutta Math. Soc. 39 (1947), 157–165. ˇ 13. On a problem of G. Choquet. Casopis Pˇest. Mat. Fys. 73 (1948), 69–77 (http://dml.cz/dmlcz/123062).

R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 41–43. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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14. On Jordan curves possessing the tangent line at each point. Rec. Math. (Mat. Sbornik, N. Ser.) 22(64) (1948), 3–26 (in Russian, http://www.mathnet.ru/php /archive.phtml?wshow=paper&jrnid=sm&paperid=6059&option lang=eng). 15. Sur la classe de Baire des d´eriv´ees approximatives d’une fonction quelconque. Ann. Soc. Polon. Math. 21 (1948), 306–323. 16. Sur l’ensemble des racines de l’equation W (x) = f (x). Sprawozd. Tow. Naukowego Warszawskiego, Wydz. III 41 (1948), 43–45. 17. Suppl´ement au m´emoire ”Sur l’ensemble des points singuliers d’une fonction d’une variable r´eelle admettant les d´eriv´ees de tous les orders.”. Fund. Math. 36 (1949), 319–320 (http://matwbn.icm.edu.pl/ksiazki/fm/fm36/fm36132.pdf). 18. O zbiorze punkt´ ow nier´ oz˙ niczkowalno´sci funkcji dowolnej. Dodatek do Rocznika Pol. Tow. Mat. 21 (1949), 23–26. ˇ 19. Sur les courbes dont la tangente admet sur chaque arc toutes les directions. Casopis Pˇest. Mat. Fys. 74 (1949), 233–235 (http://dml.cz/dmlcz/109429). 20. Sur la premi`ere d´eriv´ee. Trans. Amer. Math. Soc. 69 (1950), 1–54 (http://www.ams.org/journals/tran/1950-069-00/S0002-9947-1950-0037338-9 /S0002-9947-1950-0037338-9.pdf). 21. Recenzja ksi¸az˙ ki S. Banach. Wst¸ep do teorii funkcji rzeczywistych. Warszawa– Wroclaw 1951. Ann. Soc. Polon. Math. 24 (1951), 203–206. 22. Sur les courbes dont la tangente prend sur tout arc partiel toutes les directions. Czechoslovak Math. J. 1 (1951), 105–117 (http://dml.cz/dmlcz/100023). 23. Wyklady Matematyki, cz. III. Skrypt, PWN, L´od´z 1959. 24. Une s´erie de Fourier permut´ee d’une fonction de classe L2 divergente presque partout. C. R. Acad. Sci. Paris 251 (1960), 501–503 (http://gallica.bnf.fr/ark: /12148/bpt6k32030/f507.image.langEN). 25. Sur la convergence presque partout des s´eries de Fourier. C. R. Acad. Sci. Paris 252 (1961), 2366–2367 (http://gallica.bnf.fr/ark:/12148/bpt6k762d /f1008.image.langEN). 26. Osiem komentarzy do prac S. Banacha. In: S. Banach, Oeuvres, volume I. PWN, Warszawa 1967, 311–314, 317–318, 349–352 (http://matwbn.icm.edu.pl/ksiazki/or /or1/or1140.pdf). 27. O kr´ olowej nauk. Przekr´ oj 1557 (1975), 10–11, 23 (http://mbc.malopolska.pl /dlibra/docmetadata?id=60100&from=publication&tab=1). ´ Mat.-Fiz. 48 (1986), 7–25. 28. Zarys biografii. Zeszyty Nauk. PSl.,

PhD students 1. Jan Lipi´ nski Uniwersytet L´ odzki 1958 Sur les ensembles f (x) > a, ou f (x) sont des fonctions integrables au sens de Riemann 2. Tadeusz Tietz Uniwersytet L´ odzki 1958 R´ ownanie Fermi-Thomasa i jego zastosowania (Fermi-Thomas equation and its applications)

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3.

Laura Belowska Uniwersytet L´ odzki 1960 Rozwi¸azanie problemu Z. Zahorskiego o granicach aproksymatywnych (Solution of Z. Zahorski problem on approximate limits)

4.

Witold Czapli´ nski Uniwersytet L´ odzki 1960 O prawdopodobie´ nstwie bezpiecze´ nstwa konstrukcji i pewnych jego zastosowaniach w obliczeniach wytrzymalo´sci konstrukcji (On probability of safety of the constructions and some of its applications in computing strength of the constructions)

5.

Maria Kulbacka Uniwersytet L´ odzki 1960 O zbiorze punkt´ ow aproksymatywnej asymetrii (On the set of points of approximate asymmetry)

6.

Marian Kwapisz Uniwersytet L´ odzki 1960 Zmodyfikowana metoda kolejnych przybli˙ze´ n Picarda do rozwi¸azywania zwyczajnych r´ owna´ n r´ oz˙ niczkowych i r´ oz˙ niczkowo-r´ oz˙ nicowych (Modified Picard’s method of successive approximations for solving the ordinary differential equations and the difference-differential equations)

7.

Anna Matysiak Uniwersytet L´ odzki O granicach i pochodnych aproksymatywnych (On approximate limits and derivatives)

8.

Janina Staniszewska Uniwersytet L´ odzki 1962 O zbiorze punkt´ ow rozbie˙zno´sci szereg´ ow pot¸egowych ci¸aglych na obwodzie kola zbie˙zno´sci (On the set of divergence points of the power series continuous on the boundary of its disc of convergence)

9.

Teresa Bartczak Uniwersytet L´ odzki 1969 O punktach osobliwych Cauchy’ego funkcji wielu zmiennych (On singular points in the sense of Cauchy of the functions of several variables)

1960

´ aska 10. Lucjan Meres Politechnika Sl¸ 1979 O punktach osobliwych Pringsheima – Du-Bois Reymonda funkcji dwu zmiennych (On singular points in the sense of Pringsheim – Du-Bois Reymond of the functions of two variables) ´ aska 11. Jerzy Timmler Politechnika Sl¸ 1980 Uzbie˙znianie szereg´ ow wektorowych w przestrzeniach Rn mno˙znikami +1 lub −1 (Generating the convergent vector series in spaces Rn with multipliers +1 or −1)

Zygmunt Zahorski

Selected scientific achievements of Professor Zygmunt Zahorski∗ Edyta Hetmaniok, Mariusz Pleszczy´ nski and Roman Witula

“I like only difficult topics, the easy ones I leave for the beginners (who has done as well sometime something difficult anyway). In any case, to deal with problems more difficult than the ones that I have done. It is yet a pleasure (of struggle with difficulties) joined with an unpleasure when you are completely helpless.” Zygmunt Zahorski

Scientific creativity of Professor Zygmunt Zahorski was concentrated around the real analysis and trigonometric series. Below we present only three selected topics, in our opinion the most prestigious from among Professor Zahorski’s achievements. The most famous publication of Professor Zahorski, entitled Sur la premi´ere d´eriv´ee, Trans. Amer. Math. Soc. 69, no. 1 (1950), 1–54, which brought him the well-deserved fame and great esteem, belongs to the specially selected set of works of Polish mathematicians with the biggest number of citations (and what is more, Professor Zahorski is, after Stefan Banach, one of the most frequently cited Polish mathematician – according to the research carried out by Polish Mathematical Society). This paper made, without any doubt, a quantum jump in the direction of recognizing the properties of derivatives. In the context of a problem of characterization of the Lebesque sets for derivatives of continuous functions (that is the sets of form {x : f ′ (x) > a} and {x : f ′ (x) < a}) Professor Zahorski identified in this paper the famous descendent sequence of classes of the sets Mk , k = 0, 1, . . . , 5 and the respective descendent sequence of classes of functions Mk , k = 0, 1, . . . , 5. These classes are routinely called as the Zahorski classes. A well-known American expert in the field of real functions theory – Andrew Bruckner – has devoted to these classes one of the The present note is a translation of the extended version of text prepared for the members and sympathizers of the Student Association of Enthusiasts for the History of Mathematics and Informatics “Trisector”, functioning at the Faculty of Applied Mathematics at the Silesian University of Technology. The original text is available on webpage http://ms.polsl.pl/trysektor/materialy/ profZahorski.pdf. ∗

E. Hetmaniok, M. Pleszczy´ nski, R. Witula Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: {edyta.hetmaniok, mariusz.pleszczynski, roman.witula}@polsl.pl R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 45–50. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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E. Hetmaniok, M. Pleszczy´ nski and R. Witula

chapters in his famous monograph Differentiation of Real Functions, Springer, 1978. It is worth to mention that the Zahorski classes Mk and Mk are not only included in the archives of great mathematical achievements, but they are still the subject of discussions and investigations (we have been personally convinced about this as the participants of International Summer Conference on Real Functions Theory organized for many years by the Slovak Academy of Sciences, University of L´od´z, L´od´z University of Technology and Pomeranian University in Slupsk). The next important paper, considering Professor’s achievements, is the paper: Une s´erie de Fourier permut´ee d’une fonction de classe L2 divergente presque partout, C. R. Acad. Sci. Paris 251 (1960), 501–503. Professor S. Lipi´ nski wrote as follows: It contained the proof of Kolmogorov theorem: There exists a square integrable function, the (trigonometric – editorial note) Fourier series of which, after some rearrangement of terms, is divergent almost everywhere (in other words, the trigonometric orthogonal system is not absolutely convergent almost everywhere – editorial note), published without proof in paper A. Kolmogoroff, D. Menschoff, Sur la convergence des series de fonctions orthogonales, Math. Zeitschr. 26 (1927), 432–441. Despite numerous queries and requests for some advices, how to carry out the proof, Kolmogorov did not publish the proof and all the hints, given by him, were never enough to conduct the proof. Neither the function, nor the permutation of series were known. Finally Zahorski found them and consequences of this discovery are significant. Immediately after publication, Zahorski’s paper served as an exemplar to P. Uljanow: P.L. Uljanov, Raschodjaˇ sˇ ciesja rjady Furie, Usp. Mat. Nauk 16 (1961), 61–142 and P.L. Uljanov, Raschodjaˇ sˇ ciesja rjady po sisteme Haara i po bazisam, DAN SSSR 138 (1961), 556-559, who used Zahorki’s method for the Haar and Walsh orthogonal series, and next for any orthogonal systems complete in L2 .1 Afterwards, A.M. Olewski in his paper Raschodjaˇ sˇ ciesja rjady iz L2 po polnym sistemam, DAN SSSR 138 (1961), 545–548, did the same for the expansions with respect to any base in L2 .

The following paper by L. Tajkov deserves also for attention: L.W. Tajkow, O raschodimosti rjadow Furie po pieriestawlennoj trigonometriczeskoj sistemie, Usp. Mat. Nauk 18 (1963), 191–198. It has been proven there (with some surplus but also extraordinarily elegantly) the following theorem: there exist a couple of conjugated T functions F (x) and F˜ (x) belonging to p∈N Lp [0, 2π], the Fourier series of which, after some rearrangement (the same for both series) are divergent to ±∞ for every x ∈ [0, 2π]. Discussion and summary of the results (obtained more or less till 1982) concerning “divergence of the Fourier series” with respect to various orthogonal systems, by including and emphasizing the importance of Professor Zahorski’s paper, can be found in paper P.L. Uljanow, A.N. Kolmogorow i raschodjaˇsˇciesja rjady Furie, Usp. Mat. Nauk 38 (1983), 51–902 . Moreover, in the context of this paper it is worth to notice the, not highlighted there, result of Adriano M. Garsia (remarkable American specialist in combinatorics) which is, in some sense, dual to Professor Zahorski’s result. Thus, in paper Adriano M. Garsia, Existence of almost everywhere convergent rearrangements for Fourier series of L2 functions, Annals Math. 79, no. 3 1 Many valuable pieces of information devoted to tis subject can be found in monograph B.I. Golubow, A.W. Efimow, W.A. Skworcow, Walsh series and transformations. Theory and applications, Nauka, Moscow 1987. 2

Valuable reference is also the monograph by B.S. Kashin and A.A. Saakyan: Orthogonal series, Nauka, Moskow 1984, reviewed by the way by P.L. Uljanow. However the authors of this monograph were concentrated rather on just the contents of presented lecture and less on its connections with the source literature.

Selected scientific achievements of Professor Zygmunt Zahorski

47

(1964), 623–629 he proved that the Fourier series of every function of class L2 [−π, π] can be rearranged such that the obtained new series can be convergent almost everywhere in interval [−π, π]. And, what is more, Garsia proved (what is in fact the main result of cited paper) that the permutations on N, rearranging the given Fourier series to the series convergent almost everywhere, are, among others, the “almost all” permutations p on the set of natural numbers satisfying condition p ([mk , mk+1 )) = [mk , mk+1 ) ∞

for k = 1, 2, . . ., where m1 := 1 and {mk }k=1 is the “sufficiently fast” increasing sequence of natural numbers. Since we consider here a family of the simplest permutations on N (representing the composition of permutations on the successive intervals of N), thus the Garsia result turned on once more the green light on the road leading to the solution of the, presented below, Luzin hypothesis. ∞

Remark. Sequence {mk }k=1 , indicated above, can be dependent on specific function f , since in the proof of Garsia’s result it is only required that Smk (x, f ) → f (x) (almost everywhere) (traditionally, symbol Sn (x, f ) denotes here the n-th partial sum of the Fourier series of function f (x)). However it is known that (see volume II, pages 164 and 165 in Antoni Zygmund’s monograph Trigonometric series, Cambridge University Press, ∞ Cambridge 2002 ) if {nk }k=1 is the increasing sequence of natural numbers such that nk+1 /nk > q > 1 for k = 1, 2, . . ., then for every function f ∈ L2 [−π, π] we have Snk (x, f ) → f (x)

(almost everywhere).

For example, we have S2k (x, f ) → f (x) (almost everywhere) for every function f ∈ L2 [−π, π]. A. Zygmund payed also his attention to the fact that if f ∈ L2 [a, b], then Sn (x, f )− f (x) → 0 in L2 [a, b], which yet implies the existence of some increasing sequence ∞ {mk }k=1 of natural numbers such that Smk (x, f ) → f (x)

(almost everywhere).

Proof of this fact is also included in monograph W. Rudin, Real and Complex Analysis, PWN, Warsaw 1998 (in Polish), (see Theorem 3.12, the proof of which is carried out within the framework of proof of Theorem 3.11). The most serious problem, which Professor Zahorski confronted, was the attempt of proving the Luzin hypothesis from 1913, saying that the Fourier series of square integrable function is convergent almost everywhere. Many prominent mathematicians made the efforts of answering the question whether this hypothesis is true or not, however without success. Professor Zahorski failed as well in his “trials”, although he was working on this problem, with breaks, almost till the end of his life. In a rush – like he admitted himself – he published, also incorrect, proof of the Luzin hypothesis correctness, which for sure affected adversely his research. The Luzin hypothesis has been proven by Swedish mathematician Lennart Carleson in his, yet historical today,

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E. Hetmaniok, M. Pleszczy´ nski and R. Witula

paper: L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math. 116 (1966), 1–48. In 1992 Carleson was awarded for this result, among others, with the Wolf Prize (reprint of this paper, as well as the deep discussion of other Carleson’s scientific achievements can be found in excellent book edited by S.S. Chern and F. Hirzebruch: Wolf Prize in Mathematics, vol. 1, 91–142, World Scientific, Singapore 2000 ). The “obstacle” to being awarded with the Fields Medal for this first-class result was Carleson’s age. Carleson exceeded for 2 years the age limit – 36 years – imposed on the Fields Medal winners. It is worth to mention now, for the contrast with Carleson’s result, the Kolmogorov theorem: there exists function f ∈ L(0, 2π), the Fourier series of which is divergent almost everywhere (connection between Carleson and Kolmogorov results will recur at the end of this paper, in the final remark). Proof of the Kolmogorov theorem can be found, among others, in the first volume of mentioned above A. Zygmund’s monograph, on pages 310–314 (see also the cited above P.L. Uljanow paper from 1983 devoted anyway to the “connections” of A. Kolmogorov with the divergent Fourier series). Next, D.E. Menszov in the 30s proved the following theorem: every 2π-periodic, measurable and finite function f (x) can be expanded into the trigonometric series which is almost everywhere convergent in [0, 2π]. About this one, as well as about some other Menszov’s results in this subject, one can read in paper: A.N. Kolmogorow, S.M. Nikolski, W.A. Skworcow, P.L. Uljanow, Dmitrij Jewgeniewicz Mienszow (paper on the occasion of 90th birthday anniversary), Usp. Mat. Nauk 37 (1982), 209–219. Many facts and pieces of information presented in this paper have been elaborated, among others, on the basis of the following sources, in which some other Professor Zahorski’s results are also discussed: – J.S. Lipi´ nski, Zygmunt Zahorski’s papers on the real functions theory, Zesz. Nauk. ´ Mat.-Fiz., 48 (1986), 29–38 (in Polish, Scientific Notes of Silesian University PS., of Technology, series Mathematics-Physics). Let us add that this scientific note was the special issue of this journal dedicated to Professor Zahorski on the occasion of his 70th birthday anniversary and many famous mathematicians published there their works. For example, Professor J´ ozef Siciak published the paper (in Polish), in which the author generalized for the case of functions f : Rn → R of class C ∞ the set-theoretic characterization of the sets of singular points of these functions, given for case n = 1 by Professor Zahorski. Jean Schmets and Manuel Valdivia in their paper The Zahorski theorem is valid in Gevrey classes, Fund. Math. 151 (1996), 149–166 obtained the generalization of Siciak’s result for the functions of Gevrey class. – note entitled: Zygmunt Zahorski – an obituary, Real Anal. Exchange 23, no. 2 (1999), 359–362 (available online). – W. Wilczy´ nski, Zygmunt Zahorski and contemporary real analysis, paper in the current monograph: R. Witula, D. Slota, W. Holubowski (eds.), Monograph on ´ the occasion of 100th birthday anniversary of Zygmunt Zahorski, Wyd. Pol. Sl., Gliwice 2015.

Selected scientific achievements of Professor Zygmunt Zahorski

49

Final remarks 1. At the end it is worth to mention that the Carleson result, concerning the almost everywhere convergence of the trigonometric Fourier series of the square integrable function on (for definiteness) one-dimensional torus T = R/Z, possesses some significant generalizations: – first, R.A. Hunt in his paper On the convergence of Fourier series, 1968 Orthogonal Expansions and their Continuous Analogues (Proc. Conf. Edwardsville, 1967), 235–255, Southern Illinois Univ. Press, Carbondale, generalized Carleson’s result for the Fourier series of functions of class Lp , 1 < p < ∞; – next, N.Y. Antonov in his paper Convergence of Fourier series, East J. Approx. 2 (1996), 187–196, proved that the Fourier series of every function from the Lorentz space L log L log log log L(T) is convergent almost everywhere; – some generalization of Antonov’s result has been obtained, among others, in paper M.J. Carro, M. Mastylo, L. Rodriguez-Piazza, Almost everywhere convergent Fourier series, J. Fourier Anal. Appl. 18 (2012), 266–286. 2. A.M. Garsia in his, cited above, paper (1964, Annals Math.) showed the following general result: if {fk } is an orthonormal system of scalar functions in some Hilbert space and {ak } ∈ l2 , then there exists a permutation p : N → N such that the ∞ P ap(k) fp(k) converges almost surely (for shortening we will use hereafter series k=1

the convenient notation “a.s.”). Next, E.M. Nikishin in his paper The convergence of rearrangements of series of functions, Math. Zametki 1, no. 2 (1967), 129–136, English translation in Math. Notes 1, no. 2 (1967), 85–90, proved the following generalization of Garsia’s theo∞ P rem: if a series ξk of scalar random variables converges in measure to a random variable S and

k=1 ∞ P

|ξk |2 < ∞ a.s., then there exists a rearrangement

k=1

∞ P

ξp(k) of

k=1

this series convergent a.s. to S. Finally S. Levental, V. Mandrekar and S.A. Chobanyan in paper Towards Nikishin’s Theorem on the Almost Sure Convergence of Rearrangements of Functional Series, Funct. Anal. Appl. 45, no. 1 (2011), 33-45 proved the following theorem. Theorem. Let {kn }∞ n=1 be an increasing sequence of positive integers. If the sekn ∞ P P ξl of a series ξl of random variables (taking quence of partial sums Skn = l=1

l=1

the values from a normed space, in general) converges a.s. to a random variable ∞ P S, then there exists a rearrangement ξp(l) of the series convergent a.s. to S, l=1

provided that X

kn+1

ξl rl → 0 a.s. for n → ∞,

l=kn

where {rl }∞ l=1 is a sequence of Rademacher random variables independent of {ξl }∞ . l=1

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E. Hetmaniok, M. Pleszczy´ nski and R. Witula

Moreover, they showed that under this condition (and only under this condition) the set of {kn }−simple permutations p : N → N (which means that k1 := 1, ∞ P p([kn , kn+1 )) = [kn , kn+1 ), n ∈ N), for which the series ξp(l) converges a.s. to S l=1

has the full measure (the respective measure is defined in the final part of Professor W. Wilczy´ nski’s paper included in this monograph). It should be also emphasized that a random variable taking the values from a normed space is understood in the following way. Let (Ω, M, p) denote the underlying probability space. Let (X, ||·||) be a Banach space over R or C. A mapping ξ : Ω → X is called a random variable taking the values from X if it is Bochner measurable, i.e. ξ = lim ξn p − −a.s. where ξn : Ω → X is measurable, and takes only finitely many values for every n ∈ N. 3. Let Df := {x ∈ [0, 2π] : {Sn (x; f )} diverges} for f ∈ C(T). It is easy to prove that Df is a Gδσ set. By Carleson’s theorem Df ´ has measure zero. Professor J. Sladkowska-Zahorska in paper Sur l’ensemble des points de divergence des series de Fourier des fonctions continues, Fund. Math. 49 (1961), 271–294, proved the following partial converse. Theorem. Let B ⊆ (0, 2π) be an Fσ set of logarithmic measure zero (i.e. for each S εP> 0 there is a sequence {In } of intervals with B ⊂ In , |In | < Ln < 1 and 1/| log Ln | < ε) and let A ⊆ B be a Gδσ set. Then there exists f ∈ C(T) with Df = A. M. Ajtai and A.S. Kechris in their paper The set of continuous functions with the everywhere convergent Fourier series, Trans. Amer. Math. Soc. 302, no. 1 (1987), 207–221, used this fact to give the proof of the following theorem. Theorem. The set EC ⊂ C(T) of continuous functions with everywhere convergent Fourier series is a complete coanalytic set. In particular, it is not a Borel set in C(T).

Professor Zygmunt Zahorski – a memoir Bronislaw Szl¸ek

I have met Professor Zygmunt Zahorski in 1970 when he came to Gliwice from L´od´z and started to work in the Institute of Mathematics at the Silesian University of Technology. It is very tough for me to write about a person who has created the mathematical school. This fact as well as my respectively short acquaintance with Professor are enough to make me weigh carefully my words, especially when the noble Person, I am talking about, is dead and cannot correct my faults. Professor Zahorski was an outstanding Person, a scientist in the style of epoch which passed away a long time ago. He was characterized by the extraordinary reliability and criticism – too much quite often – for His own achievements. He was also titanicly hard-working and assiduous in pursuing his scientific goals. After obtaining the scientific results of world-wide importance (proof of the Kolmogorov hypothesis on the trigonometric series and some other results), after developing the mathematical school, for several years Professor was hardly involved in a very tough and ambitious mathematical problem concerning the orthogonal series. Although the problem has been solved by another mathematician of equally large calibre – L. Carleson, Professor did not finished his research. He wanted to achieve the set goal independently, especially because he followed a completely different path of investigations. He worked on this problem almost till the last days of his life. He did not manage to solve this, posed to himself, problem. Professor Zahorski’s work had a whiff of some kind of gambling, race against the passing time. He suffered hardly from a heart disease and on 24th October 1980 he survived a clinical death. This experience made him realize – so brutally – fragility of human life. Hence, his multiplied effort to achieve the desired scientific result arose. Unfortunately, he did not make it, he did not manage. Poor health, terrible living ordeals so often striking the people from Professor’s generation, they did not cloud over Professor’s good mood. He was a very cheerful person of specific, subtle sense of humor with a great deal of self-irony. Wonderful story-teller! He possessed the extraordinary easiness of speaking. Each of his stories B. Szl¸ek – retired senior lecturer in Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 51–52. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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B. Szl¸ek

contained a lot of digressions. Very often the listeners could have an impression that Professor lost himself in the chain of digressions, that he lost the clue. Not at all! He always came back to his main subject and all these digressions appeared to be an essential and natural supplement of his basic talk. It seems to me that one of the measures for greatness and exceptionality of a person are the anecdotes connected to this person. There is a lot of anecdotes devoted to Professor. Professor raised a numerous group of mathematicians occupying important positions at various Polish universities. All of them show the gratitude to their Master. When Institute of Mathematics in Gliwice and Polish Mathematical Society, section in Gliwice, celebrated some jubilees sacrificed to Professor (70th birthday anniversary, 80th birthday anniversary, recognition as the honorary member of Polish Mathematical Society), Professor’s alumni and collaborators from L´od´z, Gdynia, Warszawa, Wroclaw, Krak´ow and Katowice participated in these ceremonies. They also accompanied Professor on his last journey to the place of his eternal rest. Professor Z. Zahorski contributed also significantly in developing the silesian mathematical community and especially in creating the scientific centre in Gliwice. Faculty of Mathematics and Physic at the time, Faculty of Applied Mathematics at present, owes greatly its serious development to the inspiration of Professor Zygmunt Zahorski.

´ Zygmunt Zahorski and Janina Sladkowska-Zahorska

On passing away of Zygmunt Zahorski∗ Henryk Fast

I was bonded with Professor Zygmunt Zahorski by a long-standing friendship and similar interests in mathematics. I visited him always when I was in L´od´z. I met him still during my early student times and got to know him much better in the years to come. I knew him as a warm unpretentious men, totally devoid of the artificial poise so frequently found among important academics. Easily accessible, he was never too busy to stop and tell you about a mathematical problem which excited him at that moment and was equally ready to listen to your ideas and to engage into impromptu looking for solutions. And not only one could talk with him about mathematics but about personal matters as well. I asked him in my time to be one of two reviewers of my PhD dissertation, however some complications, which I don’t even remember, prevented him to do that. Among the Polish mathematical establishment of that time he was a maverick of a kind. He didn’t seem to lean toward any of the “inner groups” and was the only one among the mathematicians of stature who have not held a position at the Mathematical Institute of the Academy. By many he was considered an eccentric and perhaps, by the standards of the times, he was. I recall one of the characteristic anecdotes told later about his meeting with Kuratowski in Warsaw. He went there to collect a substantial amount of cash owed to him for something. He chattered and drank coffee unconcerned that it continued until vary late at night. When he finally left the last trains to L´od´z was already gone and he had to have a class early in the morning. Without hesitation he hired a taxi to take him from Warsaw to L´ od´z. And for the fare. . . he had to pay all the money for which he went to Warsaw. . . He was eager to attack problems of gigantic difficulty with unbelievable persistence. Years he has devoted the classical, open for many years Lusin’s problem about trigonometric series. Unfortunately without final success which belonged finally to Carleman. This memory is based on the letter to the Editorial Board of Wiadomo´sci Matematyczne [Wiadom. Mat. 40 (2014), 272-273]. Author and editors of this monograph would like to thank the Editorial Board of Wiadomo´sci Matematyczne for the permission for reprinting this letter. ∗

H. Fast e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 53–54. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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H. Fast

When in United States I was a member of some group connected with journal Real Analysis Exchange and I participated in traditional annual conferences organized by this group (held in United States as well as in other countries, in particular in Poland), I was a witness of congratulations and appreciations for Professor Zahorski’s results in the subject of differentiability. Especially from my friend Andy Bruckner, Professor in Californian University in Santa Barbara, author of a famous work concerning this subject.

From left: Henryk Fast, Borge Jessen and Zygmunt Zahorski

My memories of Professor Zygmunt Zahorski Jerzy Mioduszewski

Professor Zygmunt Zahorski moved to Gliwice in the early 1970s. He used to be in Katowice at least once a month to take part in so-called “4 o’clock Thursdays” meetings in Wieczorka Street and occasionally was a guest at the sessions that took place in Wisla both organized by Professor Mikusi´ nski. It was probably in Wisla when I met Professor Zahorski and it must have been then and there that he gave me a copy of his work written in Tˆ ohoku in 1941. That is how I got to know about a mysterious theorem by Luzin-Menchoff which in the work from Tˆohoku had its own over 4-page long proof whose significance I had not understood yet. I will mention again this yellowed copy of Professor’s work later. I used to meet Professor Zahorski earlier in Wroclaw and met him at my doctoral thesis defence as he was the reviewer of Zbigniew Ziele´zny’s doctorate at the same session of the Faculty Board. I was impressed by his straightforwardness which I was able to witness later as well. It could have been 1960 when I was invited to give a lecture at the session of the L´ od´z Branch of the Polish Mathematical Society. Professor Zahorski was a chairman and after my lecture he made some comments from which I got to know a lot about reparametrization of of the domains of functions. It was a rather far analogy to what I had previously said however such remarks were valuable especially in a face-to-face conversation. There is one more thing I remember from this particular morning. There were four people in a small room, then Doctor Stanislaw Lipi´ nski whom I was having a conversation with excused himself and left, the two other men were still engrossed in what they were talking about. Suddenly I realised that only one of them was talking and that it was Professor Zahorski. I understood that he was giving a lecture so I left the room unobtrusively. I had heard before about how passionate and selfless Professor Zahorski was when it came to teaching mathematics and now I was able to confirm how true the rumours were. Professor’s work from Tˆ ohoku included a theorem which, as I could have only suspected back then, showed how to smooth out a given function by a reparametrisation of its domain. I made some of my students interested in the topic. We needed to go back to Hobson to realise how old the problem was. One of my students went through J. Mioduszewski Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 55–56. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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J. Mioduszewski

Koepcke’s individual construction of nowhere mononotone everywhere differentiable function and then the next necessary step turned out to be Zahorski’s work together with a work by Kaplan and Slobodnik. I congratulated my student on his achievement and let him take the set of copies of the above mentioned works. However, he did not take them saying: “You wil need them more.” He was right as I returned to Zahorski’s work many times if just for pleasure’s sake. I noticed that famous Urysohn’s lemma is one simple but very particular case of lemmas that Zahorski, Maximoff and mysterious Bogomolowa had to analyse before they reached their final theorems about derivatives. Nowadays reparametrization of Cantor-Lebesgue function’s domain to make it everywhere differentiable may be found in more than one Springer magazines, however, the original key is in French 54-page long work by Zahorski published by the Transacttions AMS in 1950. Let us go back to Tˆ ohoku. Zygmunt Zahorski, before he became a doctor, had sent two works from Soviet Lvov in June 1941. One of them was sent to Japan and the other one to Matematiceskij Sbornik in Moscow. They are known to have reached their recipients in mid-July 1941. One can read in Sbornik: Zygmunt Zahorski Lw´ow, and Lvov in Tˆohoku. When Professor Zahorski dedicated his work to somebody he did it in a very complex way making his dedication into a unique souvenir. His remarks often include valuable comments. Doctor Roman Witula has in his keeping a typed copy of Professor’s Zahorski unpublished work written a few years after World War II. On the margin there is a note: “I have just got to know that Menachem Wojdyslawski was murdered by hitlerian beasts in Cz¸estochowa’s ghetto.” Zahorski got to know Wojdyslawski when he himself was Banach’s assistant. We know Wojdyslawski’s pre-war works in the field of topology, however in Lvov he and Zahorski worked together on problems in differential geometry and the above mentioned work deals with one of these particular problems. Lvov became a dangerous place to live in after June 1941 however, for Menachem Wojdyslawski it was not possible to come back to Warsaw and that is why probably he chose Cz¸estochowa’s ghetto as the seemingly safest refuge. Polish Mathematical Society gave Professor Zahorski its honorary membership. He died a few years later and was buried in Gliwice’s cemetary in a place that resembles a park rather than a necropolis.

Gliwice PhD students of Professor Zahorski and scientific backgrounds of their thesis Edyta Hetmaniok, Mariusz Pleszczy´ nski and Roman Witula

Professor Zahorski promoted in Gliwice two doctors. These are, in turn: – Jerzy Timmler, Gliwice 1980, Uzbie˙znienie szereg´ ow wektorowych w przestrzeniach Rn mno˙znikami +1 lub −1 (Generating the convergent vector series in spaces Rn with multipliers +1 or −1). ´ atkowski, A. Wakulicz. Reviewers: B. Jasek, T. Swi¸ – Lucjan Meres, Gliwice 1979, O punktach osobliwych Pringsheima – Du Bois Reymonda funkcji dwu zmiennych (On singular points in the sense of Pringsheim – Du Bois Reymond of the functions of two variables). ´ atkowski. Reviewers: J. Lipi´ nski, T. Swi¸ Let us discuss now briefly the contents of these thesis by extending slightly the objects of considerations with some additional facts and pieces of information.

1. Jerzy Timmler’s PhD dissertation For looking over the contents of J. Timmler’s PhD dissertation few essential concepts need to be introduced, inventor of which was Ernst Steinitz1 (1913, see [9, 19]). Definition 1.1. A unitP vector v ∈ Rn , bound at the origin, will P be called a direction of convergence of series uk of vectors from Rn if the series |v ◦ uk | is convergent, where ◦ denotes the inner product in Rn . 1 Ernst Steinitz, born in 1871 in Siemianowice Sl¸ ´ askie – Siemianowitz at this time (his father Sigismund Steinitz was born and worked in Gliwice (Gleiwitz at this time) which is of great symbolical importance for us – the authors). Great part of his scientific path is connected with Wroclaw (Breslau at this time).

E. Hetmaniok, M. Pleszczy´ nski, R. Witula Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: {edyta.hetmaniok, mariusz.pleszczynski, roman.witula}@polsl.pl R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 57–62. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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n Definition 1.2. If a unit vector P v ∈ R , bound at the origin, is not a direction of convergence of the given series uk , then it is called a direction of divergence of this series.

Definition 1.3. P A unit vector v ∈ Rn , bound at the origin, is called a principal direction of series uk if each open circular cone with vertex at the origin, containing vector v, is such that the sum of absolute values of terms uk , belonging to this region, is equal to ∞. Definition 1.4. Let u = {uk } be a sequence of vectors from Rn such that X lim uk = O and kuk k = ∞. By S(u) we denote the set of sums of all convergent series

∞ P

mk uk , where mk ∈

k=1

{−1, 1} for every k ∈ N. J. Timmler’s PhD dissertation is mostly devoted to the discussion of sets S(u) in case of n ≥ 2. Let us recall that from the Riemann Derangement Theorem it follows that S(u) = R always in case of n = 1. Dvoretzky and Hanani in [4] proved that S(u) is nonempty in case of n = 2. Timmler generalized this result in his PhD dissertation for the cases of any n ∈ N. Hanani in paper [7] proved, in case of n = 2, that each sequence u having at least one pair of linearly independent principal directions possesses S(u) = R2 . Furthermore, Timmler generalized this result in his PhD dissertation for the case of n = 3. More precisely, he proved by using the barycentric coordinates method that if the sequence u has at least one triple of linearly independent principal directions, then S(u) = R3 . Let us notice here that the simple adaptation of the proof of Theorem 2 from Jasek’s paper [9] enables to obtain the following results. P Theorem 1.5. If series uk of vectors from Rn is not absolutely convergent, then it has at most n directions of convergence. Moreover, the set of principal directions of this series is closed. Remark 1.6. Good supplement for the presented above subject matter can be given by Bronislaw Jasek’s2 papers [10, 11, 9]. Since majority of these results is located in the field of complex series, therefore we felt an irresistible desire to generalize these results for the series in Rn . It does not seem to be so easy, however it is very tempting for sure. Additionally we note that A. Sowa in [18] some interesting result on P obtained the property of set S(u) for the series uk in R3 possessing two principal directions and a direction of convergence all linearly independent in R3 . 2 Bronislaw Jasek (born in 1930). He defended his PhD thesis in 1962 at the Wroclaw University (supervised by Edward Marczewski). Docent in the Institute of Mathematics at the Wroclaw University of Technology since 1968. He hold the position of Dean in the Faculty of Fundamental Problems of Technology at the Wroclaw University of Technology (1968–1975), and then the position of Director of the Institute of Mathematics at the Wroclaw University of Technology. He is retired now for several years. He lives in Wroclaw. In this moment we would like to express our thanks to Professor Zbigniew Skoczylas for the information about Professor Bronislaw Jasek.

Gliwice PhD students of Professor Zahorski. . .

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In J. Timmler’s PhD dissertation the following theorem is proven as well. Theorem 1.7. For every n ∈ N there exists a positive real number C(n) such that n for every finite sequence {vk }N k=1 of vectors from R , with lengths ≤ 1, the multipliers εk ∈ {−1, 1}, k = 1, 2, . . . , N exist such that

r

X

εk vk ≤ C(n),

k=1

for every r = 1, . . . , N . n Corollary 1.8. For every sequence {vk }∞ k=1 of vectors from R , with lengths kvk k → ∞ 0, there exists a sequence {εP k }k=1 of multipliers, where εk ∈ {−1, 1} for every k = 1, 2, . . ., such that the series εk vk is convergent in Rn .

Remark 1.9. The above Theorem and Corollary have been proven independently by many authors (see, for example, Witula problem 1.2.14 in [6], Calabi and Dvoretzky [2], J. Timmler proved this theorem in [21] only in case of n = 3 but by applying the very interesting geometric considerations). √ Moreover, we know that C(1) ≤ 1, C(2) ≤ 3 (see [4], simultaneously let us notice that in paper [4] there were presented only the ideas of proof of this inequality, whereas the full proof was published by J. Timmler in paper [20]) and C(3) ≤ 12 (see [21]). Corollary 1.8 holds also in space R∞ (Y. Katznelson and O.C. McGehee [12]), however it is not true in the case of every infinitely dimensional Banach space (see [3] p. 157, Theorem 8). Remark 1.10. In reference to Corollary 1.8 let us recall the following concept (see [2]). A set S of complex numbers is called a sum factor set if for any sequence {an }∞ n=1 of complex numbers such that lim an = 0 and

X

|an | = ∞,

and for any complex number s there exists a sequence {ξn }∞ n=1 , ξn ∈ S, n ∈ N, for ∞ P which ξn an = s. n=1

H. Hornich proved in [8] that for every k ≥ 3 the set Sk of k-th roots of unity is the sum factor set. It is obvious that the set {−1, 1} is not the sum factor set. We know [2] that a bounded set M ⊂ C is the sum factor set if and only if O ∈ int(conv M ). Remark 1.11. There exists some subtle connection between Theorem 1.7 and Corollary 1.8 and the respective theorem and corollary concerning the arrangements of finite and infinite series. We present them now, but by omitting the proof of technical nature of the following theorem. n Theorem 1.12. For every n ∈ N and finite sequence {uk }N k=1 of vectors from R , N P such that uk = O, there exists a permutation p of set {1, 2, . . . , N } such that k=1

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m

v N

X

u

uX up(k) ≤ t kuk k2

k=1

(1)

k=1

for every m = 1, 2, . . . , N . Corollary 1.13. For every n ∈ N and for every sequence {uk }∞ k=1 of vectors from Rn , such that 2 n {uk }∞ (2) k=1 ∈ lR (R ) and

N X

uk = O

k=1

for infinitely many indices N ∈ N, there exists a permutation p of N such that ∞ X

up(k) = O.

k=1

Proof. (of Corollary 1.13) First we choose by induction an increasing sequence {Ni }∞ i=1 of positive integers satisfying the following conditions Ni X uk = O, (3) k=1 ∞ X

kuk k2 ≤ 4i ,

k=Ni

for every i ∈ N. By Theorem 1.12, for every i ∈ N there exists a permutation pi of set {1 + Ni , 2 + Ni , . . . , Ni+1 } such that

m

v Ni+1

X

u

u X upi (k) ≤ t kuk k2 ≤ 2i , (4)

k=1+Ni

k=1+Ni

for every m = 1 + Ni , 2 + Ni , . . . , Ni+1 . The desired permutation p of N could be defined by formula  for every k = 1, 2, . . . , N1 , k p(k) = pi (k) for every k = 1 + Ni , 2 + Ni , . . . , Ni+1 ,  and i ∈ N. Immediately from (3) and (4) it follows that

∞ P

up(k) = O.

⊔ ⊓

k=1

Remark 1.14. In the course of elaborating Theorem 1.12 and Corollary 1.13 a discussion between the authors started about the possibility of substituting condition (2) (resulting directly from inequality (1) of Theorem 1.12) with the weaker condition of the form lim uk = O. We had no any suitable proof, however completely accidentally k→∞

Gliwice PhD students of Professor Zahorski. . .

61

in paper [5] the author – Wilhelm Gross – proved (by induction but with application of some very nice geometrical considerations) the following fact (which enables to substitute condition (2) of Corollary 1.13 with condition lim uk = O): k→∞

n Theorem 1.15. For every n ∈ N and finite sequence {uk }N k=1 of vectors from R N P such that uk = O, there exists a permutation q of set {1, 2, . . . , N } such that k=1

m

X

uq(k) ≤ (2n − 1) max{kuk k : k = 1, 2, . . . , N },

k=1

for every m = 1, 2, . . . , N . Remark 1.16. In paper [12] (see Lemma 4) it is proven that for the series discussed ∞ P in Corollary 1.13 the set of sums of convergent series of the form up(k) , where p is a permutation of N, is the linear subspace of Rn .

k=1

2. Lucjan Meres’ PhD dissertation The second PhD student of Professor Zahorski, Lucjan Meres, found in his PhD dissertation the complete characterization of the set of singular points of the functions of two variables not possessing the Cauchy singularities (i.e. points singular in the sense of Cauchy). Later, in paper [15], Meres generalized this characterization for the functions of several variables (not possessing the Cauchy singularities). Results from the Meres’ PhD dissertation have been also partly published in papers [13, 14]. Generalization of the Meres theorems and the appropriate Zahorski theorem (which, in fact, constituted the groundwork of Meres’ research) has been given by J´ozef Siciak in paper [17] (professor W. Wilczy´ nski referred to this one as well in his paper included in this monograph). J. Schmets and M. Valdivia in paper [16] proved, in order to extend Siciak’s result, that for every γ ∈ (0, ∞) and n ∈ N the Zahorski theorem has a solution f : Rn → R belonging to the Gevrey class Γγ , that is to the set of f ∈ C∞ (Rn ) for which there exists a, b > 0 such that kDα f kRn ≤ a b|α| (|α|!)γ for every α ∈ Nn0 . Remark. All cited here papers published in the Scientific Notes of Silesian University ´ askiej in Polish) will be very soon of Technology (Zeszyty Naukowe Politechniki Sl¸ scanned and available online.

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Bibliography 1. Ash J.M.: Uniqueness of representation by trigonometric series. Amer. Math. Monthly 96 (1989), 873–885. 2. Calabi E., Dvoretzky A.: Convergence – and sum – factors for series of complex numbers. Trans. Amer. Math. Soc. 70 (1951), 177–194. 3. Dvoretzky A.: Some results on convex bodies and Banach spaces. In: Proc. Internat. Sympos. Linear Spaces. Jerusalem Academic Press, Jerusalem 1961, 123–160. 4. Dvoretzky A., Chojnacki Ch. (Hanani H.): Sur les changements des signes des termes d’une s´ erie a termes complexes. C. R. Acad. Sci. Paris 225 (1947), 516–518 (in the original the authors’ names were misprinted as Arye Dvoretzky and Hanani Chojnacki – the correct names of the authors are Aryeh Dvoretzky and Chaim Chojnacki - Haim Hanani). 5. Gross W.: Bedingt konvergente Reihen. Monatsh. Math. Phys. 28 (1917), 221–237. 6. Grzymkowski R., Witula R.: Complex functions and integral Laplace transformation – in examples and problems. Wyd. Prac. Komputerowej J. Skalmierskiego, Gliwice 2010 (in Polish). 7. Hanani H.: On sums of series of complex numbers. Pacific J. Math. 3 (1953), 695–709. ¨ 8. Hornich H.: Uber beliebige Teilsummen absolut konvergenter Reihen. Monatsh. Math. Phys. 49 (1941), 316–320. 9. Jasek B.: Transformations of complex series. Colloq. Math. 9 (1962), 265–275. 10. Jasek B.: Complex series and connected sets. Dissertationes Math. 52 (1966), 1–47. 11. Jasek B.: On a certain transformation of complex series. Ann. Polon. Math. 22 (1969/1970), 277–289. 12. Katznelson Y., McGehee O.C.: Conditionally convergent series in R∞ . Michigan Math. J. 21 (1974), 97–106. 13. Meres L.: On a certain family of functions of class C ∞ in the space E 2 . Demonstratio Math. 15 (1982), 19–37. 14. Meres L.: On the singular points of Pringsheim-Du Bois Reymond of a function of two variables. Demonstratio Math. 15 (1982), 297–310. 15. Meres L.: Some singular Pringsheim-Dubois-Reymond functions of many variables. Zeszyty ´ Mat.-Fiz. 43 (1985), 49–65. Nauk. PSl., 16. Schmets J., Valdivia M.: The Zahorski theorem is valid in Gevrey classes. Fund. Math. 151, no. 2 (1996), 149–166. ´ Mat.-Fiz. 48 (1986), 17. Siciak J.: Regular and singular points of C ∞ functions. Zeszyty Nauk. PSl., 127–145 (in Polish). 18. Sowa A.: Some properties of set of the points available in R3 space means of multipliers +1 and ´ Mat.-Fiz. 52 (1989), 129–136 (in Polish). −1. Zeszyty Nauk. PSl., 19. Steinitz E.: Bedingt konvergente Reihen und konvexe Systeme I. J. Reine Angew. Math 143 (1913), 128–175. ´ Mat.20. Timmler J.: Proof of some theorem given by Dvoretzky and Hanani. Zeszyty Nauk. PSl., Fiz. 30 (1978), 295–300 (in Polish). ´ Mat.-Fiz. 48 (1986), 21. Timmler J.: Some property of vector series in R3 space. Zeszyty Nauk. PSl., 203–216 (in Polish).

P´ al Erd¨ os and Zygmunt Zahorski Edyta Hetmaniok, Damian Slota and Roman Witula

Probably some Readers ask themselves what connected these two great mathematicians (who were of the same age, coincidentally). Taking into considerations the interests of P. Erd¨ os (see for example [1, 10], and everyone is quite well informed that, above all, it is about the discrete mathematics and the number theory) our suspicions fall on the trigonometric series and that is pretty right. Professor Zahorski formulated in 1957 in Colloquium Mathematicum (see [12]) the problem of determination of the best possible estimation from above of the following integral Z2π | cos k1 x + cos k2 x + . . . + cos kn x| dx, 0

where 0 < k1 < k2 < . . . < kn are integers. He observed that by applying the Schwartz inequality √ and by using the orthogonality of trigonometric system we get the estimation c n. Moreover, Zahorski conjectured that c log kn is valid as well. P. Erd¨os solved Zahorski’s problem and the conjecture. In paper [4] published in 1960 in Colloquium Mathematicum he obtained the following two results. At first, Erd¨os proved that for every ε > 0 there exists c = c(ε) > 0 such that Z2π X n 1 2 cos k x dx > c n 2 −ε , 0

k=1

for every n ∈ N. Secondly, Erd¨ os proved the existence of increasing sequence {ki }∞ i=1 of natural numbers such that Z2π X n p √ p cos ki x dx = π kn + o( kn ) 0

i=1

E. Hetmaniok, D. Slota, R. Witula Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: {edyta.hetmaniok, damian.slota, roman.witula}@polsl.pl R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 63–66. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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√ which proves that O( kn ) is the best estimation (so Zahorski’s conjecture was not true). We note that from Zahorski’s observation it follows that lim sup

kn < ∞. n

Four redactional remarks 1. Let n ∈ N, a1 , a2 , . . . , an−1 ∈ R if n > 2 and at least one ak 6= 0. Then the function f (x) = cos nx + an−1 cos(n − 1)x + . . . + a2 cos 2x + a1 cos x takes the positive as well as the negative values (the elementary proof is given in [9], see Theorem 1.7 and Remark 1.8). This fact follows also easily from the classical Sturm-Hurwitz Theorem on the lower bound of the number of roots of trigonometric polynomial (see [5, 7, 11]). Theorem (C. Sturm, 1836, A. Hurwitz, 1903). Let f (x) =

N X

(ak cos kx + bk sin kx),

k=n

where n, N ∈ N and ak , bk are real numbers. Then the number of sign changes of f is at least equal to 2n. 2. Let n ∈ N. Let us put cn := min

(

n X

)

| cos kx| : x ∈ R .

k=1

Then cn = ⌊n/2⌋ for every n > 2 with exceptions √ 4 X 3 π 0 and for all N ∈ N, N ≥ 3, we have  1/8 Z2π X N log N i nj x c e dx > k , j log log N j=1

(1)

0

whenever nj are the distinct integers (also negative) and cj ∈ C satisfy condition |cj | ≥ 1. Corollary 1. For every increasing sequence of nonnegative integers n1 < n2 < ... < nN we have  1/8 Z2π X N log N cos n x dx > k , (2) j log log N j=1 0

for some k > 0 and N ∈ N, N ≥ 2. Corollary 2. For every increasing sequence {nj }∞ j=1 of nonnegative integers we obtain Z2π X N lim cos nj x dx = ∞. N →∞ j=1 0

Harold Davenport in [3] proved that power 1/8 in the right hand side of (1) (and, in consequence, in (2) as well) could be replaced by 1/4 and the constant k by number 1/8. 4. In 2014 year Ferenc M´ oricz published in Notices of the AMS a very interesting and beautiful paper [8] dedicated to the memory of P´al Erd¨os on the 100th anniversary of his birthday. Subject matter, discussed in this paper, especially the generalized Rademacher-Menshov maximal moment inequality is directly connected with the object of our note. Extremely intriguing is also the description by M´oricz of some Erd¨ os theorem on the convergence a.e. of trigonometric series satisfying the so-called (B2 ) Erd¨ os condition, which is the generalization of the Kolmogorov Theorem [6] on the convergence a.e. of the lacunary trigonometric series. It suggests another connotation to the Kolmogorov – Zahorski mathematical relations (especially in reference to other Kolmogorov Theorem proved by Zahorski in paper [13]).

Bibliography 1. Bollobas B.: To prove and conjecture: Paul Erd¨ os and his mathematics. Amer. Math. Monthly 105, no. 3 (1998), 209–237. 2. Cohen P.J.: On a conjecture of Littlewood and idempotent measures. Amer. J. Math. 82 (1960), 191–212.

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3. 4. 5. 6.

Davenport H.: On a theorem of P.J. Cohen. Mathematica 7 (1960), 93–97. Erd¨ os P.: About an estimation problem of Zahorski. Colloq. Math. 7, fasc. 2 (1960), 167–170. Katriel G.: From Rolle’s Theorem to the Sturm-Hurwitz Theorem. ArXiv:math/0308159. Kolmogorov A.N.: Une contribution ` a l’´ etude de la convergence des s´ eries de Fourier. Fund. Math. 5 (1924), 96–97. Martinez-Maure Y.: Les multiherissons et le theoreme de Sturm-Hurwitz. Arch. Math. 80 (2003), 79–86. M´ oricz F.: Moment inequalities for maxima of partial sums in probability with applications in the theory of orthogonal series. Notices Amer. Math. Soc. 61, no. 6 (2014), 576–585. Palarz M.: Trigonometric polynomials. Graduation dissertation supervised by R. Witula, Pol. ´ Gliwice 2003 (in Polish). Sl., S´ os V.T.: Paul Erd¨ os, 1913–1996. Aequationes Math. 54 (1997), 205–220. Tabachnikov S.: Proofs (not) from the Book. Math. Intelligencer 36, no. 2 (2014), 9–14. Zahorski Z.: Problem 168. Colloq. Math. 4 (1951), 241. Zahorski Z.: Une s´ erie de Fourier permut´ ee d’une fonction de classe L2 divergente presque partout. C. R. Acad. Sci. Paris 251 (1960), 501–503.

7. 8. 9. 10. 11. 12. 13.

Jan Lipi´ nski and Zygmunt Zahorski

Profesora Zygmunta Zahorskiego „wykład” o pochodnych – przygotowany do publikacji przez sprawcę zamieszania Romana Witułę Zygmunt Zahorski Roman Wituła – komentarze i uwagi

Pani profesor Janinie Śladkowskiej-Zahorskiej – chciałem złożyć szczere wyrazy wdzięczności za pomoc: w zgromadzeniu bibliografii i uzupełnieniu faktografii. Panu profesorowi Jerzemu Mioduszewskiemu (opiekunowi całego przedsięwzięcia) podziękowania za tradycyjnie ogromny: zapał i zaangażowanie, a także inspirującą dyskusję i dodatkową literaturę, które pozwoliły nadać pracy ostateczny kształt.

Wstęp Problem, a raczej pytanie, z jakim zwróciłem się do profesora Zahorskiego zimą 1986/87, dotyczyło dosyć ogólnie przeze mnie rozumianego „opisu” mnogościowotopologicznego pochodnych. Byłem wówczas świeżo upieczonym magistrem matematyki i moja wiedza w tym temacie była raczej „standardowa”. Po krótkim okresie czasu profesor przekazał mi list – odpowiedź, z którym postanowiłem podzielić się wreszcie z innymi, mam nadzieję, że równie ciekawymi jego treści jak ja wówczas. Uwaga. Na marginesach odpowiednich stron zamieszczone są komentarze uzupełniające i uaktualniające wybrane treści z listu profesora. Ponadto, po liście profesora zamieszczono jeszcze zestaw kilku dłuższych informacji uzupełniających oraz bibliografię.

Roman Wituła Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 67–80. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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Z. Zahorski, R. Wituła

List Profesora Otóż, moja praca z Transaction of the American Mathematical Society wydrukowana w roku 1950, a zrobiona w zasadzie w latach 1939-42, zaczyna się słowami „charakteryzacja przy pomocy własności topologicznych i metrycznych klasy funkcji ciągłych, mających wszędzie pierwszą pochodną, nie jest znana”. Nie ważne tu, czy charakteryzować funkcję ciągłą, czy jej (na ogół w wielu punktach nieciągłą) pochodną. Ja dążyłem do charakteryzacji pochodnej wszędzie istniejącej. Przeciwnie niż w przypadku funkcji analitycznych, tu trudniejsze są funkcje nieograniczone, więc postawiłem takie etapy: I – pochodna ograniczona, II – pochodna skończona, III – pochodna (w niektórych punktach) nieskończona – znany był od dawna (ale chyba > 1900) dowód, że tych ostatnich musi być zbiór miary 0, więcej, ma to miejsce bez Jest to wynik założeń istnienia pochodnej (w innych punktach), a nawet nie potrzeba ciągłości, zaś Banacha [7]. pochodną wystarczy rozważać jednostronną: zbiór wszystkich punktów, w których istnieje prawostronna pochodna nieskończona dowolnej funkcji (nawet niemierzalnej) jest miary Lebesgue’a 0. Co do dowodu Dla funkcji ciągłej Weierstrassa nigdzie nieróżniczkowalnej, w każdym punkcie tego faktu, to wynika on, jedna pochodna Diniego = +∞, inna −∞, ale pochodne Diniego nie są pochodnymi, przykładowo, z oryginalnego podobnie jak granica górna i dolna ciągu nie są (na ogół) granicą ciągu. dowodu Weierstrassa Jeśli idzie o pochodną prawie wszędzie istniejącą (tj. z wyjątkiem zbioru punktów nieróżniczkowalności tej miary (L) 0) problem jest łatwiejszy i został rozwiązany w roku 1912 (a wydrukofunkcji, zamieszczonego wany w 1915 czy 1916), w pracy doktorskiej Łuzina po rosyjsku („Całka i szereg po raz pierwszy w liście trygonometryczny”), co do nazwy – to również za czasów carskich doktorat był w RoWeierstrassa do Du Bois sji II stopniem naukowym, odpowiadającym naszej habilitacji. Prace habilitacyjne są Raymonda (1875). W tym zresztą wszędzie różne, nawet na tej samej uczelni – ta była epokowa, nie z tym jedtemacie wydaje się być jeszcze nym problemem, ponadto postawiono tam kilka innych problemów, co najmniej jedna istotne tw. Banacha hipoteza (że szereg Fouriera funkcji całkowalnej z kwadratem jest zbieżny prawie wszę(1931), którego treść dzie), po przykładach Kołmogorowa z roku 1922 szeregu Fouriera funkcji całkowalnej przytoczymy w punkcie 3 w I potędze, rozbieżnego prawie wszędzie – drukowane w polskim czasopiśmie Fund. informacji uzupełniających Math. chyba w 1923 i takiegoż szeregu też dla funkcji z L1 , rozbieżnego wszędzie, na końcu pracy. 1926, druk. w C. R. Akademii Paryskiej w tymże roku, hipoteza Łuzina stała się Zob. prace [33, 55] mało prawdopodobną. Tak wybitny specjalista w szeregach trygonometrycznych jak Antoni Zygmund mówił w roku 1960, że to na pewno jest fałszywe, bo w L2 typową jest zbieżność w metryce L2 (twierdzenie Riesza-Fischera 1904) a nie prawie wszędzie. Mieńszow podał przed 1936 przykłady układów ortogonalnych ograniczonych z rozbieżnością pr. wsz. w L2 , a Kołmogorow postawił więcej niż hipotezę w 1926, wydrukowaną w 1927 w niemieckim czasopiśmie Mathematische Zeitschrift (w pracy wspólnej z Mieńszowem), że może to być nawet układ trygonometryczny, ale odpowiednio przestawiony. Mianowicie napisał, że umie podać taki przykład, ale nie podał tam, ani nigdzie później dowodu, ani funkcji – lub co równoważne, współczynników szeregu, ani sposobu przestawienia wyrazów (permutacji w ciągu nieskończonym). Wielu matematyków (rosyjskich, węgierskich, amerykańskich i zapewne innych) próbowało to zrobić, udało mi się to w roku 1960, może po trzech tygodniach pracy (i trochę w 1954), ale przydała mi się do tego wieloletnia bezskuteczna praca

Zob. pracę [56].

Profesora Zygmunta Zahorskiego „wykład” o pochodnych. . .

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właśnie nad hipotezą Łuzina. W 1954, też dość szybko, zdawało mi się, że mam Zapewne chodzi tu o tw. konstrukcję, ale zauważyłem błąd przed samym referatem na PTM, referat, co prawda, pani G. C. Young, zrobiłem, ale z rzeczy całkiem innej i znanej, choć nie wszystkim uczestnikom – o tzw. zob. [47, 17] (podobny błąd twierdzeniu Younga o symetrii (dla dowolnej funkcji punktu na osi x o wartościach rodzajowy występuje w liczbowych, zbiory granic f (xn ) prawostronnych, gdy xn → x, xn > x i lewostronnych, „nieśmiertelnej” monografii xn < x, są identyczne i jedna z nich = f (x), czy wszędzie? Nie koniecznie, ale z [38], gdzie na str. 181 wyjątkiem zbioru x-ów najwyżej przeliczalnego – to lepiej niż miary 0. powinno być „Twierdzenie Skrócony, ale jasny dla specjalistów dowód, oczywiście z funkcją i permutacją, Denjoy-YoungSaksa” oraz zgłosiłem w 1960 w C. R. Acad. Paris (drukują w ciągu trzech tygodni, ale 1–5 stron, „Twierdzenie SierpińskiegoYoung”. anonsy bez dowodu lub bardzo skrócone dowody). Profesorowi Zygmundowi, który był wtedy parę dni w Warszawie (stale mieszka Zob. pracę [58]. w Chicago), powiedziałem, że od 1945 roku wierzę w hipotezę Łuzina i nie wprawia mnie w wątpliwość prawdziwość hipotezy Kołmogorowa, że dla układu przestawionego może być inaczej, a w normalnym porządku 0, 1, 2, 3, 4, . . . właśnie tak, jak przewidywał Łuzin. Czego anons ogłosiłem w 1961 roku też w C. R. Acad. Paris, ale przed zredagowaniem na czysto i zreferowaniem w Instytucie PAN w Warszawie u prof. S. Mazura – zmarł w listopadzie 1981 roku. W czasie redagowania znalazłem błąd, zawiadomiłem Mazura, a że notę w C. R. już wydrukowali, ogłosiłem w Math. Reviews za pośrednictwem prof. Zygmunda, że rozwiązania nie ma, omyłka. Tym niemniej hipoteza Łuzina okazała się prawdziwą, udowodnił to w roku 1966 szwedzki matematyk Zob. pracę [18]. L. Carleson, chyba z 10 lat młodszy ode mnie. Pracował podobno ok. 7 lat w dobrych warunkach – na amerykańskim stypendium na Uniwersytecie Stanforda w Kalifornii. Miał jeden z głównych referatów na międzynarodowym Kongresie w Moskwie w 1966, przewodniczył na tym referacie Kołmogorow. Sam Łuzin nie dożył dowodu swojej hipotezy, zmarł w Moskwie 28 lutego 1950. A oto wynik Łuzina (udowodniony konstrukcyjnie w wyżej wymienionej jego pracy – zrobiono przedruk w 1950 roku z różnymi komentarzami i pracami autorów da- Zob. pracę [39]. jących rozwiązania niektórych problemów Łuzina lub podobnych; miałem tę książkę, ale zginęła w czasie przeprowadzki do Gliwic w 1970 roku). Na to, aby g(x) była prawie wszędzie pochodną funkcji ciągłej potrzeba i wystarcza, Dowód tego tw. można znaleźć aby g(x) była mierzalna (L) i prawie wszędzie skończona. także w

monografii [14].

Jednak owej funkcji ciągłej Łuzin nie nazywa pierwotną czy całką nieoznaczoną funkcji g. Bo jest wysoki stopień nieoznaczoności – różne funkcje „pierwotne” nie Pierwszy przykład różnią się o stałą. Wprawdzie jedną z „pierwotnych” konstruuje, ale to wcale nie jest funkcji, których różnica jednoznaczne. Korzysta w tym ze swojego twierdzenia, że dla funkcji mierzalnej na nie jest stała na danym odcinku istnieje zbiór domknięty miary różniącej się o mniej niż ε od długości tego przedziale, ale mających odcinka (ale oczywiście na ogół mniejszą niż ta długość), na którym f jest relatywnie wszędzie w tym przedziale ciągła; można, opuszczając punkty izolowane, mówić, że jest to zbiór doskonały, to równe pochodne jest domknięty i w sobie gęsty. podał Hans Hahn [27]. Inny Ktoś referował ten wynik na moim seminarium na Uniwersytecie Łódzkim, tym przykład takich funkcji podał niemniej niewiele pamiętam i jeślibym nawet potrafił zrekonstruować funkcję „pier- S. Ruziewicz (zob. [45, 46]). wotną” Łuzina, to zajęłoby mi to ze trzy miesiące czasu solidnej roboty. Idzie to jakimiś funkcjami określanymi na zbiorach podobnych do zbioru Cantora, ale dodatniej miary. Konieczność tych warunków jest względnie łatwa: prawie wszędzie skończona, bo pochodna, jak wyżej zaznaczyłem, może być +∞ lub −∞ tylko na zbiorze miary 0.

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Chodzi zapewne o pracę [57].

Wersja angielska monografii Saksa to [47]. Profesor Zahorski odwołuje się tu oczywiście do polskiej wersji [48] tej książki.

Z. Zahorski, R. Wituła

A mierzalność p.w. pochodnej funkcji ciągłej (a nawet każdej z czterech pochodnych Diniego) udowadnia się dość łatwo, chyba nawet bez założenia ciągłości, pisałem coś o tym w Roczniku PTM (Annales) w roku 1952, ale niezbyt pamiętam co, było to zapewne wtedy nowe. Przy okazji: jeśli nawet pochodna istnieje wszędzie, to gdy np. jest = +∞ na zbiorze mocy continuum tró ×udno (czemu było ó, chyba z pośpiechu, wprawdzie ortografii nie szanuję, ale i nie zwalczam) mówić o funkcji pierwotnej, np. dość łatwo jest zrobić dwie funkcje ciągłe, mające wszędzie równe pochodne, skończone poza zbiorem Cantora, +∞ na zbiorze Cantora, nieróżniące się o stałą. Wszelkie uogólnienia funkcji pierwotnej, np. w drugiej połowie „Zarysu teorii całki” Saksa [47] (wydanie polskie 1930, francuskie – przekład, różniący się tylko jednym rozdziałem w samym środku, zaś angielskie, chyba z 1937 zupełnie inne i obszerniejsze, nieznane mi bliżej, bo nie znam angielskiego), zakładają zawsze, że pochodna jest niemal wszędzie skończona, to jest z wyjątkiem zbioru co najwyżej przeliczalnego, a prawie wszędzie znaczy z wyjątkiem zbioru miary 0 – który jak wiadomo może być nieprzeliczalny, a nawet mocy continuum. Nie dlatego, żeby pochodna musiała być niemal wszędzie skończona, jak widać ze wspomnianego przykładu ze zbiorem Cantora, lecz dlatego, że inaczej trudno mówić o funkcjach pierwotnych różniących się o stałą, jest to warunek wystarczający, a mniej krępującego nie znam. Przy tym gdy pochodna jest ograniczona, do znalezienia funkcji pierwotnej wystarcza całka Lebesgue’a. Nawet gdy pochodna nie istnieje na zbiorze mocy continuum; trzeba tylko, aby istniała prawie wszędzie, bo funkcja spełniająca warunek Lipschitza – nawet mocniej, różnica dwóch funkcji monotonicznych – ciągłych lub nie – ma prawie wszędzie pochodną skończoną całkowalną (L), choć ta całka (jako funkcja górnej granicy) na ogół różni się od funkcji różniczkowalnej o dwa składniki – tzw. funkcję nieciągłości – tu są one I rodzaju i tylko dla takich się ją określa, oraz – nawet gdy jest ciągła, o tzw. funkcję osobliwości. Dopiero gdy jest absolutnie ciągła (nie chodzi o |f |, choć gdy f jest absolutnie ciągła, to |f | też), to funkcja osobliwości jest = 0 dla każdego x. Ale funkcje Lipschitza są absolutnie ciągłe, a całka funkcji mierzalnej ograniczonej oczywiście spełnia warunek Lipschitza, przy tym każda funkcja mierzalna ograniczona jest całkowalna (L) (po odcinkach długości skończonej), więc też wszystko się zgadza. Niektóre funkcje nieograniczone też są całkowalne (L), oczywiście spośród funkcji mierzalnych (L) i zawsze, obojętnie czy f jest ograniczona czy nie, a nawet dla funkcji nieskończonych na zbiorze mocy continuum i miary 0 (gdyby był miary > 0, to całka Rx d f (t)dt = f (x) prawie wszędzie. Całki (L) dotyczy pierwsza L nie istnieje), jest dx a

połowa wyżej wym. „Zarysu teorii całki” Saksa. Druga połowa dotyczy całek znacznie ogólniejszych niż Lebesgue’a, mianowicie Perrona i Denjoy. Bo niestety, gdy f ′ (x) jest nieograniczona, nawet wszędzie istniejąca i wszędzie skończona (oczywiście mieRx rzalna) może nie być całkowalną (L). Wtedy, z całek oznaczonych f ′ (t)dt = F (x), a

rozwiązuje sprawę dopiero wyższa całka Denjoy, równoważna całce Perrona, czego dowód jest w II połowie Zarysu Saksa. Całki te były zdefiniowane nie tylko niezależnie, ale zupełnie inaczej i dopiero później inni autorzy udowodnili dwa twierdzenia – chyba I, że definicja (węższa) Denjoy obejmuje całkę Perrona, a II – że i na odwrót. Definicje opisowe Denjoy są podobne do opisowej definicji Lebesgue’a, z tą różnicą że zamiast pojęcia funkcji absolutnie ciągłej, AC, używa się ACG (abs. ciągłe uogól-

Profesora Zygmunta Zahorskiego „wykład” o pochodnych. . .

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nione) w definicji węższej całki D, zaś ACG∗ (abs. ciągłe uog. w sensie szerszym) w definicji szerszej całki D∗ . W definicji D używa się pochodnej aproksymatywnej, dla D∗ – zwykłej (dla ∗ „powinno” być bardziej udziwnione, ale tutaj jest tradycyjnie na odwrót). Definicja opisowa jest jednak nie wiele warta bez konstruktywnej, bo nie gwarantuje istnienia rzeczy definiowanych, a ta jest dla całek D i D∗ „przerażająca”, są to właściwie konstrukcje całek coraz wyższych klas porządkowych skończonych lub przeliczalnych (podobnie jak hierarchia funkcji Baire’a i zbiorów Borela) wychodząc od całki Lebesgue’a jako nr 0. Otóż przejścia do wyższych klas odbywają się przy pomocy 1) procesu Cauchy’ego – robienia całek niewłaściwych dla punktów izolowanych. Wiadomo, że całka L obejmuje całkę R. Ale tylko właściwą, lub niewłaściwą bezwględnie zbieżną. Warunkowo zbieżna całka niewłaściwa R wychodzi poza całkę L (ale mieści się w I klasie całek D); 2) procesu Harnacka – który wolę pominąć, wtedy, kiedy przeszkadza całkowalności w niższej klasie zbiór doskonały nigdziegęsty, taki że w przedziałach jego dopełnienia całkowalność jest, a w całym odcinku (z nich złożonym plus ów zbiór nigdziegęsty) – nie ma. Ale po samym tym zbiorze też jest. Perron obszedł to zupełnie inaczej, przez tzw. funkcje zwyższające i zniżające, chyba coś analogicznego jest w równaniach różniczkowych zwyczajnych, którymi się też zajmował, zresztą szukanie funkcji pierwotnej jest najprostszym równaniem różniczkowym – tyle że tam nie brak innych komplikacji, więc aby nie było ich za dużo, zakłada się ciągłość pochodnej, nieładnie powiedziane, szuka się rozwiązań o klasie C1 , z pochodną ciągłą. Rosjanie nazywają szerszą całkę Denjoy całką Chinczina, który podał taką definicję niemal jednocześnie, ok. 1916. W pisowni zagranicznej figuruje on jako Khintchine, diabli wiedzą jakiej narodowości, kiedy to zwyczajnie . Ale nie wiem, czy znaczenie słowa jest takie samo jak po polsku, bo po rosyjsku chińczyk to kitajec. Całki Perrona i Denjoy (węższa) służą nie tylko do szukania funkcji pierwotnych dla pochodnych istniejących wszędzie, lub (w klasie ACG) prawie wszędzie, zresztą istnieją funkcje mierzalne niecałkowalne nawet w sensie Denjoy szerszym, lecz do Rx d szukania całek oznaczonych funkcji całkowalnych w tym sensie, przy tym dx f (t)dt = a

f (x) prawie wszędzie dla całki węższej,

d dx

apr. dla szerszej, obie dają dla

Rx

f (t)dt =

a

F (x) zwykłą ciągłość względem x, a nawet więcej – ACG lub ACG∗ . Całka Denjoy szersza daje pierwotną od pochodnej aproksymatywnej, co lepiej pominąć. Nawiasem mówiąc, Denjoy zdefiniował pewną całkę oznaczoną wychodzącą poza D i D∗ specjalnie dla szeregów trygonometrycznych. Definicji tej nie znam, nigdy jej nie widziałem i wątpię, czy potrafiłbym rozwiązać ten problem (który być może ma więcej niż jedno rozwiązanie – np. czy każda definicja szersza byłaby dobra, węższa raczej nie, ale i to wątpliwe). O problemie tym wspomina i Łuzin w swojej pracy doktorskiej, ale go nie rozwiązał i może – nie pamiętam – szkicowo tylko sprecyzował. Chodzi o to: ∞ P Załóżmy, że szereg trygonometryczny a20 + (an cos nx + bn sin nx) jest wszędzie n=1

zbieżny do funkcji skończonej f (x), oczywiście I kl. Baire’a. Według twierdzenia dowiedzionego jeszcze przez Cantora w 1870 roku, współczynniki an , bn są określone jednoznacznie przez funkcję f (ściślej Cantor dowiódł, że gdy f (x) = 0 dla każdego x, to wszystkie an , bn = 0, co oczywiście równoważne – ale dowód choć tzw. elementarny

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Z. Zahorski, R. Wituła

nie jest zbyt łatwy – jest on np. w „Zasadach rachunku różniczkowego i całkowego” S. Kowalewskiego, Niemca, w przekładzie I. Rolińskiego. Gdy w roku 1928 w VII klasie podstawówki czytałem Kowalewskiego, nie wiedziałem że nieboszczyka Rolińskiego poznam osobiście po roku 1948, w Łodzi; był to zasłużony nauczyciel szkół średnich i popularyzator, oraz jak widać tłumacz, po wojnie prof. Wyż. Szk. Pedag., a po przyłączeniu jej do Uniw. Łódzkiego również prof. U.Ł., własnych wyników nie miał, ale był to całkiem fajny gość, zrównoważony, mądry, z humorem i tzw. dobrymi manierami, nie despotyczny względem studentów i pracowników niższej rangi, no i szwagier ówczesnego biskupa łódzkiego Rozwadowskiego). Problem: jak zdefiniować całkę, aby współczynniki an , bn dawały się wyznaczyć znanymi wzorami Eulera-Fouriera z całką według tej definicji? Zob. także Po Cantorze uogólniano to twierdzenie o jednoznaczności, najpierw gdy wiadomo, punkt 5 informacji uzu- że f (x) = 0 z wyjątkiem zbioru najwyżej przeliczalnego, że nie można było rezygnować pełniających. ze zbioru miary > 0 to jasne, ale przy zbiorach miary 0 wyszły kłopoty – dla jednych, tzw. U , twierdzenie było prawdziwe, dla innych, M , fałszywe. Co do zbiorów Szereg prac o zbiorach U (unicite – jednoznaczność) i M napisali A. Rajchman, U oraz M , to proponujemy doc. Uniw. Warsz., zabity w Dachau chyba w 1941, Zygmund, pani N. Bari, Rosjanka zaglądnąć do [59, 9] oraz o francuskim czy włoskim nazwisku, autorka monografii o szeregach tryg. konkuren[32]. cyjnej z monografią Zygmunda, ale po rosyjsku – nie wiem, czy amerykanie ją przeIstnieje angielskie tłumaczyli, sama autorka w roku 1961, mając lat 60 i prawie niewidoma, ale często tłumaczenie, zob. [9]. chcąca chodzić bez prowadzenia, w czasie Zjazdu Matem. (krajowego) w Leningradzie wlazła pod tramwaj czy pociąg elektryczny, no i przede wszystkim Mieńszow, chyba jeszcze żyje, ale lat ma co najmniej 85. Wiadomo, że gdy f jest całkowalna (L), to współczynniki wyrażają się całkami (L), więc w szczególności gdy całkowalna (R) – całkami (R). Ale co będzie, gdy f nie jest całkowalna (R)? Denjoy podał ok. 1923 roku potrzebną tu definicję całki (i udowodnił wzory). Czy dotyczą one i wyjątku przeliczalnie wielu punktów – nie wiem, a nieprzeliczalne groziłyby czymś gorszym może niż zbiory M , co nie znaczy że nie trzeba próbować. Przypuszczam, że do dziś nie jest znana charakteryzacja zbiorów U (miary 0) lub co równoznaczne, zbiorów M miary 0, sam się tą problematyką nie zajmowałem. Zob. punkt 2 Sam Euler swoje wzory udowadniał, ale gruntownie błędnie, o co nie można mieć informacji uzupełniających. pretensji, ani nie było porządnej definicji całki, a nawet pochodnej (= iloraz „nieskończenie małych” przyrostów, mówiono wtedy, nie stosując pojęcia granicy) ani wystarczająco szerokiego pojęcia funkcji (wprowadził je Dirichlet w 1837 roku i podobno jednocześnie Łobaczewski) – pierwszą porządną definicję całki oznaczonej podał Cauchy i to tylko dla funkcji ciągłych, po roku 1800, a Riemann przeniósł ją ok 1850 na te funkcje nieciągłe, dla których się da stosować, od czasów Lebesgue’a scharakteryzowanych jako ograniczone i prawie wszędzie ciągłe. Euler był ścisły w pracach z teorii liczb całkowitych, zaś w analizie, wtedy z konieczności nieścisłej, zrobił bardzo dużo i miał Bardzo nosa – uzyskiwał wyniki na ogół poprawne mimo nieścisłości, pierwszorzędna intuicja. dobrymi nawiązaniami Wobec błędności jego dowodu teoria szeregów tryg. poszła w 2 kierunkach: 1) rezygnado historii szeregów cja z dowodu, co można poprawnie zrobić przez przeniesienie do definicji: szereg z tak Fouriera i zagadnienia obliczanymi przy pomocy funkcji f współczynnikami nazywany szeregiem Fouriera „pierwszeństwa” są prace funkcji f (tradycyjnie nazwy bywają niesłuszne; szeregi Fouriera wprowadził ten au[28, 12, 20, 60]. tor w 1822 w książce o równaniu przewodnictwa ciepła (cząstkowym), d’Alembert przed 1800 rozważając (też różniczkowe cząstkowe) równanie drgań struny, zaś Euler

Autorem owych „Zasad ...” jest Gerhard Kowalewski (1876-1950). Zobacz też punkt 4 informacji uzupełniających.

Profesora Zygmunta Zahorskiego „wykład” o pochodnych. . .

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ok. 1750 rozważając zjawisko okresowe np. astronomiczne; czyli Euler najwcześniej, Fourier najpóźniej), nie troszcząc się (jak chciał Euler), czy jest zbieżny i to właśnie do f . Nie oznacza to wykręcenia się sianem od problemu, tylko przenosi się on gdzie indziej – później można badać, czy jest on zbieżny do f i nawet jeśli nie, to jak z niego znaleźć f – np. okazało się (dowód Lebesgue’a), że tzw. pierwsze średnie arytmetyczne sum częściowych dążą prawie wszędzie do f i to w L1 – wcześniej Fejer udowodnił to w punktach ciągłości, i jednostajną zbieżność I średnich na całej osi x, gdy f jest wszędzie ciągła i okresowa z okresem 2π; obydwa te twierdzenia należą do analizy elementarnej, rzecz w tym, że funkcja z L1 może nie mieć ani jednego punktu ciągłości, ale prawie wszystkie punkty są jej tzw. punktami Lebesgue’a – definicję pomijam i w nich właśnie jest ta zbieżność. W L2 ma miejsce (dla każdego rozwinięcia ortogonalnego, nie tylko tryg.) zbieżność w sensie odległości całkowej → 0, tj. w metryce przestrzeni Hilberta L2 , wtedy wprawdzie ciąg sum częściowych jest tylko zbieżny wg miary (def. pomijam), a nie prawie wszędzie, ale można z niego wyjąć podciąg zbieżny prawie wszędzie – w układzie tryg. było dawno wiadomo, że wystarczy tu S2n , teraz (od 1966) wiadomo, że . . . sam Sn , tj. cały ciąg S. Chodzi tu o wynik Drugi kierunek poszedł w stronę jednoznaczności. Przecież Riemann zrobił właści- L. Carlesona [18]. Więcej wie poprawny dowód twierdzenia Eulera i to dla najprostszej funkcji, równej wszędzie informacji można znaleźć 0, bo nawet dla takiej brakowało i ten Eulera byłby i tu błędny. Ostatnie słowo dotąd w [29] (zob. również [10, 11, stanowi wspomniana praca Denjoy. On sam zmarł niedawno w wieku lat 90, ostatnio 26, 22]). widziałem go w Bułgarii (Warna) w 1967. Teraz znów powrót do właściwego tematu. Otóż w wymienionej na początku Klasy Mk i M zwane są, pracy [56] zdefiniowałem 6 klas zbiorów Mk , k = 0, 1, . . . , 5 i 5 klas funkcji Mk , co koczywiste, klasami k = 1, 2, . . . , 5, oraz klasę J = funkcje I kl. Baire’a przechodzące w każdym prze- Zahorskiego; z kolei prof. dziale przez wszystkie wartości pośrednie, co nazywa się własnością Darboux. Mają Zahorski użył litery M (odp. ją (analiza elementarna) wszystkie funkcje ciągłe, ale nie tylko – mają ją i funkcje M) od inicjału imienia swojej aproksymatywnie ciągłe, też nie tylko – również pochodne istniejące wszędzie, nawet ówczesnej sympatii. jeśli nie są apr. ciągłe. Dla pochodnych dowód jest łatwy. Wystarczy dowieść, że gdy Piękne f ′ (a) > 0, f ′ (b) < 0, (lub odwrotnie), a < b, to istnieje ξ ∈ (a, b), że f ′ (ξ) = 0. Zupeł- wykorzystanie własności dla na analogia z twierdzeniem Rolle’a, dziwne że wiele podręczników anal. elementarnej Darboux pochodnych znaleźć można to pomija. W rozważanym przypadku wziąć max absolutne f w [a, b], nie może być w pracy [54]. przyjęte ani w a, ani w b, więc w ξ wewnątrz. Ale wtedy f ′ (ξ) = 0 (często nazy- Otóż udowodniono „równowane twierdzeniem Fermata w analizie – oczywiście jego twierdzenie, że każda liczba tam ważność” fundamentalnenaturalna jest sumą co najwyżej czterech kwadratów liczb naturalnych jest dużo efek- go twierdzenia rachunku towniejsze). W przypadku gdy odwrotnie, należy rozważać minimum absolutne. całkowego oraz ′ Równie łatwe jest należenie do I kl. Baire’a. Gdy f (x) istnieje wszędzie, to twierdzenia Lagrange’a o f ′ (x) = lim n(f (x+ n1 )−f (x)), a funkcje fn (x) = n(f (x+ n1 )−f (x)) są, przy każdym wartości średniej. n→∞ ustalonym n, ciągłe. Te dwie konieczne własności pochodnej istniejącej wszędzie były oczywiście od dawna znane. Lebesgue podał  bardzo prosty przykład,  że one nie cha1 dla x = 0 0 dla x = 0 rakteryzują pochodnej: funkcje f (x) = , g(x) = sin x1 dla x 6= 0 sin x1 dla x 6= 0 obie są I kl. Baire’a z własnością Darboux, gdyby więc były pochodnymi (i to jak  1 dla x = 0 widać ograniczonymi), to pochodną byłaby też funkcja f (x) − g(x) = 0 dla x 6= 0 co niemożliwe, bo nie ma własności Darboux. A więc f , lub g, (lub obie) nie jest pochodną. Funkcja f − g ∈ I kl., co musi oczywiście mieć miejsce. Przykład ten zacytowałem, bo w dalszym ciągu jego niewielkie modyfikacje służą mi do innych (też

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Zob. np. [38].

Z. Zahorski, R. Wituła

łatwych) kontrprzykładów. Bezpośredni dowód, że f nie jest pochodną, byłby niewiele trudniejszy, ale po co. Otóż znane jest od dawna niezbyt trudne twierdzenie, że f ∈ I kl., gdy dla każdego a ∈ R zbiory {x : f (x) > a} i {x : f (x) < a} są klasy Fσ . Nawet wystarczy to wiedzieć dla gęstego na osi y zbioru wartości a, choćby przeliczalnego. Dlatego zbiory wszystkich moich klas należą do klasy Fσ , zbiór pusty zaliczam do nich wszystkich, aby nie robić wyjątków (nie można zaprzeczyć, że nie należy, tzw. prawdziwość w sposób pusty). Niebanalna jest definicja, dopiero gdy zbiór jest niepusty. Jest M0 ⊃ M1 ⊃ M2 ⊃ M3 ⊃ M4 ⊃ M5 i wszystkie te zawierania oznaczają część właściwą (co podaję przed definicją, ale uzasadnić można dopiero po definicji). Def. E ∈ M0 , gdy każdy punkt x ∈ E jest obustronnym punktem skupienia dla E; E ∈ M1 , gdy każdy x ∈ E jest obustronnym punktem kondensacji dla E, tj. w każdym jednostronnym sąsiedztwie x, (x − δ, x) oraz (x, x + δ), δ > 0 leży nieprzeliczalna część zbioru E (tj. mocy continuum, bo to zbiór Borela); E ∈ M2 , gdy owa część ma miarę > 0; E ∈ M3 (można prościej, ale zrobione tak, żeby ująć i M4 , gdzie nie da się uniknąć komplikacji – czy będę pamiętał te kilka kwantyfikatorów, choć sam wymyśliłem ten warunek, to wątpliwe, a nie chce mi się iść do szafy po odbitkę, ale spróbuję bez odbitki – gdy istnieją ciągi: zbiorów domkniętych {Fn } i liczb {ηn }, ηn > 0, takich, że dla każdych x ∈ Fn i ε > 0 istnieje δ > 0, takie że 1 )∩E| gdy hh1 > 0, |h + h1 | < δ, hh1 < ε, to |(x+h,x+h+h > ηn (w liczniku | · | oznacza |h1 | miarę Lebesgue’a); tu przedział (x + h, x + h + h1 ) jest zapisany bez zwykłej umowy, że pierwsza liczba oznacza lewy koniec; tak jest gdy h > 0, gdy h < 0 odwrotnie, ale aby nie robić wyjątków w całej pracy się tej umowy nie stosuje). Nie założę się, czy tak, czy coś równoważnego, czy może całkiem źle – chcę tylko zaznaczyć złożoność warunku; E ∈ M4 , gdy E ∈ M3 i dla każdego n, ηn > 0, E ∈ M5 , gdy każdy x ∈ E = 1. jest jego punktem gęstości. To zaś znaczy, że lim+ |(x−h,x+h)∩E| 2h h→0

Funkcję f zaliczam do klasy Mk , gdy dla każdego a ∈ R zbiory {x : f (x) > a} i {x : f (x) < a} należą do Mk . Przeciwnie niż dla zbiorów, udowadniam, że M0 = M1 = J (a przypomnijmy, że kl. J to jest I kl. Baire’a z war. Darboux), a więc klasa M0 jest zbędna. Oczywiście Mk ⊃ Mk+1 i dla k > 1 jest to część właściwa, przeciwnie niż dla k = 0. Otóż udowodniłem tam, że klasa M4 dokładnie charakteryzuje zbiory {f ′ (x) > a} (ewent. < a) dla pochodnej ograniczonej, dla {f ′ (x) > a} ma to miejsce nawet tylko przy f ′ ograniczonych z góry – jest to jedno z dwóch najtrudniejszych twierdzeń tej pracy. Konieczność jeszcze znośna, ale wystarczalność, ładne kilka stron bez „wody”, bo wodę można dawać na wykładzie czy w nieudanym bo przewodnionym skrypcie (dla zaoczniaków, a wyobrażałem sobie, że przygotowanie zaoczniaków jest niemal zerowe – no to właśnie nie należy dawać za dużo wody, bo nieszczęśnicy będą wkuwać na pamięć i się zagubią); naprawdę dobrzy dydaktycy dają trochę wody nawet w podręcznikach dla normalnych studiów, np. Mostowski, ale musi jej być jak najmniej – ale prace drukowane pisałem bez wody. No, nawet dowód na jedną stronę bywa trudny (a na 10 stron może być łatwiejszy). Główny pomysł jest taki jak (w nieznanej mi bliżej) w pracy Carlesona o hipotezie Łuzina, choć pisałem ją dużo lat wcześniej, nie wiem, czy on ją czytał, zresztą nie opatentowałem tego pomysłu, bo to proste i może i przede mną ktoś stosował. Jest to analogia do podziałów np. majątku rolnego w kolejnych sprawach spadkowych, dla uproszczenia zawsze na połowy, ale z zachowaniem dwóch warunków: 1) nie wolno dzielić obszarów 6 4 hektarów, 2) nie wolno dzielić, gdyby dochodowość, np. jakość

Profesora Zygmunta Zahorskiego „wykład” o pochodnych. . .

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gruntu choć jednej połówki po podziale spadła poniżej wyznaczonej z góry liczby stałej dla wszystkich podziałów. Wtedy po skończonej ilości podziałów obszar rozpadnie się na części na ogół nierówne polowo, takie że żadnej dzielić dalej nie wolno. U mnie tę zasadę podziału stanowi utrzymanie średniej gęstości danego zbioru powyżej pewnej liczby, u Carlesona coś bardziej złożonego, ale obaj dzielimy odcinki punktem środkowym na połowy. Można oczywiście zamiast dwóch rozpatrywać i więcej zakazów dzielenia, ale w tych pracach nie było to potrzebne. Później wynikają u mnie dwa oszacowania – jedno dla tych odcinków, gdzie drugi zakaz nie działał (maksymalna ilość podziałów, odcinki możliwie najkrótsze) i drugie, dla odcinków, które wcześniej przestały być dzielone – każdy z tych zbiorów może być pusty, ale nie oba. Co wynika u Carlesona – nie wiem, bo nie chcę czytać, aby nie utrudniać sobie pracy i bez tego wystarczająco trudnej. Żadne „prawo” nie zmusza mnie do jej robienia, ale chcę ją robić, zresztą od czerwca 1981 do października 1984 robiłem coś innego i to trudniejszego (chyba, bo starsze i nierozwiązane dotąd, choć to nie świadczy o większej trudności). Od października 1984 do lutego 1986 znowu szeregi trygonometryczne, a od lutego 1986 tamto drugie – na zmiany. Co dalej jest w tej pracy? Twierdzenia, że f ′ skończona ∈ M3 , nieskończona w pewnych punktach ∈ M2 , ale i dla niej zbiory {a < f ′ (x) < b} dla a i b zarówno skończonych, jak i niesk. są M3 . A więc wszystkie warunki mocniejsze niż wcześniej znany M1 , ale to nie są jednak charakteryzacje, tylko warunki konieczne. A co z M5 ? Jest to charakteryzacja klasy A – funkcji aproksymatywnie ciągłych, Zob. punkt 6 informacji uzuktóre jak widać są I kl. Baire’a, oczywiście nie na odwrót. To dołączone dla kompletu, pełniających. bo było znane – Maksimow chyba w 1936 w japońskim czasopiśmie Tˆohoku Math. Journal. A czy pochodna ograniczona ∈ M5 ? Jak widać nie musi. Czy na odwrót, funkcja ∈ M5 = A musi być pochodną? Jeśli jest ograniczona, tak, jeśli nieograniczona nie musi. Ponadto, przykład wzorowany na wspomnianym Lebesgue’a z sin x1 (dotyczył klasy J, najszerszej) świadczy, że przynależność funkcji ograniczonej do klasy M4 nie gwarantuje, że jest ona pochodną. Czyli M4 nie stanowi charakteryzacji pochodnych ograniczonych, choć M4 stanowi charakteryzację zbiorów {f ′ (x) > a} dla nich. Z tego wynika, że warstwami, dystrybuantą, czyli funkcjami a 7→ {g(x) > a} klasa pochodnych ograniczonych scharakteryzować się nie da, wiem od prof. Lipińskiego, że jakiś Amerykanin napisał, że w tej pracy stawiono problem takiej ich charakteryzacji. Nic podobnego, pytanie o klasę „M4 21 ” dotyczy jakiejś innej (nie wiem jakiej) charakteryzacji, że nie dystrybuantą, jest tam wyraźnie napisane. No bo (poprzestając na funkcjach ograniczonych) ich przynależność do M5 wystarcza, aby były pochodnymi, ale nie jest konieczna – jest to warunek za mocny. Zaś przynależność do M4 jest konieczna, ale nie wystarcza – warunek za słaby (i tu z ogr. M4 ⊃ kl. pochod. ogr. ⊃ M5 z ogr. oba zawierania właściwe). Neugebauer, Amerykanin, scharakteryzował Wspomniane tu twierdzenie wprawdzie pochodne (nie wiem, czy tylko ograniczone), ale jest to warunek niewiele Neugebauera charakteryzująróżniący się od definicji pochodnej, czyli od warunku banalnego wspomnianego tu na ce pochodne (wraz z początku. Widziałem, ale nie pamiętam, coś więcej o tym wie Lipiński (Uniw. Gdań- dowodem) i wiele więcej na ski). Ma być kl. pochod. ogr = M4 12 ogr. (bo funkcja ∈ M5 ograniczona być nie ten temat znaleźć można np. w musi, do M4 też nie musi). Zbiorów M4 12 oczywiście nie potrzeba. monografiach oraz [25]. Jest to najbardziej wyczerpująca odpowiedź na Pana pytanie, jaką mogę dać. Zresz- [14] Zobacz również 6 tą od ponad 30 lat nie zajmuję się tą problematyką i w ogóle funkcjami rzeczywistymi. punkt informacji Zajmuję się szeregami trygonometrycznymi, a właściwie jednym problemem w nich uzupełniającej. – zbieżnością i to prawie wszędzie w zbiorze, zbieżnością w punkcie, mimo że zna-

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ne warunki są za mocne (wystarczające, ale nie konieczne), zajmować się nie warto, ona „ma miejsce, gdy ma miejsce”, przecież zbyt złożone warunki niewiele są warte. A czasem zajmuję się czymś innym dla odpoczynku, czy odmiany, a może i dlatego, że ten kot co jednej dziury pilnował, to zdechł. Z poważaniem, Z.Z.

Kilka informacji uzupełniających (R.W.) 1. List profesora Zahorskiego znakomicie komponuje się z [61] (Zeszyt Naukowy powstał z okazji 70. rocznicy urodzin Profesora; zawiera m.in. biografię – którą zamieszczamy wraz z tłumaczeniem na język angielski także w niniejszej monografii – oraz spis artykułów Profesora; autorami prezentowanych tam prac jest zacne, międzynarodowe grono byłych studentów, kolegów oraz kontynuatorów idei Profesora; zdecydowanie cenna pozycja naukowa zasługująca na rozgłos); stanowi też autorskie spojrzenie na zagadnienia: klas Zahorskiego, twierdzenie Carlesona etc. Ponadto, list ten uzupełnia poglądy na podane kwestie innego polskiego autorytetu z dziedziny funkcji rzeczywistych – prof. Jana S. Lipińskiego, zamieszczone w [61] w jego artykule przeglądowym. 2. W roku 2007 obchodzono uroczyście trzysetną rocznicę urodzin L. Eulera. Stworzyło to nową okazję do studiowania jego dzieł. W związku z tym i nie tylko „w tym związku” zdecydowanie lepiej rozumiemy dziś metody dowodzenia tego genialnego matematyka. Ten zdawałoby się daleki od współczesnego formalizmu twórca (pisze o tym prof. Zahorski) w wielu momentach swojej twórczości przekłada się na język zupełnie dziś poprawny, czy to czysto formalny czy też z użyciem granic (dobrym przykładem tego zjawiska mogą tu być prace [1, 2, 6, 36, 37]). 3. Twierdzenie Banacha, wspomniane na marginesie strony 68, brzmi następująco: Zbiór tych f ∈ C[0, 1], dla których w każdym punkcie x ∈ [0, 1) mamy D+ f (x) = −∞

lub

D+ f (x) = ∞

i równocześnie w każdym punkcie x ∈ (0, 1] mamy D− f (x) = −∞

lub

D− f (x) = ∞

jest rezydualny. Włoski matematyk Pier Mario Gandini, w pracy [24], uogólnił to twierdzenie na przestrzenie C([0, 1]n ), dodatkowo, rozszerzając odpowiedni zbiór rezydualny do dopełnienia pewnego zbioru σ-porowatego. 4. Istnieje przynajmniej jeszcze jeden powód, dla którego warto wspomnieć o Gerhardzie Kowalewskim. Otóż, jest on autorem następującego twierdzenia o wartości średniej dla układu równań utworzonego z n całek.

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Twierdzenie ([34]). Niech x1 , . . . , xn ∈ CR [a, b]. Istnieją liczby t1 , . . . , tn ∈ [a, b] oraz nieujemne liczby rzeczywiste λ1 , . . . , λn takie, że: n X

λk = b − a

k=1

oraz

Zb a

xr (t)dt =

n X

λk xr (tk ),

r = 1, . . . , n.

k=1

W pracy [35] Kowalewski uogólnił to twierdzenie zastępując miarę liniową dt miarą wagową F (t)dt, gdzie F ∈ C[a, b], F jest stałego znaku na (a, b) oraz n X

k=1

λk =

Zb

F (t)dt.

a

Dopiero w 2008 roku, Slobodanka Janković i Milan Merkle w pracy [31] rozszerzyli to twierdzenie na dowolne przedziały I ⊂ R wprowadzając w miejsce miary F (t)dt dowolną skończoną dodatnią miarę µ określoną na σ-ciele borelowskim przedziału I (odpowiednio funkcje xk należą wówczas do zbioru CR (I) ∩ Lµ (I), k = 1, . . . , n). Ponadto, jak zauważają ci autorzy, z wyjątkiem dwóch cytowań, wyniki Kowalewskiego pozostawały zupełnie nieznane – jakże niesłusznie. Powyższe wyniki wraz z dowodami przedstawiono również w książce [30]. 5. Historia twierdzeń o jednoznaczności dla szeregów trygonometrycznych (wliczając w to również wielokrotne szeregi trygonometryczne) trwa nadal (zob. prace [3, 4, 5, 55, 59]). Warto tu wspomnieć jeszcze na nieco młodszy w stosunku do twierdzenia Cantora, wynik Du Bois Reymonda [13] z 1876 roku, cytowany i dowodzony zarówno w [59], jak i w [9]: Twierdzenie A. Jeśli szereg trygonometryczny ∞  a0 X + an cos(nx) + bn sin(nx) 2 n=1

jest zbieżny wszędzie do sumy skończonej f (x), całkowalnej na [0, 2π], to jest on szeregiem Fouriera funkcji f (x). Zwróćmy uwagę (za Niną Bari [9]), że oryginalnie, Du Bois Reymond, odnosił ten wynik do całkowalności w sensie Riemanna (jego wynik pozostawał w mocy także w przypadku, gdy zaniedbamy zbieżność szeregu trygonometrycznego na pewnym zbiorze przeliczalnym). Stosowne rozszerzenie twierdzenia A na funkcje całkowalne w sensie Lebesgue’a zawdzięczamy samemu Lebesgue’owi (ale twierdzenie A nadal nosi nazwę twierdzenia Du Bois Reymonda). Jest oczywiste, że twierdzenie A implikuje dyskutowany tu wynik Cantora z 1870 roku. Na zakończenie wspomnijmy jeszcze, że podobne twierdzenia o jednoznaczności dla szeregów Haara i Walsha dyskutowali i rozstrzygnęli m.in. Rosjanie W.A. Skwor-

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cow (zob. [50, 51, 52]) oraz M.G. Płotnikow (zob. [42, 43, 44]). Temat jednoznaczności wielokrotnych szeregów trygonometrycznych dyskutowany jest też w rozdziale trzecim monografii V.L. Shapiro [49]. 6. W klasie funkcji ograniczonych f : (0, 1) → R charakteryzacji pochodnych dokonał, wspomniany w końcówce listu profesora, I. Maximoff (zob. [40, 41]): Twierdzenie B. Funkcja ograniczona f : (0, 1) → R jest równoważna w sensie Lebesgue’a pochodnej wtedy i tylko wtedy, gdy f jest I klasy Baire’a i równocześnie spełnia warunek Darboux. Twierdzenie to wraz z pięknym dowodem bazującym na koncepcji Davida Preissa znaleźć można w rozdziale czwartym monografii [25]. Nomen omen dowód ten wykorzystuje efektywnie, napomknięte w liście profesora twierdzenie Neugebauera charakteryzujące pochodne (ograniczone). Autorzy monografii [25] kwestionują też poprawność oryginalnego, tj. podanego przez I. Maximoffa dowodu twierdzenia B (jeśli jest to prawda, to autorem pierwszego poprawnego dowodu tego twierdzenia byłby wspomniany już D. Preiss). Ponadto, jak wykazano w [25] za Goffmanem i Neugebauerem charakteryzacja z twierdzenia B zachodzi też w obrębie klasy funkcji ograniczonych f : (0, 1) → R, równoważnych w sensie Lebesgue’a pochodnej aproksymatywnej. Więcej przekrojowych informacji poświęconych charakteryzacjom pochodnych znaleźć można w pracach [15, 23] oraz [19]. Zwłaszcza w tej ostatniej znaleźć można następujący ciekawy wynik Krzysztofa Chrisa Ciesielskiego [19, 53]: Twierdzenie C. Żadna z następujących klas funkcyjnych nie jest topologizowalna: klasa Λ wszystkich pochodnych klasy Zahorskiego Mk , k = 1, ..., 5, klasa wszystkich funkcji spełniających warunek Darboux, klasa wszystkich funkcji mierzalnych oraz klasa wszystkich funkcji mających własność Baire’a.

Bibliografia 1. Andrews G.A.: Euler’s „De Partitio Numerorum”. Bull. Amer. Math. Soc. 44 (2007), 561–573. 2. Apostol T.M.: An elementary view of Euler’s Summation Formula. Amer. Math. Monthly 106 (1999), 409–418. 3. Ash J.M., Rieders E., Kaufman R.P.: The Cantor – Lebesgue property. Israel J. Math. 84 (1993), 179–191. 4. Ash J.M.: Uniqueness of representation by trigonometric series. Amer. Math. Monthly 96 (1989), 873–885. 5. Ash J.M.: Uniqueness for higher dimensional trigonometric series. Cubo 4 (2002), 97–120. 6. Ayoub R.: Euler and the Zeta function. Amer. Math. Monthly 81 (1974), 1067–1086. 7. Banach S.: Sur les ensembles de points o´ u la d´eriv´ee est infinie. Comptes Rendus de l’Acad´emie de Sciences 173 (1921), 457–459. 8. Bari N.K.: The problem of uniqueness of expansion of a function into a trigonometric series. Uspekhi Mat. Nauk 4, no. 3 (1949), 3–68 (in Russian). 9. Bary N.K.: A Treatise on Trigonometric Series, vol. 1 and 2. Pergamon Press, New York 1964. 10. Benedetto J.J.: Harmonic Analysis and Applications. CRC Press, Boca Raton 1997. 11. Benedetto J.J., Czaja W.: Integration and Modern Analysis. Birkh¨ auser, Boston 2009. 12. Bochner S.: Fourier series came first. Amer. Math. Monthly 86 (1979), 197–199.

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13. Du Bois Reymond P.: Beweis dass die koeffizienten der trigonometrischen Rechen. . . Abh. Akad. Wiss. M¨ unchen 12 (1876), 117–166. 14. Bruckner A.M.: Differentiation of Real Functions. Springer, Berlin 1978. 15. Bruckner A.M.: The problem of characterizing derivatives revisited. Real Anal. Exchange 21 (1995–96), 112–133. 16. Bruckner A.M., Leonard J.L.: Derivatives. Amer. Math. Monthly 73 (1966), 24–56. 17. Bruckner A.M., Thomson B.S.: Real variable contribution of G.C. Young and W.H. Young. Expo. Math. 19 (2001), 337–358. 18. Carleson L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116 (1966), 135–157. 19. Ciesielski K.: Set theoretic real analysis. J. Appl. Anal. 3 (1997), 143–190. 20. Coppel W.A.: J.B. Fourier – On the occasion of his two hundredth birthday. Amer. Math. Monthly 76 (1969), 468–483. 21. Djaczenko M.J.: Some problems of the theory of multiple trigonometric series. Uspiekhi Mat. Nauk 47 (1992), 97–162. 22. Feffermann C.: Pointwise convergence of Fourier series. Ann. of Math. (2) 98, no. 3 (1973), 551–571. 23. Freiling Ch.: On the problem of characterizing derivatives. Real Anal. Exchange 23, no. 2 (1997-1998), 805–812. 24. Gandini P.M.: An extension of a theorem of Banach. Arch. Math. 67 (1996), 211–216. 25. Goffman C., Nishiura T., Waterman D.: Homeomorphisms in Analysis. Amer. Math. Soc., Providence 1997. 26. Grafakos L.: Modern Fourier Analysis. Springer, Berlin 2009. ¨ 27. Hahn H.: Uber den Fundamentalsatz der Integralrechnung. Monatsh. Math. Phys 16 (1905), 161–166. 28. Halmos P.R.: Fourier series. Amer. Math. Monthly 85 (1978), 33–34. 29. Hetmaniok E., Pleszczyński M., Wituła R.: Selected scientific achievements of Professor Zygmunt Zahorski. In this monograph: Monograph on the occasion of 100th birthday anniversary of Zygmunt Zahorski, Wituła R., Słota D., Hołubowski W. (eds.), Wyd. Pol. Śl., Gliwice 2015, 45–49. 30. Hetmaniok E., Słota D., Wituła R.: Mean Value Theorems. Silesian University of Technology Press, Gliwice 2012 (in Polish). 31. Janković S., Merkle M.: A mean value theorem for systems of integrals. J. Math. Anal. Appl. 142 (2008), 334–339. 32. Kechris A.S., Louveau A.: Descriptive Set Theory and the Structure of Sets of Uniqueness. London Math. Soc. Lecture Note Ser. 128, Cambridge Univ. Press, Cambridge 1987. 33. Kolmogoroff A.: Une s´erie de Fourier-Lebesgue divergente presque partout. Fund. Math. 4 (1923), 324–328. 34. Kowalewski G.: Ein Mittelwertsatz f¨ ur ein System von n Integralen. Z. Math. Phys. (Schl¨ omilch Z.) 42 (1895), 153–157. 35. Kowalewski G.: Bemerkungen zu dem Mittelwertsatze f¨ ur ein System von n Integralen. Z. Math. Phys. (Schl¨ omilch Z.) 43 (1896), 118–120. 36. Laugwitz D., Rodewald B.: A simple characterization of the Gamma Function. Amer. Math. Monthly 94 (1987), 534–536. 37. Laugwitz D.: On the historical development of infinitesimal mathematics. Amer. Math. Monthly 104 (1997), 447–455. 38. Łojasiewicz S.: Introduction to Real Functions Theory. PWN, Warsaw 1973 (in Polish). 39. Luzin N.N.: Integral and trigonometric series, editing and comments – N.K. Bari and D.E. Menshov. Goztiechizdat, Moscow 1951 (in Russian). 40. Maximoff J.: On continuous transformation of some functions into an ordinary derivative. Ann. Scuola Norm. Super. Pisa 12 (1943), 147–160. 41. Maximoff J.: Sur les fonctions d´eriv´ees. Bull. Sci. Math. 64 (1940), 116–121. 42. Plotnikov M.G.: On uniqueness sets for multiple Walsh series. Mat. Zametki 81, no. 2 (2007), 265–279 (English transl. in Math. Notes 81, no. 1–2 (2007), 234–246). 43. Plotnikov M.G.: Several properties of generalized multivariete integrals and theorems of the du Bois-Reymond type for Haar series. Sb. Math. 198, no. 7 (2007), 967–991.

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44. Plotnikov M.G.: Quasi-measures, Hausdorff p-measures and Walsh and Haar series. Izv. RAN: Ser. Mat. 74, no. 4 (2010), 157–188 (English transl. in Izv. Math. 74, no. 4 (2010), 819–848). 45. Ruziewicz S.: On unapplicability of the fundamental theorem of integral calculus for functions possessing the infinite derivatives. Mathematical-Physical Works 31 (1920), 31–33 (in Polish). 46. Ruziewicz S.: Sur les fonctions qui ont la mˆeme d´eriv´ee et dont la diff´erence n’est pas constante. Fund. Math. 1 (1920), 148–151. 47. Saks S.: Theory of the Integral. Dover Publications, New York 1937 (1933). 48. Saks S.: Zarys teorii całki. Wyd. Kasy im. Mianowskiego, Warsaw 1930. 49. Shapiro V.L.: Fourier Series in Several Variables with Applications to Partial Differential Equations. CRC Press, Boca Raton 2011. 50. Skvortsov V.A.: On uniqueness theorem for a multidimensional Haar series. Izv. Akad. Nauk. Arm. SSR Mat. 23, no. 3 (1988), 293–296 (English transl. in Soviet. J. Contemp. Math. Anal. 23, no. 3 (1988), 104–108). 51. Skvortsov V.A.: Uniqueness sets for multiple Haar series. Mat. Zametki 14, no. 6 (1973), 789–798 (English transl. in Math. Notes 14, no. 6 (1973), 1011–1016). 52. Skvortsov V.A., Talalyan A.A.: Some uniqueness questions of multiple Haar and trigonometric series. Mat. Zametki 46, no. 2 (1989), 104–113 (English transl. in Math. Notes 46, no. 2 (1989), 646–653). 53. Tartaglia M.: Sulla caratterizzazione delle derivate. Pubbl. Dip. Mat. Stat., Napoli 1988. 54. Tong J., Hochwald S.: An application of the Darboux property of derivatives. Int. J. Math. Edu. Sci. Technology 34 (2003), 150–153. 55. Uljanow P.L.: A.N. Kolmogorov and divergent Fourier series. Uspiekhi Mat. Nauk 38 (1983), 51–90. 56. Zahorski Z.: Sur la premi`ere d´eriv´ee. Trans. Amer. Math. Soc. 69 (1950), 1–54. 57. Zahorski Z.: On the set of not-differentiability points of any function. Report on the V Congress of Polish Mathematicians in Kraków, 29-31 of May 1947. Supplement to the PTM Annual 21 (1949), 23–26. 58. Zahorski Z.: Une s´erie de Fourier permut´ee d’une fonction de classe L2 divergente presque partout. C. R. Acad. Sci. Paris 251 (1960), 501–503. 59. Zygmund A.: Trigonometric Series, vol. I. Cambridge Univ. Press, 1959 (Russian transl. was used, Mir Press, Moscow 1965, edited by N.K. Bari). 60. Zygmund A.: Notes on the history of Fourier series, Studies in harmonic analysis. Studies in Mathematics 13, J.M. Ash (ed.), Math. Assoc. of America (1976), 1–19. 61. Zeszyty Nauk. P.Śl., ser. Mat.-Fiz 48, Gliwice 1986.

Professor Zygmunt Zahorski’s “lecture” on derivatives – prepared for publication by Roman Witula, the turmoil maker Zygmunt Zahorski Roman Witula – comments and remarks

´ My sincere gratitude to Professor Janina Sladkowska-Zahorska for her valuable help in collecting the bibliography and in completing the factual materials. My sincere gratitude to Professor Jerzy Mioduszewski (supervisor of this venture) for his traditionally giant enthusiasm and commitment as well as for inspiring discussion and supplementary literature, thanks to which the paper could get the final form.

Introduction Problem, or rather a question which I posed to Professor Zahorski in the winter 1986/87 concerned the set-theoretic and topological “description” of derivatives, quite generally understood by me. At that time I was a newly fledged master in mathematics and my knowledge in this matter was rather “conventional”. After a short period of time Professor gave me in response a letter which I wanted finally to share with others, in the hope that the Readers will be as curious about its contents as me at that time. Remark. In margins of respective pages one can find some comments completing and updating selected parts of Professor’s letter. Additionally, after Professor’s letter a collection of few longer pieces of supplementary information and the bibliography are given.

Roman Witula Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 81–95. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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Professor’s letter Well, my paper published in Transaction of the American Mathematical Society in 1950, but prepared actually in 1939-42, starts with the following words: “characterization of the class of continuous functions, possessing everywhere the first derivative, with the aid of their topological and metrical properties is unknown”. It is not important whether the continuous function or its derivative (in general, discontinuous in many points) will be characterized. I aimed at characterization of derivative existing everywhere. On the contrary as in case of analytical functions, in this case the unbounded functions are more difficult, therefore I put the following stages: I – bounded derivative, II – finite derivative, III – infinite (in some points) derivative – for a long time (however > 1900) there is known a proof that the set III is a null set and, which is more, it happens without the assumption of derivative existence (in This is the other points) and even the continuity is not needed and it is enough to consider the Banach result [7]. one-sided derivative: the set of all points in which the right-sided infinite derivative of any function (even the unmeasurable function) exists is the null measure set. Proof of this In case of the continuous nowhere differentiable Weierstrass function in each point fact results, for instance, from one of the Dini derivatives is = +∞, the other is = −∞, but the Dini derivatives the original Weierstrass’ are not the derivatives, similarly as the upper and lower limits of a sequence are not proof of this function (in general) the limit of this sequence. nowheredifferentiability, Considering the derivative existing almost everywhere (i.e. except the (L) null meapresented for the first time sure set of points), the problem is easier and was solved in 1912 (and published in in the Weierstrass’ 1915 or 1916) in the Luzin’s PhD dissertation in Russian (“Integral and trigonometric letter to Du Bois Raymond series”) – to be more precise one should note that PhD degree was in Russia, even in (1875). Essential in the tsarist times, the second kind scientific degree corresponding with our habilitation. this matter seems to be In any case, the postdoctoral dissertations are different everywhere, even in the same also the Banach university – this one was of epochal matter considering not only this one problem, theorem (1931) which will be but also several other problems put there. At least one Luzin’s hypothesis (claiming cited in third item of that the Fourier series of square integrable function is convergent almost everywhere) supplementary information appeared to be unlikely true, after Kolmogorov’s examples of the Fourier series of given at the end of this a function integrable in the first power, divergent almost everywhere, presented in paper. 1922 and published in Polish journal Fundamenta Mathematicae probably in 1923 See papers and such defined series for the L1 function, divergent everywhere, presented in 1926 [33, 55] and published in C. R. Acad. Sci. Paris in the same year. The outstanding authority in the matter of trigonometric series, Professor Antoni Zygmund, said in 1960 that it is certainly false since in L2 the convergence in L2 is typical (Riesz-Fischer Theorem, 1904), not the almost everywhere convergence. Before 1936 Menshov gave the examples of orthogonal bounded systems with the almost everywhere divergence in L2 and Kolmogorov put in 1926 something more than hypothesis, published in 1927 in German journal Mathematische Zeitschrift (in paper joint with Menshov), saying that it can be even the trigonometric system, but properly rearranged. Namely he wrote that he could give such example, however he has never given, there or anywhere else, any proof nor function – or, which is equivalent, any coefficients of such series nor the way of rearranging the terms (permutation in the infinite sequence). Many mathematicians (Russians, Hungarians, Americans and probably the others) tried to do that. I succeeded in 1960, maybe after three weeks of work (and a little bit in 1954), and the many-years unsuccessful efforts to prove the Luzin’s hypothesis appeared to be See paper [56].

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very useful for this achievement. In 1954, quite quickly as well, I thought that I had a construction, however I noticed a mistake right before giving a talk for the Polish Mathematical Society. I gave indeed the talk, but on the completely different and known, however not to all of the audience, subject – on the so called Young Theorem Probably it concerns Mrs about symmetry (for any function of point on the x axis of numerical values the sets G.C. Young, see [47, 17]. of right-sided limits of f (xn ), for xn → x, xn > x, and left-sided limits for xn < x, are identical and one of them is equal to f (x). Everywhere? Not necessarily, however with the exception of the at most countable set of xs – which is better than the null set.) Abbreviated, but still clear for the specialists, proof, with a function and permutation certainly, I submitted in 1960 to C. R. Acad. Sci. Paris (they publish within three weeks but only 1–5 pages, notices with no proofs or with very shortened proofs). See paper [58]. I said to Professor Zygmund, who was then for few days in Warsaw (he lives in Chicago) that since 1945 I believe in the Luzin’s hypothesis and I have no doubts caused by the truth of Kolmogorov’s hypothesis that it can be different for the rearranged systems, whereas for the normal order 0, 1, 2, 3, 4, . . . it is exactly as predicted by Luzin. I announced this in 1961, in C. R. Acad. Sci. Paris as well, but before final elaborating and reporting it in the Institute of Polish Mathematical Society in Warsaw for Professor S. Mazur – he passed away in November 1981. During the edition I have found an error. And, since the note was already published in C. R., I announced in Mathematical Reviews, through the mediation of Professor Zygmund, that there is no solution, a mistake. Nevertheless, the Luzin’s hypothesis appeared to be true and has been proven in 1966 by Swedish mathematician L. Carleson, I think about See paper [18]. 10 years younger than me. Supposedly he worked about 7 years in good conditions – on American scholarship at the Stanford University in California. He gave one of the main talks at the International Congress in Moscow in 1966, chairman for this talk was Kolmogorov. Luzin did not live to see the proof of his hypothesis, he passed away in Moscow on 28 February 1950. And this is the Luzin’s result (construction proof was given in his above mentioned work – reprint, containing various comments and works of authors giving the solutions See paper [39]. of some Luzin’s problems or similar problems as well, was made in 1950. I had this book but I lost it somehow during my move to Gliwice in 1970). It is necessary and sufficient for function g(x) to be almost everywhere the derivative Proof of this theorem can be of a continuous function, that g(x) is (L) measurable and almost everywhere finite. also found in monograph [14].

However Luzin does not call this continuous function as antiderivative nor indefinite integral of g. It is because we have here a high rank of uncertainty – different First example of functions, “antiderivatives” do not differ in constant. Indeed, he constructs one of the “an- difference of which is not tiderivatives”, but it is not at all unique. He uses here his theorem, in which he claims constant on the given interval that for a function measurable on the interval there exists a closed set of the measure even though they have differing less than ε from the length of this interval (certainly, in general smaller than everywhere in this interval this length) where f is relatively continuous. One can say, by omitting the isolated equal derivatives, points, that this is the perfect set, i.e. closed and dense-in-itself. was given by Hans Hahn Somebody has presented this result on my seminar at the University of L´od´z, [27]. Another example however I do not recall too much of it and even if I could reconstruct this Luzin’s presented S. Ruziewicz “antiderivative”, it would take probably about three months of my good work. It is (see [45, 46]).

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done by the use of some functions defined on sets similar to the Cantor set, but of positive measure. Necessity of these conditions is relatively simple: it is finite almost everywhere because the derivative, as I have noticed above, can be +∞ or −∞ only in the null measure set. Measurability almost everywhere of the derivative of a continuous function (and even each of four Dini derivatives) is quite easy to prove, probably even Probably paper without the assumption about continuity. I wrote something about that in Annales [57] is concerned. of Polish Mathematical Society in 1952, unfortunately I do not remember too much, however it was probably new at that time. By the way: even if the derivative exists everywhere, then if for example it is equal to +∞ on the set of cardinality of the continuum, it is hard to speak about the antiderivative, for example, it is quite easy to construct two continuous functions possessing everywhere equal derivatives, finite outside of the Cantor set and equal +∞ on the Cantor set, not differing in a constant. Every generalization of antiderivative function, for example presented in the second English version part of “Outline of the Theory of Integral” by Saks [47] (Polish edition from 1930, of Saks’ monograph French – translation differing in only one chapter, right in the middle, and English is [47]. Professor edition, probably from 1937, completely different and more extensive, not known Zahorski refers here certainly closely to me since I do not speak English), assumes always that the derivative is to the Polish version [48] of finite almost everywhere, it means with the exception of at most countable set, and this book. the expression almost everywhere means with the exception of the null measure set, which obviously can be uncountable or even of cardinality of the continuum. Not because the derivative would must be finite almost everywhere, like it can be seen in the mentioned example with the Cantor set, only because in the other case it is hard to talk about the antiderivative functions differing in a constant. This is the sufficient condition and I do not know any other less inconvenient. If the derivative is finite, then for finding the antiderivative the Lebesgue integral is sufficient. Even if the derivative does not exist on the set of cardinality of the continuum, it is only needed that it exists almost everywhere, since the function satisfying the Lipschitz condition – or even more, the difference of two monotonic functions, continuous or not – possesses almost everywhere finite (L) integrable derivative, even though this integral (as the function of upper limit) in general differs from the differentiable function by two elements – the so called discontinuity function – they are of the first kind here and only for such ones it is defined, and – even if it is continuous – the so called singularity function. Only if it is absolutely continuous (it does not concern |f |, even though if f is absolutely continuous, then |f | as well), then the singularity function is equal to 0 for each x. But the Lipschitz functions are absolutely continuous, the integral of a measurable bounded function certainly satisfies the Lipchitz condition and each measurable bounded function is also (L) integrable (over intervals of finite length), thus everything is correct. Some unbounded functions are also (L) integrable, of course functions from among (L) measurable functions, and always, no matter if f is bounded or not, and even for functions infinite on the set of cardinality of the continuum and null measure (if the set would be of measure > 0, then the L integral would not exist), here is Rx d f (t)dt = f (x) almost everywhere. First part of the above mentioned “Outline dx a

Professor Zahorski refers certainly to the Polish version of this book (see [48]).

of the Theory of Integral” by Saks concerns the (L) integral. Second part is devoted to the Perron and Denjoy integrals, the more general integrals than the Lebesgue integral. Because, unfortunately, when f ′ (x) is unbounded, even if it exists and is

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85

finite everywhere (measurable, of course), it can be not (L) integrable. Then, considRx ering the definite integrals f ′ (t)dt = F (x), the problem is solved only by the higher a

Denjoy integral, equivalent to the Perron integral, proof of which can be found in the second part of Saks’ “Outline”. Both these integrals have been defined not just independently, but completely differently and only later the other authors proved two theorems – the first, I think, that Denjoy’s definition (more specific) contains the Perron’s definition, and vice versa which is claimed by the second theorem. Denjoy’s descriptive definitions are similar to the Lebesgue’s definition, with such difference that instead of the idea of absolutely continuous (AC) function the idea of absolutely continuous generalized (ACG) function is applied in the definition of the narrow D integral and the idea of absolutely continuous generalized in the wider sense function (ACG∗ ) is used in the definition of the wide D∗ integral. In D definition the approximative derivative is used, whereas D∗ definition uses the ordinary derivative (for ∗ it “should” be more weird, here it is traditionally in opposite). However, the descriptive definition is not much worth without the constructive definition, because it does not guarantee the existence of defined elements. In case of D and D∗ integrals the constructive definition is “terrifying”. There are actually the constructions of integrals of more and more high ordinal classes, finite or countable (similar to the hierarchy of Baire’s functions and Borel’s sets) starting with the Lebesgue integral as number 0. So, transferring to higher classes is executed through: 1) the Cauchy process of creating the improper integrals for isolated points. It is a well known fact that the L integral includes the R integral. But only the proper integral or the improper absolutely convergent integral. Conditionally convergent improper R integral comes beyond the L integral (but is included in the first class of D integrals); 2) the Harnack process – which I prefer to omit, when the integrability in the lower class is disturbed by some perfect nowhere dense set, such that the integrability occurs in the intervals of its compliment, whereas in the entire line segment (composed of these intervals plus this nowhere dense set) the integrability does not occur. But in this set itself the integrability occurs as well. Perron treated this in the completely different way, by means of the so called raising and descending functions, probably something analogous exists for the ordinary differential equations, in which he was involved as well. Besides, seeking the antiderivative is the simplest version of a differential equation – only with a number of other complications, therefore to avoid the situation of too big amount of complications the continuity is assumed, not nicely said, so the solution from C1 class with the continuous derivative is sought. The Denjoy integral is called by Russians the Chinczin integral. Chinczin gave such definition almost in the same time, in about 1916. In foreign spelling the name is Khintchine, devil knows of which nationality. In Russian it is simply , but I do not know whether the meaning of this word is the same as in Polish, since the Chinese in Russian sounds as “kitajec”. The name Chinczin Perron and Denjoy (narrow) integrals serve not only for seeking the antideriva- sounds as “Chi´ nczyk”, tive functions for derivative existing everywhere or (in ACG class) almost every- which in Polish language means where, besides the measurable non-integrable functions exist even in the wider Den- the Chinese. joy sense, but for seeking definite integrals of functions integrable in this sense, and Rx d d f (t)dt = f (x) almost everywhere for the narrow integral, approximative dx for dx a

86

the wide integral, both give for

Z. Zahorski and R. Witula

Rx

f (t)dt = F (x) the ordinary continuity with respect

a

to x, and even more – ACG or ACG∗ . The wide Denjoy integral gives the antiderivative function for the approximative derivative, which is better to omit. Incidentally, Denjoy has defined some definite integral coming beyond D and D∗ , particularly for the trigonometric series. I do not know this definition, I have never seen it and I doubt whether I could solve this problem (having probably more than one solution – for example, whether each wide definition would be good enough, the narrow definition rather not, however even this is questionable). This problem has been mentioned also by Luzin in his PhD dissertation, but Luzin did not solve it and maybe - I do not know exactly - he only schematically specified it. It is about the following: ∞ P Let us assume that the trigonometric series a20 + (an cos nx + bn sin nx) is n=1

everywhere convergent to a finite function f (x), certainly of the first Baire class. According to the theorem proved by Cantor in 1870 roku, coefficients an , bn are uniquely determined by function f (more precisely, Cantor proved that if f (x) = 0 for each x, then all an , bn = 0, which is certainly equivalent – proof of this theorem, although is called as “elementary”, is not so easy and can be found, for example, in Author of these “Principles of differential and integral calculus” by S. Kowalewski, German, translated “Principles . . . ” is Gerhard into Polish by I. Roli´ nski. When in 1928 in 7th grade of primary school I was reading Kowalewski (1876-1950). Kowalewski’s book, I could not know that I will meet, the deceased already, Roli´ nski See also item 4 of personally after 1948 in L´ od´z. He was the honored teacher in secondary schools, the supplementary information. popularizer and, as one can see, the translator, after war the professor in Pedagogical Academy in L´od´z and after joining it with University of L´od´z he became the professor of this university. He did not have his own results but he was a really cool fellow, levelheaded and wise, with the sense of humor and good manners, not despotic to students and workers of lower rank. He was also the brother-in-law of Rozwadowski who was the bishop of L´od´z at that time). Problem: how to define an integral, such that coefficients an , bn could be determined by using the known Euler-Fourier formulae with the integral according to this definition? See also item 5 After Cantor, the uniqueness theorem was generalized, first in a case when f (x) = 0 of supplementary with the exception of at most countable set, it was obvious that one could not resign information. from the set of measure > 0, but for the null measure sets some troubles arose – for some, the so called U , the theorem was true, for others, the so called M , the theorem was false. For sets U and Many works concerned the sets U (unicite – uniqueness) and M have been written M , we propose to take a look by A. Rajchman, docent in the University of Warsaw, killed in Dachau probably in into [59, 9] and [32]. 1941, Zygmund, Mrs N. Bari, Russian with French or Italian surname, the author of monograph devoted to the trigonometric series, competitive with the Zygmund’s There exists an monograph, but in Russian – I do not know whether Americans translated it (Bari in English translation [9]. 1961 in the age of 60, being almost blind but wanting often to walk without assistance, during the (domestic) Forum of Mathematicians in Leningrad, got run over by a tram or electric train.) and, first of all, Menshov, I think he still lives but is at least 85 years old. It is known that if f is (L) integrable, then the coefficients are expressed by means of (L) integrals, thus, in particular, if f is (R) integrable, then by means of (R) integrals.

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But what if f is not (R) integrable? Denjoy gave in about 1923 the required definition of integral (and he proved the formulae). Whether they concern also the exception of countably many points – I do not know, and the uncountable would threaten with something maybe worse than the M sets, which does not mean that one cannot try. I suppose that till today the characterization of U (null measure) sets is not known or, what is equivalent, of the M (null measure) sets. I did not deal with this subject by myself. Euler himself proved his formulae, but in completely bad way, what one cannot See item 2 of supplementary have any grudge against him for, because there was no proper definition of an integral, information. or even of a derivative (equal to the quotient of “infinitely small” increments, it was said at that time, not using the concept of limit), or sufficiently wide concept of a function (they were introduced only in 1837 by Dirichlet and supposedly at the same time by Lobaczevski) – the first proper definition of a definite integral was given by Cauchy, and he has done this only for continuous functions, after 1800, and Riemann transformed it, in about 1850, for these discontinuous functions, for which it could be applied to, characterized since Lebesgue’s days as bounded and continuous almost everywhere. Euler was precise in works on the integer numbers theory, however in analysis, not precise by necessity these days, he did a lot and he had a good nose for it – he obtained results correct in general, despite some inaccuracies, first-class intuition. In view of incorrectness of his proof, the trigonometric series theory went in two directions: 1) resignation of the proof, which can be correctly done by transferring to definition: series with coefficients computed in such a way with the aid of function f Very good positions called the Fourier series of function f (the names are traditionally undeserved – the referring to the history of Fourier series have been introduced by this author in 1822 in the book concerning Fourier series and the the heat conduction equation (partial differential equations), d’Alembert before 1800 “priority” problem are considering (partial differential equation as well) the vibrating string equation, and papers [28, 12, 20] and Euler in about 1750 considering the periodic phenomena, for example astronomical, [60]. it means that Euler was the earliest, Fourier the latest), not taking care (as Euler wanted) if it is convergent and if exactly to f . It does not mean that the problem was get off lightly, it was only transformed to some other problem – one can investigate later if it is convergent to f , and even if it is not, how f can be found by using it – for example, it turned out (Lebesgue’s proof) that the so called first arithmetic means of partial sums converge almost everywhere to f and, which is more, in L1 – Fejer proved this earlier in the continuity points and the uniform convergence of the first means on the whole x axis in case when f is continuous everywhere and periodic with period 2π. Both of these theorems make a part of elementary analysis, the thing is that some functions from L1 can have not a one continuity point, but almost all points are their so called Lebesgue points – I omit the definition, and in these points exactly this convergence holds. In L2 (for each orthogonal expansion, not only the trigonometric one) the convergence → 0 holds in the sense of integral distance, i.e. in the metric of Hilbert space L2 . Then admittedly the sequence of partial sums is It concerns the L. Carleson only convergent in measure (I omit the definition) and not almost everywhere, but result [18]. More the subsequence convergent almost everywhere may be selected – in the trigonometric information can be found system it has been known for a long time that S2n is enough, now (since 1966) we in [29] (see also [10, 11, 26, know that . . . Sn itself, meaning the whole sequence S. 22]).

88

Mk and Mk classes are called, which is obvious, the Zahorski classes; and Professor Zahorski used letter M (respectively M) from the initial of name of his sweetheart at that time. Beautiful application of the Darboux property for derivatives can be found in paper [54], where the “equivalence” between the fundamental theorem of integral calculus and the Lagrange mean value theorem is proved.

See for example [38].

Z. Zahorski and R. Witula

Second direction went into the uniqueness. Yet Riemann did practically correct proof of the Euler theorem and for the simplest function, everywhere equal to “0”, because even for this function the proof was missing and the one made by Euler would be incorrect here as well. Last word till now is given by the mentioned Denjoy’s work. He died recently at the age of 90. I saw him lately in Bulgaria (Varna) in 1967. Let me return to the actual subject. Well, in paper [56], mentioned at the beginning, I defined six classes of sets Mk , k = 0, 1, . . . , 5, five classes of functions Mk , k = 1, 2, . . . , 5 and class J = functions of the first Baire class taking all the intermediate values in each interval, which is called the Darboux property. This property is possessed (elementary analysis) by all the continuous functions but not only – it is also possessed by the approximatively continuous functions and also not only – by the derivatives existing everywhere, even if they are not approximatively continuous, as well. Proof for the derivatives is easy. It is sufficient to prove that if f ′ (a) > 0, f ′ (b) < 0, (or conversely), a < b, then there exists ξ ∈ (a, b) such that f ′ (ξ) = 0. This is the complete analogy of the Rolle Theorem, so it is strange that many manuals for elementary analysis omit this fact. In considered case the absolute maximum of f in [a, b] should be taken, but it cannot be taken in a or in b either, therefore in ξ within the interval. But then f ′ (ξ) = 0 (often called as the Fermat Theorem in analysis – certainly his theorem saying that each natural number is a sum of at most four squares of natural numbers is much more spectacular). In the opposite case, the absolute minimum should be considered. Equally easy is belonging to the first Baire class. If f ′ (x) exists everywhere then ′ f (x) = lim n(f (x+ n1 )−f (x)) and functions fn (x) = n(f (x+ n1 )−f (x)) are, for each n→∞ fixed n, continuous. These two necessary properties of the everywhere existing derivative have been certainly known for the long time. Lebesgue gave a very  simple exam1 for x = 0 , ple that they do not characterize the derivative: functions f (x) = sin x1 for x 6= 0  0 for x = 0 g(x) = are both of the first Baire class with the Darboux property, sin x1 for x 6= 0 so if they would  be the derivatives (bounded, as it can be seen), then the function 1 for x = 0 f (x) − g(x) = would be derivative as well, which is impossible, since it 0 for x 6= 0 does not have the Darboux property. Thus, f , or g, (or both) is not the derivative. Function f − g belongs to the first class, which obviously must happen. I cited this example because I still use its tiny modifications for creating other (simple, as well) counterexamples. Direct proof of the fact that f is not a derivative would be not much difficult, but what for. Well, for a long time not very difficult theorem saying that f belongs to the first class if for each a ∈ R sets {x : f (x) > a} and {x : f (x) < a} are of the Fσ class has been known. It is even enough to know that for the set of value a, dense on the y axis, even countable. Therefore the sets of all my classes belong to the Fσ class and the empty set I include to all of them, in order to avoid the exceptions caused by it (one cannot contradict that it does not belong, the so called truth in empty way). The definition is not trivial only if the set is nonempty. We have M0 ⊃ M1 ⊃ M2 ⊃ M3 ⊃ M4 ⊃ M5 and all these inclusions denote proper subsets (which I say before the definition, but which can be justified only after the definition). Definitions: E ∈ M0 if each point x ∈ E is the both-sided limit point for E; E ∈ M1 if each point x ∈ E is the both-sided condensation point for E, that is,

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in each one-sided neighbourhood x, (x − δ, x) and (x, x + δ), δ > 0 the uncountable part of set E is included (it means of cardinality of the continuum, since it is the Borel set); E ∈ M2 if this part is of the measure > 0; E ∈ M3 (it can be easier, however it is done like that for the purpose to enclose also M4 where complications cannot be avoided – I am full of doubts whether I could recall these few quantifiers, even though I invented this condition on my own, and I rather do not feel like approaching the bookcase for reprint, however I will try without the reprint – if there exist the sequences: of closed sets {Fn } and of numbers {ηn }, ηn > 0, such that for every x ∈ Fn and ε > 0 there exists δ > 0, such that if hh1 > 0, |h + h1 | < δ, hh1 < ε 1 )∩E| then |(x+h,x+h+h > ηn (|·| in numerator denotes the Lebesgue measure), interval |h1 | (x + h, x + h + h1 ) is written here without the usual agreement that the first number denotes the left bound; it is like this if h > 0, in opposite if h < 0, but for not making the exceptions in the entire paper this agreement is not applied). I will not bet if it is like this, or equivalently, or completely wrong – I just want to show complexity of this condition; E ∈ M4 if E ∈ M3 and for each n, ηn > 0; E ∈ M5 if each point = 1. x ∈ E is its density point, which means that lim |(x−h,x+h)∩E| 2h

h→0+

I include function f to class Mk , if for each a ∈ R sets {x : f (x) > a} and {x : f (x) < a} belong to Mk . Unlike for the sets, I prove that M0 = M1 = J (let us recall that class J is the first Baire class with the Darboux condition), therefore class M0 is redundant. Certainly Mk ⊃ Mk+1 and for k > 1 it is the proper subset, on the contrary as for k = 0. Well, I proved there that class M4 precisely characterizes the sets {f ′ (x) > a} (alternatively < a) for the bounded derivative, for {f ′ (x) > a} it takes place even just for the bounded from above f ′ – it is one of two most difficult theorems in this paper. Necessity is even tolerable, but sufficiency – few fine pages with no “waffle”, since one can waffle during the lecture or in the unsuccessful, because “over-waffled”, script (for external students – I always thought that preparation of external students is miserable, so it is exactly why one should not “waffle” too much to them – these wretches learn by heart and get lost immediately); really good teachers waffle a little bit even in manuals for normal students, for example Mostowski, but as little as possible – however the printed papers I wrote without waffling. It happens that even the 1-page proof is tough (and the 10-page proof can be easier). The main idea is the same as in (not known closely to me) Carleson’s paper devoted to the Luzin hypothesis, even though I wrote it many years earlier, I do not know whether he read it, anyway I did not patent this idea, since it is simple and maybe used by somebody else before me. This is the analogy to division of, for example, agricultural property in the successive inheritance cases, for simplification, always by half but satisfying two conditions: 1) one can never divide regions 6 4 hectares, 2) one can never divide if profitability, for instance the soil quality, of at least one half would decrease below a constant number predetermined for every divisions. Then, after the finite number of divisions, the region will break down into the parts unequal, in general, with respect to the area and such that none can be divided any more. In my work this rule of division is represented by preserving the mean density of a given set above some number, in Carleson’s work by something more complex, however, both of us divide the intervals in half with the aid of a middle point. One can of course consider more than two restrictions of division, but it was not necessary in these papers. Later two estimations resulted in my work – one for these segments for which the second prohibition did not

90

Z. Zahorski and R. Witula

work (maximal numbers of divisions, the shortest possible segments) and the second one for the segments stopped to be divided earlier – each of these sets can be empty, but not both of them. What resulted in Carleson’s work – I do not know and I do not want to read it for not making my work harder, since it is already hard enough. None of the “laws” forces me to do this work, but I want to do it. Anyway, from June 1981 till October 1984 I was dealing with something else and something more difficult (I think, because it is older and unsolved till now, however it is not the evidence of its higher difficulty). From October 1984 till February 1986 the trigonometric series again and from February 1986 the other thing – alternately. What next in this paper? Theorems that the finite f ′ belongs to M3 , infinite in some points ∈ M2 , but even in this case the sets {a < f ′ (x) < b} for a and b, finite as well as infinite, are in M3 . Thus, all the conditions are stronger than the previously given M1 , but still these are not the characterizations, only the necessary conditions. See item 6 of And what about M5 ? This is the characterization of A class – class of the approxsupplementary information. imatively continuous functions which are, as it can be seen, of the first Baire class, not conversely. This can be added to the package, since it was known – given probably by Maximoff in 1936 in Japanese journal Tˆ ohoku Math. Journal. Does the bounded derivative belong to M5 ? As it can be seen, it does not have to. Or conversely, does the function from M5 = A must be a derivative? If it is bounded, yes, if unbounded, not necessarily. Moreover, the example patterned on the mentioned Lebesgue’s example with sin x1 (concerned the class J, the widest one) shows that belonging of the bounded function to M4 class does not guarantee that it is a derivative. That is, M4 does not give the characterization of bounded derivatives, even though M4 gives the characterization of sets {f ′ (x) > a} for these derivatives. This implies that the class of bounded derivatives cannot be characterized by layers, with the aid of distribution function, i.e. functions a 7→ {g(x) > a}. I know from Professor Lipi´ nski that some American wrote that in this paper the problem of such characterization of bounded derivatives had been posed. Nothing of that kind. Question about “M4 21 ” class concerns some other (I do not know which one) characterization, not by means of distribution function which is written there clearly. It is because (confining to the bounded functions) their belonging to M5 is sufficient for them to be derivatives, however it is not necessary – this condition is too strong. And belonging to M4 is necessary, however not sufficient – the condition is too weak (bounded M4 ⊃ class of bounded derivatives ⊃ M5 with restrictions, both inclusions are proper). Neugebauer, Mentioned here the American, characterized indeed the derivatives (I just do not know whether they the Neugebauer theorem were only bounded), but he gave the condition not differing too much from the definicharacterizing the derivatives tion of derivative, that is from the trivial condition mentioned here at the beginning. (with the proof) and I saw this, but I do not remember. Professor Lipi´ nski (University of Gda´ nsk) knows many more information in something more about that. Class of bounded derivatives should be = M4 1 bounded 2 this subject can be found, (since the function belonging to M does not have to be bounded, belonging to M4 5 for example, in mono1 sets are not needed of course. does not have to be either). The M 42 graphs [14] and [25]. See This is the most exhaustive answer I can give you for your question. Anyway for also item 6 of supplementary over 30 years I have not dealt with this subject-matter or, in general, with the real information. functions either. I work on trigonometric series, as a matter of fact on one problem concerning them – the convergence almost everywhere in a set, convergence in a point, even though the known conditions are too strong (sufficient, not necessary ones), it is not worth to deal with them, the convergence “takes place when it takes place”,

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after all too complicated conditions are not worth too much. And sometimes I work on something else, for a rest or for a change, or even because the cat looking after one hole, dies. Sincerely yours, Z.Z.

Some additional pieces of information (R.W.) 1. Professor Zahorski’s letter perfectly matches with [61] (Scientific Note created on the occasion of 70 anniversary of Professor’s birthday, containing, among others, Professor’s biography – which is included, in Polish and English version, in this monograph – and list of his works; papers presented there have been prepared by the very noble, international group of Professor’s former students, colleagues and continuators of Professor’s idea; definitely valuable scientific publication, deserving fame). It also gives the author’s take on the concept of Zahorski’s classes, on the Carleson theorem, etc. Moreover, this letter completes the opinions on the given issues of the other Polish authority in the field of real functions – Professor Jan Lipi´ nski, included in [61], in his survey. 2. In 2007 the 300 anniversary of L. Euler’s birthday was celebrated. This event gave a new occasion to study his works. In connection with this and not only “in this connection”, we understand today much better the methods of proving of this genial mathematician. Even though this author seems to be far from the present-day formalism (Professor Zahorski writes about this), many elements of his creation can be translated on the language perfectly correct today, the purely formal as well as by the usage of limits (a good example of this phenomenon can be papers [1, 2, 6, 36, 37]). 3. The Banach theorem, mentioned on the margin of page 82, sounds as follows: Set of all functions f ∈ C[0, 1], for which in each point x ∈ [0, 1) we have D+ f (x) = −∞

or

D+ f (x) = ∞

and concurrently in each point x ∈ (0, 1] we have D− f (x) = −∞

or

D− f (x) = ∞,

is residual. Italian mathematician Pier Mario Gandini, in paper [24], generalized this theorem for spaces C([0, 1]n ), additionally extending the adequate residual set to the complement of σ-porous set. 4. There exists at least one more reason for which one should mention Gerhard Kowalewski. He is the author of the following theorem on the mean value for the system of equations formed from n integrals.

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Theorem ([34]). Let x1 , . . . , xn ∈ CR [a, b]. There exist numbers t1 , . . . , tn ∈ [a, b] and nonnegative real numbers λ1 , . . . , λn such that n X

λk = b − a

k=1

and

Zb a

xr (t)dt =

n X

λk xr (tk ),

r = 1, . . . , n.

k=1

In paper[35] Kowalewski generalized this theorem by substituting the linear measure dt by the weight measure F (t)dt, where F ∈ C[a, b], F is of the constant sign on (a, b) and Zb n X λk = F (t)dt. k=1

a

Only in 2008, Slobodanka Jankovi´c and Milan Merkle in paper [31] extended this theorem for any intervals I ⊂ R by introducing in place of measure F (t)dt any finite positive measure µ defined on the Borel σ-field of interval I (functions xk belong then to the set CR (I) ∩ Lµ (I), k = 1, . . . , n, respectively). Moreover, as it is noticed by these authors, except two citations the Kowalewski’s results remained completely unknown – how unfairly. The above results, together with the proofs, are also presented in monograph [30]. 5. History of the uniqueness theorems for trigonometric series (including the multiple trigonometric series) still lasts (see papers [3, 4, 5, 55, 59]). It is worth to mention additionally the Du Bois Reymond result [13] from 1876, a little bit younger in relation to the Cantor result, cited and proved in [59] as well as in [9]: Theorem A. If trigonometric series ∞  a0 X + an cos(nx) + bn sin(nx) 2 n=1

is convergent everywhere to a finite sum f (x) integrable on [0, 2π], then it is the Fourier series of function f (x). Let us notice (after Nina Bari [9]) that, originally, Du Bois Reymond referred this result to integrability in the Riemann sense (his result holds true also in the case when we neglect the convergence of the trigonometric series on some countable set). Appropriate extension of Theorem A for the functions integrable in Lebesgue sense we owe to Lebesgue himself (however Theorem A is still called the Du Bois Reymond theorem). Obviously, Theorem A implies the discussed here Cantor’s result from 1870. At the end let us also recall that the similar uniqueness theorems for the Haar and the Walsh series have been discussed and solved, among others, by Russians W.A. Skvorcov (see [50, 51, 52]) and M.G. Plotnikov (see [42, 43, 44]). The subject

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of uniqueness of the multiple trigonometric series is also discussed in the third chapter of V.L. Shapiro’s monograph [49]. 6. The characterization of derivatives in the class of bounded functions f : (0, 1) → R was achieved by I. Maximoff (see [40, 41]), mentioned in the ending of Professor’s letter: Theorem B. Bounded function f : (0, 1) → R is equivalent in the Lebesgue sense to the derivative if and only if f is of the first Baire class and, concurrently, satisfies the Darboux condition. This theorem, together with the beautiful proof based on the David Preiss conception, can be found in fourth chapter of monograph [25]. Funnily enough, this proof effectively uses the Neugebauer theorem characterizing the (bounded) derivatives, mentioned in Professor’s letter. Authors of monograph [25] call also in question the correctness of the original, i.e. given by I. Maximoff, proof of Theorem B (if it is true, then the author of the first correct proof of this theorem would be the already mentioned D. Preiss). Moreover, as it was shown in [25] after Goffman and Neugebauer, the characterization from Theorem B holds also in the class of bounded functions f : (0, 1) → R, equivalent in the Lebesgue sense to the approximative derivative. More outlined information devoted to the characterizations of derivatives can be found in papers [15, 23] and [19]. Especially in the latter the following interesting result achieved by Krzysztof Chris Ciesielski is given [19, 53]: Theorem C. Neither of the following function classes is topologicable: class Λ of all derivatives of the Zahorski class Mk , k = 1, ..., 5, class of all functions satisfying the Darboux condition, class of all measurable functions and class of all functions possessing the Baire property.

Bibliography 1. Andrews G.A.: Euler’s ,,De Partitio Numerorum”. Bull. Amer. Math. Soc. 44 (2007), 561–573. 2. Apostol T.M.: An elementary view of Euler’s Summation Formula. Amer. Math. Monthly 106 (1999), 409–418. 3. Ash J.M., Rieders E., Kaufman R.P.: The Cantor – Lebesgue property. Israel J. Math. 84 (1993), 179–191. 4. Ash J.M.: Uniqueness of representation by trigonometric series. Amer. Math. Monthly 96 (1989), 873–885. 5. Ash J.M.: Uniqueness for higher dimensional trigonometric series. Cubo 4 (2002), 97–120. 6. Ayoub R.: Euler and the Zeta function. Amer. Math. Monthly 81 (1974), 1067–1086. 7. Banach S.: Sur les ensembles de points o´ u la d´ eriv´ ee est infinie. Comptes Rendus de l’Acad´ emie de Sciences 173 (1921), 457–459. 8. Bari N.K.: The problem of uniqueness of expansion of a function into a trigonometric series. Uspekhi Mat. Nauk 4, no. 3 (1949), 3–68 (in Russian). 9. Bary N.K.: A Treatise on Trigonometric Series, vol. 1 and 2. Pergamon Press, New York 1964. 10. Benedetto J.J.: Harmonic Analysis and Applications. CRC Press, Boca Raton 1997. 11. Benedetto J.J., Czaja W.: Integration and Modern Analysis. Birkh¨ auser, Boston 2009. 12. Bochner S.: Fourier series came first. Amer. Math. Monthly 86 (1979), 197–199. 13. Du Bois Reymond P.: Beweis dass die koeffizienten der trigonometrischen Rechen. . . Abh. Akad. Wiss. M¨ unchen 12 (1876), 117–166.

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14. Bruckner A.M.: Differentiation of Real Functions. Springer, Berlin 1978. 15. Bruckner A.M.: The problem of characterizing derivatives revisited. Real Anal. Exchange 21 (1995–96), 112–133. 16. Bruckner A.M., Leonard J.L.: Derivatives. Amer. Math. Monthly 73 (1966), 24–56. 17. Bruckner A.M., Thomson B.S.: Real variable contribution of G.C. Young and W.H. Young. Expo. Math. 19 (2001), 337–358. 18. Carleson L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116 (1966), 135–157. 19. Ciesielski K.: Set theoretic real analysis. J. Appl. Anal. 3 (1997), 143–190. 20. Coppel W.A.: J.B. Fourier – On the occasion of his two hundredth birthday. Amer. Math. Monthly 76 (1969), 468–483. 21. Djaczenko M.J.: Some problems of the theory of multiple trigonometric series. Uspiekhi Mat. Nauk 47 (1992), 97–162. 22. Feffermann C.: Pointwise convergence of Fourier series. Ann. of Math. (2) 98, no. 3 (1973), 551–571. 23. Freiling Ch.: On the problem of characterizing derivatives. Real Anal. Exchange 23, no. 2 (19971998), 805–812. 24. Gandini P.M.: An extension of a theorem of Banach. Arch. Math. 67 (1996), 211–216. 25. Goffman C., Nishiura T., Waterman D.: Homeomorphisms in Analysis. Amer. Math. Soc., Providence 1997. 26. Grafakos L.: Modern Fourier Analysis. Springer, Berlin 2009. ¨ 27. Hahn H.: Uber den Fundamentalsatz der Integralrechnung. Monatsh. Math. Phys 16 (1905), 161–166. 28. Halmos P.R.: Fourier series. Amer. Math. Monthly 85 (1978), 33–34. 29. Hetmaniok E., Pleszczy´ nski M., Witula R.: Selected scientific achievements of Professor Zygmunt Zahorski. In this monograph: Monograph on the occasion of 100th birthday anniversary ´ Gliwice 2015, of Zygmunt Zahorski, Witula R., Slota D., Holubowski W. (eds.), Wyd. Pol. Sl., 45–49. 30. Hetmaniok E., Slota D., Witula R.: Mean Value Theorems. Silesian University of Technology Press, Gliwice 2012 (in Polish). 31. Jankovi´ c S., Merkle M.: A mean value theorem for systems of integrals. J. Math. Anal. Appl. 142 (2008), 334–339. 32. Kechris A.S., Louveau A.: Descriptive Set Theory and the Structure of Sets of Uniqueness. London Math. Soc. Lecture Note Ser. 128, Cambridge Univ. Press, Cambridge 1987. 33. Kolmogoroff A.: Une s´ erie de Fourier-Lebesgue divergente presque partout. Fund. Math. 4 (1923), 324–328. 34. Kowalewski G.: Ein Mittelwertsatz f¨ ur ein System von n Integralen. Z. Math. Phys. (Schl¨ omilch Z.) 42 (1895), 153–157. 35. Kowalewski G.: Bemerkungen zu dem Mittelwertsatze f¨ ur ein System von n Integralen. Z. Math. Phys. (Schl¨ omilch Z.) 43 (1896), 118–120. 36. Laugwitz D., Rodewald B.: A simple characterization of the Gamma Function. Amer. Math. Monthly 94 (1987), 534–536. 37. Laugwitz D.: On the historical development of infinitesimal mathematics. Amer. Math. Monthly 104 (1997), 447–455. 38. Lojasiewicz S.: Introduction to Real Functions Theory. PWN, Warsaw 1973 (in Polish). 39. Luzin N.N.: Integral and trigonometric series, editing and comments – N.K. Bari and D.E. Menshov. Goztiechizdat, Moscow 1951 (in Russian). 40. Maximoff J.: On continuous transformation of some functions into an ordinary derivative. Ann. Scuola Norm. Super. Pisa 12 (1943), 147–160. 41. Maximoff J.: Sur les fonctions d´ eriv´ ees. Bull. Sci. Math. 64 (1940), 116–121. 42. Plotnikov M.G.: On uniqueness sets for multiple Walsh series. Mat. Zametki 81, no. 2 (2007), 265–279 (English transl. in Math. Notes 81, no. 1–2 (2007), 234–246). 43. Plotnikov M.G.: Several properties of generalized multivariete integrals and theorems of the du Bois-Reymond type for Haar series. Sb. Math. 198, no. 7 (2007), 967–991. 44. Plotnikov M.G.: Quasi-measures, Hausdorff p-measures and Walsh and Haar series. Izv. RAN: Ser. Mat. 74, no. 4 (2010), 157–188 (English transl. in Izv. Math. 74, no. 4 (2010), 819–848).

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45. Ruziewicz S.: On unapplicability of the fundamental theorem of integral calculus for functions possessing the infinite derivatives. Mathematical-Physical Works 31 (1920), 31–33 (in Polish). 46. Ruziewicz S.: Sur les fonctions qui ont la mˆ eme d´ eriv´ ee et dont la diff´ erence n’est pas constante. Fund. Math. 1 (1920), 148–151. 47. Saks S.: Theory of the Integral. Dover Publications, New York 1937 (1933). 48. Saks S.: Zarys teorii calki. Wyd. Kasy im. Mianowskiego, Warsaw 1930. 49. Shapiro V.L.: Fourier Series in Several Variables with Applications to Partial Differential Equations. CRC Press, Boca Raton 2011. 50. Skvortsov V.A.: On uniqueness theorem for a multidimensional Haar series. Izv. Akad. Nauk. Arm. SSR Mat. 23, no. 3 (1988), 293–296 (English transl. in Soviet. J. Contemp. Math. Anal. 23, no. 3 (1988), 104–108). 51. Skvortsov V.A.: Uniqueness sets for multiple Haar series. Mat. Zametki 14, no. 6 (1973), 789– 798 (English transl. in Math. Notes 14, no. 6 (1973), 1011–1016). 52. Skvortsov V.A., Talalyan A.A.: Some uniqueness questions of multiple Haar and trigonometric series. Mat. Zametki 46, no. 2 (1989), 104–113 (English transl. in Math. Notes 46, no. 2 (1989), 646–653). 53. Tartaglia M.: Sulla caratterizzazione delle derivate. Pubbl. Dip. Mat. Stat., Napoli 1988. 54. Tong J., Hochwald S.: An application of the Darboux property of derivatives. Int. J. Math. Edu. Sci. Technology 34 (2003), 150–153. 55. Uljanow P.L.: A.N. Kolmogorov and divergent Fourier series. Uspiekhi Mat. Nauk 38 (1983), 51–90. 56. Zahorski Z.: Sur la premi` ere d´ eriv´ ee. Trans. Amer. Math. Soc. 69 (1950), 1–54. 57. Zahorski Z.: On the set of not-differentiability points of any function. Report on the V Congress of Polish Mathematicians in Krak´ ow, 29-31 of May 1947. Supplement to the PTM Annual 21 (1949), 23–26. 58. Zahorski Z.: Une s´ erie de Fourier permut´ ee d’une fonction de classe L2 divergente presque partout. C. R. Acad. Sci. Paris 251 (1960), 501–503. 59. Zygmund A.: Trigonometric Series, vol. I. Cambridge Univ. Press, 1959 (Russian transl. was used, Mir Press, Moscow 1965, edited by N.K. Bari). 60. Zygmund A.: Notes on the history of Fourier series, Studies in harmonic analysis. Studies in Mathematics 13, J.M. Ash (ed.), Math. Assoc. of America (1976), 1–19. ´ ser. Mat.-Fiz 48, Gliwice 1986. 61. Zeszyty Nauk. P.Sl.,

´ and Zygmunt Zahorski Janina Sladkowska-Zahorska

From left: Wladyslaw Wilczy´ nski, Zygmunt Zahorski and Jan Lipi´ nski

Zygmunt Zahorski and Jan Lipi´ nski

Zygmunt Zahorski and contemporary real analysis Wladyslaw Wilczy´ nski

Professor Zygmunt Zahorski was an eminent specialist in real analysis. His papers, concerning mainly different classes of real functions of a real variable, are precise and sophisticated (in the positive meaning of the word). Numerous mathematicians throughout the world are working on real functions theory using his ideas and results. Among most widely known are: his first pupil professor Jan Stanislaw Lipi´ nski and American mathematician Andrew M. Bruckner from Santa Barbara, California. Since the number of papers inspired by the results, ideas or techniques of professor Zahorski exceedes hundreds, in the sequel we shall concentrate on some chosen publications. An essential part of mathematical activity of professor Lipi´ nski was strictly connected with the kind of problems considered by professor Zahorski. His achievements deserve a separate presentation. Fortunately, quite recently Paul Humke has done an excellent work (see [30]). The most frequently quoted and the most influential paper of professor Zygmunt Zahorski is the monumental treatise on the first derivative ([6]). A lot of very interesting facts on Dini derivatives is included in the habilitation thesis, which, however, was never published. In the paper [6] professor Zahorski tried to characterize sets of the form {x : f (x) < a} and {x : f (x) > a}, when f is a real function of a real variable possessing a continuous primitive function. To this end he considered two hierarchies: of sets M0 ⊃ M1 ⊃ M2 ⊃ M3 ⊃ M4 ⊃ M5 and of functions M0 ⊃ M1 ⊃ M2 ⊃ M3 ⊃ M4 ⊃ M5 . Let E ⊂ R be a non-empty Fσ set. We say that E belongs to the class – M0 if each point of E is a point of bilateral accumulation of E, – M1 if each point of E is a point of bilateral condensation of E, – M2 if each one sided neighbourhood of each x ∈ E intersects E in a set of positive measure, – M3 if there exists a sequence {Kn }n∈N S of closed sets and a sequence {ηn }n∈N of numbers from [0, 1) such that E = n Kn and for each x ∈ Kn and each c > 0 there exists a number ǫ(x, c) > 0 such that if h and h1 satisfy hh1 > 0, hh1 < c, |h + h1 | < ǫ(x, c), then W. Wilczy´ nski Faculty of Mathematics and Computer Science, University of L´ od´ z, 90-238 L´ od´ z, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 99–108. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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λ(E ∩ (x + h, x + h + h1 )) > ηn , h1 – M4 if E fulfills M3 and each ηn is positive, – M5 if each point of E is a point of density of E. Let f be an extended real valued function defined on some interval I. We say that f ∈ Mk if {x : f (x) < a} ∈ Mk and {x : f (x) > a} ∈ Mk for each a ∈ R, k = 0, 1, . . . , 5. The above sets usually are called the associated sets of f . Professor Zahorski has proved that all inclusions between classes of sets and of functions are strict with one exception: M0 = M1 and this is a class of Darboux Baire one functions. Also in [6] it is proved that M5 is the class of approximately continuous functions. It is interesting that approximately continuous functions were considered already by A. Denjoy around 1915 and the density topology was introduced about 40 years later by O. Haupt and Ch. Pauc ([29]). Recall that the real function of a real variable is approximately continuous if it is continuous when the ordinary topology is used on the range and the density topology is used on the domain. One can find more information in [55]. Main results of the paper [6] are connected with the conditions M2 , M3 and M4 . Namely, professor Zahorski has proved that if f is a derivative (possibly infinite) of a continuous function, then all associated sets of f belong to M2 , if f is a finite derivative, then all associated sets belong to M3 and if f is a bounded derivative, then all associated sets are in M4 . None of these necessary conditions is sufficient. More detailed information can be found in [14] and [17], where is also an exhaustive list of references. Now it is time to explain why professor Zahorski in [6] has studied the characterization of sets associated with the derivative and not the characterization of the derivative in terms of associated sets. Numerous classes of functions are characterized in that way, for example a function is continuous if and only if the associated sets are open, a function is Baire one if and only if they are Fσ , a function is measurable if and only if they are measurable and so on. Observe that all mentioned classes are closed with respect to superpositions from outside with a homeomorphism, i.e. if f is in the class and h : R −→ R is a homeomorphism, then h ◦ f is in the same class. The class onto

of derivatives (bounded or unbounded, finite or assuming infinite values) is far from possessing this property. This was observed by G. Choquet in [21] who proved that if h is a nonlinear homeomorphism h : R −→ R then there exists a bounded derivative onto f0 such that h ◦ f0 is not a derivative. Professor Zahorski was obviously aware of this fact. A. Bruckner in [15] deeply studied the bad behaviour of the derivative with this respect and proved that there exists a bounded derivative f such that for each nowhere linear (linear on no interval) homeomorphism h : R −→ R and each interval onto I ⊂ R there exists a point x ∈ I such that h ◦ f is not the derivative of its integral at x. The decisive step in characterizing the associated sets of the derivative has been made by D. Preiss in [42]. His paper is devoted to the characterization of the triple S, G, E of subsets of the real line for which there exists a function f : R → R differentiable at each point such that E = {x : f ′ (x) > 0}, G = {x : f ′ (x) = +∞} and S is the set of points of discontinuity of f . Preiss has defined the class of sets M ∗ (the definition is rather complicated and uses the condition similar to M4 of Zahorski) and

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has proved that M ∗ is the class of associated sets for not necessarily finite derivatives whose primitives need not be continuous and M2∗ = M ∗ ∩ M2 is the class of associated sets for not necessarily finite derivatives whose primitives are continuous and M3∗ = M ∗ ∩ M3 is the class of associate sets for finite derivatives. His results have also application to the approximative derivatives. C. Neugebauer in [37] has proved the characterization of derivatives in terms of the behaviour of interval functions. His theorems say that a function f : I0 → R (where I0 is some interval) is Darboux Baire one if and only if it fulfills the following condition C1 : for each interval I ⊂ I0 there exists a point xI ∈ Int I such that I → x implies f (xI ) → f (x) (here I → x means that x ∈ I and λ(I) → 0) and a function f : I0 → R is the derivative if and only if it fulfills the condition C1 and moreover if I = I1 ∪ I2 , Int I1 ∩ Int I2 = ∅, then (for xI from C1 ) f (xI1 ) · λ(I1 ) + f (xI2 ) · λ(I2 ) f (xI ) = . λ(I) The last equality means that f (xI ) · λ(I) is an additive interval function. The theorem of Neugebauer shows how much Darboux Baire one functions differ from the derivatives. D. Preiss and M. Tartaglia in [43] have given an interesting characterization of derivatives in terms of the set of derivatives (a sort of circular characterization according to Ch. Freiling ([27])). They proved that f is a derivative if and only if for each set E ⊂ R there is a derivative g such that f −1 (E) = g −1 (E). Continuing this way K. Ciesielski in [22] has proved a general theorem stating that numerous families F of real functions (including the family ∆ of all derivatives) can be characterized as a family of the form C(D, A) = {f ∈ RR : f −1 (A) ∈ D for every A ∈ A}, where A is some family of subsets of R and D = {f −1 (A) : f ∈ F and A ∈ A}. His main theorem reads as follows: Let F , R be such that card(R) ≤ ∁+ , card(F ) ≤ ∁, where ∁ denotes the cardinality of continuum and ∁+ is the next cardinal number, F contains all constant functions and card(g(R)) = ∁ for any non-constant function g which is a difference of two functions from F . Then there exists a family A ⊂ 2R of cardinality less or equal to card(R) such that F ∩ R = R ∩ C(D, A), where D = {f −1 (A) : f ∈ F and A ∈ A} as before. If we take F = ∆ and R – the family of Borel functions, we conclude that there exists a family A ⊂ 2R such that card(A) ≤ ∁ and ∆ = R ∩ C(D, A), where D = {f −1 (A) : f ∈ ∆ and A ∈ R}. Ciesielski also proved in [22] that there exists a Bernstein set B ⊂ R such that ∆ = DB1 ∩ C(D0 , {B + c : c ∈ R}) = C(D, A), S where A = c∈R {(−∞, c), (c, ∞), B + c)}, D0 = {f −1 (B + c) : f ∈ ∆ and c ∈ R} and D = {f −1 (A) : f ∈ ∆ and A ∈ A}, so the family consisting of all translations of a single Bernstein set is sufficient. Recall the definition of the Kurzweil-Henstock integral. Let I be a closed interval, I1 , . . . , In – a partition of I and x1 , . . . , xn – a sequence of points such that xi belongs to the interval Ii for each i. Such system of intervals and points is called a tagged partition of I. Suppose that f isPany function defined on I then each tagged partition n yields a Riemann sum given by i=1 f (xi )·λ(Ii ). If δ is a positive function defined on I and for each i ∈ {i, . . . , n} we have λ(Ii ) < δ(xi ), then the tagged partition is called δ-fine (such positive function is usually called gauge function). A function f : R → R is Kurzweil-Henstock integrable if and only if for each closed interval I and for each ǫ > 0 there exists a gauge δ : I → R+ such that any two δ-fine tagged partitions of I have Riemann sums which differ by less than ǫ · λ(I). The LH-integral is then

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defined to be the limit of the corresponding Riemann sums as ǫ → 0. It is known that derivatives are KH-integrable. Ch. Freiling in [27] has observed that in fact a function f is a derivative if and only if it is KH-intergrable. The paper [27] is a good source of informations about possible characterizations of derivatives. Let us come back for a moment to the class M3 . C. Weil in [53] has introduced the property Z and has proved that if a function has the Darboux and Denjoy property, then the property Z implies the Zahorski property M3 . Moreover, derivatives, approximate derivatives, Lp -derivatives all have the property Z. P.S. Bullen and D.N. Sarkhel have made a step further – they have defined the property Z ∗ (stronger than Z) in the following way: The function f on I is said to have the property Z ∗ if for every c ∈ I and ǫ > 0, η > 0 there is a neighbourhood Ic of c such that the following conditions Z + and Z − hold: Z + : if f (x) ≥ f (c) − ǫ a.e. on a closed interval J ⊂ Ic , then λ(A) − λ(B) ≤ ≤ η · ρ(c, J) (ρ(c, J) – a distance between c and J), where A = {x ∈ J : f (x) ≥ f (c) + ǫ}, B = {x ∈ J : f (c) − ǫ ≤ f (x) < f (c)}, Z − : if f (x) ≤ f (c) + ǫ ≤ η · ρ(c, J),

a.e. on a closed interval J ⊂ Ic , then λ(A) − λ(B) ≤

where A = {x ∈ J : f (x) ≤ f (c) − ǫ}, B = {x ∈ J : f (c) < f (x) ≤ f (c) + ǫ}. The main result of [19] says that k-th Peano derivative, k-th approximate Peano derivative and k-th Lp -derivative all have the property Z ∗ (for k ≥ 1). Still in the paper [6] one can find the following theorem: Theorem. Let f be a function fulfilling on an interval I the following conditions: (i) f is a Darboux function, (ii) f ′ exists (finite or not) possibly except a denumerable set of points, (iii) f ′ (x) ≥ 0 a.e. Then f is continuous and nondecreasing on I. In 1939 G. Tolstov ([49]) has proved the theorem which is an improvement of the theorem of Goldowski-Tonelli: Theorem. Let f be a function fulfilling on an interval I the following condition: i) f is approximately continuous, ′ ii) fap exists (finite or not) possibly except a denumerable set of points, ′ iii) fap (x) ≥ 0 a.e. Then f is continuous and nondecreasing on I. Observe that the condition (i) in Zahorski’s theorem is weaker than in Tolstov’s while conditions (ii) and (iii) are stronger because they involve ordinary derivative instead of the approximate derivative. Professor Zahorski asked if it is possible to prove a theorem which implies both Tolstov’s theorem and Zahorski’s theorem. However, there exists a non-monotone function which is Darboux and fulfills condition ii) and iii) of Tolstov (see [17], p. 45), so simply taking the weaker condition from each pair does not work. From the second condition of Zahorski it follows that f is Baire one

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´ atkowski in [48] and A. Bruckner in [11] and [12] have proved that if function. T. Swi¸ f is Darboux Baire one function on I and fulfills conditions (ii) and (iii) of Tolstov, then f is continuous and nondecreasing on I. In fact, A. Bruckner has proved the more general scheme: Let P be a function-theoretic property sufficiently strong to imply (a) any Darboux Baire one function which satisfies property P on an interval I is of generalized bounded variation on I, (b) any continuous function of bounded variation which satisfies property P on I is nondecreasing on I. Then any Darboux Baire one function which satisfies property P on I is continuous and increasing on I. Using this result E. Lazarow and W. Wilczy´ nski have proved a similar theorem for the category analogue of the approximate derivative ([36]). For further informations on monotonicity conditions see [23]. E. Lazarow has proved in [35] that a finite Iapproximate derivative is Baire one. ´ atkowski real valued functions R. Pawlak in [40] when studying Darboux and Swi¸ of two real variables has introduced the hierarchy of Zahorski classes on arcs in R2 and has proved, among others, that the same inclusions and equalities between these classes, Darboux Baire one functions and approximately continuous functions of two variables hold as in the case of functions of one variable. The problem of characterization of the set of points of nondifferentiability of a continuous functions has been solved by professor Zahorski in [2] and [3]. He has shown that this set is the union of a Gδ set with a Gδσ set of Lebesgue measure zero and that any set of this form is the set of points of nondifferentiability for some continuous function. For a continuous function of bounded variation the term Gδ can be dropped from the statement. This theorem has been extended by A. Brudno ([18]) to arbitrary functions. The construction of a continuous function with prescribed set of points of differentiability has been simplified by S. Piranian ([41]). Observe that classical constructions of Bolzano, Weierstrass or van der Waerden are dealing with only one set of points of nondifferentiability of a continuous function, namely, the set equal to R. The beautiful result of professor Zahorski for a long time was not commonly recognized. In the book Real and Abstract Analysis, Springer-Verlag 1969 by E. Hewitt and K. Stromberg one can find on page 266 the following sentences: “(17.13) Question. Suppose that λ(A) = 0, A ⊂ [a, b]. Is it possible to find a monotone function f on [a, b] such that f ′ exists exactly on A′ ∩ (a, b)? The complete answer seems to be unknown”. F.M. Filipczak in [24] has studied the set of points of differentiability from slightly another point of view. He proved that if E, F, G and H are subsets of an interval I such that E ⊂ F ⊂ G ⊂ H ⊂ I, E is of type Fσ , H is simultaneously of type Fσ and Gδ , λ(H \ E) = 0, H \ F is countable (so F and G are Gσ ’s), then there exists a real function defined on I such that E is the set where f ′ exists finite, F is the set of points of continuity of f and G is the set where f ′ exists finite or not. In [25] and [26] F.M. Filipczak has established the Borel class of symmetric derivatives of approximately continuous functions and has proved that the set of points of symmetric nondifferentiability is characterized exactly as in the case of ordinary nondifferentiability. His theorem reads as follows: If E is the set of the form

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E = A ∪ B, where A ∈ Gδ , B ∈ Gδσ and λ(B) = 0, then there exist a continuous function f such that f ′ exists and is finite for x ∈ / E, a symmetric derivative Df and unilateral derivatives do not exist for x ∈ E. Professor Zygmunt Zahorski was interested also in the behaviour of more regular functions, namely, functions belonging to the class C ∞ . If f ∈ C ∞ is a real function of 2 h ′ a real variable and T f (x, h) := f (x)+ 1! f (x)+ h2! f ′′ (x)+ . . . associated Taylor series, then there are three possibilities: either the radius of convergence of T is positive and the series is convergent to f in some neighbourhood of 0, or the radius of convergence equals zero, or the radius of convergence is positive but the series does not converge to f . In the first case we say that x is a regular point (or a point of analyticity), in the second we say that x is a singular point in the sense of Pringsheim (or a point of divergence), in the third we say that x is a singular point in the sense of Cauchy (or a point of false convergence). The paper [4] contains an elegant and complete characterization of three sets. The theorem of Zahorski says that if f ∈ C ∞ , then the set A of regular points is open, the set D of points singular in the sense of Pringsheim is of type Gδ , the set F of points singular in the sense of Cauchy is the set of the first category of type Fσ and that if A, D, F are three disjoint sets such that R = A ∪ D ∪ F , A is open, D is Gδ and F is Fσ of the first category, then there exists a function f ∈ C ∞ for which A is the set of regular points, D – the set of points singular in the sense of Pringsheim and F – the set of points singular in the sense of Cauchy. The proof has been simplified by H. Salzmann and K. Zeller in [45]. J. Siciak in [46] using the method of these authors, has obtained an analogous result for functions of several variables. Professor Zahorski has obtained also interesting results in differential geometry. In [5] he has proved among others that if K is a rectifiable curve in R2 , then there exists a parametric representation for K each of whose coordinate functions has a bounded derivative. Essentially the same result has been obtained by G. Choquet in [21]. More informations on this topic one can find in [13]. The paper [7] contains an unexpected construction of the very winding curve – the tangent line assumes all directions on each subarc of the curve. Moreover, the tangent line does not exist on dense set of points – it follows from the properties of the derivative. W. Wilczy´ nski in [54] has presented a construction of a continuous function f defined on the unit circle K = {(x, y) : x2 + y 2 ≤ 1} the graph of which is a rectifiable surface. The normal line to this surface takes every direction (from the upper semi-sphere) on each part of the surface. In this case also the set of non-differentiability of f must be dense of K. Another important result concerning derivatives is contained in [1]. Professor Zahorski has constructed an everywhere differentiable continuous function with an infinite derivative on an arbitrary given Gδ set of Lebesgue measure zero. Earlier it was known that these exists a continuous function with an infinite derivative on an arbitrary Gδ null set and with finite Dini derivatives elsewhere. V. Tzodiks in [50] has proved the following related result: A necessary and sufficient conditions for two sets E1 and E2 to be sets where f ′ = +∞ and f ′ = −∞, where f is a finite function, are: E1 and E2 be Fσδ ’s of measurable zero such that there exist disjoint Fσ sets H1 and H2 with E1 ⊂ H1 and E2 ⊂ H2 . T. Nishiura in [38] has used Zahorski classes of sets in the theory of absolute measurable spaces and absolute null spaces. A separable metrizable space is a Zahorski space if it is empty or it is the union of a countable sequence of topological copies

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of the Cantor set. A subset of a separable metrizable space is a Zahorski set if it is a Zahorski subspace of this space. A Zahorski measure determined by the set E, where E is a Zahorski set in a separable metrizable space X is a continuous, complete, finite Borel measure µ on X such that µ(X \ E) = 0 and µ(E ∩ U ) > 0 if U is an open set such that E ∩ U 6= ∅. The reader can observe immediately the analogy with the classes M1 and M2 . T. Nishiura has shown the relationship between Zahorski set and Lusin set and has expressed the opinion ([38], p. 193): “Zahorski spaces appear in a very prominent way in many proofs”. According to the opinion of Professor Zygmunt Zahorski (see his biography in ´ askiej Matematyka-Fizyka, z. 48, Gliwice 1986, p. 19) Zeszyty Naukowe Politechniki Sl¸ among his publications there is only one “of essential good quality”. He did not say which one he had in mind. The international mathematical community duly appreciates numerous theorems of Zahorski, their influence in the development of the theory of real functions and orthogonal expansions, so it is really difficult to say what is his greatest achievement. I believe that it may be the construction of the rearrangement of terms of a Fourier series ([8]). In 1927 A. Kolmogorov in the paper [33] common with D. Menshov stated the following theorem: There exists a function f ∈ L2 [0, 2π] whose Fourier series after some rearrangement of terms diverges almost everywhere. In spite of efforts of Kolmogorov himself and of his students the proof was still unattainable until 1960, when professor Zahorski accomplished the construction of the series and the rearrangement. Later P.L. Ulyanov ([51]) observed that similar construction works for the Walsh and the Haar system and A.M. Olevskiˇi ([39]) and P.L. Ulyanov ([52]) proved that for any complete, orthogonal, normal system there exists a function f ∈ L2 whose Fourier series with respect to this system after some rearrangement of terms diverges almost everywhere. Professor Zahorski set a high value on his rearrangement result. He used to mention a theorem of A.M. Garsia [28] which says that the existence of such permutation is highly improbable. Namely, let f ∈ L2 [0, 2π] and let {mk }k∈N be an increasing sequence of positive integers such that Smk (x, f ) → f (x) almost everywhere. (Here Sm (x, f ) denotes the m-th partial sum k→∞

of the Fourier series f ). The existence P of a sequence {mk }k∈N is assured by the folP∞ ∞ lowing theorem (see [9], p. 178-181): if k=1 m1k < +∞ and k=n m1k = O( m1n ) and f ∈ L2 [0, 2π], then Smk (x, f ) → f (x) almost everywhere. Consider the permutation k→∞

σ = {σ1 , σ2 , . . . } of the natural numbers related to the sequence {mk }k∈N in the following way: if mk−1 < i ≤ mk , then mk−1 < σi ≤ mk (we assume m0 = 0). Let Pk be the set of all permutations of {mk−1 + 1, mk−1 + 2, . . . , mk }. The set P of all permutations σ of the natural numbers described above can be naturally identified with the 1 direct product of {Pk }k∈N . If µk is the measure on Pk such that µk (p) = (mk −m k−1 )! for p ∈ Pk , then µ = ⊕n µn is the measure on P. The theorem of A.M. Garsia says precisely: If f ∈ L2 [0, 2π], {mk }k∈N is the above mentioned sequence and independently for each k ∈ N we permute at random the terms of the Fourier series of f whose indices are between mk−1 + 1 and mk , then with probability µ equal to one the resulting rearranged series will converge almost everywhere. So professor Zahorski have done something which was almost impossible. To be honest it is necessary to mention the result of R. Bilyen, R. Kallman and P. Lewis ([10]). Suppose that G is a set of all permutations of the set of natural numbers. If we put

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d(σ, σ ′ ) =

∞ X

2−n (dn (σ, σ ′ ) + dn (σ −1 σ



−1

)),

where

n=1

dn (σ, σ ′ ) =

|σn − σn′ | , 1 + |σn − σn′ |

then (G, d) is a Polish space. The main result in [10] says that Pit∞{fn }n∈N is a sequence of Borel functions defined on some interval I such that n=1 fn diverges almost P∞ everywhere, then the set {σ ∈ G : n=1 fσn diverges a.e.} is residual in (G, d). So from the point of view of Baire category it should be “easy” to find a rearrangement destroying the convergence. N.N. Lusin in 1913 conjectured that each function in L2 [0, 2π] has an a.e. convergent Fourier series. Professor Z. Zahorski for many years struggled with this problem and perhaps this experience helped him in finding the rearrangement. The problem of Lusin was finally solved by L. Carleson ([20]) and soon after appearing of his paper R. Hunt ([31]) was able to extend this result to all spaces Lp [0, 2π] for p > 1. A.N. Kolmogorov in [32] gave an example of a function in L1 [0, 2π] with an a.e. divergent Fourier series. This leaves only a narrow place for improving the result of Hunt. Moreover, in the example of Kolmogorov the function is in the class L log log L. Sj¨olin in [47] has proved that each function in the space L log L log log L has also an a.e. convergent Fourier series. For more informations concerning this topic see [44].

Bibliography ¨ 1. Zahorski Z.: Uber die Menge der Punkte in welchen die ableitung unendlich ist. Tˆ ohoku Math. J. 48 (1941), 321–330. 2. Zahorski Z.: On the set of points of nondifferentiability of a continuous function. Rec. Math. (Mat. Sbornik, N. Ser.) 9(51) (1941), 487–510 (in Russian). 3. Zahorski Z.: Sur l’ensemble des points de non-d´ erivabilit´ e d’une fonction continue. Bull. Soc. Math. France 74 (1946), 147-178. 4. Zahorski Z.: Sur l’ensemble de points singuliers d’une fonction d’une variable r´ eelle admettant les deriv´ ees de tous les ordres. Fund. Math. 34 (1947), 183–245. 5. Zahorski Z.: On Jordan curves possessing the tangent line at each point. Rec. Math. (Mat. Sbornik, N. Ser.) 22(64) (1948), 3–26 (in Russian). 6. Zahorski Z.: Sur la premi` ere d´ eriv´ ee. Trans. Amer. Math. Soc. 69 (1950), 1–54. 7. Zahorski Z.: Sur les courbes dont la tangente prend sur tout arc partiel toutes les directions. Czechoslovak Math. J. 1 (1951), 105–117. 8. Zahorski Z.: Une s´ erie de Fourier permut´ ee d’une fonction de classe L2 divergente presque partout. C. R. Acad. Sci. Paris 251 (1960), 501–503. 9. Bari N.: Trigonometric Series. Izd. Fiziko-Matematieskoj Literatury, Moskwa 1961 (in Russian). 10. Bilyen R.G., Kallman R.R., Lewis P.W.: Rearrangement and category. Pacific J. Math. 121, no. 1 (1986), 41–46. 11. Bruckner A.M.: A theorem on monotonicity and a solution to a problem of Zahorski. Bull. Amer. Math. Soc. 71 (1965), 713-716. 12. Bruckner A.M.: An affirmative answer to a problem of Zahorski and some consequences. Mich. Math. J. 13 (1966), 15-26. 13. Bruckner A.M.: Creating differentiability and destroying derivatives. Amer. Math. Monthly 85, no. 7 (1978), 554–562. 14. Bruckner A.M.: Differentiation of Real Functions. Lecture Notes in Mathematics 659, SpringerVerlag, London 1978.

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15. Bruckner A.M.: On destruction of derivatives via changes of scale. Bull. of the Inst. of Math. Academia Sinica 9, no. 3 (1981), 407–415. ´ Mat.16. Bruckner A.M.: Some indirect consequences of a theorem of Zahorski. Zeszyty Nauk. PSl., Fiz. 48 (1986), 47–54. 17. Bruckner A.M., Leonard J.L.: Derivatives. Amer. Math. Monthly 73, no. 4 (1966), 22–56 (Part II, Papers in Analysis, Number 11 of the Herbert Ellsworth Slaught Memorial Papers). 18. Brudno A.: Continuity and differentiability. Rec. Math. (Mat. Sbornik) (13), 55 (1943), 119–134 (in Russian). 19. Bullen P.S., Sarkhel D.N.: A continuity property of derivatives. Canad. Math. Bull. 39, no. 1 (1996), 10–20. 20. Carleson L.: On convergence and growth of partial sums of Fourier series. Acta Math. 116 (1966), 135–157. 21. Choquet G.: Application des propri` et´ es descriptives de la fonction contingent ` a la th´ eorie des fonctions de variable e´ eelle et ` a la g´ eometric diff´ erentielle des vari´ et´ es cart´ esiennes. J. Math. Pures Appl. (9), 26 (1947), 115–126. 22. Ciesielski K.: Characterizing derivatives by preimages of sets. Real Anal. Exchange 23, no. 2 (1997-1998), 553–565. 23. Ene V.: Real Functions-Current Topics. Lecture Notes in Mathematics 1603, Springer-Verlag, Berlin 1995. 24. Filipczak F.M.: Sur la structure de l’ensemble des points de discontinuit´ e des fonctions qui admettant une d´ eriv´ ee aux points de continuit´ e. Fund. Math. 40 (1967), 59–79. 25. Filipczak F.M.: Sur les deriv´ ees symetriques de fonctions approximativement continues. Fund. Math. 34 (1975/1976), 249–256. 26. Filipczak F.M.: Sur la structure de l’ensemble des points o` u une fonction continue n’admet pas de deriv´ ee symetrique. Dissertationes Math. 130 (1976), 1–48. 27. Freiling Ch.: On the problem of characterizing derivatives. Real Anal. Exchange 23, no. 2 (19971998), 805–812. 28. Garsia A.M.: Existence of almost everywhere convergent rearrangements for Fourier series of L2 functions. Annals of Math. 79, no. 3 (1964), 623–629. 29. Haupt O., Pauc Ch.: La topologie de Denjoyenvisag´ ee comme vraie topologic. C. R. Acad. Sci. Paris 234 (1952), 390–392. 30. Humke P.D.: A modest review of a great deal of work. In: Traditional and Present-day Topics in Real Analysis (dedicated to Professor Jan Stanislaw Lipi´ nski), M. Filipczak, E. WagnerBojakowska (eds.), L´ od´ z Univ. Press, L´ od´ z 2013, 11–26. 31. Hunt R.A.: On the convergence of Fourier series. Orthogonal expansious and their continuous analogues. In: Proceedings of the Conference held at Southern Illinois University, Edwardsville 1967. Southern Illinois University Press, Carboundale 1968, 235–255. 32. Kolmogorov A.N.: Une s´ erie de Fourier-Lebesgue divergente presque partout. Fund. Math. 4 (1923), 324–328. 33. Kolmogorov A.N., Menshov D.E.: Sur la convergence des s´ eries de fonction orthogonales. Math. Z. 26 (1927), 432–441. ´ Mat.-Fiz. 34. Lipi´ nski J.: Zygmunt Zahorski work on theory of real functions. Zeszyty Nauk. PSl., 48 (1986), 29–36 (in Polish). 35. Lazarow E.: On the Baire class of I-approximate derivatives. Proc. Amer. Math. Soc. 100, no. 4 (1987), 669–674. 36. Lazarow E., Wilczy´ nski W.: I-approximate derivatives. Radovi Matematiˇ cki 5 (1989), 15–27. 37. Neugebauer C.J.: Darboux functions of Baire class one and derivatives. Proc. Amer. Math. Soc. 13 (1962), 838–843. 38. Nishiura T.: Absolute Measurable Spaces. Encyclopedia of Mathematics and Its Applications 120, Cambridge University Press, Cambridge 2008. 39. Olevskiˇi A.M.: Divergent series in L2 with respect to complete systems. Dokl. Akad. Nauk SSSR 141 (1961), 28–31 (in Russian). 40. Pawlak R.J.: On Zahorski classes of functions of two variables. Rev. Roum. Math. Pur. Appl. (1) 35 (1990), 53–71. 41. Piranian G.: The set of nondifferentiability of a continuous function. Amer. Math. Monthly 73, no. 4 (1966), 57–61 (Part II. Papers in Analysis, Number 11 of the Herbert Ellsworth Slaught Memorial Papers).

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42. Preiss D.: Level sets of derivatives. Trans. Amer. Math. Soc. 272, no. 1 (1982), 161–184. 43. Preiss D., Tartaglia M.: On characterizing derivatives. Proc. Amer. Math. Soc. 123 (1995), 2417–2420. 44. de Reyna J.A.: Pointwise Convergence of Fourier Series. Lecture Notes in Math. 1785, SpringerVerlag, Berlin 2002. 45. Salzmann H., Zeller K.: Singularitaten unendlich oft differenzierbaren Funktionen. Math. Z. 62 (1955), 354–357. ´ 46. Siciak J.: Regular and singular points of the function of the class C ∞ . Zeszyty Nauk. PSl., Mat.-Fiz. 48 (1986), 127–146 (in Polish). 47. Sj¨ olin P.: An inequality of Paley and convergence a.e. of Walsh-Fourier series. Arkiv Math. 7 (1969), 551–570. ´ atkowski T.: On the conditions of monotonicity of functions. Fund. Math. 59 (1966), 189– 48. Swi¸ 201. 49. Tolstov G.: Sur quelques propri´ et´ es des fonctions approximativement continues. Rec. Math. (Mat. Sbornik) 5 (1939), 637–645. 50. Tzodiks V.: On sets of points where the derivative is equal to +∞ and −∞ respectively. Rec. Math. (Mat. Sbornik) (43) 85 (1957), 429–450 (in Russian). 51. Ulyanov P.L.: Divergent Fourier series of class Lp , p ≥ 2. Dokl Akad. Nauk SSSR 137 (1961), 786–789 (in Russian). 52. Ulyanov P.L.: Divergent series obtained using the Haar system and using bases. Dokl. Akad. Nauk SSSR 138 (1961), 556–559 (in Russian). 53. Weil C.W.: A property for certain derivatives. Indiana Univ. Math. J. 23 (1973/1974), 527–536. 54. Wilczy´ nski W.: A non-parametric surface having the property of Zahorski. Bull. Acad. Pol. Sc. 22, no. 3 (1974), 251–256. 55. Wilczy´ nski W.: Density topologies. Chapter 15 in Handbook of Measure Theory. Edited by E. Pap, Elsevier, North Holland 2002, 675–702.

´ From left: Janina Sladkowska-Zahorska, Zygmunt Zahorski, Ernest Plonka, Bogdan Koszela, Ewa Lazarow, Wladyslaw Wilczy´ nski

On the topological entropy of discontinuous functions. Strong entropy points and Zahorski classes Ewa Korczak-Kubiak, Anna Loranty and Ryszard J. Pawlak

Abstract. The basis of our considerations are the issues dealing with entropy of discontinuous functions (among others almost continuous functions and functions belonging to a fixed Zahorski class). Particular emphasis is on the local aspects of entropy including problems regarding strong entropy points (also almost fixed points). Keywords: entropy, Darboux function, Zahorski classes, almost fixed point, strong entropy point, almost continuity, Γ -approximation, f -bundle, manifold. 2010 Mathematics Subject Classification: 26A18, 26A15, 54H20, 54C60, 54C70, 37B40, 37E15.

1. Introduction – historical outline The theory of (discrete) dynamical systems is extensive and strongly expanding field of mathematics that uses a lot of facts from various branches of mathematics including real analysis. It is interesting, therefore, to consider the issues associating the discrete dynamical systems theory and real analysis. In this chapter we will present some issues concerning entropy of functions belonging to different Zahorski classes. The results presented in this part are mainly based on the paper [22] and [23]. If we give statements from other publications related to this topic, this will be marked by giving references to the relevant article. We start with a short historical overview of the problems presented in this chapter. First, we present some intuitive description of problems connected with information system and information flow. We will not present the issues of information systems in details (basic facts on this subject can be found, among others in [24, 25]). However, assume that we have a set X of elements (information) divided into a finite number of disjoint subsets {A1 , A2 , . . . , Ak }, which are distinguished on the basis of fixed E. Korczak-Kubiak, A. Loranty, R.J. Pawlak Faculty of Mathematics and Computer Science, L´ od´ z University, Banacha 22, 90-238 L´ od´ z, Poland, e-mail: {ekor, loranta, rpawlak}@math.uni.lodz.pl R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 109–123. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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attributes (this partition is denoted by P ). Suppose also that we have a probability k P measure1 µ on X, so µ(Ai ) = 1. Then we may (see [26]) assign to the partition P i=1

the number (the entropy of partition) defined in the following way: H(P ) := −

k X

µ(Ai ) · log µ(Ai ).

i=1

Roughly speaking, if partition P describes a state of information flow, the number H(P ) may be regarded as a “measure of uncertainty”. If H(P ) = 0, then situation is defined precisely – measure is focused on some set Ai0 from the partition P (i.e. µ(Ai0 ) = 1). Moreover, we can say that the higher the entropy of partition is, the greater uncertainty is (in this case, the measure is more evenly distributed over the different sets of the partition). After a given period of time, elements of X change the values of their attributes and thereby they “move” to the other sets. Perhaps a new partition of X (onto sets measurable with respect to µ) is created. These changes are described by a certain function – let us denote it by φ. After the next unit of time, the elements “move” again and we obtain a new partition of X. The changes are described by the function φ. It means that in comparison to initial state these changes are described by the function φ2 = φ ◦ φ. Going further in this way, we obtain the dynamics of the function φ. The entropy of this function (the definition one can find at the end of this section) determines the level of uncertainty of dynamics of function φ. If it is 0, then we can talk about a certain stability of this dynamics. If it is greater than 0, we can say that this dynamics is chaotic and the number qualified as the entropy can be considered as a certain kind of “measure of chaos”. In the sixties of the twentieth century R.L. Adler, A.G. Konheim and M.H. McAndrew [1] introduced the notion of the topological entropy of a continuous function f : X (i.e f : X → X) defined on a compact space X. In 1971 T. Goodman [11] proved the variational principle determining the relationship between the topological entropy and the entropy with respect to measure (cf. Theorem 7.7). Earlier, in 1969, L.W. Goodwyn [12] proved that for a fixed invariant measure, the entropy of function with respect to this measure is not greater than its topological entropy. In the case of functions important from the point of view of the real analysis theory (e.g. functions belonging to a fixed Zahorski class) their entropy with respect to a measure is difficult to use because of the necessity of constructing and selecting invariant measures. Since each Zahorski class contains also discontinuous functions, we can not directly apply the original definition. Fortunately, in [7] it has been shown that there is a possibility of using existing definitions in the case of discontinuous functions. It is worth noting that until now the topological entropy has been mainly related to Darboux-like functions ([7, 22, 23, 21, 17, 14]). Now, we shortly recall definition of a topological entropy given for continuous function by R. Bowen [3] and E. Dinaburg [8] and extended to an arbitrary function ˇ by Ciklov´ a [7]. 1 Assumptions concerning this measure and the corresponding functions are described in details in Section 7.

On the topological entropy of discontinuous functions. . .

111

Let (X, ρ) be a compact metric space, f : X be a function, ε > 0 and n ∈ N. A set M ⊂ X is (n, ε)-separated if for each x, y ∈ M , x = 6 y there is 0 6 i < n such that ρ(f i (x), f i (y)) > ε. Let sn (ε) = max{card(M ) : M ⊂ X is (n, ε)-separated set}. The topological entropy of the function f is the number   1 h(f ) = lim+ lim sup log (sn (ε)) . n ε→0 n→∞ A detailed analysis of behaviour of various functions pointed out the desirability of searching subsets of a domain of a function, or even points, on which the behaviours of a function having significant impact on the value of its topological entropy, are focused. Description of such analysis will be presented in sections 3 and 4. This analysis led us to distinguish an important object – “a strong entropy point of a function” (the respective definition is presented in Section 4). Of course, not every function (even if we assume its continuity) has a strong entropy point. For this reason, issues related to approximating functions by other functions with a strong entropy point seemed to be interesting ([22, 23, 14]). At the same time it is interesting to approximate a function by functions with “a lot of continuity points”. Therefore, uniform convergence is not a very convenient tool. Moreover, it is worth noting that if f belongs to any Zahorski class, then f is almost continuous i.e each open set containing the graph of this function contains a graph of a continuous function (see Proposition 6.1). For these reasons, it is quite natural to consider Γ -approximation. More information about this kind of approximation is presented in Section 5.

2. Preliminaries Throughout the paper N and R denote the set of positive integers and real numbers, respectively. We use the letter λ to denote the Lebesgue measure in R. The symbol D(X, Y ) (B1 (X, Y )) denotes the family of all functions f : X → Y which are Darboux functions (of first Baire class). In both the above notations if X = Y we will write only one X, e.g. D(X) instead of D(X, X). Moreover, if we wish to consider the intersection of two classes of functions, we shall write them next to each other (e.g. DB1 (X, Y ) or DB1 (X)). Furthermore, we write D and B1 if X = Y = R. Let f : X . Then we define f 0 (x) = x and f i (x) = f (f i−1 (x)) for any i ∈ N. If A ⊂ X and n ∈ N then f −n (A) = {x ∈ X : f n (x) ∈ A}. A point x0 ∈ X is a fixed point of a function f if f (x0 ) = x0 . A set of all fixed points of f we denote by Fix(f ). We say that a space X has the fixed point property if Fix(f ) 6= ∅ for any continuous function f : X . If x0 ∈ X, f m (x0 ) = x0 for some m ∈ N and f n (x0 ) 6= x0 for any n ∈ {1, . . . , m − 1}, then we call a point x0 a periodic point of f of prime period m. The symbol Perm (f ) stands for a set of all periodic S points of f of prime period m. W say that x0 ∈ X is a periodic point of f if x0 ∈ Pern (f ). For any x ∈ X the n∈N

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orbit of f at the point x is the set {x, f (x), f 2 (x), f 3 (x), . . . }. If x ∈ Perm (f ) for some m ∈ N then the orbit of f at the point x is called a periodic orbit of f of period m. We say that functions f, g : X are conjugate via a homeomorphism φ : X if φ ◦ f = g ◦ φ. Let f : X → Y , A ⊂ X and B ⊂ Y . We say that a set A f -covers a set B (denoted by A → B) if B ⊂ f (A). Moreover, the restriction of f to the set A is denoted by f

f ↾ A. The symbol card(A) stands for cardinality of A. Let (X, ρ) be a metric space. The symbol dist(x, A), where x ∈ X and A ⊂ X, stands for a distance from the point x to the set A. If x0 ∈ X and r > 0, then we use the symbol B(x0 , r) to denote an open ball with the centre at x0 and the radius r. We say that a topological space X is an m-dimensional topological manifold with boundary if X is a second countable Hausdorff space and every point q ∈ X has a neighborhood that is homeomorphic to the m-dimensional upper half space Hm = {(x1 , . . . , xm ) ∈ Rm : xm > 0} (see [15]). We say that α ∈ R is a left (right ) range of f : R at x0 ∈ R if f −1 (α) ∩ (x0 − δ, x0 ) 6= ∅ (f −1 (α) ∩ (x0 , x0 + δ) 6= ∅) for any δ > 0. The symbols R+ (f, x0 ) and R− (f, x0 ) stand for the sets of all left and all right ranges of f at x0 , respectively. Let f : [0, 1] be a Darboux function. A point x0 ∈ (0, 1) (x0 = 0, x0 = 1) is an almost fixed point of f (for short x0 ∈ Fixa (f )) iff x0 ∈ int(R− (f, x0 ))∪int(R+ (f, x0 )) (x0 ∈ int(R+ (f, x0 )), x0 ∈ int(R− (f, x0 ))). For any finite family {I1 , . . . In } of closed intervals contained in [0, 1] we define a matrix Mf (I1 , . . . In ) = [aik ]i,k6n in the following way: aik = 1 if Ii → Ik and f

aik = 0 otherwise. A maximal absolute value of an eigenvalue of this matrix will be denoted by σ(Mf (I1 , . . . In )).

3. Bundles connected with function In this section we will concentrate on a metric space (X, ρ). We write it X for short. We start with the example of the function (having some special property) defined on the square I 2 = [0, 1] × [0, 1] with the Euclidian metric. In the square I 2 we consider two rectangles P1 with vertices a1 , a4 , b4 , b1 and P2 with vertices c, d, e, k. We divide P1 and P2 into rectangles as it is shown in Figure 1. We define the function f : I 2 in the following way: f ((0, 0)) = (0, 0) and f (t) = t if t belongs to the rectangle P with vertices x, q, r, b4 . Moreover, we define the function f on P1 as follows: we “stretch” each rectangle from the partition of P1 onto the rectangle P1 in such a way that the line segment with endpoints a3 and b3 is converted to the line segment with endpoints a1 and b1 , the line segment with endpoints a2 and b2 is converted to the line segment with endpoints a4 and b4 . On the segments a1 b1 and a4 b4 function f is the identity. Analogously we define the function f on the rectangle P2 . Next we extend f to continuous function defined on I 2 . It is easy to see that h(f ) > 0. However, one can show that h(f ↾ P ) = 0 and h(f ↾ P2 ) > h(f ↾ P1 ).

On the topological entropy of discontinuous functions. . .

b1 = (0,1) b2

a1

a2

(1,1) = r

b3

b4

a3

a4

k

e

c

d

x

p = (0,0)

113

(1,0) = q

Fig. 1

The natural question arises: Is it possible to identify subsets of domain which have a particular impact on the value of the entropy of the whole function? This question is interesting because, as the next example will show, sometimes even small changes have a significant impact on the value of the entropy of a function. This time the example will regard discontinuous functions, however it is not difficult to show analogous example for a continuous function. So, let us consider two functions g, k : [0, 1] whose graphs are presented in Figure 2.

1 2

1 2

g 1 4

k

1 4

1 4

1 2

1 2

Fig. 2

We see that g(0) = k(1) = 14 and the graphs of g and k are more and more “dense” near the point 1 and 0, respectively (their graphs are of the same type as that of sin( x1 )). Notice that “behaviours” of these functions are similar to each other, but specificity of their behaviours is different on different sets. As a result, h(g) = 0,

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whereas h(k) = +∞. In that sense, the function g may be regarded as “predictable” and the function k as “strongly chaotic”. Similar considerations are presented in [23], where one can find other examples of functions slightly differing from each other, but the entropy of one of them is 0 and of the second one is greater then 0. Simultaneously, in consideration of the complexity of entropy definition, one can ask another question: Is there a simple way to indicate the sets which have a decisive impact on the value of entropy of a fixed function? The next part of the chapter is devoted to the answer to the above question. Some basis of these considerations one can find in [2, 18, 20]. Following [23], we will consider the concept of f -bundle, which is some generalization of the notion horseshoe (e.g. [2]). Let f : X . A pair (F , J) = Bf , where F is a family of pairwise disjoint (nonsingletons) continuums in X and J ⊂ X is a connected set such that A → J for any f

A ∈ F is called an f -bundle. Moreover, if we additionally assume that A ⊂ J for all A ∈ F then such an f -bundle is called an f -bundle with dominating fibre. By the cardinality of Bf (denoted by card(Bf )) we will mean the cardinality of the family F. S Let f : X , ε > 0, n ∈ N and Bf = (F , J) be an f -bundle. A set M ⊂ F is (Bf , n, ε)-separated if for each x, y ∈ M , x 6= y there is 0 6 i < n such that f i (x), f i (y) ∈ J and ρ(f i (x), f i (y)) > ε. If B

sn f (ε) = max{card(M ) : M ⊂ X is (Bf , n, ε)-separated set}, then the entropy of the f -bundle Bf is defined in the following way:    1 Bf h(Bf ) = lim lim sup log sn (ε) . ε→0 n→∞ n Theorem 3.1 ([23]). Let f : X be an arbitrary function and Bf = (F , J) be an f -bundle with dominating fibre. Then h(Bf ) ≥ log(card(Bf )) if Bf is finite and h(Bf ) = +∞ otherwise. Recall again the example of the function f : I 2 presented before. From now on, the symbol P (t1 , t2 , t3 , t4 ) will stand for a rectangle with vertices t1 , t2 , t3 , t4 . Then, for the function f , one can consider the following f -bundles with dominating fibres: (Fi , Ji ) (i = {1, 2, 3}): F1 = {P (x, q, r, b4 )} and J1 = P (x, q, r, b4 ), F2 = {P (a3 , a4 , b4 , b3 ), P (a1 , a2 , b2 , b1 )} and J2 = P (a1 , a4 , b4 , b1 ); F3 consists of the grey rectangles (see Figure 3) and J3 = P (c, d, e, k). Theorem 3.1 implies2 h((F1 , J1 )) > 0, h((F2 , J2 )) > log 2 and h((F3 , J3 )) > log 3.

2 In considered rectangles P (a , a , b , b ) and P (c, d, e, k) one can find f -bundles of cardinality 1 4 4 1 greater then those presented. However it would require more complicated notation unnecessary for our further considerations.

On the topological entropy of discontinuous functions. . .

b1 = (0,1) b2

a1

a2

b3

b4

a3

a4

k

e

c

d

p = (0,0)

x

115

(1,1) = r

(1,0) = q

Fig. 3

We say that a map f is chaotic in the sense of Li and Yorke if f has orbits of arbitrarily large periods and there exists an uncountable set B (called a scrambled set) such that for every x, y ∈ B such that x 6= y and every periodic point z we have 1. lim sup |f n (x) − f n (y)| > 0; n→∞

2. lim inf |f n (x) − f n (y)| = 0; n→∞

3. lim sup |f n (x) − f n (z)| > 0. n→∞

Theorem 3.2 ([23]). Let f : [0, 1] be a continuous function. If there exists an f -bundle Bf = (F , J) with dominating fibre such that card(Bf ) > 2, then f is chaotic in the sense of Li and Yorke.

4. Strong entropy points In this section X still stands for a metric space (X, ρ). Let us modify in the following way the function f : I 2 considered previously. At first, we have considered two rectangles P (a1 , a4 , b4 , b1 ) and P (c, d, e, k), their partitions onto the smaller rectangles and the function “stretching” each of the rectangles from partitions onto the whole initial rectangle. In this way, the f -bundles (F2 , J2 ) and (F3 , J3 ) described above, have been created. Assume that we consider next rectangles whose lengths of sides converge to zero, partitions onto more and more parts and function f “stretching” each of the rectangles from partitions onto the whole initial rectangle. In this way we can create a sequence of f -bundles (see Figure 4). It is easy to notice that the entropy of each of successive bundles will be greater and greater. The observation with the above example may be an intuitive illustration of the following definitions.

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b1 = (0,1) b2

a1

a2

b3

b4

a3

a4

k

e

c

d

x

p = (0,0)

(1,1) = r

(1,0) = q

Fig. 4

Let f : X . We shall say that a sequence of f -bundles Bfk = (Fk , Jk ) converges to S a point x0 (Bfk −→ x0 ), if for any ε > 0 there exists k0 ∈ N such that Fk ⊂ B(x0 , ε) k→∞

and B(f (x0 ), ε) ∩ Jk 6= ∅ for any k > k0 . Having the above notion we can define multifunction Ef : X ⊸ R ∪ {+∞} in the following way: Ef (x) = {lim sup h(Bfn ) : Bfn −→ x}. n→∞

n→∞

Obviously for an arbitrary function f : X and a point x0 ∈ X we have Ef (x0 ) 6= ∅. An important issue in considering multivalued functions is, in some sense, “regularity” of their values. The most essential kind of such “regularity” deals with closedness. The next theorems regard simple observations connected with this issue. Theorem 4.1 ([23]). If f : X is an arbitrary function, then Ef (x) is a closed set for any x ∈ X. Theorem 4.2 ([23]). If f : X is a continuous function, then Ef is a closed multifunction. Return again to the function f : I 2 described at the begining of this section. In each case of considered f -bundles, their entropy is finite, but according to Theorem 3.1 h(f ) = +∞. Notice that in this case the entropy of the function is focused around the point (0, 0). Thus let us adopt the definition which may mean the entropy of a function at a point. For any function f : X and x ∈ X the entropy of f at point x, denoted by ef (x), is defined to be sup Ef (x). Theorem 4.3 ([23]). If f : X is an arbitrary function and x ∈ X, then ef (x) 6 h(f ). Theorem 4.4 ([23]). If f : X is a continuous function, then the function ef (x) is an upper semicontinuous selection.

On the topological entropy of discontinuous functions. . .

117

In this section we focus our attention also on strong entropy points. We shall say that a point x0 ∈ X is a strong entropy point of a function f : X if h(f ) ∈ Ef (x0 ) and x0 ∈ Fix(f ). The family of all strong entropy points of a function f will be denoted by Es (f ). First we note the obvious statements. Theorem 4.5 ([23]). A point x0 ∈ X is a strong entropy point of a function f : X if and only if ef (x0 ) = h(f ) and x0 ∈ Fix(f ). The above observations imply immediately the following statement. Theorem 4.6. Let f : X be an arbitrary function and x0 ∈ X. (a) If x0 ∈ Fix(f ) and ef (x0 ) = +∞, then x0 ∈ Es (f ). (b) If x0 ∈ Fix(f ) and +∞ ∈ Ef (x0 ), then x0 ∈ Es (f ). Notice that in our example of the function f : I 2 , ef ((0, 0)) = +∞, so the entropy of f at the point (0, 0) coincides with the entropy of the whole function and what is more, the point (0, 0) is a fixed point of f . It means that (0, 0) is a strong entropy point of f . The function f : I 2 considered above is continuous. The natural question arises whether the similar considerations may be referred to discontinuous functions. The answer will be contained in Sections 5 and 9. We will finish this part of the paper with the theorem showing that the notion of strong entropy point is interesting from the point of view of dynamical systems. Theorem 4.7. Let functions f : X and g : X be conjugate. Then Es (f ) 6= ∅ if and only if Es (g) 6= ∅.

5. Almost continuous functions defined on m-dimensional manifold In this section we will consider approximations of some functions by functions having strong entropy point. From the point of view of known facts regarding discrete dynamical systems and topology, the issue of approximation by use of continuous functions is very importatnt. On the other hand, we aim to combine these considerations with real functions theory regarding, among others, Zahorski classes. Approximation of discontinuous functions by continuous functions eliminates the possibility of considering uniform approximation (i.e. approximation by use of topology of uniform convergence). Simultaneously, a definition of strong entropy points indicates that our considerations should be directed towards classes of functions having a fixed point. Taking into account all the above facts, it is appropriate to consider almost continuous functions and to investigate graph-approximation (Γ -approximation). Let (X, τ ) be a topological space and K be some class of functions from X into itself. We shall say that a function f : X is Γ -approximated by functions belonging to K if for each open set U ⊂ X × X containing the graph of f , there exists g ∈ K such that the graph of g is a subset of U .

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In this section we will focus on almost continuous functions in the sense of Stallings [27], namely on functions f : X which are Γ -approximated by continuous functions. It is known that the family of all functions f : [0, 1] which are almost continuous and of first Baire class is equal to the family DB1 ([0, 1]) (see [4]). However, there exist topological spaces (X, τX ), (Y, τY ) and an almost continuous function f : X → Y such that f ∈ B1 (X, Y ) and f 6∈ D(X, Y ) (see [18]). Moreover, we have the following fact. Theorem 5.1 ([27]). If X is a Hausdorff space with the fixed point property then each almost continuous function f : X has a fixed point. The above theorem plays an important role in the proof of the following claim. Theorem 5.2 ([23]). Let X be a compact, m-dimensional manifold with boundary having the fixed point property and f : X be a function. The following conditions are equivalent: (1) The function f is almost continuous. (2) The function f can be Γ -approximated by continuous functions having a strong entropy point. (3) The function f can be Γ -approximated by continuous functions having infinite topological entropy. (4) The function f can be Γ -approximated by discontinuous but almost continuous functions having a strong entropy point. (5) The function f can be Γ -approximated by discontinuous but almost continuous functions having infinite topological entropy.

6. The Zahorski classes Working on the issues regarding derivatives, Zygmunt Zahorski distinguished a hierarchy of classes of functions. Following [29], let us start with definitions of some classes of sets. All these classes of sets consist of some subsets of R. The class M0 consists of the empty set and all nonempty sets E of type Fσ such that every point of E is a point of bilateral accumulation of E. The family of all nonempty sets E of type Fσ such that every point of E is a point of bilateral condensation of E complemented by the empty set constitutes the class M1 . A set E belongs to the class M2 if it is empty or if it is a nonempty set of type Fσ and for each x ∈ E and any ε > 0 sets (x, x + ε) ∩ E and (x − ε, x) ∩ E have a positive measure. The class M3 consists of all nonempty sets ESof type Fσ such that there exists a sequence {Kn }n∈N of closed sets such that E = Kn and a sequence {ηn }n∈N of numbers such that n∈N

0 6 ηn < 1 (n ∈ N) and for each n ∈ N, each x ∈ Kn and each c > 0 there exists a number ε(x, c) > 0 such that if h and h1 satisfy conditions h · h1 > 0, hh1 < c, |h + h1 | < ε(x, c), then λ(E ∩ (x + h, x + h + h1 )) > ηn . |h1 |

On the topological entropy of discontinuous functions. . .

119

In addition, we assume that the empty set belongs to the class M3 . A slight change in the definition of the class M3 leads us to the class M4 . More specifically, in this case, we replace the above condition 0 6 ηn < 1 (n ∈ N) with the condition 0 < ηn < 1 (n ∈ N). We say that E belongs to the class M5 if it is empty or if it is a nonempty set of type Fσ and for each x ∈ E we have lim+

h→0

λ(E ∩ [x − h, x + h]) = 1, 2h

(1)

that is every point of E is a density point of E. It is worth adding that to check whether a nonempty set E of type Fσ belongs to the class M3 it is enough to show the following condition: for each x ∈ E and each sequence {In }n∈N of closed intervals converging to x (i.e. lim dist(x, In ) = 0) and n→∞

λ(In ) n→∞ dist(x,In )

not containing x such that λ(In ∩ E) = 0 for each n ∈ N, we have lim

= 0.

Using the above hierarchy of sets we can define some classes of functions. Let i ∈ {0, 1, . . . , 5}. We say that a function f : R belongs to the class Mi , if sets {x : f (x) > α} and {x : f (x) < α} belong to the class Mi for any α ∈ R. Certainly one can define similar classes for functions f : [0, 1] . Moreover, to simplify notation, let the symbol M6 stand for the family of all continuous functions. It is easy to see that M0 ⊃ M1 ⊃ M2 ⊃ M3 ⊃ M4 ⊃ M5 ⊃ M6 . It is well known that all the above inclusions, except the first one from the left, are proper and M0 = M1 = DB1 ([29, 5]). Therefore, further instead of writing: the function f is a Darboux function and of first Baire class, we will write briefly: f ∈ M1 . In [29] one can also find the property saying that the class M5 coincides with the family of approximately continuous functions i.e the family of functions f having the following properties (see [9, 10]): for any x ∈ R there exists a Lebesgue measurable set Ex such that x is a density point of Ex (see condition (1)) and f (x) = lim f (t). t→x, t∈Ex

Additionally, it is shown there that each derivative (a function f is a derivative if there exists a function g such that f = g′) belongs to the class M3 and each bounded derivative is in M4 . Furthermore, it is known that each bounded function from the class M5 is a derivative (see [10]). Moreover, we have the following, commonly known, facts. Proposition ([29]). For any set E ∈ M4 there exists a bounded derivative f : R such that f (x) = 0 for x 6∈ E and f (x) ∈ (0, 1) for x ∈ E. Proposition ([5]). If E ∈ M4 and x ∈ E, then d(E, x) = lim inf h→0+

λ(E∩[x−h,x+h]) 2h

> 0.

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In view of the preceding sections it is also worth noting: Proposition 6.1. Each function f from the class Mi , for i ∈ {0, 1, . . . , 6}, is almost continuous. In the case of discrete dynamical systems, functions whose domain and range is the same compact space are predominantly studied. In our considerations we will deal with the unit interval. So, from now on, we will focus on functions from the unit [0,1] interval into itself. Let i ∈ {0, 1, . . . , 6}. The symbol Mi denote the family of all functions f : [0, 1] such that f ∈ Mi . [0,1]

Theorem 6.2. If f ∈ Mi

(i ∈ {0, 1, . . . , 6}), then Fix(f ) 6= ∅.

Taking into account Proposition 6.1 and Theorem 5.2 we can immediately show [0,1] that each function from Mi is Γ -approximated by continuous functions ξ : [0, 1] . Furthermore, one can prove the following theorem, useful in various considerations. [0,1]

Theorem 6.3. If f ∈ Mi (i ∈ {0, 1, . . . , 6}), then f is Γ -approximated by continuous functions ξ : [0, 1] such that Fix(ξ) ∩ (0, 1) 6= ∅.

7. A topological entropy of discontinuous function from the unit interval into itself As it has been already mentioned in the introduction, the problems connected with topological properties of dynamical systems (including topics related to topological entropy) can be also considered in the case of discontinuous functions. In order to unify considerations we limit our further considerations only to the [0,1] functions from the class M1 and finner classes of functions. From now on we [0,1] will assume that all functions belong to the class M1 . Now, we present some results related to issue connected with topological entropy of such functions. Theorem 7.1 ([22]). Let f be a function and n ∈ N \ {1}. If {I1 , . . . , In } is a family of pairwise disjoint closed intervals, then h(f ) > log σ(Mf (I1 , . . . , In )). Theorem 7.2 ([22]). A topological entropy of a function f equals 0 if and only if h(f n ) = 0 for each n ∈ N. Theorem 7.3 ([22]). If f is a turbulent function3 then h(f ) > 0. Theorem 7.4 (Itinerary Lemma, [28]). For every function f and any family {I1 , . . . , In } of closed intervals such that I1 → I2 → . . . → In → I1 there exists x0 ∈ I1 such f

f

f

f

that x0 ∈ Fix(f n ) and f i (x0 ) ∈ Ii+1 for i ∈ {1, . . . , n − 1}.

3 A function f is turbulent if there exist compact subintervals J, K ⊂ [0, 1] with at most one common point such that J ∪ K ⊂ f (J) ∩ f (K).

On the topological entropy of discontinuous functions. . .

121

Theorem 7.5 (Sharkovski˘ı’s Theorem, [28]). Let 0, for any k ∈ N.

The proof of the condition (b) is analogous.

⊔ ⊓

The next theorem is a simple consequence of the above proposition, Theorem 2.3 and the fact that if f (x) 6 g(x) for x > 0, then Tf ⊂ Tg . Theorem 3.6. Let f ∈ A. f (x) x

6 lim sup f (x) x < ∞, then Tf = Tf k = Td .

f (x) x x→0+ lim inf f (x) x x→0+

6 lim sup f (x) x < ∞, then Tf k ⊂ Tf $ Td .

(a) If 0 < lim inf x→0+

(b) If 0 = lim inf (c) If 0
0 ([5], Proposition 3). Observe, that for any α > 0 the function f (x) = xα and any its iteration fulfill △2 (f ∈ A only for α > 1).

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If a function f fulfills △2 then the topology Tf is invariant under multiplication by nonzero numbers ([5], Theorem 4). If Tf ⊂ Td then △2 is the necessary and sufficient condition for this invariantness. If Tf 6⊂ Td then there exists a function g ∈ / △2 such that Tf = Tg ([5], Theorem 6). One can ask, if it is possible that for a function f ∈ △2 there exists a number k ∈ N such that f k has not such property. The answer is negative due to the following theorem. Theorem 3.8. If f, g ∈ A fulfill the condition △2 , then their composition g ◦ f also fulfills △2 . Proof. Assume that f ∈ △2 . Hence there exist numbers M > 0, δ1 > 0 such that f (2x) f (x) < M for x ∈ (0, δ1 ). By the monotonicity of the function g we have g(f (2x)) < g(M f (x)) for each x ∈ (0, δ1 ). Since g ∈ △2 , we have that for each α > 0, lim sup g(αx) g(x) < ∞. x→0+

g(Mx) g(x)

Put α = M . Then there exist K > 0 and δ2 > 0 such that < K for x ∈ (0, δ2 ). From the assumption lim f (x) = 0 it follows that there exists δ3 > 0 such that x→0+

f (x) < δ2 whenever x ∈ (0, δ3 ). Hence for δ = min(δ1 , δ2 , δ3 ) and x ∈ (0, δ), g(f (2x)) g(M f (x)) 6 < K. g(f (x)) g(f (x)) Therefore, lim sup x→0+

g(f (2x)) < ∞. g(f (x)) ⊔ ⊓

Corollary 3.9. If f ∈ △2 , then f k ∈ △2 for any k ∈ N. It is easy to check that if f ∈ A and 0 < lim inf x→0+

f (x) f (x) 6 lim sup < ∞, x x x→0+

(1)

then f ∈ △2 if and only if f k ∈ △2 . Moreover, if (1) holds then Tf = Tf k = Td so this case is not interesting, because the density topology is invariant under multiplication by nonzero numbers. The next examples show that if one of the inequalities of (1) is not fulfilled we do not have the equivalence f ∈ △2 ⇐⇒ f k ∈ △2 . We will use the functions similar to the functions defined in [7], Lemma 1. Example 3.10. There exists a function g ∈ A such that 0 = lim inf x→0+

g(x) x


0 is a strictly decreasing sequence tending to zero such that lim bn+1 bn = 0. Put n→∞

Similarity and topologies generated by iterations of functions

 2 x   bn g(x) = bn+1   b1

133

√ for x ∈ ( an bn , bn ), √ for x ∈ [bn+1 , an bn ], for x > b1 .

Obviously, g is nondecreasing, g(x) 6 1 for any x > 0 and lim g(x) = 0. Moreover, x→0+

lim inf x→0+

g(x) g(an ) 6 lim = lim n→∞ an n→∞ x

r

bn+1 = 0. bn

Hence g ∈ A and lim sup g(x) x < ∞. It does not fulfil condition △2 . Indeed, let x > 0. x→0+

Then x ∈ [bn+1 , bn ) for a certain n ∈ N. Therefore, s √ g(2x) g(2 an bn ) bn √ lim sup > lim > lim = ∞. n→∞ n→∞ g(x) b g( an bn ) x→0+ n+1 From the definition of g we obtain  4 √ x   b3n for x ∈ [ an bn , b√n ), g 2 (x) = bn+1 for x ∈ [bn+1 , an bn ),   b1 for x > b1 . From Proposition 3.1 it follows that g 2 ∈ A. We will show that g 2 ∈ △2 . Notice, that √ 4 for x > an bn we have g 2 (x) 6 bx3 . Take x > 0. Then there is natural number n such n 4 √ 4 that x ∈ (bn+1 , bn ]. If x, 2x ∈ [ an bn , bn ) then g 2 (x) = xb3 and g(2x) 6 (2x) b3n . Hence n √ √ g2 (2x) g2 (2x) g2 (x) 6 16. If x, 2x ∈ [bn+1 , an bn ) then g2 (x) = 1. If x ∈ [ an bn , bn ) and 2x > bn then

g2 (2x) g2 (x)

=

g2 (2x) g2 (bn )

·

g2 (bn ) g2 (x)

6

g2 (2bn ) g2 (bn )

· 1 6 16. Hence g 2 ∈ △2 .

⊔ ⊓

Example 3.11. There exists a function g ∈ A such that 0 < lim inf x→0+

g(x) x


0 be a strictly decreasing sequence tending to zero such that lim aan+1 =0 n n→∞ √ and bn = an an+1 , n ∈ N. Put  2 x   an g(x) = an−1   a0

√ for x ∈ (an , an bn ], √ for x ∈ ( an bn , an−1 ], for x > a0 .

The function g belongs to A and it does not fulfill condition △2 . Indeed, √ r g(2 an bn ) an−1 an−1 √ > = b an g( an bn ) n

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M. Filipczak and M. Terepeta

and lim sup x→0+

Moreover,

√ g(2x) g(2 an bn ) √ > lim = ∞. n→∞ g( an bn ) g(x)

√ r g(x) g( an bn ) an+1 lim sup > lim √ = lim 4 = ∞. n→∞ n→∞ x an an b n x→0+

From the definition of g we obtain  4 √ x  n , an bn ],  a3n for x ∈ (a p g 2 (x) = an for x ∈ ( an+1 bn+1 , an ],   a1 for x > a1 . The function g 2 is continuous, g 2 ∈ A. Analogously as in previous example we show that g 2 ∈ △2 . ⊔ ⊓ Example 3.12. There exists a function g ∈ A such that 0 = lim inf x→0+

lim sup x→0+

g(x) x

g(x) x


0 and (bn )n>0 be sequences from the previous example. Put  2 p   x  for x ∈ b2n , a2n−1 b2n−2 ,  a2n−1  p g(x) = a2n a2n+1 b2n+1 , b2n , for x ∈   a0 for x > a1 . The function g has all required properties: belongs to A, it does not fulfill condition g(x) △2 and 0 = lim inf g(x) x < lim sup x = ∞. From the definition of g we obtain x→0+

x→0+

 x4   a32n−1 2 g (x) = a2n   a0

p p for x ∈ [ a2n−1 b2n , a2n−1 b2n−1 ), p p for x ∈ [ a2n+1 b2n+1 , a2n−1 b2n ), √ for x > a1 b1 .

The function g 2 is continuous, g 2 ∈ A. Analogously as in previous example we show that g 2 ∈ △2 . ⊔ ⊓ In Examples 3.10 and 3.11 we have constructed the functions which fulfill the condition lim sup g(2x) / △2 ), but their iterations satisfy △2 . We can not g(x) = ∞ (g ∈ x→0+

exchange the the upper limit in this condition into the limit as it is shown by the next theorem. f (2x) x→0+ f (x)

Theorem 3.13. Let f, f k ∈ A. If lim

= ∞, then f k ∈ / △2 for any k ∈ N.

n) Proof. From the assumption, lim ff(2x = ∞ for any sequence (xn )n∈N such that n→∞ (xn ) lim xn = 0, in particular

n→∞

f (2f (xn )) = ∞. n→∞ f (f (xn )) lim

(2)

Similarity and topologies generated by iterations of functions

Fix the sequence (xn )n∈N . There is n0 ∈ N such that

135 f (2xn ) f (xn )

> 2 for n > n0 . Hence

f 2 (2xn ) > f (2f (xn )).

(3)

Therefore, for each sequence (xn )n∈N we obtain f 2 (2xn ) (3) f (2f (xn )) > lim =∞ 2 n→∞ f 2 (xn ) n→∞ f (xn ) lim

f 2 (2x) 2 x→0+ f (x)

and finally we obtain lim f k (2x) k x→0+ f (x)

lim

= ∞, so f 2 ∈ / △2 . We can prove by induction that

= ∞ for any k ∈ N, which finishes the proof.

⊔ ⊓

4. Similarity between topologies generated by function and its iteration Now we will focus on a problem: do there exist topologies generated by a function and its iteration which are not similar? Theoretically we know that there are f -density topologies which are not similar, but in practice it is not easy to indicate specific functions for which we can obtain such result. To examine the interior of sets in density topology we often construct Cantor-like sets of positive measure. It is not difficult to construct a set of this kind which has no f -density points. Example 4.1. There exists a set of positive measure without superdensity points. By induction we will define a central Cantor set E ⊂ [0, 1] of positive measure which has no f -density points for f (x) = x2 . (1) (1)  From the interval [0, 1] we remove concentric open interval denoted by a1 , b1  (1) (1) of the length 41 . Put E1 = a1 , b1 . Suppose that for certain n > 2 we have 2n−2 S (i) (i)  constructed the sets E1 , E2 , . . . , En−1 such that En−1 = an−1 , bn−1 . The set [0, 1] \

n−1 S

i=1

Ek consists of 2

n−1

closed intervals and from each such interval we remove (i) (i)  concentric interval of the length 41n which we denote by an , bn , i = 1, . . . , 2n−1 . 2n−1 ∞ S S (i) (i)  1 Put En = an , bn . Then λ(En ) = 2n−1 · 41n = 2n+1 . The set E = [0, 1] \ En k=1

i=1

is of positive measure since λ(E) = 1 −

∞ P

n=1

1 2n+1

=

1 2.

n=1

Fix x0 ∈ E. Then for any natural n there is a number in ∈ {1, . . . , 2n−1 } such (i ) (i ) that the distance between x0 and the interval (an n , bn n ) is the smallest. By putting a(in ) +b(in )

cn = n 2 n we obtain a sequence (cn )n∈N converging to x0 . Without loosing the generality, we may assume that (cn )n∈N is decreasing to x0 . Observe, that cn −x0 6 21n  (i ) (i )  and λ E ′ ∩ [x0 , cn ] > 12 · λ (an n , bn n ) = 12 · 41n . Hence

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λ (E ′ ∩ [x0 , cn ]) > (cn − x0 )2 λ E ′ ∩[x0 ,cn ] (cn −x0 )2 n→∞

and lim inf



1 2

1 4n  1 2 2n

·

=

1 . 2

> 21 . Therefore, x0 is not a superdensity point of E.

⊔ ⊓

In general it is rather difficult to describe an interior of a set in f -density topologies (see [10]). Theorem 4.2 ([10], Theorem 3 and 10). (a) For any f ∈ A and any set A ⊂ R there exists a countable ordinal α > 1 such that intTf (A) = A ∩ Φα f (B) where B is a measurable kernel of A. (b) Let f ∈ A and lim inf + x→0

f (x) x

= 0. For each n ∈ N there exists a perfect nowhere

dense set A such that intTf (A) $ A∩Φkf (A) for k < n and intTf (A) = A∩Φnf (A). However, if λ (Φf (A)△A) = 0, then we know that the interior of A is not empty (see Remark 2.1). To answer the question from the beginning of this section, firstly we will construct Cantor-like set of positive measure which has nonempty interior in Tf for a power function f (x) = xα , (α > 1) and secondly, we will show that this set has the empty interior for another power function. Theorem 4.3. For f (x) = xα , α > 1, there exists a perfect nowhere dense set E of positive measure such that almost every point of E is its f -density point. Proof. Analogously as in Example 4.1 we will define a central Cantor set E ⊂ [0, 1] of positive measure which has desired properties. Following denotations of this example, (1) (1)  from the interval [0, 1] we remove concentric open interval denoted by E1 = a1 , b1 of the length 21α . Suppose that for certain n > 2 we have the sets E1 , E2 , . . . , En−1 such 2n−2 n−1 S S (i) (i)  that En−1 = an−1 , bn−1 . From each of closed intervals of the set [0, 1] \ Ek i=1 k=1 (i) (i)  1 1 we remove concentric interval of the length 2n−1 · n2(2n−1)α and denote it by an , bn , 2n−1 S (i) (i)  1 . i = 1, . . . , 2n−1 . By putting En = an , bn we have λ(En ) = n2(2n−1)α Let E = [0, 1] \

∞ S

n=1

i=1

En . Let us notice that for any n ∈ N λ(En+1 )
k and i = 1, . . . , 2n−1 there exists a number zn,k > bn such that 1 λ(En ) = . (5) (i) α (i) k zn,k − an

Similarity and topologies generated by iterations of functions

137

(i)

We can observe that if z > zn,k then 1 λ(En ) < . (i) α k z − an It is evident, that and For k ∈ N define the sets

(6)

 α1 (i) zn,k − a(i) n = k · λ(En )

(7)

 α1 (i) zn,k − b(i) . n < k · λ(En )

(8)

n−1

∞ 2[ [  (i) (i)  bn , zn,k . Ak = n=k i=1



We will show that λ lim sup Ak = 0. For any k ∈ N we have k→∞

n−1

λ(Ak ) = =

n−1

∞ 2X X

∞ 2X  (8) X 1 (i) zn,k − b(i) 6 k · λ(En ) α = n

n=k i=1 ∞ X n−1

n=k i=1 ∞ X

n=k

Hence for any m > 2

1

1

· kα ·

2

1 α

n ·

22n−1

∞ ∞  [  X λ Ak 6 k=m

From the fact that the sequence

k=m ∞ S

6

n=k

1 1 = k−1 . 2n 2

1 1 = m−2 . 2k−1 2

Ak is decreasing we obtain

k=m ∞ [ ∞ ∞    \   [  λ lim sup Ak = λ Ak = lim λ Ak = 0. k→∞

Let N = lim sup Ak ∪ k→∞

m=1 k=m

 S ∞ 2n−1 S

m→∞

k=m

 (i) {bn } and x ∈ E \ N . We will show that x is a left-

n=1 i=1

hand f -density point of E for f (x) = xα , α > 1. Fix ε > 0 and take k0 ∈ N such that 2 n0 −1 } k0 < ε and x ∈ E \ Ak0 . Then there are numbers n0 > k0 and i0 ∈ {1, . . . , 2 (i )

such that bn00 < x. We denote n  (i0 ) o : n 6 n0 , i = 1, . . . , 2n−1 , b(i) t0 = min x − b(i) . n n ∈ bn0 , x For any t ∈ (0, t0 ) we denote by nt the smallest number n and by it the biggest (i ) (i )  (i ) (i ) (i ) (i ) number i for which antt , bntt ⊂ (x − t, x) and bntt − antt 6 bn00 − an00 . Then (i ) nt > n0 and x > zntt,k0 , so

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λ(Ent ) x− Since x − t
1 and function f (x) = xα there exists a set Eα of positive measure such that intTf (Eα ) = Eα ∩ Φf (Eα ) 6= ∅. Proposition 4.5. Let f (x) = xα , g(x) = x3α , α > 1. Then Tf 6∼ Tg . Proof. Let f (x) = xα , α > 1 and E be the set constructed in the proof of Theorem 4.3. We will show that no point of this set is its g-density point, so E has the empty interior in the topology Tg for g(x) = x3α . Fix x0 ∈ E. Analogously as in Example 4.1 we have the sequence (cn )n∈N   decreasing to x0 such that cn − x0 6 1 n2(2n−1)α

1 2n

and λ (E ′ ∩ [x0 , cn ]) >

1 2

(i )

. Hence λ (E ′ ∩ [x0 , cn ]) > g(cn − x0 )

1 n2(2n−1)α+n  1 3α 2n

=

(i )

· λ (an n , bn n ) =

2n(α−1)+α n

1 2

·

1 2n−1

·

Similarity and topologies generated by iterations of functions

139

λ(E ′ ∩[x0 ,cn ]) and lim g(cn −x0 ) = ∞. Therefore, x0 ∈ / Φg (E) and consequently, E ∩Φg (E) = ∅. n→∞ Thus E has the empty interior in Tg . ⊔ ⊓

Observe that if the interior of a set is empty for a function f1 (x) = xα1 , then it is also empty for any function f2 (x) = xα2 with α2 > α1 . Corollary 4.6. For any function f (x) = xα , α > 1, there is n ∈ N such that Tf n 6∼ Tf . If α > 3, then Tf 2 6∼ Tf . Notice that for α = 2, Theorem 4.3 gives us the set which almost all points are its superdensity points, but Proposition 4.5 does not resolve the problem if the second iteration of f (x) = x2 generates the topology which is similar or not to the superdensity topology. Nevertheless, using Corollary 4.6 we can construct a decreasing sequence (Tn )n∈N of topologies generated by functions, such that Ts ' Tn for any n ∈ N and Ti is not similar to Tj for any i 6= j. Comparing this result with Corollary 2.6 we can say that there are a lot of f -density topologies dissimilar to superdensity topology, of both kinds: the smaller and larger than Ts . At the end let us compare topologies generated by functions of the form f (x) = xα with Hashimoto topology T ∗ . Remark 4.7. Any nowhere dense set has an empty interior in Hashimoto topology T ∗ . Consequently, by Theorem 4.3, if f (x) = xα and α > 1 then Tf is not similar to T ∗ . The latter remark does not establish that no f -density topology with lim inf ∗

x→0+

f1 (x) x

=

0 is similar to T . For example, the function ( 1 x x for x ∈ (0, 1) g (x) = 1 for x > 1 belongs to A and for any α > 1 the set Eα , constructed in Theorem 21 satisfies equality: Φg (Eα ) = ∅. Therefore Tg is smaller than (and not similar to) any Tfα where fα (x) = xα and α > 1. We do not know if Tg is similar to T ∗ .

Bibliography 1. Aniszczyk B., Frankiewicz R.: Nonhomeomorphic density topologies. Bull. Polish Acad. Sci. Math. 34, no. 3–4 (1986), 211–213. 2. Bartoszewicz A., Filipczak M., Kowalski A., Terepeta M.: Similarity between topologies. Centr. Eur. J. Math. 12 (2014), 603–610. 3. Filipczak M., Filipczak T.: A generalization of the density topology. Tatra Mt. Math. Publ. 34 (2006), 37–47. 4. Filipczak M., Filipczak T.: On the comparison of density type topologies generated by functions. Real Anal. Exchange 36 (2010), 341–352. 5. Filipczak M., Filipczak T.: Remarks on f -density and ψ-density. Tatra Mt. Math. Publ. 34 (2006), 141–147. 6. Filipczak M., Filipczak T.: On f -density topologies. Topology Appl. 155 (2008), 1980–1989.

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7. Filipczak M., Filipczak T.: On ∆2 condition for density-type topologies generated by functions. Topology Appl. 159 (2012), 1838–1846. 8. Filipczak M., Filipczak T.: Density type topologies generated by functions. Properties of f density. In: Traditional and Present-day Topics in Real Analysis (dedicated to Professor Jan Stanislaw Lipi´ nski), M. Filipczak, E. Wagner-Bojakowska (eds.), L´ od´ z Univ. Press, L´ od´ z 2013, 411–430. 9. Filipczak M., Hejduk J.: On topologies associated with the Lebesgue measure. Tatra Mt. Math. Publ. 28 (2004), 187–192. 10. Filipczak M., Wagner-Bojakowska E.: The interior operation in f -density topology. Tatra Mt. Math. Publ. 35 (2007), 51–64. 11. Taylor S.J.: On strengthening the Lebesgue Density Theorem. Fund. Math. 46 (1959), 305–315. 12. Terepeta M., Wagner-Bojakowska E.: Ψ-density topology. Rend. Circ. Mat. Palermo, Ser. II 48 (1999), 451–476. 13. Lukeˇs J., Maly J., Zajiˇ cek L.: Fine Topology Methods in Real Analysis and Potential Theory. Springer-Verlag, Berlin 1986. 14. Wilczy´ nski W.: Density topologies. In: Handbook of Measure Theory, Elsevier Science B.V., North Holland, Amsterdam 2002, 675–702.

On density points on the real line with respect to sequences tending to zero Jacek Hejduk, Anna Loranty and Renata Wiertelak

Abstract. We present the results connected with density points on the real line with respect to sequences tending to zero. The first part deals with the family od sets having the Baire property and convergence with respect to the σ-ideal of the first category sets. The second part is devoted to the family of Lebesgue measurable sets and convergence with respect to the σ-ideal of the Lebesgue null measure sets. Keywords: I-density topologies, deep I-density topologies, comparison of topologies, J -density topology, J -approximately continuous function. 2010 Mathematics Subject Classification: 54A10, 26A15, 54A20.

1. Introduction Through the paper we will use the standard notation: R will be the set of real numbers, Q will be the set of rational numbers and N the set of positive integers. By Ba and L we will denote the family of Baire sets and Lebesgue measurable sets, respectively. Moreover, I will stand for the σ-ideal of the first category sets in R and L for the σ-ideal of the Lebesgue null measure sets. By λ(A) we shall denote the Lebesgue measure of a measurable set A and by |I| the length of an interval I. Furthermore, Tnat will denote the natural topology on R and hsi – an unbounded and non-decreasing sequence {sn }n∈N of positive real numbers. We shall say that a family F of subsets of R is invariant if for every P ∈ F, x ∈ R and m ∈ R \ {0} we get that P + x ∈ F and mP ∈ F, where P + x ={a + x : a ∈ P }, mP ={ma : a ∈ P }. J. Hejduk, A. Loranty, R. Wiertelak Faculty of Mathematics and Computer Science, L´ od´ z University, Banacha 22, 90-238 L´ od´ z, Poland, e-mail: {hejduk, loranta, wiertelak}@math.uni.lodz.pl R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 141–154. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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According to paper [11] we shall say that 0 is a density point of a set A ∈ Ba with respect to category if the sequence {fn }n∈N = {χnA∩[−1,1] }n∈N converges with respect to the σ-ideal I to the characteristic function χ[−1,1] . It means that every subsequence of the sequence {fn }n∈N contains a subsequence converging to the function χ[−1,1] everywhere except for a set of the first category. Basing on this concept we consider more general approach. For J = [a, b] we put 1 (a + b), 2 h(A, J)(x) = χ 2 (A−s(J))∩[−1,1] (x). s(J) =

|J|

By J we shall denote a sequence of non-degenerate and closed intervals {Jn }n∈N tending to zero, that means lim s(Jn ) = 0

n→∞

and

lim |Jn | = 0.

n→∞

These conditions are equivalent to the following one: diam{Jn ∪ {0}} −→ 0. n→∞

From now on, the family of all sequences of intervals tending to zero will be denoted by ℑ. Moreover, we will identify sequences which differ a finite numbers of their terms. To shorten notation, we will write J instead of {Jn }n∈N . We say that a sequence of intervals J = {[an , bn ]}n∈N ∈ ℑ, is right-side (leftside) tending to zero if there exists n0 ∈ N such that bn > 0 (an < 0) for n > n0 and   min{0, an } max{0, bn } lim =0 lim =0 . n→∞ n→∞ bn an Sequence of intervals J ∈ ℑ is one-side tending to zero if it is right-side or left-side tending to zero. Definition 1.1. Let P be a proper σ-ideal of subsets of R and J ∈ ℑ. The point 0 is a P(J )-density point of a set A ⊂ R if P

h(A, Jn )(x) −→ χ[−1,1] (x), n→∞

it means ∀ {nk }k∈N

∃ {nkm }m∈N

∃Θ ∈ P

∀x ∈ /Θ

h(A, Jnkm )(x) −→ χ[−1,1] (x), m→∞

P

where the symbol −→ stands for a convergence with respect to σ-ideal P. n→∞

It is obvious that 0 is a P(J )-density point of a set A ⊂ R if and only if   2 ∈ P. ∀ {nk }k∈N ∃ {nkm }m∈N lim sup [−1, 1] \ (A − s(Jnkm )) |Jnkm | m→∞

On density points on the real line with respect to sequences tending to zero

143

We shall say that a point x0 ∈ R is a P(J )-density point of a set A ⊂ R if and only if 0 is a P(J )-density point of the set A − x0 . For every set A ⊂ R let us denote ΦP(J ) (A) = {x ∈ R : x is a P(J )-density point of A}. Now, let us consider invariant σ-algebra S of subsets of R, P – a proper, invariant σ-ideal contained in S and J ∈ ℑ. As the consequence of definition of a P(J )-density point we get the following theorem. Theorem 1.2. For every sets A, B ∈ S we have: 1. ΦP(J ) (∅) = ∅, ΦP(J ) (R) = R; 2. A △ B ∈ P ⇒ ΦP(J ) (A) = ΦP(J ) (B); 3. ΦP(J ) (A ∩ B) = ΦP(J ) (A) ∩ ΦP(J ) (B). If an operator ΦP(J ) satisfies an additional condition ΦP(J ) (A) \ A ∈ P for any A ∈ S, then it is an almost lower density operator on (R, S, P). Whereas, if it fulfills an additional condition ΦP(J ) (A) △ A ∈ P for any A ∈ S, then it is a lower density operator on (R, S, P). Putting TP(J ) = {A ∈ S : A ⊂ ΦP(J ) (A)}, by conditions 1 and 2 we get that TP(J ) is a topology if S coincides with the family of all subsets of real line and P is any proper σ-ideal. However, it does not have to be topology.  For example for S equals the family of Borel sets on the real line, P = {∅}  and J = n1 , n1 n∈N we have that TP(J ) is not a topology (see [6]). In our further considerations we will concentrate only on two σ-algebras: Ba, L and corresponding to them σ-ideals: I, L. It is worth noting that, although in both cases the properties of the topologies generated by the respective density points are similar, then methods of their proving in each case are different.

2. The case of family of sets having the Baire property In this section we will focus on the σ-ideal of the first category sets in R, so in Definition 1.1 we will consider σ-ideal I instead of σ-ideal P. Inthis way  we will obtain an I(J )-density point. Taking the special sequence J = − n1 , n1 n∈N , we get that x0 is an I(J )-density point of a set A ∈ Ba nh if and only io if x0 is an I-density point of A (see [11]). Moreover, for hsi and J =

− s1n , s1n

n∈N

, the notion of an

I(J )-density point of a set A ∈ Ba is equivalent to the notion of an hsi-I-density point of A (see [7]). It should be emphasized that for an operator ΦI(J ) the analogue of Lebesgue Density Theorem holds: Theorem 2.1 ([12]). For every sets A ∈ Ba and J ∈ ℑ we have: A △ ΦI(J ) (A) ∈ I.

144

J. Hejduk, A. Loranty and R. Wiertelak

Proof. Let A ∈ Ba, then there exist an open set G and a set P ∈ I such that A = G △ P . We will show that A \ ΦI(J ) (A) ∈ I. Let us take a point x ∈ G. Then there exists a number n0 ∈ N such that x + Jn ⊂ G for n > n0 , hence Jn ⊂ G − x. So we have  2 2 A − (x + s (Jn )) ⊃ ((G \ P ) − (x + s (Jn ))) = |Jn | |Jn |   2 = (G − x) − s(Jn ) \ P − (x + s(Jn )) ⊃ |Jn |   2 2 (Jn − s(Jn )) \ (P − (x + s(Jn ))) = [−1, 1] \ P − (x + s(Jn )) . ⊃ |Jn | |Jn |  If P ∈ I then |J2n | P − (x + s(Jn )) ∈ I. Hence for x ∈ G we obtain that I

h(A − x, Jn )(x) −→ χ[−1,1] (x), n→∞

so that A \ ΦI(J ) (A) ⊂ A \ G ∈ I. To finish the proof we must show that ΦI(J ) (A) \ A ∈ I. Observe that ΦI(J ) (A) ⊂ R \ ΦI(J ) (R \ A). Then ΦI(J ) (A) \ A ⊂ (R \ ΦI(J ) (R \ A)) ∩ (R \ A) = (R \ A) \ ΦI(J ) (R \ A) ∈ I. ⊔ ⊓

2.1. An I(J )-density topology and its properties By Theorem 1.2 and Theorem 2.1 we have that operator ΦI(J ) is a lower density operator on (R, Ba, I). It is well known that for any measurable space (X, S, P), where S is a σ-algebra of subsets of X and P ⊂ S is a proper σ-ideal, if an operator φ : S → S is a lower density operator on (X, S, P) and a pair (S, P) has the hull property, then the family T = {A ∈ S : A ⊂ φ(A)} is a topology (see [9]), so we have Theorem 2.2 ([12]). The family TI(J ) = {A ∈ Ba : A ⊂ ΦI(J ) (A)} is a topology on R, which will be called I(J )-density topology. Moreover, Tnat

TI(J ) .

Since for any J ∈ ℑ, an operator ΦI(J ) is a lower density operator, so by Theorem 25.3 in [9] we obtain immediately the following theorem. Theorem 2.3. Let J ∈ ℑ. (i) (R, TI(J ) ) is a Baire space; (ii) (R, TI(J ) ) is neither a first countable, nor a second countable, nor a separable, nor a Lindel¨ of space; (iii) A ∈ I if and only if A is a closed and discrete set with respect to a topology T ; (iv) a set A ⊂ R is compact with respect to a topology TI(J ) if and only if A is finite.

On density points on the real line with respect to sequences tending to zero

145

(v) I is equal to the family of all meager sets with respect to a topology TI(J ) ; (vi) A ∈ Ba if and only if A is a union of two sets - one of them is open with respect to a topology TI(J ) and a second one is closed with respect to a topology TI(J ) ; (vii) Ba coincides with the family of all Borel sets (Baire sets) with respect to a topology TI(J ) . Moreover, we have that Theorem 2.4 ([13]). Let J ∈ ℑ. Then [a, b) ∈ TI(J ) ((a, b] ∈ TI(J ) ) for a < b if and only if the sequence J is right-side (left-side) tending to zero. This theorem yields to the following Theorem 2.5. If the sequence J ∈ ℑ is right-side (left-side) tending to zero, then (R, TI(J ) ) is not connected. From Definition 1.1 we have the following property. Property 2.6 ([12]). Let J ∈ ℑ, then 0 is an I(J )-density point of the set [ Ak = {0} ∪ int(Jn ), n>k

for every k ∈ N. Moreover, Ak ∈ TI(J ) . The next theorem shows that we have obtained an essential extension of I-density points. Theorem 2.7 ([12]). For every sequence J ∈ ℑ there exists a sequence K = {Kn }n∈N of intervals tending to zero such that TI(J ) \ TI(K) 6= ∅



TI(K) \ TI(J ) 6= ∅.

Proposition 2.8 ([12]). Let J = {Jn }n∈N and K = {Kn }n∈N be sequences tending to zero. If for every n ∈ N there exists k(n) ∈ N such that Jn = Kk(n) then TI(K) ⊂ TI(J ) . The succeeding theorems gives us an examples of situation when the topologies generated by sequences of intervals are identical. Theorem 2.9 ([12]). Let J ∈ ℑ and l ∈ N. If we divide every interval Jn on equal l intervals and order them in a sequence K, then TI(J ) = TI(K) . Theorem 2.10 ([12]). Let J = {Jn }n∈N and K = {Kn }n∈N be sequences of intervals tending to zero. If λ(Jn △ Kn ) lim = 0, n→∞ λ(Jn ∩ Kn ) then TI(J ) = TI(K) . The following theorems show some properties of the family of I(J ) type topologies. Theorem 2.11 ([12]). Let TH = {V \ P : V ∈ Tnat ∧ P ∈ I}. Then \ TI(J ) = TH . J ∈ℑ

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Theorem 2.12 ([12]). Let T ∗ be the topology generated by T ∗ = 2R

[

and

S

J ∈ℑ

TI(J ) . Then

TI(J ) 6= T ∗ .

J ∈ℑ

Theorem 2.13 ([12]). For any sequence J ∈ ℑ, the space (R, TI(J ) ) is Hausdorff but not regular.

2.2. I(J )-approximately continuous functions The class of approximately continuous function was defined by Denjoy in [3]. The category analogue of approximate continuity was presented by Poreda, WagnerBojakowska and Wilczy´ nski in [11]. A function f : R → R is I(J )-approximately continuous, if it is continuous with respect to the I(J )-density topology on the domain and the natural topology on the range. Theorem 2.14 ([13]). Let J be a sequence of intervals tending to zero. Then every I(J )-approximately continuous function is of the first Baire class. In the proof of the last theorem of this subsection, the following easy fact is needed. Conclusion 2.15 ([13]). If a sequence J ∈ ℑ is not one-side tending to zero, then for any nonempty set U ∈ TI(J ) , δ > 0 and x ∈ ΦI(J ) (U ) we have that U ∩ (x, x + δ) 6= ∅

and

U ∩ (x − δ, x) 6= ∅.

Theorem 2.16 ([13]). Let J ∈ ℑ. Every I(J )-approximately continuous function is Darboux function if and only if the sequence J is not one-side tending to zero. Proof. Necessity. Let the sequence J ∈ ℑ is right-side tending to zero and define function f (x) = x − k for x ∈ [k, k + 1). This is an I(J )-approximately continuous function but it is not a Darboux function. If the sequence J ∈ ℑ is left-side tending to zero, then we consider function g(x) = x − k

for x ∈ (k, k + 1].

Sufficiency. Let the sequence J ∈ ℑ be not one-side tending to zero and f be an I(J )-approximately continuous function. Fix an x ∈ R and for each n ∈ N define the set   1 1 Vn = f (x) − , f (x) + . n n By I(J )-approximate continuity of the function f there exists a set Un ∈ TI(J ) such that f (Un ) ⊂ Vn and x ∈ Un is an I(J )-density point of the set Un . Conclusion 2.15 implies that for any n ∈ N there exist

On density points on the real line with respect to sequences tending to zero

147

    1 1 yn1 ∈ Un ∩ x − , x and yn2 ∈ Un ∩ x, x + . n n   Hence we obtain two sequences yn1 n∈N and yn2 n∈N such that yn1 −→ x n→∞

and

and

yn2 −→ x n→∞

lim f (yn1 ) = lim f (yn2 ).

n→∞

n→∞

Obviously, by Theorem 2.14 we have that f is of the first Baire class. Thus, by Young’s criterion (Theorem 1.1 in [1]), we conclude that function f is Darboux function. ⊓ ⊔

3. The case of the family of Lebesgue measurable sets The notion of a density point connected with the Lebesgue measure was introduced at the beginning of 20th century. We say that x0 ∈ R is a density point of a Lebesgue measurable set A if λ(A ∩ [x0 − h, x0 + h]) lim = 1. (1) 2h h→0+ Moreover, we have that the condition (1) can be replaced with the following one: lim

λ(A ∩ [x0 − n1 , x0 + n1 ]) 2 n

n→∞

= 1.

(2)

This observation led to the generalization of a density point introduced in 2004 by M. Filipczak and J. Hejduk ([5]). They considered in (2) a sequence hsi instead of the sequence {n}n∈N . Thus we say that x0 ∈ R is an hsi-density point of a set A ∈ L if lim

n→∞

1 sn , x0 2 sn

λ(A ∩ [x0 −

+

1 sn ])

= 1.

(3)

Moreover we have that the condition (2) is equivalent the condition (3) iff sn lim inf sn+1 > 0 (see [5]). n→∞

One can ask what will happen if we replace the sequence {[x0 − s1n , x0 + s1n ]}n∈N by a sequence J + x0 , where J ∈ ℑ? In this case we also obtain a new kind of a density poin – a J -density point. This notion was introduced in 2013 by R. Wiertelak. Definition 3.1. Let A ∈ L, J ∈ ℑ. We shall say that the point 0 is a J -density point of a set A if λ(A ∩ Jn ) lim = 1. (4) n→∞ |Jn | Note that in the general case of a P(J)-density point we consider a special sequence of characteristic functions converges to χ[−1,1] with respect to σ-ideal P (see Definition 1.1). It is worth adding that in the case of a J -density point we can also consider such sequence. More specifically, an equivalent formulation of (4) is:

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χ

λ

2 (A−s(Jn ))∩[−1,1] |Jn |

(x) −→ χ[−1,1] (x), n→∞

λ

where the symbol −→ denotes a convergence with respect to the Lebesgue measure. n→∞ Indeed, the following equivalences are obvious. λ(A ∩ Jn ) λ((A − s(Jn )) ∩ (Jn − s(Jn ))) = 1 ⇔ lim =1⇔ n→∞ n→∞ |Jn | |Jn | 2 lim λ((A − s(Jn )) ∩ (Jn − s(Jn ))) = 2 ⇔ n→∞ |Jn | 2 2 lim λ( (A − s(Jn )) ∩ (Jn − s(Jn ))) = 2 ⇔ n→∞ |Jn | |Jn | 2 λ lim λ( (A − s(Jn )) ∩ [−1, 1]) = 2 ⇔ χ |J2 | (A−s(Jn ))∩[−1,1] (x) −→ χ[−1,1] (x). n n→∞ n→∞ |Jn | lim

In addition, it is worth noting that the last condition saved in the last line above is equivalent to the following ∀ {nk }k∈N

∃ {nkj }j∈N

χ |J 2

nk | j

(A−s(Jnk ))∩[−1,1] (x) j

−→ χ[−1,1] (x) L a.e.

j→∞

where L a.e. means that in this case we consider L-almost everywhere convergence. Thus in the case of the Lebesgue measure we can check whether a point 0 is a J density point of a set A ∈ L, as in the case of I(J)-density point. However, it appears that in the case of the Lebesgue measure the condition (4) is easier to check and it is more often applied. Obviously, a point x0 ∈ R is a J -density point of a set A ∈ L if 0 is a J -density point of a set A − x0 or equivalently if lim

n→∞

λ(A ∩ (x0 + Jn )) = 1. |Jn |

If for any A ∈ L and J ∈ ℑ we put ΦJ (A) = {x ∈ R : x is a J -density point of the set A}, then we obtain that ΦJ (A) ∈ L for any A ∈ L (see [10]) and operator ΦJ : L → L has properties presented in Theorem 1.2 for P = L. It is also worth noting that theorem analogous to Theorem 2.1 is not true for every sequence J ∈ ℑ. In [2] there is a construction of a set A ∈ L and a sequence J ∈ ℑ such that ΦJ (A) △ A 6∈ L. However, if we consider a subfamily ℑα ⊂ ℑ such that for any sequence J ∈ ℑα we have diam(Jn ∪ {0}) α(J ) = lim sup < ∞, |Jn | n→∞ then the analogue of Lebesgue Density Theorem holds. Theorem 3.2 ([10]). If J ∈ ℑα and A ∈ L then ΦJ (A) △ A ∈ L.

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Proof. We only need to show that A \ ΦJ (A) ∈ L for any bounded set A. Moreover, there is no loss of generality in assuming that the sequence {|Jn |}n∈N is decreasing and diam{{0} ∪ Jn } < 2α(J )|Jn |. (5) First, we will prove that for any 0 < ε < 1   λ(A ∩ (Jn + x)) Eε = x ∈ A : lim inf < 1 − ε ∈ L. n∈N |Jn |

(6)

Suppose, contrary to our claim, that the outer Lebesgue measure of E, denoted by λ∗ (E), is greater than 0. Thus one can find a set G ∈ Tnat such that Eε ⊂ G and (1 − ε)λ(G) < λ∗ (Eε ). Let E be the family of all closed intervals I ⊂ G such that λ(A ∩ I) < (1 − ε)|I| and I = Jn + x for some x ∈ Eε and n ∈ N. Observe that (i) every neighbourhood of each x ∈ Eε contains an interval I ∈ E; S (ii) for any sequence {In } of disjoint intervals from E the inequality λ∗ (Eε \ In ) > 0 holds. The property (i) is obvious. The property (ii) results from the following fact  [  X X λ∗ Eε ∩ In 6 λ(A ∩ In ) 6 (1 − ε) |In | = n∈N

n∈N

n∈N

[  = (1 − ε)λ In 6 (1 − ε)λ(G) < (1 − ε)λ∗ (Eε ). n∈N

Now, we will construct inductively a sequence {In }n∈N of disjoint intervals from E. We start by putting   k0 = min i ∈ N : ∃ Ji + x ∈ E x∈Eε

and choosing interval I1 from E such that |I1 | = |Jk0 |. Assume that intervals Ii for i ∈ {1, 2, . . . , n} have been chosen. Let En be the subset of E which consists of all intervals that are disjoint from I1 , . . . , In . Properties (ii) and (i) imply that En 6= ∅. Define   kn = min i ∈ N :



x∈Eε

Ji + x ∈ E n .

and choose an intervalSIn+1 from En with length |Jkn |. Putting B = Eε \ In we obtain, by (ii), that λ∗ (B) > 0. Hence there is N ∈ N n∈N

such that

∞ X

|In |
N let Kn be the interval S concentric with In such that |Kn | = (4α(J ) + 1)|In |. The inequality (7) implies that Kn does not cover the set B, so there exists n∈N

a point

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x∈B\

[

Kn .

(8)

n>N

Therefore x ∈ Eε \

N S

In . From (ii) and (i) it follows that there exists an interval

n=1

Ix ∈ EN such that Ix = Jnx + x for some nx ∈ N. It is clear that Ix ∩ In 6= ∅ for some n > N . Putting n0 = min{n ∈ N : In ∩ Ix 6= ∅} we obtain that |Jnx | = |Ix | 6 |In0 |. The condition (5) implies dist(x, In0 ) 6 diam{x ∪ Ix } = diam{{0} ∪ Jnx } < 2α(J )|Jnx | 6 2α(J )|In0 |, where dist(x, In0 ) denotes the distance between the point x and the interval In0 . Thus x ∈ Kn0 , contrary to (8). From (6) and the inclusion [ A \ ΦJ (A) ⊂ Eε ε∈(0,1)∩Q

we get immediately that A △ ΦJ (A) ∈ L.

⊔ ⊓

Therefore for any J ∈ ℑα an operator ΦJ is a lower density operator on (R, L, L). In addition, it is worth noting that when we compare the above proof with proof of Theorem 2.1, it is easy to observe differences in the methods that are used in them. We see at once that in the case of J -density points we need to take other action than in the case of I(J )-density points. Obviously, one can ask what will happen if we consider any J ∈ ℑ. In this case we can prove the following fact. Theorem 3.3 ([10]). If J ∈ ℑ and A ∈ L then ΦJ (A) \ A ∈ L. Therefore for any J ∈ ℑ an operator ΦJ is an almost lower density operator on (R, L, L).

3.1. A J-density topology and its property In this section we will focus our attention on topology generated by J -density points. As in the case of I(J )-density topology (see Section 2.1) we have that family TJ = {A ∈ L : A ⊂ ΦJ (A)} is a topology for any J ∈ ℑα . What is more, in [9] one can find the property that for any measurable space (X, S, P), where S is a σ-algebra of subsets of X and P ⊂ S is a proper σ-ideal, if an operator φ : S → S is an almost lower density operator on (X, S, P) and a pair (S, P) has the hull property, then the family T = {A ∈ S : A ⊂ φ(A)} is a topology, so we immediately obtain

On density points on the real line with respect to sequences tending to zero

151

Theorem 3.4 ([10]). Let J ∈ ℑ. The family TJ = {A ∈ L : A ⊂ ΦJ (A)} is a topology called a J -density topology. Moreover, as in the case of I(J )-density topology, it is easy to see that Tnat TJ . Furthermore, since for any J ∈ ℑ, an operator ΦJ is an almost lower density operator, so by Theorem 25.27 in [9] we obtain immediately the following claim Theorem 3.5. Let J ∈ ℑ. (i) (R, TJ ) is neither a first countable, nor a second countable, nor a separable, nor a Lindel¨ of space; (ii) A ∈ L if and only if A is a closed and discrete set with respect to a topology TJ ; (iii) a set A ⊂ R is compact with respect to a topology TJ if and only if A is finite. If J ∈ ℑα , then an operator ΦJ is a lower density operator, so in this case to the properties presented in Theorem 3.5 we can add another (cf. Theorem 25.3 in [9]) Theorem 3.6. Let J ∈ ℑα . (a) (R, TJ ) is a Baire space; (b) L is equal to the family of all meager sets with respect to a topology TJ ; (c) A ∈ L if and only if A is a union of two sets - one of them is open with respect to a topology TJ and a second one is closed with respect to a topology TJ ; (d) L coincides with the family of all Borel sets (Baire sets) with respect to a topology TJ . One can ask about the connection between the density topology Td and a J density topology. If we consider an unbounded and nondecreasing sequence {sn }n∈N of positive numbers and a sequence J = {Jn }n∈N , where Jn = [− s1n , s1n ] for n ∈ N, then we have that Td ⊂ TJ (see [4]). In general, such a relationship does not have to take place. Indeed, we have Theorem 3.7 ([10]). If J ∈ ℑ \ ℑα , then there exists an open set A such that 0 ∈ Φd (A) and 0 6∈ ΦJ (A). From the above theorem we can deduce at once Theorem 3.8. If J ∈ ℑ \ ℑα , then Td \ TJ 6= ∅. Moreover, we can show that there exists a sequence J ∈ ℑ\ℑα such that TJ \Td 6= ∅ and Td \ TJ 6= ∅. However, if we consider any sequence J ∈ ℑα , then Td \ TJ = ∅. We can describe the relationship between the density topology and TJ in the following way. Theorem 3.9 ([10]). Let J ∈ ℑ. The following conditions are equivalent: (a) α(J ) < +∞; (b) Td ⊂ TJ .

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It should be added that there are sequences J , K ∈ ℑα such that Td 6= TJ and Td = TK . Interesting is also the question about the relationship between the J -density topologies for different sequences J ∈ ℑ. The question whether the claim analogous to Theorem 2.7 is true in the case of the J - density topology is still open. However, we have Theorem 3.10. There exist sequences K, J ∈ ℑ such that TJ \ TK 6= ∅, TK \ TJ 6= ∅, TJ 6= Td and TK 6= Td . 1 1 To prove this it suffices to consider sequences J = {[− (2n−1)! , (2n−1)! ]}n∈N and 1 1 K = {[− (2n)! , (2n)! ]}n∈N (see [4]). The next theorem shows some connection between J -density topology and hsidensity topology associated with hsi-density points.

Theorem 3.11 ([10]). If J ∈ ℑα , then there exists a sequence K ∈ ℑ of symmetrical intervals such that TK ⊂ TJ . Moreover, from Theorem 3.8 and 3.9 it may be concluded Theorem 3.12. If J ∈ ℑα and K ∈ ℑ \ ℑα , then TJ \ TK 6= ∅. Furthermore, Theorem 8 in [5] gives Theorem 3.13. Let T ∗ be the topology generated by

S

TJ . Then

J ∈ℑ

T ∗ = 2R

and

[

TJ 6= T ∗ .

J ∈ℑ

We end this section with two properties connected with separation axioms for J density topology. The second one will show the differences between the J -density topology and I(J )-density topology for J ∈ ℑα . Since for any J ∈ ℑ, we have that Tnat ⊂ TJ , so Property 3.14 ([8]). For any J ∈ ℑ a space (R, TJ ) is a Hausdorff space. Moreover, we have Property 3.15 ([8]). For any J ∈ ℑα a space (R, TJ ) is regular and it is not normal. The question whether for any sequence J ∈ ℑα a space (R, TJ ) is completely regular is still open. Just like the question whether for any J ∈ ℑ \ ℑα a space (R, TJ ) is regular, completely regular or normal.

3.2. J-approximately continuous functions It was mentioned in Section 2.2 that the notion of approximately continuous functions was introduced by Denjoy. He considered this class of functions in conjunction with the density points. Now, we will concentrate on the analogue concept in connection with J -density points.

On density points on the real line with respect to sequences tending to zero

153

Let f : R → R and J ∈ ℑ. We say that f is J -approximately continuous at a point x0 ∈ R if there exists a set Ax0 ∈ L such that x0 ∈ ΦJ (Ax0 ) and f (x0 ) = x→x lim , f (x). 0

x∈Ax0

Obviously, we say that f : R → R is a J -approximately continuous function if it is J -approximately continuous at each point x ∈ R. It is easy to see that, if f : R → R is a continuous function, then it is J -approximately continuous for any sequence J ∈ ℑ. If f : R → R is an approximately continuous function, then it is J -approximately continuous for any sequence J ∈ ℑα . For any J ∈ ℑ \ ℑα there exists an approximately continuous function f : R → R which is not J -approximately continuous (see [8]). Theorem 3.16 ([8]). Let J ∈ ℑ. The family of all J -approximately continuous functions f : R → R is closed under addition and multiplication. Moreover, if g : R → R is J -approximately continuous, then the function g1 is J -approximately continuous, whenever g(x) 6= 0 for any x ∈ R. Now, we will focus our attention on sequences J belonging to ℑα . The question whether the following statements are true also for sequences J ∈ ℑ \ ℑα is still open. Theorem 3.17 ([8]). Let J ∈ ℑα . If f : R → R is a J -approximately continuous function, then f is of the first Baire class. The relationship between the J -approximately continuous functions and J -density topology for J ∈ ℑα can be explained in the following theorem. Theorem 3.18 ([8]). Let f : R → R and J ∈ ℑα . The function f is J -approximately continuous if and only if for any β ∈ R the sets {x ∈ R : f (x) < β} and {x ∈ R : f (x) > β} belong to the topology TJ . In addition, there is a relationship between these functions and Lebesgue measurable functions analogous to the case of approximately continuous functions. Theorem 3.19 ([8]). Let f : R → R. The following conditions are equivalent: (i) f is a Lebesgue measurable function, (ii) there exists B ∈ L such that for any sequence J ∈ ℑα and any x ∈ R \ B the function f is J -approximately continuous at a point x, (iii) there exists a sequence J ∈ ℑα and there exists BJ ∈ L such that the function f is J -approximately continuous at each point x ∈ R \ BJ .

Bibliography 1. Bruckner A.M.: Differentiation of Real Functions. Lecture Notes in Math. 659, Springer Verlag, Berlin 1978. 2. Cs¨ ornyei M.: Density theorems revisited. Acta Sci. Math. 64 (1998), 59–65. 3. Denjoy A.: M´ emoires sur les d´ eriv´ es des fonctions continues. Journ. Math. Pure et Appl. 1 (1915), 105–240. 4. Filipczak M., Filipczak T., Hejduk J.: On the comparison of the density type topologies. Atti Sem. Mat. Fis. Univ. Modena LII (2004), 1–11.

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5. Filipczak M., Hejduk J.: On topologies associated with the Lebesgue measure. Tatra Mt. Math. Publ. 28 (2004), 187–197. 6. Hejduk J.: Density Topologies with Respect to Invariant σ-Ideal. L´ od´ z Univ. Press, L´ od´ z 1997. 7. Hejduk J., Horbaczewska G.: On I-density topologies with respect to a fixed sequence. Reports on Real Analysis, Conference at Rowy (2003), 78–85. 8. Hejduk J., Loranty A., Wiertelak R.: J -approximately continuous functions (to appear). 9. Hejduk J., Wiertelak R.: On the abstract density topologies generated by lower and almost lower density operators. In: Traditional and Present-day Topics in Real Analysis (dedicated to Professor Jan Stanislaw Lipi´ nski), M. Filipczak, E. Wagner-Bojakowska (eds.), L´ od´ z Univ. Press, L´ od´ z 2013, 431–447. 10. Hejduk J., Wiertelak R.: On the generalization of density topologies on the real line (to appear in Math. Slovaca at 2014). 11. Poreda W., Wagner-Bojakowska E., Wilczy´ nski W.: A category analogue of the density topology. Fund. Math. 125 (1985), 167–173. 12. Wiertelak R.: A generalization of density topology with respect to category. Real Anal. Exchange 32, no. 1 (2006/2007), 273–286. 13. Wiertelak R.: About I(J)–approximately continuous functions. Period. Math. Hungar. 63, no. 1 (2011), 71–79.

Maximal classes for some families of Darboux-like and quasicontinuous-like functions J´an Bors´ık

Abstract. The paper is a survey concerning maximal additive, multiplicative and latticelike classes for certain families of functions similar to quasicontinuous or Darboux functions. Keywords: maximal addditive class, maximal multiplicative class, quasicontinuous functions, Darboux functions. 2010 Mathematics Subject Classification: 54C30, 54C08.

1. Introduction Let X be a topological space and let F be a nonempty family of real functions defined on X. For F , we define the maximal additive class Madd (F ) as Madd (F ) = {f : X → R; f + g ∈ F for every g ∈ F }, the maximal multiplicative class Mmult (F ) as Mmult (F ) = {f : X → R; f · g ∈ F for every g ∈ F }, the maximal class with respect tu maximum Mmax (F ) as Mmax (F ) = {f : X → R; max(f, g) ∈ F for every g ∈ F }, the maximal class with respect to minimum Mmin (F ) as Mmin (F ) = {f : X → R; min(f, g) ∈ F for every g ∈ F }, J. Bors´ık Mathematical Institute, Slovak Academy of Sciences, Greˇs´ akova 6, 04001 Koˇsice, Slovakia, Katedra fyziky, matematiky a techniky FHPV, Preˇsovsk´ a univerzita v Preˇsove, ul. 17. novembra 1, 08001 Preˇsov, Slovakia, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 155–168. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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and the maximal latticelike class Mlatt (F ) as Mlatt (F ) = {f : X → R; max(f, g) ∈ F and min(f, g) ∈ F for every g ∈ F }. The notion of maximal classes for certain family of functions might be used for the first time in the Bruckner’s monograph [8]. The maximal additive class is always nonempty because the zero constant function 0 (i.e. f (x) = 0 for all x ∈ X) belongs to Madd (F ) for each family F . Similarly, the maximal multiplicative family is always nonempty because the constant function 1 (f (x) = 1) belongs to Mmult (F ). The classes Mmax (F ), Mmin (F ) and Mlatt (F ) can be empty (e.g for quasicontinuous functions with closed graph). Further, if the constant function 0 belongs to F then Madd (F ) ⊂ F, and if the constant function 1 belongs to F then Mmult (F ) ⊂ F. If F is closed under addition then F ⊂ Madd (F ), and if F is closed under multiplication then F ⊂ Mmult (F ). So, if the family F is closed under addition and contains the constant 0 function then Madd (F ) = F , and similarly, if the family F is closed under multiplication and contains the constant 1 function, then Mmult (F ) = F . This is true also conversely. If F = −F (where −F = {f ; −f ∈ F}) then Mmin (F ) = −Mmax (F ). Moreover, Mlatt (F ) = Mmax (F ) ∩ Mmin (F ). Therefore, if we have knowledge of the maximal classes Mmax (F ) and Mmin (F ) then we have knowledge of Mlatt (F ), too. However, for some families (e.g. for quasicontinuous almost continuous (in the sense of Stallings) functions), we have knowledge of Mlatt (F ), however, a characterization of maximal classes with respect to maximum or minimum is an open problem. Moreover, we can define maximal classes with respect to composition of functions: Mout (F ) = {f ; f ◦ g ∈ F for every g ∈ F} and Min (F ) = {f ; g ◦ f ∈ F for every g ∈ F }. Maximal classes with respect to composition were investigated e.g. in [3, 29, 37, 54, 55]. The main problem with characterizations of maximal classes is that the operator of maximal classes is not monotone. Hence, a characterizations of maximal classes can be unexpected and interesting. Unfortunately, the results concerning maximal classes are scattered throughout the literature. In this paper, we will deal with maximal classes for some families of functions that are generalizations of continuity. Unless explicitly written, X and Y will be topological spaces. For a subset A of X, denote by Cl(A) the closure of A and by Int(A) the interior of A. In the results described below, unless explicitly written, we will assume that the functions are defined on R.

2. Darboux and similar functions Darboux functions The oldest results on maximal classes are know for Darboux functions. Maximal additive and maximal multiplicative classes were characterized by T. Radakovi´c in 1931. A function f defined on R is Darboux if whenever a < b and c is any number between f (a) and f (b), there is a number z ∈ (a, b) such that f (z) = c. Let C denote the family of all continuous functions, D the family of all Darboux functions and Const

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the family of all constant functions. Further, let usc denote upper semicontinuous functions and lsc lower semicontinuous functions. Theorem 2.1 ([48]). Madd (D) = Mmult (D) = Const. A characterization for maximum and minimum classes was given by J. Farkov´a. Theorem 2.2 ([11]). Mmax (D) = D ∩ usc, Mmin (D) = D ∩ lsc and Mlatt (D) = C. J. Jastrz¸ebski characterized maximal additive families for some subclasses of Darboux functions. Denote the family of Darboux functions whose upper and lower boundary functions are continuous by DC , the family of functions which take on every real value in every interval by D∗ , and the family of functions which take on every real value c-times in every interval by D∗∗ . Further, let ConstI be a family of S functions such that there exixts a sequence of open intervals (Ik ) such that k∈N Ik is dense in R and f ↾ Ik is constant, and let ConstcI be a family of functions such that S there is a sequence (Ik ) of open intervals and a sequence (Ak ) of sets such that k∈N Ik is dense in R, Ak ⊂ Ik , the cardinality of Ak is less that c and f ↾ (Ik \ Ak ) is constant ([24]). Theorem 2.3 ([24]). Madd (DC ) = C ∩ ConstI , Madd (D∗ ) = ConstI and Madd (D∗∗ ) = ConstcI . Darboux Baire one functions Denote the family of all Baire one functions by B1 . The maximal additive class for Darboux Baire one functions was characterized by A. M. Bruckner and J. Ceder, see also [8]. Theorem 2.4 ([9]). Madd (D ∩ B1 ) = C. In [8], it is also shown that C Mmult (DB1 ). The problem of characterizing the maximal multiplicative family was solved by R. Fleissner. Theorem 2.5 ([13]). Mmult (D ∩ B1 ) = M. Here, M stands for the Fleissner family of all functions with the following property: if x0 is a right-hand (left-hand) point of discontinuity of f , then f (x0 ) = 0, and there exists a sequence (xn ) converging to x0 such that xn > x0 (xn < x0 ) and f (xn ) = 0. Maximum and minimum classes were characterized again by J. Farkov´a. Theorem 2.6 ([11]). Mmax (D ∩ B1 ) = D ∩ usc, Mmin (D ∩ B1 ) = D ∩ lsc and Mlatt (D) = C. The characterization of maximal additive and multiplicative classes was extended for functions defined on the Euclidean space Rm by L. Miˇs´ık in [38] and later, for functions defined on some Banach spaces, in [39]. Let X be a topological space and let B be a base for the topology in X. A real function defined on X is called B-Darboux if for each A ∈ B, every x, y ∈ Cl(A) and each c between f (x) and f (y) there exists a point z ∈ A such that f (z) = c. Denote the family of such functions by DB ([40]).

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Theorem 2.7 ([39]). Let X be a finite-dimensional strictly convex Banach space and let B be the system of all sets a+Ur , where a ∈ X , Ur = {x ∈ X; kxk < r} and r > 0. Then, Madd (DB ∩ B1 ) = C and Mmult (DB ∩ B1 ) = MM , where MM is the family of all B-Darboux Baire one functions with the property: if x ∈ X and B ∈ B are such that x ∈ Cl(B) \ B and if there is a sequence (Cn ) of elements of B such that x ∈ Cl(Cn ) \ Cn , Cn+1 ⊂ Cn ⊂ B, lim diam(Cn ) = 0 and sup inf f (Cn ) < inf sup f (Cn ), n→∞

n

n

then f (x) = 0, and there exists a sequence (xk ) of points of B such that f (xk ) = 0 for all k. Here, diam(C) is the diameter of the set C.

Connectivity and functionally connected functions A function f : X → R is connectivity if the graph of f restricted to C is a connected subset of X × R for each connected subset C of R. A function f : R → R is functionally connected if for each a < b and each continuous function g : [a, b] → R with (f (a) − g(a))(f (b) − g(b)) < 0, there is a point c ∈ [a, b] with f (c) = g(c) ([25]). Denote by Con the family of all connected functions and by F con the family of all functionally connected functions. Characterizations of maximal classes for these families are similar, however, a characterization of maximal classes for maximum and minimum seems to be an open problem. Theorem 2.8 ([25]). Madd (Con) = Madd (F con) = C. Theorem 2.9 ([26]). Mlatt (Con) = Mlatt (F con) = C, Mmult (Con) = Mmult (F con) = M, Mmax (F con) = D ∩ usc and Mmin (F con) = D ∩ lsc. Extendable functions A function f is extendable if there exists a connectivity function F : R × [0, 1] → R such that F (x, 0) = f (x) for every x ∈ R. Denote by Ext the family of all extendable functions. Theorem 2.10 ([27]). Madd (Ext) = Mlatt (Ext) = C and Mmult (Ext) = M.

Functions with perfect road A function f : R → R has a perfect road if for every x ∈ R there exists a perfect set P having x to be a bilateral limit such that f ↾ P is continuous at x ([35]). Denote functions with perfect road by P R. Maximal additive and multiplicative classes were characterized by K. Banaszewski. Theorem 2.11 ([2]). Madd (P R) = C and Mmult (P R) = M.

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Young (peripherally continuous) functions A function f : R → R is a Young function (peripherally continuous function) if for each x ∈ R there exist sequences (xn ) and (yn ) such that xn < x, yn > x, lim xn = lim yn = x and lim f (xn ) = lim f (yn ) = f (x). Denote the family of all n→∞ n→∞ n→∞ n→∞ Young functions by P C. Maximal classes were characterized by K. Banaszewski. Theorem 2.12 ([2]). Madd (P C) = Mmax (P C) = Mmin (P C) = Mlatt (P C) = C and Mmult (P C) = M.

CIVP functions and SCIVP functions A function f : R → R has the Cantor intermediate value property if for every x, y ∈ R and for each Cantor set K between f (x) and f (y) there exists a Cantor set C between x and y such that f (C) ⊂ K ([15]). A function f has the strong Cantor intermediate value property if for every x, y ∈ R and for each Cantor set K between f (x) and f (y), there exists a Cantor set C between x and y such that f (C) ⊂ K and the restriction f ↾ C is continuous. Denote the family of functions with the Cantor intermediate value property by CIVP and with the strong Cantor intermediate value property by SCIVP. Theorem 2.13 ([4]). Assume CH. Then, Madd (CIV P ) = Mmult (CIV P ) = Mlatt (CIV P ) = Madd (D ∩ CIV P ) = Mmult (D ∩ CIV P ) = Mlatt (D ∩ CIV P ) = Const. Theorem 2.14 ([14]). Assume CH. Then, Madd (SCIV P ) = Mmult (SCIV P ) = Const.

Almost continuous (Stallings) functions A function f is almost continuous (in the sense of Stallings) if for every open set G ⊂ R × R containing the graph of f , there is a continuous function g such that the graph of g lies in G ([52]). Denote the family of all almost continuous functions (in the sense of Stallings) by ACS. For these functions we known all maximal classes. Theorem 2.15 ([43]). Madd (ACS) = C. Theorem 2.16 ([26]). Mmult (ACS) = Mlatt (ACS) = C. The maximal classes for maximum and minimum were characterized only recently (it is an affirmative answer to a conjecture in [26]). Theorem 2.17 ([34]). Mmax (ACS) = D ∩ usc and Mmin (ACS) = D ∩ lsc.

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3. Quasicontinuous and similar functions Quasicontinuous functions Whereas the great part of results on maximal classes for Darboux-like functions concerns functions defined on R, the great part of results on maximal classes for quasicontinuous-like functions concerns functions defined on topological spaces. Let X be a topological space and let C(f ) denote the set of all continuity points of f . Recall that a function f : X → R is said to be quasicontinuous at a point x if for each neighbourhood U of x and each ε > 0, there is an open nonempty set G ⊂ U such that f (G) ⊂ (f (x) − ε, f (x) + ε) ([28]). A function f is quasicontinuous if it is such at every point. Denote the family of all quasicontinuous functions by Q and the family of bounded functions by b. Further, denote by Q(f ) the set of all quasicontinuity points of f . The maximal additive class and the maximal lattice-like class for quasicontinuous functions was decribed by Z. Grande and L. Soltysik in [22]. Theorem 3.1 ([22]). Let X be a topological space. Then Madd (Q) = Mlatt (Q) = C. The maximal multiplicative family for a complete metric space X was characterized by Z. Grande and L. Soltysik in [22] and later, by Z. Grande, for an arbitrary topological space X. Let QM stand for the family of all quasicontinuous functions f : X → R with the property: if f is discontinuous at x, then f (x) = 0 and x ∈ Cl(C(f )∩f −1 (0)). Further, let QbM denote the family of all quasicontinuous functions with the property: if f is discontinuous at x then f (x) = 0. Theorem 3.2 ([21]). Let X be a topological space. Then, Mmult (Q) = QM and Mmult (b ∩ Q) = QbM . Denote the family of all functions for which the set X \ C(f ) is nowhere dense by C ∗ , and the family of all functions for which the set X \ Q(f ) is nowhere dense by Q∗ . Whereas Madd (Q) 6= Mmult (Q), for the family Q∗ , these maximal classes coincide. Theorem 3.3 ([7]). Let X be a Baire space. Then, Madd (Q∗ ) = Mmult (Q∗ ) = C ∗ . Maximal classes for maximum and minimum were described by T. Natkaniec. Theorem 3.4 ([42]). Let X be a topological space. Then, Mmax (Q) = Mmin (Q) = C.

Upper and lower quasicontinuous functions A function f : X → R is upper (lower) quasicontinuous at x ∈ X if for every positive ε > 0 and every neighbourhood U of x there exists a nonempty open set G ⊂ U such that f (y) < f (x) + ε (f (y) > f (x) − ε) for each y ∈ G ([10]). Let QE denote the family of all functions which are both upper and lower quasicontinuous at each x ∈ X. Notice that QE is a nowhere dense set in Q in the topology of the uniform convergence (for X = R). Maximal additive and lattice-like classes were characterized by E. Stro´ nska in [53]. In her paper, we can find some necessary and some sufficient conditions for the maximal multiplicative class, however, its characterization is still open.

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Theorem 3.5 ([53]). Let X be a topological space. Then, Madd (QE ) = Mmax (QE ) = Mmin (QE ) = Mlatt (QE ) = C.

Symmetrically quasicontinuous functions A function f : X × Y → R (X and Y are topological spaces) is said to be quasicontinuous at (x, y) with respect to first (second) coordinate if for every neighbourhoods U , V and W of x, y and f (x, y), respectively, there are nonempty open sets G and H such that x ∈ G ⊂ U , H ⊂ V (G ⊂ U , y ∈ H ⊂ V ) and f (G × H) ⊂ W . A function f is symmetrically quasicontinuous at (x, y) if it is quasicontinuous both with respect to the first and the second coordinate ([46]). Denote by Qsx , Qsy and Qss the family of all functions which are quasicontinuous with respect to first coordinate, quasicontinuous with respect to second coordinate, symmetrically quasicontinuous at each point, respectively. Further, let Qss0 denote the family of all functions from Qss such that f (x, y) 6= 0 for each (x, y) ∈ X × Y . For x ∈ X, a function fx : Y → R, fx (y) = f (x, y) is the x-section of f ; similarly, the y-section f y : X → R is defined as f y (x) = f (x, y). Maximal additive classes for Qsx and Qsy are characterized for arbitrary topological spaces X and Y . Theorem 3.6 ([19]). Let X and Y be topological spaces. Then, Madd (Qsx ) = {g ∈ Qsx ; sections gx are continuous}. Similarly, Madd (Qsy ) = {g ∈ Qsy ; sections g y are continuous}. The investigation of the maximal additive class for Qss is more complicated. Let (x, y) ∈ X × Y be a point. We say that a closed set A ⊂ X × Y belongs to the family S(x, y) [(P (x, y)] if we have: Ax = {y} [Ay = {x}], x ∈ Cl((Int(A))y ) [y ∈ Cl((Int(A))x ) ], and for each point (u, v) ∈ A\{(x, y)}, we have u ∈ Cl(Int(A))v ) and v ∈ Cl(Int(A))u ), where Ax = {t ∈ Y ; (x, t) ∈ A} and Ay = {t ∈ X; (t, y) ∈ A}. Theorem 3.7 ([19]). Let X and Y be topological spaces such that, for each point (x, y) ∈ X × Y , the families S(x, y) and P (x, y) are nonempty. Then, Madd (Qss ) = {f ∈ Qss ; f is separately continuous}. There is an open problem whether above the theorem holds for arbitrary topological spaces. The Euclidean plane (X = Y = R) satisfies the assumptions of this theorem. Moreover, in his paper, we can find some necessary and some sufficient conditions for the maximal multiplicative classes for families Qsx , Qsy and Qss , however, their characterization is open. For the family Qss0 we have a characterization. Theorem 3.8 ([19]). Let X and Y be topological spaces such that for each point (x, y) ∈ X × Y the families S(x, y) and P (x, y) are nonempty. Then, Mmult (Qss0 ) = {f ∈ Qss0 ; f is separately continuous}.

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Strongly quasicontinuous functions For a measurable set E ⊂ R let ℓ(E) stand the Lebesgue measure of E. For a measurable set E and x ∈ R, the numbers dl (E, x) =

lim inf

t→0+ ,k→0+

du (E, x) = lim sup t→0+ ,k→0+

ℓ(E ∩ [x − t, x + k]) , k+t ℓ(E ∩ [x − t, x + k]) k+t

are called the upper and lower density of E at x, respectively. If dl (E, x) = du (E, x), we call this number the density of E at x and denote it by d(E, x). If d(E, x) = 1, we say that x is a density point of E. The family Td = {A ⊂ R; A is measurable and every point x ∈ A is a density point of A} is a topology called the density topology. A function f : R → R is called approximately continuous at x ∈ R if there is a measuarble set E containing x such that d(E, x) = 1 and the restriction f ↾ E is continuous at x. Let Dap (f ) be the set of all points at which f is not approximately continuous. A function f is approximately continuous if Dap (f ) = ∅. Equivalently, a function f is approximately continuous if it is continuous as the application from R equipped with the density topology Td . Of course, approximately continuous functions are closed under addition and multiplication and so maximal classes are obvious. Let A denote the family of all approximately continuous functions and Cae the family of functions continuous almost everwhere (i.e. such that ℓ(D(f ) = 0). A function f : R → R is strongly quasicontinuous if for every x ∈ R, for every set A ∈ Td containing x and for every ε > 0 there is an open interval I such that I ∩ A 6= ∅ and |f (t) − f (x)| < ε for all t ∈ A ∩ I. A function f : R → R is s1 -strongly quasicontinuous (s2 -strongly quasicontinuous) if for every x ∈ R, for every set A ∈ Td containing x and for every ε > 0 there exists an open interval I such that I ∩ A 6= ∅, I ∩ A ⊂ C(f ) (I ∩ A ⊂ R \ Dap (f )) and |f (t) − f (x)| < ε for all t ∈ A ∩ I. Denote the families of strongly quasicontinuous functions, s1 -strongly quasicontinuous functions and s2 -strongly quasicontinuous functions by Qs , Qs1 and Qs2 , respectively ([18]). Maximal families were investigated by E. Stro´ nska. Theorem 3.9 ([55]). Madd (Qs ) = Mmax (Qs ) = Mmin (Qs ) = Mlatt (Qs ) = Qs ∩ A ∩ Cae , Madd (Qs1 ) = Mmax (Qs1 ) = Mmin (Qs1 ) = Mlatt (Qs1 ) = Qs1 ∩ A ∩ Cae and Madd (Qs2 ) = Mmax (Qs2 ) = Mmin (Qs2 ) = Mlatt (Qs2 ) = Qs2 ∩ A ∩ Cae . Let Tae be the family of all sets A ∈ Td such that ℓ(A \ Int(A)) = 0. The family Tae is also topology. Let MQ denote the family of all functions with this property: if f is not Tae -continuous at x ∈ R (where f is considered as the application from R equipped with the topology Tae ) then f (x) = 0 and du ({t ∈ R; f (t) = 0}, x) > 0. Theorem 3.10 ([55]). Mmult (Qs ) = Qs ∩ MQ , Mmult (Qs1 ) = Qs1 ∩ MQ and Mmult (Qs2 ) = Qs2 ∩ MQ . The results were later extended for functions f : Rm → R by E. Stro´ nska in [54].

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4. Darboux quasicontinuous and similar functions Darboux quasicontinuous functions Maximal classes for functions both Darboux and quasicontinuous were characterized by T. Natkaniec. Theorem 4.1 ([41]). Madd (D ∩ Q) = Mmult (D ∩ Q) = Const, Mmax (D ∩ Q) = D ∩ usc, Mmin (D ∩ Q) = D ∩ lsc and Mlatt (D ∩ Q) = C.

Darboux cliquish functions A function f : X → R is said to be cliquish if for each point x and for each neighbourhood U of x and each ε > 0 there is an open nonempty set G ⊂ U such that |f (y) − f (z)| < ε for all y, z ∈ G. A function f defined on R is cliquish if the set of continuity points of f is dense in R. Denote the family of cliquish functions by Cliq. The family Cliq is closed under addition and multiplication, so maximal families are obvious. Further, let Bα denote the family of all functions in Baire class α and L the family of all measurable functions. Maximal classes for Darboux cliquish functions were investigated by A. Maliszewski. A function f is in honorary Baire class two if there is a function g in Baire class one which equals f for all but countably many arguments. Denote the family of all honorary Baire class two functions by B2h . The maximal additive class for Darboux honorary Baire class two functions was characterized by I. Pokorn´ y in [47], the maximal multiplicative class by A. Maliszewski in [33]. Theorem 4.2 ([47]). Madd (B2h ) = Const. Theorem 4.3 ([33]). Let α > 2. Then, Madd (D ∩ Cliq) = Madd (L ∩ D ∩ Cliq) = Madd (D ∩ Cliq ∩ Bα ) = Madd (D ∩ Bα ) = Const. Theorem 4.4 ([33]). Let α > 2. Then, Mmult (D ∩ Cliq) = Mmult (L ∩ D ∩ Cliq) = Mmult (D ∩ Cliq ∩ Bα ) = Mmult (D ∩ Bα ) = Mmult (D ∩ B2h ) = Const.

Darboux almost continuous (Stallings) functions T. Natkaniec characterized maximal additive, multiplicative and lattice-like classes for these families. A problem to characterize the maximal classes with respect to maximum (minimum) remains open. Theorem 4.5 ([41]). Madd (ACS ∩ Q) = Mlatt (ACS ∩ Q) = C and Mmult (ACS ∩ Q) = M.

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Connectivity quasicontinuous functions Similarly, maximal additive, multiplicative and lattice-like families were characterized by T. Natkaniec. And, similarly, a problem of characterizing the maximal maximum (minimum) class is open. Theorem 4.6 ([41]). Madd (Con∩Q) = Mlatt (Con∩Q) = C and Mmult (Con∩Q) = M. ´ atkowski and extra strong Swi¸ ´ atkowski functions Strong Swi¸ ´ atkowski function if whenever a < b and A function f : R → R is a strong Swi¸ c is between f (a) and f (b), there is x ∈ (a, b) ∩ C(f ). A function f is an extra ´ atkowski function if f ([a, b]) = f ([a, b] ∩ C(f )) for all a < b. Let Ss destrong Swi¸ ´ atkowski functions and Ses the family of all extra note the family of all strong Swi¸ ´ atkowski functions. Evidently, Ses ⊂ Ss ⊂ D ∩ Q. Maximal families were strong Swi¸ characterized by P. Szczuka. Theorem 4.7 ([56]). Madd (Ss ) = Madd (Ses ) = Mmult (Ss ) = Mmult (Ses ) = Mmax (Ss ) = Mmax (Ses ) = Mmin (Ss ) = Mmin (Ses ) = Mlatt (Ss ) = Mlatt (Ses ) = Const.

5. Other generalizations of continuity Functions with closed graph A function f : X → R has closed graph if the graph of f is a closed subset of X × R. Let U denote the family of functions with closed graph. Maximal additive and multiplicative families were characterized by R. Menkyna. Theorem 5.1 ([36]). Let X be a topological space. Then, Madd (U) = C. Theorem 5.2 ([36]). Let X be a locally compact normal topological space. Then, Mmult (U) = {f ∈ C; f −1 (0) is an open set }.

Quasicontinuous functions with closed graph Maximal families for these families were characterized by W. Sieg. Theorem 5.3 ([51]). Let X = R. Then, Madd (Q ∩ U) = C, Mmult (Q ∩ U) = {f ∈ C; f −1 (0) is an open set }, and Mmax (Q ∩ U) = Mmin (Q ∩ U) = Mlatt (Q ∩ U) = ∅. It remains open to question whether this theorem holds for an arbitrary topological space.

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Graph continuous A function f : X → R is graph continuous if there exists a continuous function g : X → R such that the closure of the graph of f contains the graph of f . Let Gr be the family of graph continuous functions. Maximal families were characterized by K. Sak´alov´a. Theorem 5.4 ([50]). Let X be a connected Hausdorff topological space. Then, Madd (Gr) = Gr ∩ Cliq.

Simply continuous functions A function f : X → R is simply continuous if for each open set V in R, the set f −1 (V ) is the union of an open set and a nowhere dense set in X ([5]). Let S denote the family of all simply continuous functions. We have Q ⊂ S and, if X isSBaire, S ⊂ Cliq. Further, let ConstG be the family of all functions for which the set G(f ) is dense in X, where G(f ) = {G ⊂ X; G is open and f is constant on G}. (Therefore, for X = R we have the Jastrz¸ebski family ConstI .) Theorem 5.5 ([7]). Let X be a Baire space such that the family of all connected open sets is a π-base for X and there is a dense set of first category in X. Then, Madd (S) = Mmult (S) = ConstG . This theorem does not hold for an arbitrary topological space X. For maxima and minima we have Theorem 5.6 ([6]). Assume that X is a topological space with the following property: S (*) if (Xn ) is a partition of X such that n∈M Xn is simply open for each M ⊂ N, and G is a nonempty open set in X, then G ∩ Int Xn 6= ∅ for some n ∈ N. Then Mmax (S) = Mmin (S) = Mlatt (S) = S. If X is either a Baire space, or has a locally countable π-base, then it possesses the property (*), however, there are topological spaces which are neither Baire nor have countable π-base, but still possess the property (*). Moreover, there are topological spaces which do not satisfy condition (*). It is an open problem whether this theorem is true for an arbitrary topological space.

Almost continuous functions (Husain) A function f : X → R is precontinuous or almost continuous (in the sense of Husain) if for each point x ∈ X and for every ε > 0 we have x ∈ Int(Cl(f −1 (f (x) − ε, f (x) + ε))). Let ACH denote the family of all functions almost continuous in the sense of Husain. The maximal families were characterized by Z. Grande. Theorem 5.7 ([16]). Let X be a topological space. Then, Madd (ACH) = Mmult (ACH) = Mmax (ACH) = Mmin (ACH) = Mlatt (ACH) = C.

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ρ-upper continuous and similar functions A function f : R → R is 1-upper continuous if for every x ∈ R there exists a measurable set E containing x such that du (E, x) = 1 and the restriction f ↾ E is continuous at x. For 0 < ρ < 1, a function f is ρ-upper continuous if there is a measurable set E containing x such that d(E, x) > ρ and the restriction f ↾ E is continuous at x. Let C1 denote the family of all 1-upper continuous functions and Cρ denote the family of all ρ-upper continuous functions. Maximal classes for families C1 and Cρ were investigated by S. Kowalczyk and K. Nowakowska. Theorem 5.8 ([31]). Let 0 < ρ < 1. Then Madd (Cρ ) = A. The family A is only a proper subset of Madd (C1 ). A measurable set E is called sparse at x ∈ R if for every measurable set F ⊂ R, if du (F, x) < 1, then du (E ∪F, x) < 1 ([12]). A function f : R → R is T ∗ -continuous if for each x ∈ R and for each ε > 0, the complement of the set {y ∈ R; |f (y) − f (x)| < ε} is sparse at x. Let CT ∗ denote the family of all T ∗ -continuous functions. Theorem 5.9 ([31]). Madd (C1 ) = CT ∗ . For 0 < ρ < 1, let Zρ be the family of all measurable functions f : R → R such that for each x ∈ Dap (f ) we have f (x) = 0, and, for each measurable set E such that f −1 (0) ⊂ E and du (E, x) > ρ, we have lim du (E ∩{y ∈ R; |f (y)| < ε}, x) > ρ. Let Z1 ε→0+

be the family of all measurable functions such that for each point x at which f is not T ∗ -continuous we have f (x) = 0 and, for each measurable set E such that f −1 (0) ⊂ E and du (E, x) = 1 and for each ε > 0 we have du (E ∩ {y ∈ R; |f (y)| < ε}, x) = 1. The family A is a proper subset of Zρ and the family CT ∗ is a proper subset of Z1 . Theorem 5.10 ([31]). Mmult (C1 ) = Z1 and Mmult (Cρ ) = Zρ . Similar functions are defined in [45]. Let 0 < λ 6 ρ < 1. A function f : R → R is called [λ, ρ]-continuous if for each x ∈ R there exists a measurable set E containing x such that dl (E, x) > λ, du (E, x) > ρ and f ↾ E is continuous at x. Let C[λ,ρ] be the family of all [λ, ρ]-continuous functions. Maximal additive families for these functions were characterized in [32]. Theorem 5.11 ([32]). Madd (C[λ,ρ] ) = A. For 0 < λ 6 ρ < 1, let Z[λ,ρ] be the family of all measurable functions f : R → R such that for each x ∈ Dap (f ) we have f (x) = 0, and for each measurable set E such that f −1 (0) ⊂ E, dl (E, x) > λ and du (E, x) > ρ we have lim+ dl (E ∩ {y ∈ ε→0

R; |f (y) − f (x)| < ε}, x) > λ and lim+ du (E ∩ {y ∈ R; |f (y) − f (x)| < ε}, x) > ρ. The ε→0

family A is a proper subset of Z[λ,ρ] . Theorem 5.12 ([32]). Mmult (C[λ,ρ] ) = Z[λ,ρ] . Theorem 5.13 ([32]). Mmax (C[λ,ρ] ) = Mmin (C[λ,ρ] ) = Mlatt (C[λ,ρ] ) = A. A function f : R → R is called [0]-continuous if for each x ∈ R there exists a measurable set E containing x such that dl (E, x) > 0 and f ↾ E is continuous at x. Let C[0] be the family of all [0]-continuous functions. For a function f , let T (f ) be the set of all x ∈ R such that for each measurable set E with dl (E, x) > 0

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we have lim+ dl (E ∩ {y ∈ R; |f (y) − f (x)| < ε}, x) > 0 ([30]). Let T[0] denote the ε→0

family of all functions for which T (f ) = R. Further, let W[0] be the family of all measurable functions f : R → R such that for each x ∈ / T (f ), we have f (x) = 0, and for each measurable set E such that f −1 (0) ⊂ E and dl (E, x) > 0, we have lim dl (E∩{y ∈ R; |f (y)−f (x)| < ε}, x) > 0. We have A ( T[0] ( CT ∗ and W[0] ( T[0] . ε→0+

Theorem 5.14 ([30]). Madd (C[0] ) = T[0] and Mmult (C[0] ) = W[0] . Acknowledgements. The paper was supported by Grant VEGA 2/0177/12 and APVV-0269-11.

Bibliography 1. Banaszewski D.: On some subclasses of DB1 functions. Problemy Mat. 13 (1993), 33–41. 2. Banaszewski K.: Algebraic properties of E-continuous functions. Real Anal. Exchange 18 (1992/93), 153–168 3. Banaszewski K.: On E-continuous functions. Real Anal. Exchange 21 (1995/96), 203–215. 4. Banaszewski K.: Algebraic properties of functions with the Cantor intermediate value property. Math. Slovaca 48 (1998), 173–185. 5. Biswas N.: On some mappings in topological spaces. Calcutta Math. Soc. 61 (1969), 127–135. 6. Bors´ık J.: Maxima and minima of simply continuous and quasicontinuous functions. Math. Slovaca 46 (1996), 261–268. 7. Bors´ık J.: Maximal additive and maximal multiplicative family for the class of simply continuous functions. Real Anal. Exchange 20 (1994-95), 204–211. 8. Bruckner A.M.: Differentiation of Real Functions. Springer-Verlag, Berlin 1978. 9. Bruckner A.M., Ceder J.: Darboux continuity. Jber. Deutsch. Math. Ver. 67 (1965), 93–117. 10. Ewert J., Lipski T.: Lower and upper quasicontinuous functions. Demonstratio Math. 16 (1983), 85–93. ˇ 11. Farkov´ a J.: About the maximum and the minimum of Darboux functions. Matemat. Cas. 21 (1971), 110–116. 12. Filipczak T.: On some abstract density topologies. Real Anal. Exchange 14 (1988-89), 140–166. 13. Fleissner R.: A note on Baire 1 Darboux function. Real Anal. Exchange 3 (1977-78), 104–106. 14. Gibson R.G., Natkaniec T.: Darboux like functions. Real Anal. Exchange 22 (1996-97), 671–687. 15. Gibson R.G., Roush F.: The Cantor intermediate value property. Topology Proc. 7 (1982), 55–62. 16. Grande Z.: Algebraic structures generated by almost continuous functions. Math. Slovaca 41 (1991), 231–239. 17. Grande Z.: On almost continuous additive functions. Math. Slovaca 46 (1996), 203–2011. 18. Grande Z.: On some special notions of approximate quasi-continuity. Real Anal. Exchange 24 (1998/99), 171–184. 19. Grande Z.: On the maximal additive and multilplicative families for the quasicontinuities of Piotrowski and Vallin. Real Anal. Exchange 32 (2006-2007), 511–518. 20. Grande Z.: On the maximal families for the class of strongly quasicontinuous functions. Real Anal. Exchange 20 (1994-95), 631–638. 21. Grande Z.: On the maximal multiplicative family for the class of quasicontinuous functions. Real Anal. Exchange 15 (1989-90), 437–441. 22. Grande Z., Soltysik L.: Some remarks on quasi-continuous real functions. Problemy Mat. 10 (1990), 79–86. 23. Husain T.: Almost continuous mappings. Prace Mat. 10 (1960), 1–7. 24. Jastrz¸ebski J.: Maximal additive families for some classes of Darboux functions. Real. Anal. Exchange 13 (1987-88), 351–354. ´ Mat-Fiz 25. Jastrz¸ebski J., J¸edrzejewski J.M.: Functionally connected functions. Zesz. Nauk. PSl., 48 (1986), 73–80.

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26. Jastrz¸ebski J., J¸edrzejewski J.M., Natkaniec T.: On some subclasses of Darboux functions. Fund Math. 138 (1991), 165–173. 27. Jastrz¸ebski J., Natkaniec T.: On sums and products of extendable functions. Real Anal. Exchange 24 (1998/99), 589–598. 28. Kempisty S.: Sur les fonctions quasicontinues. Fund. Math. 19 (1932), 184–197. 29. Kowalczyk S.: Compositions of ρ-upper continuous functions. Math. Slovaca (to appear) 30. Kowalczyk S., Nowakowska K.: A note on the [0]-lower continuous functions. Tatra Mt. Math. Publ. (submitted). 31. Kowalczyk S., Nowakowska K.: Maximal classes for ρ-upper continuous functions. J. Appl. Anal. 19 (2013), 69–89. 32. Kowalczyk S., Nowakowska K.: Maximal classes for the family of [λ, ρ]-continuous functions. Real Anal. Exchange 36 (2010-2011), 307–324. 33. Maliszewski A.: On the sums and the products of Darboux cliquish functions. Acta Math. Hung. 73 (1996), 87–95. 34. Maliszewski A.: The maximal class with respect to maximums for the family of almost continuous functions. Real Anal. Exchange 32 (2006-2007), 313–318. 35. Maximoff I.: Sur les fonctions ayant la propri´ et´ e de Darboux. Prace Mat.-Fyz. 43 (1936), 241– 265. 36. Menkyna R.: The maximal additive and multiplicative families for functions with closed graph. Acta Math. Univ. Comenianae 52-53 (1987), 149–153. 37. Menkyna R.: The maximal families with respect to the composition of functions with a closed graph. Acta Math. Univ. Comenianae 54-55 (1988), 185–190. 38. Miˇs´ık L.: Maximal additive and maximal multiplicative families for the family of all intervalDarboux Baire one functions. Real Anal. Exchange 5 (1980), 285–302. 39. Miˇs´ık L.: Maximal additive and maximal multiplicative families for the family of B-Darboux Baire one functions. Math. Slovaca 31 (1981), 405–415. ¨ 40. Miˇs´ık L.: Uber die Funktionen der ersten Klasse mit der Eigenschaft von Darboux. Mat.-fyz. ˇ cas. SAV 14 (1964), 44–48. 41. Natkaniec T.: On quasi-continuous functions having Darboux property. Math. Pannon. 3 (1992), 81–96. 42. Natkaniec T.: On the maximum and the minimum of quasi-continuous functions. Math. Slovaca 42 (1992), 103–110. 43. Natkaniec T.: Two remarks on almost continuous functions. Problemy Mat. 10 (1988), 71–78. 44. Neubrunn T.: Quasicontinuity. Real Anal. Exchange 14 (1988/89), 259–306. 45. Nowakowska K.: On a family of [λ, ρ]-continuous functions. Tatra Mt. Math. Publ. 44 (2009), 129–138. 46. Piotrowski Z., Vallin R.W.: Conditions which imply continuity. Real Anal. Exchange 29 (2003/04), 211–217. 47. Pokorn´ y I.: On Darboux functions in honorary Baire class two. Acta Math. Hungar. 58 (1991), 1–2. ¨ 48. Radakoviˇ c T.: Uber Darbouxsche und stetige funktionen. Monatsh. Math. Phys. 38 (1931), 117– 122. 49. Sak´ alov´ a K.: On the maximal multiplicative family for the class of graph continuous functions. Demonstratio Math. 32 (1999), 615–620. 50. Sak´ alov´ a K.: The maximal additive family for the class of graph continuous real functions. Tatra Mt. Math. Publ. 19 (2000), 183–186. 51. Sieg W.: Maximal classes for the family of quasi-continuous functions with closed graph. Demonstratio Math. 42 (2009), 41–45. 52. Stallings J.: Fixed point theorems for connectivity maps. Fund. Math. 47 (1959), 249–263. 53. Stro´ nska E.: Maximal families for the class of upper and lower semi-quasicontinuous functions. Real Anal. Exchange 27 (2001-2002), 599–608. 54. Stro´ nska E.: On the maximal families for some classes of strongly quasicontinuous functions on Rm . Real Anal. Exchange 32 (2006-2007), 3–18. 55. Stro´ nska E.: On the maximal families for some special classes of strongly quasi-continuous functions. Real Anal. Exchange 23 (1997-98), 743–752. ´ atkowski functions. Real Anal. Exchange 56. Szczuka P.: Maximal classes for the family of strong Swi¸ 28 (2002-2003), 429–437.

Permutations preserving the convergence or the sum of series – a survey Roman Witula

Abstract. Presented paper is a survey of many authors’ achievements in the research subject matter concerning the permutations preserving the convergence or the sum of series and the algebraic properties of the families of such permutations. The convergence classes of divergent permutations will be also discussed. This survey has been treated widely, however not exhaustive. Keywords: convergent permutation, divergent permutation, convergence class. 2010 Mathematics Subject Classification: 40A05, 05A99.

1. Introduction In classical Riemann Derangement Theorem about rearrangement of the conditionally convergent series the series is fixed and the rearranged series vary. In case when we fix the permutation p on N and we variate the selection of conditionally convergent series a number of new problems appear, including the problem concerning the form of sets of limit points of the series rearranged by p, which is a dual issue to the problem described in the above Riemann Derangement Theorem. This new problem was solved by Nash-Williams and White [37] just in 1999. Simultaneously we note that many special cases of this problem were discovered earlier [29, 66, 63]. Some other problems turned out to be interesting and essential as well, for example the problem of combinatoric description of the convergent (divergent) permutations or the permutations preserving the sum. Discussion of such problems is the object of this paper, frame of which was given by my habilitation thesis [72]. Remark 1.1. This survey does not contain any constructions of permutations on N distinguished and discussed in this paper, which can make the considered issues quite R. Witula Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 169–190. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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rough for the Readers not initiated into the subject matter. Remedy for this technical problem can be the series of papers, perfectly supplementing this deficiency, which I recommend to all the interested Readers (see [44, 52, 53, 60, 73, 79]).

2. Basic ideas and distinguished sets of permutations Family of all permutations on N = {1, 2, 3, ...}, it means the bijections of the set of natural numbers on itself, will be denoted by P. Permutation p ∈ P preserving the convergence of all rearranged by p convergent series of real terms will be called the convergent permutation. In other words, permutation P P p ∈ P is convergent if for each convergent series an of real terms, the series ap(n) rearranged by permutation p is convergent as well. Family of all convergent permutations will be denoted by symbol C. For the contrast, permutations belonging to set D := P \ C will be called the divergent permutations. Thus, P permutation p ∈ P is the divergent permutation if there exists a convergent series an of real terms which is rearranged by permutation p P into a divergent series ap(n) . Let us also introduce, after Kronrod [34], the following families of permutations (I have distinguished these families independently in 90’s, without knowing the Kronrod’s work): – CC and CD are the subfamilies of C composed of permutations p called the twosided or one-sided convergent permutations in dependence on that, whether the inverse permutation p−1 is convergent or divergent, respectively, – DC and DD are, by analogy to CC and CD, the subfamilies of D of permutations p called the two-sided or one-sided divergent permutations in dependence on that, whether permutation p−1 is convergent or divergent. Moreover, we indicate permutations p ∈ PP preserving the sum, P it means satisfying the condition: for each convergent real series an P if the seriesP ap(n) rearranged by p is convergent, then it preservesPthe sum, that is ap(n) = an . Henceforward we will denote by the same symbol an the series, it means the appropriate sequence of partial sums as well as its sum, if only the given series is convergent. The proper interpretation will depend on the context of discussion. Family of permutations preserving the sum will be denoted by symbol S. Permutations belonging to set I := P \ S will be called the substantially singular permutations. Thus, P each substantially singular permutation p ∈ P rearranges some convergent series an of real P terms into P series P ap(n) convergent as well, but of changed value of the sum, it means ap(n) 6= an . We distinguish also some important for further discussion subfamily F of family S. We say that permutation p ∈ P belongs to F, if there exists a finite partition N1 , N2 , . . . , Nn(p) of the set of natural numbers such that the restrictions p Ni , i = 1, 2, . . . , n(p), are the increasing maps. One more auxiliary idea will be essential for further discussion. We say that the finite and nonempty set A ⊂ N is a union of k mutually separated intervals, in short: k msi, if there exists partition I1 , I2 , . . . , Ik of set A composed from k intervals (segments) of natural numbers, it means from the sets of successive natural numbers such that dist(Ii , Ij ) > 2 for any i, j = 1, 2, . . . , k, i 6= j. For convenience we will also say the given set A ⊂ N or a few of sets A1 , . . . , Ar ⊂ N

Permutations preserving the convergence or the sum of series – a survey

171

are the unions of at most n msi for some n ∈ N, if each of them is a union of k(A) msi or, respectively, of k(A1 ) msi, k(A2 ) msi,. . ., k(Ar ) msi and all these numbers are not greater than n. Number n plays here a role of majorant of the given set of numbers. We say that the given sequence {An } of finite and nonempty subsets of N is increasing if An < An+1 for every n ∈ N and we write A < B if a < b for any a ∈ A, b ∈ B, where A, B ⊂ N are the nonempty sets. Notation A ⊂ B will be used only in case of strict inclusion.

3. Selected algebraic properties of the distinguished sets of permutations We say that family A ⊂ P is algebraically small if P \ G(A) 6= ∅, where G(A) is a group of permutations generated by A. Next, we say that family A ⊂ P is algebraically big if A ◦ A = P, where operation ◦ of composition of two nonempty sets of permutations A and B is defined as follows A ◦ B := {p ◦ q : p ∈ A and q ∈ B}, whereas p ◦ q(n) := p(q(n)), n ∈ N. Families A ⊂ P which are neither algebraically small nor algebraically big will be called as the sets algebraically medial. There exist the sets of generators of P which are the sets algebraically medial [70]. Let us also note that – family C is algebraically small (Pleasants [43, 44]), – families S and I are algebraically big (Kronrod [34] Pleasants [43], Witula [68], in fact we know many algebraically big subsets of family S – some of them will be presented in the further parts of this survey), – families DC i CD are semigroups (Witula [65, 74]), more precisely we have Φ◦Φ = Φ for each Φ ∈ {DC, CD, C, C−1 }, – family DD is algebraically big (Witula [65]), – I proved that (see [65, 74]): DC ◦ DD = DD ◦ DC = DC ∪ DD = D and CD ◦ DD = DD ◦ CD = CD ∪ DD = D−1 . Also the following equalities1 DDk ◦ DDl = P are satisfied for any k, l ∈ N, k, l > 2 (Witula [70]), where Ak := {pk : p ∈ A} for any nonempty A ⊂ P and pk denotes the k-fold composition of p with itself (thus, symbol of type DDk will denote the kth ∞ S power of set DD, it means (DD)k ). Moreover we have DD \ DDk 6= ∅, but it is unknown whether the equality

∞ S

k=2

k

DD = P holds. Family CC is a group with regard

k=1 1 Remark. Identified here the algebraically big sets S and DD I can decompose into countably many algebraically big subsets. However I do not know whether it is true for each algebraically big subset of family P.

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to composition of mappings and plays the role of unity with regard to operation ◦ for many from among sets discussed here. More precisely, we have [68, 74]: CC ◦ A = A ◦ CC = A for A ∈ {CC, C, C−1 , CD, D, DC, DD, S0 } (definition of family S0 will be given on page 175). One can easily verify that family CC is the maximal, with regard to inclusion, group of permutations included in C. Next, from the fact that CD is the semigroup, it results in particular that set CC contains all the torsion elements of semigroup C. The family τC of torsion elements of this semigroup is a normal quotient of group CC and, which is more, of the infinite index (more precisely, index |CC : τC | = c). We have similar result for the subgroup τ of torsion elements of group P. Certainly τC ⊂ τ and τ ∩ DD 6= ∅. Furthermore, we have here a number of unsolved problems like, for instance, whether τC (τ respectively) is the maximal, with regard to inclusion, normal subgroup in CC (in P respectively). E.H. Johnston [31] introduced interesting subgroup R of group CC, composed from permutations p ∈ P satisfying condition: sup{card(I \ p(I)) : I ⊂ N is an interval} < ∞.2 One can show that R is not the normal subgroup of group CC (Johnston proved this fact only for group P). One can also prove that CC is not the normal subgroup in semigroup C (it means, there exists τ ∈ C such that τ CCτ −1 ∩ D 6= ∅) – G.S. Stoller [56]. I have generalized this fact importantly by proving relations (see [71]): (pCCq) ∩ DD 6= ∅ and (qCCp) ∩ DD 6= ∅ for any p ∈ (CD ∪ DD) and q ∈ D. Additionally, I have proven that for any permutation p ∈ P we have pDp−1 ⊆ D ⇔ p ∈ CC

and pDDp−1 ⊆ DD ⇔ p ∈ CC,

where if p ∈ CC then the above inclusions turn into equalities (see [71, Th. 2.7]).3

4. Selected characterizations of convergent and divergent permutations A number of various characterizations of convergent, divergent and other permutations are known – see among others [22, 26, 35, 43, 49, 51, 52, 53, 57]. Let me present few of them: – p ∈ C if and only if there exists constant k = k(p) ∈ N such that the set p([1, n] ∩ N) is a union of at most k msi for each n ∈ N (in other words, if 2 Remark. R´ oz˙ a´ nski et al. [46] have completed the description of elements of subgroup R on the basis of the following equivalence relations ̺ defined on family b+ of the bounded sequences of positive P real numbers: {an }̺{bn } ⇔ for every increasing sequence of positive integers {nk } the series ank P and bnk are simultaneously convergent or divergent.

3 Remark. Existence of divergent permutations (two-sided divergent permutations, respectively) in sets τ Rτ −1 , τ ∈ C, pRq, p ∈ CD and q ∈ DC, remains problematic, where R is the group introduced by Johnston.

Permutations preserving the convergence or the sum of series – a survey

C

CC

S

CC ∩ F

G

D(1) C2 ∪ D 2 S0 \ G F ⊂ C2 ⇒ DD ∩ G 6= ∅ D2 F \ D2 6= ∅ ⇒ C2 \ D2 6= ∅ D(1) \ G

CD CD ∩ F

PD

S0

173

DC D(1) ∩ DC F ∩ DC

DD I D(1) ∩ DD F ∩ DD F ∩ I

All sets displayed in the table are nonempty. In some blocks with the aid of smaller types there are indicated selected subsets of sets presented at the top of these blocks. Case when the vertical side of some block is included in the vertical side of another block denotes the appropriate inclusion between sets displayed at the top of these blocks. Inclusions S0 ⊂ G and S ⊂ G are the hypotheses formulated by me.

lim sup t(p, n) 6 k, that is, when lim sup t(p, n) < ∞, where t(p, n) denotes the n→∞

n→∞

number of msi partitioning set p([1, n])), R.P. Agnew (1955) [2], N. Bourbaki (1951) [9], – p ∈ D ⇔ lim sup t(p, n) = ∞ (characterization dual to the previous one), n→∞

– p ∈ D if and only if for any r, s ∈ N there exists the increasing sequence of natural numbers {xn }2r n=1 spliced by p, such that x1 > s and each of two following sequences {p(xn )}rn=1 and {p(xn+r )}rn=1 is monotonic (one can also assume that min{p(xn )}2r n=1 > s), Witula (1995) [80, 66]. Remark 4.1. An increasing sequence {xn }2r n=1 of positive integers is spliced by p if the increasing sequences composed from elements of sets {p(xn ) : n = 1, . . . , r} and {p(xn+r ) : n = 1, . . . , r} are alternating. We say that increasing sequences of natural numbers: y1 , y2 , . . . , yr and z1 , z2 , . . . , zr are alternating if y1 < z1 < y2 < z2 < . . . < yr < zr or z1 < y1 < z2 < y2 < . . . < zr < yr . It should be emphasized that the idea of splicing is connected with any divergent permutation which is justified by sequences, assigned to these permutations, conglomerating intervals and growing the numbers of intervals (see Section 7). Moreover, by using properties of these sequences one can prove the characterization of divergent permutation presented in here (see Remark 7.7), in the way alternative to proofs from papers [80, 66]. – p ∈ C if and only if there exists r ∈ N such that for each increasing sequence {xn }2N n=1 of natural numbers spliced by p the inequality N 6 r holds (this characterization is dual to the previous characterization restricted only to the condition of splicing the sequence {xn }2r n=1 by p). First of the above four characterizations introduced by Bourbaki and Agnew has been generalized by me into functions f : N → N preserving convergence of the convergent

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real series rearranged by them4 (see [64]). Functions of that kind will be shortly called the convergent functions and their description, discovered by me, is the following. Function f : N → N is the convergent function if and only if there exists a natural number t = t(f ) such that for each interval I ⊂ N there exists a partition I1 , I2 , . . . , Is of interval I possessing the following properties: 1) s 6 t, 2) all restrictions f Ii , i = 1, 2, . . . , s, are injections, 3) each of sets f (Ii ), i = 1, 2, . . . , s, is a union of at most t msi. Remark 4.2. The above description implies that the convergent functions, and in particular the convergent permutations, with respect to the series in any normed space (and even in any linear-topological space) are functions (permutations, respectively) preserving the Cauchy condition. It is important, among others, since the convergent permutations preserve also the sums of convergent series rearranged by them, whereas the convergent functions, which are not the permutations of the set of natural numbers, do not possess this property any more! Remark 4.3. Characterization of the convergent functions, introduced by me, and the above remark can be transferred without changes onto the vector series in the normed spaces, as well as generally in the linear-topological spaces, over the fields of characteristic zero. Remark 4.4. Professor Lech Drewnowski in paper [18, Lemmas 4.4, 4.5 and Prop. 4.6] discusses many equivalent characterizations of convergent functions and, what is the most essential, he proves that they characterize the functions preserving convergence of the series and, independently, the functions transforming the convergent series (bounded series, respectively) into the bounded series and the functions transforming the Cauchy sequences into the Cauchy sequences, etc. in the linear-topological spaces and even in the normed F -spaces. Remark 4.5. M.A. Sarig¨ ol [51] has presented the characterization of permutations p ∈ P preserving the property of bounded variation of scalar sequences (these are exactly the permutations, the inverse permutations of which are convergent). Drewnowski in paper [18] has generalized this characterization onto functions f : N → N and in case of permutations also onto the sequences of bounded variation in the linear-topological spaces and in the normed F -spaces. In particular, I have noticed here an intriguing result (not recorded either in paper [51] or in paper [18]): if p ∈ P and lim sup |p−1 (n + 1) − p−1 (n)| < ∞, then p ∈ C (see [18, Prop. 6.3]). n→∞

5. Family S0 of permutations preserving the sum Till the last year I was convinced that the combinatoric characterization of permutations preserving the sum was still unknown. It was the reason for preparing paper [68] in which I have distinguished the following family S0 of permutations 4 Function f : N → N will be called the function preserving convergence of the (real) series if for P P each convergent series an of real terms the series af (n) is convergent as well.

Permutations preserving the convergence or the sum of series – a survey

175

on N. We say that permutation p ∈ P belongs to S0 if there exists a natural number k = k(p) such that for each n ∈ N the nonempty finite sets An , Bn ⊂ N exist and satisfy the conditions: 1) p(An ) = Bn , 2) [1, n] ⊂ An , 3) each of sets An and Bn is a union of at most k msi. By using the Cauchy condition one can easily verify that S0 ⊆ S and I formulated the conjecture that the equality S0 = S holds, however I could not prove it (in next subsection it will appear that in 1999, and this date is not a mistake, this conjecture was proved true by Nash-Williams and White – see [37]). I based this conjecture on the following reason. All the known till now examples of permutations preserving the sum are the permutations belonging to S0 . One can find here the really nontrivial constructions, like for example given in paper [29] the construction of permutation p ∈ S0 such that lim t(p; n) = lim t(p−1 ; n) = ∞, (1) n→∞

n→∞

where for each n ∈ N symbol t(p; n) denotes the number of msi partitioning set p([1, n]). Remark 5.1. Condition (1) is connected with one more interesting characterization, P this time an into P of permutations p ∈ P rearranging some convergent real series series ap(n) divergent to ∞. Hu and Wang in the mentioned paper [29] have proven that the necessary and sufficient condition, under which this fact is happening for given permutation p ∈ P, is that lim t(p; n) = ∞.5 n→∞

I have proven the above characterization of Hu-Wang independently in paper [66], and I am convinced that the proof given by me is more intelligible. In paper [68] I have proven many algebraic and combinatoric properties of family S0 . Let me begin with the condition guaranteeing that p ∈ S0 , namely: lim inf t(p; n) < ∞. It is not in the least the necessary condition, which results from n→∞

the previously given example by Hu-Wang (see condition (1)). I present now the collection of selected algebraic relations for family S0 (on the basis of Theorem 2.4 in paper [68]): −1 (i) S−1 =S 0 = S0 which is compatible with equality S (both equalities result from the definitions of families S0 and S, respectively).

Remark 5.2. If G ⊂ P is a group and G 6= P then (P \ G)−1 = P \ G, G ◦ (P \ G) = (P \ G) ◦ G = P \ G, but either S0 or S are not groups, since both of them are the algebraically big sets. (ii) C ∪ C−1 ⊂ S0 and S0 ∩ DD 6= ∅ (the comment is required for the second relation resulting easily from inclusion D(1) ⊂ S0 , where D(1) is the family of such permutations p ∈ D for which p([1, n] ∩ N) = [1, n] ∩ N for infinitely many n ∈ N, it means fulfilling condition 5 Despite of the appearances, constructing such permutation is not difficult – essential detail of this construction is described in item (v) of Lemma 7.1 in Section 7.

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lim inf t(p, n) = 1. Family D(1) is algebraically big, which is already proven by n→∞

G.S. Stoller [56]. Additionally I have proven that D(1) ◦ I = I ◦ D(1) = P – Theorem 2.7 in paper [68]). (iii) C ◦ S0 = S0 ◦ C−1 = S0 and (CD ◦ S0 ) ∪ (S0 ◦ DC) ⊂ S0 ⊂ (DC ◦ S0 ) ∩ (S0 ◦ CD) (let us recall that family CC is unity with regard to operation ◦ for many of discussed here families of permutations. Bigger with regard to inclusion families C or C−1 , as it can be seen, satisfy the rule of one-sided unities, since we have C ◦ S = S ◦ C−1 = S,

CD ◦ C = CD, C−1 ◦ DC = DC,  where DC ∪ CD ⊂ C ◦ CD ∩ C−1 ◦ DC . Moreover, I have proven that S ◦ CC = S and I ◦ CC = I). (iv) CD ◦ S0 ◦ DC ⊂ S0 ⊂ DC ◦ S0 ◦ CD (similarly like the second one from among relations (iii), also the above relation shows well the subtle set-theoretic difference between the occuring sets, whereas the qualitative nature of these differences needs still to be investigated), (v) D2 ⊂ S0 and C2 ∩ I 6= ∅, which implies that D3 ∩ I 6= ∅, where C1 := CD, D1 := DC, Ck+1 := Dk ◦ C, Dk+1 := Ck ◦ D for each k ∈ N. It is known that Ck ⊆ Ck+1 , Dk ⊆ Dk+1 Ck ⊆ Dk+1 ∩ Ck+1 for each k ∈ N and, what is S , Dk ∪S the most important, Ck = Dk = G, where, let us recall, G is the group of k∈N

k∈N

permutations generated by family C (see [68]). We already know that C2 \D2 6= ∅ (see [68, 71]]), however we do not know whether family C2 \ S0 is algebraically big. I have proven, independently of Kronrod, that set I = D \ S is algebraically big [68, Theorem 2.7]. Still unknown is also the answer to question whether there exists k ∈ N such that Dk = Dk+1 or Ck = Ck+1 ( let us note that such equality implies that, respectively, Dk = Dk+l = Ck+l or Ck = Ck+l = Dk+l for each l ∈ N). I have proven as well that family P \ G is algebraically big and D(1) \ G 6= ∅6 (see [77]). In particular, it implies that {g} ◦ (P \ G) = (P \ G) ◦ {g} = P \ G for any permutation g ∈ G. These equalities hold also for any group G ⊂ P (taken in place of group G), such that P \ G is algebraically big. In reference to item (i) −1 −1 let us additionally notice that equalities C−1 2k = C2k , D2k = D2k , C2k−1 = D2k−1 and D−1 2k−1 = C2k hold for every k ∈ N. Another important fact should be also emphasized, namely, that permutations p ∈ S are the only functions f : N → N preserving the sum of series P rearranged by them. P In other words, if for the given convergent real series an the P P f -rearranged series af (n) is convergent as well and af (n) = an , then function f is a permutation. Proof of this fact, by contradiction of thesis, can be immediately generalized onto the vector series in any nontrivial normed space (one should discuss separately the spaces over the fields of characteristic zero and over the fields of characteristic different than zero – see [64]).

6 In paper [71] the example (Example 1) of permutation p ∈ D(1) \ (C ∪ D ) is given, which is the 2 2 slightly weaker relation. Whereas, it is noticed there (Remark 2.4) that card(D(1) \ (C2 ∪ D2 )) = c.

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Nash-Williams and White’s exciting papers. Equality S0 = S holds Last year (more precisely in October 2013), when I was preparing my lecture for presenting my habilitation thesis in the L´ od´z University, I read again paper [37] by Nash-Williams and White concerning the form of cluster set of the sequence of partial P sums of ap(n) , i.e. the convergent real series rearranged by a given permutation p ∈ P. I noticed almost immediately (which surprised me extremely) that this paper contains the positive solution of my conjecture claiming that relation S0 = S is true. Nash-Williams and White proved the following theorem. Theorem 5.3. Permutation p ∈ P preserves the sum of rearranged series if and only if the width of p is finite (the width of permutation p ∈ P is a fundamental conception introduced by Nash-Williams and White in paper [37]) or, equivalently, if and only if p ∈ S0 . We note that a proper part of this fact, i.e. the implication: if the width of p is finite then p ∈ S0 is formulated in Proposition 2.2 of [37]. Let us emphasize that NashWilliams and White in paper [37] do not recall the definition or even the conception of permutations preserving the sum which was certainly the reason of not noticing, by myself and other readers, the characterization of permutations preserving the sum described in Section 5. We also note that next papers [38, 39], published by the same authors, concern the generalizations of theorems contained in [37] onto the series in finitely dimensional spaces. All three papers [37, 38, 39] made by Nash-Williams and White are, with no doubts, the very important events in the discussed subject matter.

6. Family of permutations decomposable into the finite sum of increasing maps Subject matter of my research was also the family of permutations F ⊂ P, original against a background of previously discussed families, composed from permutations p ∈ P for which there exists the finite partition N1 , N2 , . . . , Nn(p) of the set of natural numbers such that p Ni is the increasing map for each i = 1, 2, . . . , n(p). Definition of family F implies also another, dual characterization of permutations belonging to F. So, permutation p ∈ P belongs to F if and only if the value of k(p) := sup{card(A) : ∅ 6= A ⊂ N and p A is decreasing map} is finite (see [73, Theorem 7]). Certainly F is the group of permutations (see [73, Theorem 2]). However, the connection between family F and the convergent or divergent permutations is not noticeable directly, since it is located at the level of superpositions of convergent permutations, it means within families C2 = DC ◦ CD and D2 = CD ◦ DC. One can prove that F ⊂ C2 which implies also that DC ◦ F ◦ CD ⊂ C2 (see [73, Conclusion 1]), and one can give the example of permutation p ∈ F \ D2 (see [73,

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Example 1]). Inclusion F ⊂ C2 results from the definitely more generally formulated theorem (see [73, Theorem 4]): Theorem 6.1. Let p ∈ F and let l(p) denote the smallest natural number such that: for each, sufficiently big n ∈ N, there exists the partition N(n) of set {k ∈ N : k > n} with l(p) = card N(n), such that p N is the increasing map for each N ∈ N(n). Then there exists the convergent permutation q satisfying conditions: qp ∈ C, c∞ (q) 6 4l(p) + 1 and c∞ (qp) 6 2l(p) + 1, where c∞ (p) = lim

n→∞

 sup{c(p, I) : I ⊂ N is the bounded interval such that min I > n} ,

whereas c(p, I) denotes the number of mutually separated intervals creating the partition of set p(I). Coefficients, appearing here, can be successfully applied for more subtle formulation of many relations, also these ones previously given by me. Moreover, I would like to emphasize that within many of discussed here families of permutations, the elements of family F are spred out almost everywhere, in such sense that ([73, Theorems 3 and 6]): – F ∩ A 6= ∅ for each of four two-sided families A ∈ {CC, CD, DC, DD}, and even for A = I (see Remark 3, [73]), – A \ F 6= ∅ for each A ∈ {CC, CD, D(1) ∩ DC, D(1) ∩ DD, (D(2) \ D(1)) ∩ DC, (D(2) \ D(1))∩DD, D(1)∩C2 ∩D2 }, where D(2) := {p ∈ D : lim inf t(p, n) = 2}. However, n→∞

I do not know whether F \ A 6= ∅ for each one of sets A mentioned above. Family F can be also used for strengthening the presentation of the Riemann Derangement Theorem (by the way this result comes from the Riemann’s post¨ doctoral dissertation entitled “Uber die Darstellbarkeit einer Funktion durch eine trigonometrische Reihe”, see [25, p. 232]). In paper [73] I have presented two of such theorems (Theorems 9 and 10) generalizing the Kronrod’s results from paper [34] P (see also [45, 21]). So, I have proven that for any conditionally convergent series an of real terms and closed interval I ⊂ R ∪ {±∞} there exist permutations p ∈ F and P P q ∈ P \ F, such that the set of limit points of every series ap(n) and aq(n) is equal to I. In paper P [67] similar result can be found but for permutation p ∈ DD, and in case when an ∈ I or when I = [α, ∞] or I = [−∞, β], α < ∞, β > −∞, α, β ∈ R ∪ {±∞}, for permutation q ∈ DC with P additional condition c(q −1 ) 6 5 −1 (which may be strengthened to c(q ) 6 3, when an ∈ I). Definition of number c(ϕ), ϕ ∈ C, is given on page 182. Let us noticed that c∞ (ϕ) 6 c(ϕ), ϕ ∈ C. We note also that J.H. Smith in [55] considers the selection of permutations p in the Riemann Derangement Theorem with the predetermined decomposition of p into cycles.

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7. Sequences conglomerating intervals and growing the numbers of intervals Let p ∈ P and let σ(t(p, ◦)) denote the set of limit points of sequence {t(p, n)}∞ n=1 with regard to topology of the 2-points compactification of set R with the standard topology, in other words σ(t(p, ◦)) is derivative of this sequence. Then the following relations hold true. Lemma 7.1. (i) {t(p, n)}∞ n=1 ⊂ N and t(p, n + 1) − t(p, n) ∈ {−1, 0, 1}, for every n ∈ N, (ii) set σ(t(p, ◦)) is the closed interval ⊂ N ∪ {∞}, more precisely we have σ(t(p, ◦)) = [lim inf t(p, n), lim sup t(p, n)], n→∞

n→∞

(iii) if card(σ(t(p, ◦))) = 1 then either p is the almost identity permutation on N or σ(t(p, ◦)) = {∞}, that is lim t(p, n) = ∞, n→∞

(iv) 1 ∈ σ(t(p, ◦)) if and only if lim inf t(p, n) = 1, if we assume additionally that p n→∞

is divergent, then 1 ∈ σ(t(p, ◦)) if and only if p ∈ D(1), (v) if there exists an increasing sequence {xn }∞ n=1 of natural numbers such that lim sup(xn+1 − xn ) < ∞ and n→∞

lim (p(xn+1 ) − p(xn )) = ∞,

n→∞

then lim t(p, n) = lim t(p−1 , n) = ∞.

n→∞

n→∞

(vi) If A and B are any closed intervals of the form [k, l], [k, ∞] or [∞], where k, l ∈ N, k < l, then there exists permutation p ∈ P such that σ(t(p, ◦)) = A and σ(t(p−1 , ◦)) = B. Remark 7.2. Property (v) plus the combinatoric characterization of permutation p ∈ S0 enable to give easily the previously cited construction of permutation from paper by Hu and Wang [29]. The matter is just the idea of this construction. Let us now assign to permutation p two auxiliary sets (t(p, 0) := 0): U(p) := {u ∈ N : t(p, u) − t(p, u − 1) = 1} and V(p) := {v ∈ N : t(p, v) − t(p, v − 1) = −1}. Lemma 7.3 (see [66]). (i) If p ∈ D, then the sets U(p) and V(p) are infinite. More precisely, if at least one of these sets is finite then the other one is finite as well, whereas p is then the almost identity permutation. (p) (ii) If we denote by Ii,n , i = 1, 2, . . . , t(p, n) the sequence of msi creating the partition of set p([1, n]) for each n ∈ N then

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R. Witula (p)

(p)

card(p−1 (Ii,n ) ∩ U(p)) − card(p−1 (Ii,n ) ∩ V(p)) = 1 for each i, n ∈ N, i 6 t(p, n). (iii) card([1, n] ∩ U(p)) − card([1, n] ∩ V(p)) = t(p, n) for each n ∈ N. Let us assume additionally that permutation p ∈ P is not the almost identity permutation (in particular, it can be the divergent permutation). Then the sets U(p) and V(p) are infinite and the increasing sequences of all elements of sets p(U(p)) and ∞ p(V(p)) will be denoted by {un (p)}∞ n=1 and {vn (p)}n=1 , respectively, and called the sequences growing the numbers of intervals and conglomerating the intervals (adequately to their properties). Lemma 7.4 (see [66]). We have un (p) < vn (p) < un+1 (p)

p−1 (un (p)) < p−1 (vn (p))

and

for each n ∈ N. I have also proven that (see [66]): Theorem 7.5. If {xn }∞ n=1 is the increasing sequence of natural numbers then there exist permutations p ∈ DC and q ∈ DD satisfying conditions lim t(ϕ, n) = ∞

n→∞

and ϕ(U(ϕ)) = {x2n−1 }∞ n=1

and

ϕ(V(ϕ)) = {x2n }∞ n=1

for each ϕ ∈ {p, q}. Remark 7.6. The above theorem is the strengthened version of Theorem 5.3 from paper [66]. The proof is given in paper [75]. Remark 7.7. By using properties (ii) and (iii) from Lemma 7.3 we can prove the previously given characterizations of divergent permutations connected with the idea of splicing the sequences. For example, let p ∈ D and k ∈ N. There exists (p) n ∈ N such that t(p, n) > k. We select an element xi from each interval Ii,n for (p)

(p)

every i = 1, 2, . . . , t(p, n). Assuming that Ii,n < Ii+1,n , i = 1, 2, . . . , t(p, n) − 1, t(p,n)

we obtain that the sequence {xi }i=1 is increasing. We also select elements yi∗ ∈ (p) (p) (p) ∗ p−1 ((max Ii,n , min Ii+1,n )), i = 1, 2, . . . , t(p, n)−1 and yt(p,n) ∈ p−1 ((max It(p,n),n , ∞)). 2t(p,n)

Let {xi }i=1+t(p,n) be an increasing sequence composed from elements yi∗ , i = 2t(p,n)

1, 2, . . . , t(p, n). Certainly the sequence {xi }i=1 is spliced by p. Application of the Erd˝os-Szekeres Theorem7 [80] enables to assume additionally that sequences t(p,n) 2t(p,n) {p(xi )}i=1 and {p(xi )}i=1+t(p,n) are both monotonic. 7 Remark. In paper [80] several essential supplements for the Erd˝ os-Szekeres Theorem are also presented. For example, it has been revealed that the existence of increasing or decreasing subsequence composed from three successive elements of investigated sequence, it means the monotonic subsequence of the form {ak , ak+1 , ak+2 }, is of great importance for discussing problems of that kind. A part of discussed there problems can be described in the language of theory of “pattern avoiding permutations” (see [7, 11]).

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Remark P 7.8. Sets U(p) and V(p) can be also used for constructing the convergent real series an , the p-rearrangements of which possess the set of limit points determined in advance. For instance, in paper [66, Theorem 4.4] the following result is proven: Theorem 7.9. Let {pi }∞ lim t(pi , n) = ∞ for every i ∈ N. If permutai=1 ⊂ D and n→∞ tions pi , i ∈ N, possess the identical sequences growing the numbers of intervals and P conglomerating the intervals, then for each α ∈ R there exist the series a and n P bn of real numbers satisfying conditions: P P (1) an = bn = α, ∞ P (2) api (n) = ∞, for each i = 1, 2, . . ., n=1

(3) set of limit points of every series

∞ P

n=1

bpi (n) , i = 1, 2, . . . is equal to [α, ∞].

8. Relations between the convergence classes of permutations on N P Convergence class of permutation p ∈ P is the family P of all the convergent series an of real terms, such that the p-rearranged series ap(n) is convergent asPwell. Convergence class of given permutation p ∈ P will be denoted by symbol (p). P Certainly if p ∈ C then (p) is the family of all convergent real series. Therefore, from theoretical point of view, only the convergent classes of divergent permutations can be interesting. I have considered many various problems concerning this subject. For example, in paper [79] I have introduced the idea of strongly and weakly divergent permutations p ∈ D in dependence on that whether they fulfil the condition, respectively: lim t(p, n) = ∞,

n→∞

or

lim inf t(p, n) < ∞. n→∞

In other words, permutation p ∈ D is strongly divergent if it rearranges some convergent real series into the series divergent to ∞, whereas p ∈ D is weakly divergent if it belongs to some D(k), k ∈ N, where D(k) := {p ∈ D : lim inf t(p, n) = k}, it n→∞ P P means if for each an the sum of series an in the set of P convergent real series limitSpoints of ap(n)Scan be found. So, p ∈ D is weakly divergent if and only if p∈ D(k). Family D(k) has been introduced by Kronrod [34], however he did k∈N

k∈N

not distinguish families D(k), k ∈ N, separately. Let us also notice that the weakly divergent permutations are simultaneously the sum preserving permutations since S D(k) ⊂ S0 .8 k∈N

8

I prove in [63, Section 2] that if p ∈ D(k) for P some k ∈ N and an is a conditionally convergent series such that the set of limit points of series a is equal to [α, β] ⊂ R, then k(α − β) p(n) P P+ β 6 an 6 k(β − α) an or P+ α. Hence it follows that the intervals [α, β] with P k(β − α) + α < k(α − β) + β > an cannot be the sets of limit points of series ap(n) for any p ∈ D(k). We note also that Kronrod [34, Theorems 6, 6a, 7] shows that if p−1 is a weakly divergent permutation,

P

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Let me begin the problem concerning mutual relations between the convergence classes of strongly and weakly divergent permutations with the following fact. There exist the strongly divergent permutation p and the weakly divergent permutations q1 and q2 fulfilling relations X X X X (q1 ) = (p) and (q2 ) ⊂ (p), – see Example 2 in paper [79]. I have proven (see [79, Theorem 1]) that for each strongly P divergent P permutation p there exists a weakly divergent permutation q such that (q) ⊆ (p) and, under some additional assumptions about permutation p, there exist the weakly divergent permutations q1 and q2 fulfilling the relations as in the example given above. In paper [79] there is also proven that for each strongly divergent P permutation P p ∈ DC there exists a weakly divergent permutation q, such that (p) = (q). Moreover, an example of strongly divergent permutation p ∈ DC is given. This example essentially completes the example from paper [29], i.e. the strongly divergent permutation, the inverse permutation of which is strongly divergent as well. The next problem concerning the convergence classes is connected with a question about the possibility of restricting or expanding the convergence class of the given divergent permutation – it appears that it is always possible and not uniquely at all. I will precede the discussion by an example showing the subtleness of such operations. P In paperP [81] I have given the example of permutations p, P P Pq ∈ D(1) such that if (p)∪ (q) ⊆ (σ) then σ ∈ C. Simultaneously, set (p) ∩ (q) is the convergence class of some permutation from D(1). Analyzing the construction of these permutations p, q, I have noticed that one of the following three conditions: p, q ∈ DC or p ∈ DC and q ∈ DD or p, q ∈ DD can be additionally assumed here. We consider now the thread of restrictions – expansions of the convergence class of given permutation p ∈ D. In paper [81] – fifth section, there is introduced a family Ω ⊂ D,9 DC ⊂ Ω and DD ∩ Ω 6= ∅ such that for any permutation p ∈ Ω there exists a subset Ω(p) ⊂ Ω, card(Ω(p)) = c satisfying two basic conditions: X X (p) ⊂ (ϕ) for each ϕ ∈ Ω(p) and any two permutations ϕ, ψ ∈ Ω(p) are incomparable, which will hereafter mean that X X X X (ϕ) \ (ψ) 6= ∅ and (ψ) \ (ϕ) 6= ∅. In particular, if p ∈ DC then also Ω(p) ⊂ DC and one can assume that c(ϕ−1 ) 6 4c(p−1 ) + 1 for each ϕ ∈ Ω(p), where

P P ap(n) is convergent to some point from R ∪ {±∞} then we have Pan = ∞ (or −∞) and series

|

ap(n) | = ∞.

9

Permutation p ∈ P belongs to Ω if there exists an increasing sequence {In (p)}∞ n=1 of intervals satisfying three conditions: sequence {p−1 (In (p))}∞ n=1 is increasing as well, there exists constant k = k(p) ∈ N such that each of sets p−1 (In (p)) is a union of at most k msi and, moreover, one can −1 (I (p)) and lim c(p, J ) = ∞. indicate sequence {Jn }∞ n n n=1 of intervals such that Jn ⊂ p n→∞

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183

c(q) := sup{c(q, I) : I ⊂ N is the bounded interval} for any convergent permutation q. Study of the proof of these relations given in paper [81] implies that one can always suppose that only weakly divergent permutations belong to Ω(p). I have announced in paper [69] one more result (Theorem 6.1): if p ∈ DC then there exists family Ω(p) ⊂ DC, card(Ω(p)) = c, incomparable internally (which will denote, by virtue of definition, that any two different permutations belonging to this family are incomparable) and such that (the proof is presented in [78]): X \ X (p) ⊂ (ω). ω∈Ω(p)

Remark 8.1. There exist many unexpected relations (including the inclusions) connected with the one-sided divergent permutations. Even a single thing, that if p, q ∈ DC then X X (pq) ⊂ (p) which means in consequence that, for example X X (pn+1 ) ⊂ (pn ) for any n ∈ N and p ∈ DC and, additionally, that in family DC does not exist a permutation with minimal convergence class, with regard to inclusion between the convergence classes. Similarly, from the fact that DC ◦ DC = DC (see [69, Theorem 2.2]) it results P thatPfor each permutation p ∈ DC there exists permutation q ∈ DC such that (p) ⊂ (q), in consequence, in family DC does not exist a permutation with maximal convergence class, with regard to inclusion between the convergence classes. However, one can find some other, unsolved yet, problems concerning the existence of effects of the so called countable maximality and, respectively, countable minimality.10 One can also formulate these problems individually, within each of families DC, D, DD with regard to the relation of inclusion between the convergence classes of given permutations.

10 Let < be a binary and transitive relation on the infinite set X. We say that X possesses the effect of countable maximality if the following two conditions are satisfied

a) in set X there are no elements maximal with regard to relation 0 with limu→∞ ϕi (u) = ∞. Let Y be a real separable Banach space with the norm k · k. Let θ be the zero element in Y . Let Pk (Y ) denote the set of all nonempty and compact subsets of Y . For any A, B ∈ Pk (Y ) we denote n o dist(A, B) = max max min kx − yk, max min kx − yk , x∈A y∈B

y∈B x∈A

X = {F : N → 2Y : F (i) ∈ Pk (Y ) for every i ∈ N}. Every function from N to 2Y will be called a sequential vector multifunction. For every F ∈ X we define the functions |F | by the formula: |F |(i) = dist(F (i), θ)

for every i ∈ N.

Let now [a, b] denote a compact interval for all a, b ∈ R, a 6 b. Define Xϕ = {F ∈ X : |F | ∈ lϕ }. Let V be an abstract set of indices . Let V be a filter of subsets of V . Let 0 : N → Y be such that 0(i) = θ for every i ∈ N.

2. Preliminary We will present first some definitions and auxiliary results from the book [9].

Modular spaces Definition 2.1. Let X be a real vector space. A functional ρ : X → [0, +∞] is called a modular, if the following conditions hold for arbitrary x, y ∈ X: 1. ρ(0) = 0 and ρ(x) = 0 implies x = 0, 2. ρ(−x) = ρ(x), 3. ρ(αx + βy) 6 ρ(x) + ρ(y) for α, β > 0, α + β = 1. If in place of 3 there holds 3’. ρ(αx + βy) 6 αρ(x) + βρ(y) for α, β > 0, α + β = 1, then the modular ρ is called convex. Definition 2.2. If ρ is a modular in X, then Xρ = {x ∈ X : lim ρ(λx) = 0} λ→0

is called a modular space.

Approximation in Musielak-Orlicz sequence vector spaces of multifunctions

205

Theorem 2.3. If ρ is a modular in X, then x |x|ρ = inf{u > 0 : ρ( ) 6 u} u is an F-norm in Xρ , having the following properties: – if ρ(λx1 ) 6 ρ(λx2 ) for every λ > 0, where x1 , x2 ∈ Xρ , then |x1 |ρ 6 |x2 |ρ and moreover, – if x ∈ Xρ , then |λx|ρ is a nondecreasing function of λ > 0, – if |x|ρ < 1, then ρ(x) 6 |x|ρ . If ρ is a convex modular, then x kxkρ = inf{u > 0 : ρ( ) 6 1}, u is a norm in Xρ , which is called the Luxemburg norm. Theorem 2.4. Let ρ be a modular in X. If x ∈ Xρ and xk ∈ Xρ for k = 1, 2, . . ., then the condition |x − xk |ρ → 0 as k → ∞ is equivalent to the condition ρ(λ(x − xk )) → 0 as k → ∞ for every λ > 0. If ρ is a convex modular in X, then the same statement holds, replacing | · |ρ by k · kρ . Definition 2.5. Let ρ be a modular in X. A sequence {xn } of elements of Xρ is called modular convergent to x ∈ Xρ , if there exists a λ > 0 such that ρ(λ(xk − x)) → 0 as ρ k → ∞. We denote this writing xk → x. Theorem 2.6. The ρ-convergence in Xρ follows from norm convergence in Xρ . Norm convergence and ρ-convergence are equivalent in Xρ , if and only if, the following condition holds: if xk ∈ Xρ , ρ(xk ) → 0, then ρ(2xk ) → 0. Definition 2.7. Let ρ be a modular in X. A set A ⊂ Xρ will be called ρ-closed, if ρ xk ∈ A and xk → x imply x ∈ A. The smallest ρ-closed set containing the set A will be called the ρ-closure of A and ρ ρ denoted A . If A = Xρ , then A will be called ρ-dense in Xρ .

Musielak-Orlicz spaces Definition 2.8. Let (Ω, Σ, µ) be a measure space, where the measure µ is complete and not vanishing identically. A real function ϕ on Ω × [0, +∞), will be said to belong to the class Φ, if it satisfies the following conditions: i. ϕ(t, u) is a ϕ-function of the variable u > 0 for every t ∈ Ω, i.e. ϕ(t, u) is a nondecreasing, continuous function of u such that ϕ(t, 0) = 0, ϕ(t, u) > 0 for u > 0, ϕ(t, u) → ∞ as u → ∞, ii. ϕ(t, u) is a Σ-measurable function of t for every u > 0. Let X be the set of all real-valued, Σ-measurable and finite µ-almost everywhere functions on Ω, with equality µ-almost everywhere.

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A. Kasperski

It is easily seen that ϕ(t, |x(t)|) is a Σ-measurable function of t for every x ∈ X and that Z ρ(x) = ϕ(t, |x(t)|)dµ Ω

is a modular in X. Moreover, if ϕ(t, u) is a convex function of u for all t ∈ Ω, then ρ is a convex modular in X. Definition 2.9. The modular space Xρ will be called Musielak-Orlicz space and denoted by Lϕ : Z Lϕ = {x ∈ X : ϕ(t, λ|x(t)|)dµ → 0 as λ → 0+ }. Ω

Moreover, the set Lϕ 0

= {x ∈ X :

Z

ϕ(t, |x(t)|)dµ < ∞}



will be called the Musielak-Orlicz class. A function x ∈ X will be called a finite element of Lϕ , if λx ∈ Lϕ 0 for every λ > 0. The space of all finite elements of X will be denoted by E ϕ . If X is the space of sequences x = {ti } with real terms ti , ϕ = {ϕi }, where ϕi are ϕ-functions and ∞ X ρ(x) = ϕi (|ti |), i=1 ϕ

ϕ

ϕ

we shall write l in place of L and l is called the Musielak-Orlicz sequence space. Theorem 2.10. a. Lϕ is the set of all x ∈ X such that ρ(λx) < ∞ for some λ > 0. ϕ ϕ b. Lϕ 0 is a convex subset of L and L is the smallest vector subspace of X containing ϕ L0 . c. E ϕ is the largest vector subspace of X contained in Lϕ 0. Definition 2.11. A function ϕ will be called locally integrable, if Z ϕ(t, u)dµ < ∞ A

for every u > 0 and A ∈ Σ with µ(A) < ∞. Theorem 2.12. Let S be the set of all simple, integrable functions on Ω and let ϕ ∈ Φ be locally integrable. Then S ⊂ E ϕ . Moreover, supposing µ to be σ-finite, E ϕ is the closure of S with respect to the F-norm | · |ρ and S is ρ-dense in Lϕ . Theorem 2.13. Let µ be σ-finite. Then the Musielak-Orlicz space Lϕ is complete with respect to the F-norm | · |ρ .

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Theorem 2.14. ϕ a. If µ is σ-finite and atomless and ϕ ∈ Φ, then E ϕ = Lϕ 0 = L , if and only if, the following condition holds:

ϕ(t, 2u) 6 Kϕ(t, u) + h(t)

(∆2 )

for all u > 0 and almost every t ∈ Ω, where h is a nonnegative, integrable function in Ω and K is a positive constant. b. If ϕ ∈ Φ, where ϕ = {ϕn }, then l0ϕ = lϕ , if and only if, the following condition holds: there exist positive numbers δ, K and a sequence {an } of nonnegative ∞ P numbers with an < ∞ such that n=1

ϕn (u) < δ implies ϕn (2u) 6 Kϕn (u) + an

(δ2 )

for all u > 0, n = 1, 2, . . . Let us remark that the implication ϕ (∆2 ) ⇒ Lϕ 0 = L

holds without any assumptions on the measure µ for any ϕ ∈ Φ. Theorem 2.15. a. If µ is σ-finite and atomless and ϕ ∈ Φ, ϕ is locally integrable, then the following conditions are mutually equivalent: (1) (2) (3) (4)

ϕ Lϕ 0 = L , ϕ E = Lϕ , ϕ satisfies the condition (∆2 ), modular convergence and norm convergence are equivalent in Lϕ .

b. If ϕ ∈ Φ, where ϕ = {ϕn }, then the following conditions are mutually equivalent: (1) (2) (3) (4)

l0ϕ = lϕ , E ϕ = lϕ , ϕ = (ϕi ) satisfies the condition (δ2 ), modular convergence and norm convergence are equivalent in lϕ .

Let us remark that if ϕ ∈ Φ is a convex function of the variable u ∈ R for every t ∈ Ω, then ϕ is of the form Z|u| ϕ(t, u) = p(t, τ )dτ,

(1)

0

where p(t, u) is the right-hand derivative of ϕ(t, u) for a fixed t ∈ Ω. Definition 2.16. We shall say that a function ϕ ∈ Φ is an N -function if ϕ is a convex function of u for every t ∈ Ω and there hold the conditions:

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lim

ϕ(t, u) = 0, u

(2)

lim

ϕ(t, u) =∞ u

(3)

u→0+

u→∞

for every t ∈ Ω. Theorem 2.17. ϕ ∈ Φ is an N -function, if and only if, ϕ is of the form (1), where p(t, τ ) > 0 for τ > 0, p(t, τ ) is a right-continuous and nondecreasing function of τ > 0, p(t, 0) = 0, p(t, τ ) → ∞ as τ → ∞ for every t ∈ Ω. Remark 2.18. Let ϕ be an N -function of the form (1) and let p⋆ (t, σ) = sup{τ : p(t, τ ) 6 σ}. If p satisfies the conditions expressed in the previous theorem, then p⋆ satisfies the same assumptions. Definition 2.19. Let ϕ be an N -function of the form (1) and let p⋆ be defined by the formula p⋆ (t, σ) = sup{τ : p(t, τ ) 6 σ}. Then the function Z|u| ϕ (t, u) = p⋆ (t, σ)dσ ⋆

0

is called complementary to ϕ in the sense of Young. Evidently, ϕ⋆ is again an N -function. Theorem 2.20. Let ϕ be an N -function and let ϕ⋆ be complementary to ϕ in the sense of Young. Then they satisfy the Young inequality uv 6 ϕ(t, u) + ϕ⋆ (t, v) for u, v > 0, t ∈ Ω, and ϕ⋆ (t, v) = sup{uv − ϕ(t, u)},

ϕ(t, u) = sup{uv − ϕ⋆ (t, v)},

u>0

v>0

consequently, ϕ is complementary to ϕ⋆ in the sense of Young. We shall introduce now the Orlicz norm k · kρ,O in Lϕ if ϕ is an N -function. Theorem 2.21. Let a measure µ be σ-finite, ϕ be an N -function and ϕ locally integrable, ϕ⋆ complementary to ϕ in the sense of Young and let n Z o ϕ⋆ L1 = y : ϕ⋆ (t, |y(t)|)dµ 6 1, y measurable . Ω

Z

Then kxkρ,O = sup



y∈Lϕ 1



x(t)y(t)dµ

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is a norm in Lϕ (called the Orlicz norm) and kxkρ 6 kxkρ,O 6 2kxkρ for all x ∈ Lϕ . Theorem 2.22. Let ϕ be an N -function, ϕ⋆ complementary to ϕ in the sense of ⋆ Young, x ∈ Lϕ , y ∈ Lϕ . Then there hold the following H¨ older inequalities: Z x(t)y(t)dµ 6 kxkρ,O kykρ0 , Ω

Z x(t)y(t)dµ 6 kxkρ kykρ0 ,O , Ω

where ρ(x) =

Z

ρ0 (y) =

ϕ(t, |x(t)|)dµ,



Z

ϕ⋆ (t, |y(t)|)dµ.

Ω ⋆

Corollary 2.23. If ϕ is an N -function and ϕ is a complementary to ϕ in the sense ⋆ of Young, y ∈ Lϕ , then Z f (x) =

x(t)y(t)dµ



is a linear, continuous functional over Lϕ with the norm kf k = kykρ0 ,O . ρ

Definition 2.24. We say that functional f : Lϕ → R is ρ- continuous if xn → 0 in Lϕ implies f (xn ) → 0. Theorem 2.25. Let a measure µ be σ-finite and let an N -function ϕ satisfies the following condition: – for every u0 > 0 there exists a c > 0 such that t ∈ Ω.

ϕ(t,u) u

> c for u > u0 and all

Let the function ϕ⋆ complementary to ϕ be locally integrable. Then Z f (x) = x(t)y(t)dµ Ω ⋆

is a ρ-continuous linear functional over Lϕ for every y ∈ Lϕ . Theorem 2.26. Let a measure µ be σ-finite and let N -function ϕ be such that: – for every u0 > 0 there exists a c > 0 for which ⋆

ϕ(t,u) u

> c for u > u0 and t ∈ Ω.

Moreover, let both functions ϕ and ϕ (complementary to ϕ) be locally integrable. Then ⋆ for every linear, ρ-continuous functional f over Lϕ there exists a function y ∈ Lϕ such that Z f (x) = x(t)y(t)dµ Ω

for every x ∈ Lϕ .

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Theorem 2.27. Let an N -function ϕ and its complementary ϕ⋆ be locally integrable. Let us suppose that for every u0 > 0 there is a c > 0 for which ϕ(t,u) > c for u u > u0 and t ∈ Ω. Moreover, let us suppose that one of the following two conditions is satisfied: a. a measure µ is σ-finite and atomless and ϕ satisfies the condition (∆2 ), b. Ω = {1, 2, . . .}, µ({n}) = 1 for n = 1, 2, . . . and condition (δ2 ) holds for ϕ. Then the general form of a linear functional over Lϕ continuous with respect to the norm is Z f (x) = x(t)y(t)dµ Ω ⋆

for x ∈ Lϕ with y ∈ Lϕ , and kf k = kykρ0 , 0. Now we will present some general definitions and auxiliary results about the multifunctions from the book [1]. Let now X be a Hausdorff topological space. Denote: – – – – – – – –

2X : the collection of all subsets of X, 2X \ ∅: the collection of all nonempty subsets of X. Pf (X) = {A ⊆ X : nonempty, closed}, Pf c (X) = {A ⊆ X : nonempty, closed, convex}, Pbf (X) = {A ⊆ X : nonempty, bounded, closed}, Pbf c (X) = {A ⊆ X : nonempty, bounded, closed, convex}, Pk (X) = {A ⊆ X : nonempty, compact}, Pkc = {A ⊆ X : nonempty, compact, convex}.

If (X, d) is a metric space we denote: BX (x, a) = {y ∈ X : d(x, y) < a}. Let now (X, d) be a metric space. In what follows given any x ∈ X and A ∈ 2X \ ∅, the distance of x from A, is defined by d(x, A) = inf{d(x, a) : a ∈ A}. As usual, d(x, ∅) = +∞. Definition 2.28. If A, C ∈ 2X , we define a. h⋆ (A, C) = sup{d(a, C) : a ∈ A}, b. h⋆ (C, A) = sup{d(c, A) : a ∈ C}, c. h(A, C) = max{h⋆ (A, C), h⋆ (C, A)}, the Hausdorff distance between A and C. Directly from the definition, we can check that the following properties hold for any A, C, D ∈ 2X : h(A, A) = 0,

h(A, C) = h(C, A),

h(A, C) 6 h(A, D) + h(D, C).

Hence h is an extended pseudometric on 2X . Moreover, note that h(A, C) = 0 if and only if A = C. So Pf (X) equipped with the Hausdorff distance h becomes a metric space.

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Theorem 2.29. If (X, d) is a complete metric space, then so is (Pf (X), h). Theorem 2.30. If (X, d) is a complete metric space, then Pk (X) is a closed subset of (Pf (X), h), whence (Pk (X), h) is a complete metric space. Theorem 2.31. Pbf (X) is a closed subset of (Pf (X), h). Therefore, if (X, d) is a complete metric space, so is (Pbf (X), h). Theorem 2.32. If X is a Banach space, then Pkc (X), Pbf c (X), Pf c (X), Pk (X), Pbf (X) are complete subspaces of the metric space (Pf (X), h). Theorem 2.33. If X is a normed space and A, C, A1 , C1 , A2 , C2 ∈ 2X \ ∅, then h(λA, λC) = |λ| h(A, C)

for all λ ∈ R,

h(A1 + A2 , C1 + C2 ) 6 h(A1 , C1 ) + h(A2 + C2 ).

Measurable multifunctions Now we will present some definitions and auxiliary results about the measurable multifunction from the book [1]. Throughout this section (Ω, Σ) is a measurable space, (X, d) a separable metric space.Let us fix a multifunction F : Ω → 2X . Definition 2.34. a. F is said to be “strongly measurable” if for every C ⊆ X closed, we have F − (C) = {ω ∈ Ω : F (ω) ∩ C 6= ∅} ∈ Σ. b. F is said to be “measurable” if for every U ⊆ X open, we have F − (U ) = {ω ∈ Ω : F (ω) ∩ U 6= ∅} ∈ Σ. c. F is said to be “K-measurable” if for every K ⊆ X compact, we have F − (K) = {ω ∈ Ω : F (ω) ∩ K 6= ∅} ∈ Σ. d. F is said to be “graph measurable” if GrF = {(ω, x) ∈ Ω × X : x ∈ F (ω)} ∈ Σ × B(X). Theorem 2.35. If F is strongly measurable, then F is measurable. Theorem 2.36. A multifunction F : Ω → 2X is measurable if and only if for every x ∈ X, ω → d(x, F (ω)) = inf{d(x, x′ ) : x′ ∈ F (ω)} is a measurable R+ -valued function.

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Definition 2.37. Let Y be a metric space. A function f : Ω × X → Y is said to be a “Caratheodory function” if a. for every x ∈ X, ω → f (ω, x) is measurable, b. for every ω ∈ Ω, x → f (ω, x) is continuous. Theorem 2.38. If Y is a metric space and f : Ω × X → Y is a Caratheodory function, then f (·, ·) is jointly measurable. Theorem 2.39. If F : Ω → Pf (X) ∪ ∅ is measurable, then F is graph measurable. Theorem 2.40. If F : Ω → Pk (X), then F is strongly measurable if and only if it is measurable. Theorem 2.41. If F : Ω → Pf (X), then strong measurability ⇒ measurability ⇒ K-measurability. Theorem 2.42. If X is σ-compact and F : Ω → Pf (X), then strong measurability ⇔ measurability ⇔ K-measurability. Theorem 2.43. Let (Ω, Σ, µ) be a σ-finite and complete measure space. Let (X, d) be complete separable metric space and F : Ω → Pf (X). Consider the following statements: (a) (b) (c) (d) (e)

for every D ∈ B(X), F − (D) ∈ Σ, F is strongly measurable, F is measurable, for every x ∈ X, ω → d(x, F (ω)) is measurable, GrF ∈ Σ × B(X),

then all these statements are equivalent. Theorem 2.44. If (Ω, Σ) is a measurable space, X is a complete measure space and F : Ω → Pf (X) is measurable, then F admits a measurable selection, i.e., there exists f : Ω → X measurable such that for every ω ∈ Ω, f (ω) ∈ F (ω). Theorem 2.45. If (Ω, Σ) is a measurable space, X is a complete metric space and F : Ω → Pf (X), then the following statements are equivalent: a. F is measurable, b. there exists a sequence {fn }n>1 of measurable selectors of F such that for every ω∈Ω F (ω) = {fn (ω)}n>1 .

Decomposable sets and sets of Lp selectors Now we will present some definitions and auxiliary results about the decomposable sets and the sets of selectors from the book [1]. Let now (Ω, Σ, µ) be a σ-finite measure space, X a Banach separable space. Let L0 (Ω, X) be the space of all equivalent classes in the set of all measurable maps from Ω to X.

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Definition 2.46. A subset K of L0 (Ω, X) is said to be decomposable if for all f1 , f2 ∈ L0 (Ω, X), A ∈ Σ, we have χA f1 + χΩ\A f2 ∈ K. For 1 6 p 6 ∞, we define SFp = {f ∈ Lp (Ω, X) : f (ω) ∈ F (ω) µ − a.e.}. Lemma 2.47. If F : Ω → 2X \ ∅ is graph measurable and 1 6 p 6 ∞, then SFp 6= ∅ if and only if inf{kxk : x ∈ F (ω)} 6 h(ω) µ − a.e. for some h ∈ Lp (Ω). Theorem 2.48. If F : Ω → Pf (X) is graph measurable and SFp 6= ∅, then there exists a sequence {fn }n>1 ⊆ SFp such that F (ω) = {fn }n>1 µ − a.e. Theorem 2.49. If K is a nonempty, closed subset of Lp (Ω, X) for 1 6 p < ∞, then K = SFp for some uniquely defined measurable multifunction F : Ω → Pf (X) if and only if K is decomposable. Definition 2.50. A multifunction F : Ω → 2X \ ∅ is said to be Lp -integrably bounded (1 < p 6 ∞) and integrably bounded (for p = 1) if there exists h ∈ Lp (Ω) such that |F (ω)| := sup{kxk : x ∈ F (ω)} 6 h(ω) µ − a.e. Theorem 2.51. If F is graph measurable, then SFp is Lp (Ω, X)-bounded if and only if F is Lp -integrably bounded (1 < p 6 ∞).

Integral of multifunction Now we will present some definitions and auxiliary results about the integral of multifunction from the book [1]. Throughout out this section (Ω, Σ, µ) is a fixed σ-finite measure space and X is a separable Banach space. Let F : Ω → 2X \ ∅ be a multifunction with SF1 6= ∅. Then the set-valued Aumann integral of F is defined in the following way. Definition 2.52.

Z Ω

F (ω)dµ(ω) =

nZ

o f (ω)dµ(ω) : f ∈ SF1 .



We say that two measurable multifunctions F1 , F2 : Ω → 2X \ ∅ are equivalent if F1 (ω) = F2 (ω) µ-a.e. Denote by L1f (X) the space of all equivalence classes of multifunctions F : Ω → Pf (X) which are graph measurable and integrably bounded. Also by L1f c (X) we donote the subspace of all (equivalence classes) of graph measurable and integrably bounded multifunctions with values in Pf c (X). Since

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h(F (ω), G(ω)) 6 |F (ω)| + |G(ω)|, we deduce that h(F, G) ∈ L1 (Ω)+ . So we can define Z ∆(F, G) = h(F (ω), G(ω))dµ(ω). Ω

It is easily seen that ∆ is a metric on L1f (X) and we have Theorem 2.53. The space (L1f (X), ∆) is a complete metric space and (L1f c (X), ∆) is its closed subspace. Theorem 2.54. If F, G ∈ L1f (X), then Z Z h( F (ω)dµ(ω), G(ω)dµ(ω)) 6 ∆(F, G). Ω



1 Theorem 2.55. If F, G : Ω → Pf (X) are graph measurable with SG , SF1 6= ∅, then Z Z Z cl (F (ω) + G(ω))dµ(ω) = cl[ F (ω)dµ(ω) + G(ω)dµ(ω)]. Ω





Theorem 2.56. If F : Ω → 2X \ ∅ is a graph measurable multifunctions with SF1 6= ∅, then Z Z Z cl convF (ω)dµ(ω) = conv F (ω)dµ(ω) = cl convF (ω)dµ(ω). Ω





Theorem 2.57. If the R measure µ is nonatomic, F : Ω → Pf (X) is graph measurable and SF1 6= ∅, then cl F (ω)dµ(ω) is convex. Ω

Corollary 2.58. If µ is nonatomic, X is finite dimensional, F : Ω → Pf (X) is graph R measurable and Sf1 6= ∅, then F (ω)dµ(ω) is convex. Ω

Theorem 2.59. If µ is nonatomic, F : Ω → Pf (Rn ) is graph measurable and for n every ω ∈ Ω, F (ω) ⊆ R+ , then Z Z F (ω)dµ(ω) = convF (ω)dµ(ω). Ω



3. Main results We start from the generalization of the definition of Musielak-Orlicz sequence space of multifunctions from [5]. We use some ideas from [3, 5, 7, 8] and we generalize the main approximation theorem for lϕ from [8].

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Theorem 3.1. Let Fn ∈ Xϕ for every n ∈ N. Suppose that for every ǫ > 0 and for every a > 0 there is K > 0 such that ρ(a dist(Fn (·), Fm (·)) < ǫ for all m, n > K. Then there exists F ∈ Xϕ , such that ρ(a dist(Fn (·), F (·))) → 0 as n → ∞ for every a > 0. Proof. Let Fn ∈ Xϕ for every n ∈ N. If the assumptions of the Theorem hold, then {Fn } is a Cauchy sequence in the complete space C(Y ) with Hausdorff metric. Hence there are F (i) ∈ C(Y ) such that dist(Fn (i), F (i)) → 0 as n → ∞ for every i ∈ N. Fix ǫ > 0. Applying the Fatou lemma we easily obtain that there exists K > 0 such that ρ(a dist(Fn (·), F (·)) 6 ǫ for every n > K. We also have for every a > 0 that ρ(a|F |) 6 ρ(2a dist(Fn (·), F (·)) + ρ(2a|Fn |). So F ∈ Xϕ .

⊔ ⊓

The space Xϕ will be called a Musielak-Orlicz vector sequence space of multifunctions. Definition 3.2. A function g : V → R tends to zero with respect to a filter V, written V g(v) −→ 0, if for every ǫ > 0 there is V ∈ V such that | g(v) |< ǫ for every v ∈ V . Definition 3.3. An operator C : Xϕ → Xϕ will be called an X-linear operator if for all F, G ∈ Xϕ , a, b ∈ R, C(aF + bG)(i) = aC(F )(i) + bC(G)(i)

for every i ∈ N.

Definition 3.4. A family T = (Tv )v∈V of operators Tv : Xϕ → Xϕ , for every v ∈ V will be called (X, dist, V)-bounded, if there exist constants k1 , k2 > 0 and a function V g : V → R+ such that g(v) −→ 0, and for all F, G ∈ Xϕ there is a set VF,G ∈ V for which ρ(a dist(Tv (F )(·), Tv (G)(·))) 6 k1 ρ(ak2 dist(F (·), G(·))) + g(v) for all v ∈ VF,G and for every a > 0. d,ϕ,V

Definition 3.5. Let Fv ∈ Xϕ for every v ∈ V. Let F ∈ Xϕ . We write Fv −→ F , if for every ǫ > 0 and every a > 0 there exists V ∈ V such that ρ(a dist(Fv (·), F (·))) < ǫ for every v ∈ V . Definition 3.6. Let S ⊂ Xϕ . d,ϕ,V

SXϕ ,d,V = {F ∈ Xϕ : Fv −→ F, for some Fv ∈ S, v ∈ V}. Theorem 3.7. Let the family T = (Tv )v∈V of X-linear operators for every v ∈ V, d,ϕ,V

be (X, dist, V)-bounded. Let So ⊂ Xϕ and let Tv (F ) −→ F for every F ∈ So . Let now S be the set of all finite linear combinations of elements of the set So . Then d,ϕ,V Tv (F ) −→ F for every F ∈ SXϕ ,d,V . The proof analogous to that Theorem 4 in [5] is omitted.

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4. Applications Let now V = N and the filter V will consist of all sets V ⊂ V which are complements of finite sets. We shall say that ϕ is τ+ -bounded, if there are constants k1 , k2 > 1 and a double sequence {ǫn,j } such that ϕn+j (u) 6 k1 ϕn (k2 u) + ǫn,j for u ∈ R, n, j = 0, 1, . . . , where ǫn,j > 0, ǫn,0 = 0, ǫj =

∞ P

ǫn,j < ∞, i ∈ N, and

n=0

ǫj → 0 as j → ∞, s = supj∈N ǫj < ∞. Let Kv,j : V × V → R+ and let the family (Kv )v∈V P∞ V Kv,j be almost-singular, i.e. σ(v) = j=0 Kv,j 6 σ < ∞ for all v ∈ V and σ(v) −→ 0 for j = 1, 2, . . . Let F ∈ Xϕ . We define a family T = (Tv )v∈V of operators by the formula: i X Tv (F )(i) = Kv,i−j F (j) for every i ∈ V. j=0

Lemma 4.1. Let (Kv )v∈V be almost-singular, let ϕ = (ϕi )i∈V be τ+ -bounded and ϕi be convex for every i ∈ V, then Tv : lϕ → lϕ for every v ∈ V. The proof analogous to that of Lemma 1 in [5] is omitted. Lemma 4.2. If the assumptions of Lemma 1 hold, then the family T = (Tv )v∈V is (Xϕ , dist, V)-bounded and Tv is Xϕ -linear-operator for every v ∈ V. The proof analogous to that of Lemma 2 in [5] is omitted. Lemma 4.3. Let ϕ = (ϕi )∞ i=0 satisfy the condition (δ2 ). Let F ∈ Xϕ and F = (F (i))∞ . Let F be such that Fv (i) = F (i) for i = 0, 1, . . . , v and Fv (i) = 0 for i > v i=0 d,ϕ,V

v. Then Fv −→ F . The proof analogous to that of Lemma 3 in [5] is omitted. Now, let us denote: xj,Kv = {0, . . . , 0, Kv,1 , Kv,2 , . . .}. | {z } j−times

d,ϕ,V

Theorem 4.4. Let the assumptions of Lemmas 1 and 3 hold. If xj,Kv −→ 0 for d,ϕ,V

V

every j ∈ V, Kv,o −→ 1, then Tv (F ) −→ F for every F ∈ Xϕ . The proof analogous to that of Theorem 5 in [5] is omitted. Now, let us denote: xj,Kv = {0, . . . , 0, Kv,0 , Kv,1 , . . .}. | {z } j−times

d,ϕ,V

Theorem 4.5. Let the assumptions of Lemmas 1 and 3 hold. If xj,Kv −→ 0 for d,ϕ,V

every j ∈ V, then Tv (F ) −→ 0 for every F ∈ Xϕ . The proof analogous to that of Theorem 6 in [5] is omitted.

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5. Pk (Y )-linear functionals Now we will present some generalizations on the spaces of multifunctions of the classical Riesz theorems about a linear and continuous functional on a Banach space. We use the results of [6, 9]. Let now ϕ be an N -function and N be the set of all natural numbers. Let k · kϕ ϕ denote the Luxemburg norm in lϕ and k · kO ϕ denote the Orlicz norm in l . Definition 5.1. The mapping M : Xϕ → Pk (Y ) such that M (F + G) = M (F ) + M (G), M (aF ) = aM (F ) for all F, G ∈ Xϕ , a > 0, will be called an Pk (Y )-linear functional on Xϕ . Definition 5.2. We say that M : Xϕ → Pk (Y ) is continuous at F ∈ Xϕ if for every ǫ > 0 there is δ > 0 such that from k dist(F (·), G(·))kϕ < δ it follows that dist(M (F ), M (G)) < ǫ. If M is continuous at every F ∈ Xϕ , then we say that M is continuous on Xϕ . Let f = (fn ), where fn ∈ R for every n ∈ N. Denote ∞ X Mf (F ) = fn F (n) n=1

for every F ∈ X. Lemma 5.3. Let 1 < p < ∞, p1 + 1q = 1, f = (fk ) where fk = 0 for k = n+1, n+2, . . . Then Mf is a Pk (Y )-linear and continuous functional on Xlp . Proof. We have Mf (F ) =

n X

fk F (k)

k=1

for every F ∈ Xlp , so Mf (F ) ∈ Pk (Y ) for every F ∈ Xlp and Mf is a Pk (Y )-linear. We also have for all F, G ∈ Xlp that n X dist(Mf (F ), Mf (G)) 6 |fk | dist(F (k), G(k)) 6 kf klq k dist(F (·).G(·))klp . k=1

Theorem 5.4. Let 1 < p < ∞, continuous functional on Xlp .

⊔ ⊓ 1 p

+

1 q

= 1, f ∈ lq . Then Mf is a Pk (Y )-linear and

Proof. Let f = [f1 , . . . , fn , . . .], fn = [f1 , . . . , fn , 0, 0, . . .], f ∈ lq . It is easy to prove that for every F ∈ Xlp the sequence {Mfn (F )} is a Cauchy sequence in hPk (Y ), disti, so there is A ∈ Pk (Y ) such that dist(Mfn , A) → 0 as n → ∞. We also have for every F ∈ Xlp : ∞ X dist(Mf (F ), Mfn (F )) 6 dist(fk F (k), θ) 6 k=n+1

6

∞ X k=n+1

as n → ∞. So Mf (F ) = A.

|fk ||F (k)| 6 (

∞ X

k=n+1

1

|fk |q ) q k|F |klp → 0

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Let F, G ∈ Xlp . We have dist(Mf (F ), Mf (G)) 6 6 dist(Mf (F ), Mfn (F )) + dist(Mfn (F ), Mfn (G)) + dist(Mfn (G), Mf (G)), so Mf is Pk (Y )-linear and continuous functional on Xlp .

⊔ ⊓

Analogously we obtain the following two theorems (see also [9], Theorem 13.18): Theorem 5.5. Let f ∈ m. Then Mf is a Pk (Y )-linear and continuous functional on Xl1 . Theorem 5.6. Let ϕ and its complementary ϕ⋆ be the N-functions, ϕ = (ϕi ), such that for every u0 > 0 there is c > 0 for which ϕiu(u) > c for u > u0 and i ∈ N, (δ2 ) ⋆ holds for ϕ, f ∈ lϕ . Then Mf in Pk (Y )-linear and continuous functional on Xϕ . Denote wf (F ) =

∞ X

fn |F (n)|,

n=1

for every F ∈ Xlp . Applying the proof of Theorem 5.4 we obtain the following Theorem 5.7. Let 1 < p < ∞,

1 p

+

1 q

= 1, f ∈ lq , f (n) > 0 for every n ∈ N. Then:

wf (F + G) 6 wf (F ) + wf (G) and wf (aF ) = awf (F ) for all F, G ∈ Xlp . Moreover, for every ǫ > 0 there is δ > 0 such that from F, G ∈ Xlp , k|F | − |G|klp < δ it follows |wf (F ) − wf (G)| < ǫ.

Bibliography 1. Hu S., Papageorgiou N.S.: Handbook of Multivalued Analysis. 1. Theory. Kluwer, Dordrecht 1997. 2. Kasperski A.: Modular approximation by a filtered family of sublinear operators. Comment. Math. 27 (1987), 109–114. ˜ ϕ by a filtered family of X ˜ ϕ -linear operators. Funct. 3. Kasperski A.: Modular approximation in X Approx. Comment. Math. 20 (1992), 183–187. ˜ ϕ by a filtered family of dist-sublinear operators and 4. Kasperski A.: Modular approximation in X dist-convex operators. Math. Japonica 38 (1993), 119–125. 5. Kasperski A.: Notes on approximation in the Musielak-Orlicz sequence spaces of multifunctions. Comment. Math. Univ. Carolin. 36 (1995), 19–24. 6. Kasperski A.: Remarks on multifunctionals on some spaces of multifunctions. Math. Japonica 51 (2000), 411–417. 7. Kasperski A.: On some approximation problems in Musielak-Orlicz spaces of multifunctions. Demonstratio Math. 37 (2004), 393–406. 8. Musielak J.: Modular approximation by a filtered family of linear operators. In: Functional Analysis and Approximation, Proc. Conf. Oberwolfach, August 9-16, 1980, Birhauser-Verlag, Basel 1981, 99–110. 9. Musielak J.: Orlicz Spaces and Modular Spaces. Lecture Notes in Math. 1034, Springer-Verlag, Berlin 1983.

Nemytskij operator, R˚ adstr¨ om embedding and set-valued functions Jakub Jan Ludew

Abstract. We give several results concerning a Nemytskij operator, generated by a set-valued functions. We consider two function spaces, namely the C 1 and AC spaces of continuously differentiable, resp., absolutely continuous, set-valued functions. We prove that the situation in which the Nemytskij operator is Lipschitzian continuous is characterized by a specific form of a function which generates the operator. Keywords: Nemytskij operator, C 1 space, AC space, set-valued functions, Jensen equation. 2010 Mathematics Subject Classification: 26E25, 39B52, 47H04, 47H30, 54C60.

1. Introduction All linear spaces considered in this article are assumed to be real. In the following we shall write I instead of [0, 1]. In 1982 J. Matkowski showed (cf. [15]), that a Nemytskij operator N (which is defined by the formula φ 7→ N (φ) := g(·, φ(·)), where g is a given function) maps the function space Lip(I, R) into itself and is Lipschitzian with respect to the Lipschitzian norm if and only if its generator is of the form g(x, y) = a(x)y + b(x),

x ∈ I, y ∈ R,

for some a, b ∈ Lip(I, R). This result was extended to a lot of spaces by J. Matkowski and others (cf. e.g. [12, 13, 16]), in particular to the spaces C k (I, R) and AC(I, R) of all k-times continuously differentiable, resp., absolutely continuous, functions φ : I → R (cf. [17]). Recently Matkowski has shown (cf. [18]) that if we only assume that the operator N is uniformly continuous, then the generator g is of the form above. J.J. Ludew Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 219–237. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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Set-valued versions of Matkowski’s results were investigated for instance in papers [5, 6, 7, 9, 10, 11, 19, 23, 24] and [25]. The main goal of this paper is to examine a set-valued analogue of Matkowski’s result in the cases of the C 1 and AC spaces. If (Z, || · ||Z ) is a normed space, then by cc(Z) we denote the space of all nonempty, compact and convex subsets of Z. If A and B are subsets of Z, then we define A + B := {a + b : a ∈ A, b ∈ B} and αA := {αa : a ∈ A}, where α ∈ R. Moreover, if α, β ∈ R and A, B ∈ cc(Z), then α(A + B) = αA + αB,

α(βA) = (αβ)A,

1A = A,

and if α, β > 0, then (α + β)A = αA + βA. Let d denote the Hausdorff metric on the space cc(Z), defined by the fromula d(A, B) := inf{t > 0 : A ⊆ B + tS,

B ⊆ A + tS},

where S is a closed unit ball in the space Z. If A ∈ cc(Z), then let us define ||A||cc(Z) as follows: ||A||cc(Z) := sup{||z||Z : z ∈ A}.

(1)

Moreover, if C is a non-empty subset of a real linear space, then we shall say that C is a convex cone, if it satisfies the following two conditions: C+C ⊆ C and λC ⊆ C for all λ > 0. Lemma 1.1 ([21], Lemma 2). Let Z be a normed space. If A, B and C are non-empty, compact and convex subsets of Z, then d(A + B, A + C) = d(B, C). Lemma 1.2 ([20], Theorem 5.6, p. 64). Let Y be a vector space and let Z be a Hausdorff topological vector space. Moreover, let C be a convex cone in Y . A set-valued function F defined on C, with non-empty and compact values in Z, satisfies the Jensen equation 1  1  F (y1 + y2 ) = F (y1 ) + F (y2 ) , y1 , y2 ∈ C, 2 2 if and only if there exist an additive set-valued function A, defined on C with nonempty, compact and convex values in Z and a non-empty, compact and convex subset B of Z such that F (y) = A(y) + B, y ∈ C. Theorem 1.3 ([21]). For every normed linear space Z there exists a normed linear space (VZ , ||·||VZ ) and an isometric embedding π : cc(Z) → VZ , where cc(Z) is endowed with the Haussdorf distance d, for which π(cc(Z)) is a convex cone in VZ and the conditions π(cc(Z)) − π(cc(Z)) = VZ π(A + B) = π(A) + π(B) π(αA) = απ(A)

(2)

are satisfied for A, B ∈ cc(Z), α > 0. Moreover, VZ is essentialy unique, i.e. if VZ1 and VZ2 are normed linear spaces and π 1 : cc(Z) → VZ1 , π 2 : cc(Z) → VZ2 are embeddings which satisfy the above conditions, then there exists exactly one isometric isomorphism T : VZ1 → VZ2 for which T ◦ π 1 = π 2 .

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If E, E ′ are arbitrary non-empty sets, by F (E, E ′ ) we denote the set of all functions f : E → E ′ . Every function g : I × E → E ′ generates the so-called Nemytskij operator N : F (I, E) → F(I, E ′ ), defined by the formula (N φ)(x) := g(x, φ(x)),

φ ∈ F(I, E),

x ∈ I.

(3)

For a function A : I × C → cc(Z) we shall write Ay = A(·, y), Ax = A(x, ·), x ∈ I, y ∈ C. Thus Ay : I → cc(Z) for y ∈ C and Ax : C → cc(Z) for x ∈ I. Let Y, Z be normed linear spaces, and let C be a convex cone in Y (C is endowed with the metric induced from Y ). Consider the set L(C, cc(Z)) := {A : C → cc(Z) : A is additive and continuous}. The formula dL (A, B) :=

d(A(y), B(y)) ||y||Y y∈C\{0} sup

(4)

defines a metric in L(C, cc(Z)) (cf. [23] and [25]). Next the functional ||A||L :=

||A(y)||cc(Z) , ||y||Y y∈C\{0} sup

A ∈ L(C, cc(Z)),

(5)

is not a norm, since L(C, cc(Z)) with addition and multiplication by real scalars, defined in the usual way, is not a vector space (except the case that Z is a singlepoint space). Lemma 1.4 ([22], Lemma 5). Let Y and Z be normed linear spaces and let C be a convex cone in Y with nonempty interior. Then there exists a positive constant M0 such that for every additive and continuous set-valued function F : C → cc(Z) (in particular, for the functions the values of which are singletons) the inequality d(F (y1 ), F (y2 )) 6 M0 ||F ||L ||y1 − y2 ||Y holds. By L(Y, V ) we denote the normed space of all continuous linear operators, which act on the normed space Y and with the values in normed space V . Lemma 1.5. Let V be a real normed space and let C be a convex cone with nonempty interior in a real normed space Y . If a function A : C → V is additive and continuous then there exists exactly one linear and continuous extention A : Y → V of a function A such that ||A||L(Y,V ) 6 M0 ||A||L (for a constant M0 see Lemma 1.4). Proof. Let y ∈ Y . It is easy to observe that there exist y1 , y2 ∈ C such that y = y1 −y2 . Let us define A(y) := A(y1 ) − A(y2 ). It is easly seen that this definition is correct and, moreover, A is linear and continuous extension of A. The uniqueness of the extension of A is obvious. To complete the proof we have to show that ||A||L(Y,V ) 6 M0 ||A||L . (6)

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Let y ∈ Y and let y = y1 − y2 , for y1 , y2 ∈ C; from Lemma 1.4 we get ||A(y)||V = ||A(y1 ) − A(y2 )||V 6 M0 ||A||L ||y1 − y2 ||Y = M0 ||A||L ||y||Y , and hence (6) is verified.

⊔ ⊓

If (V, || · ||V ) is a real normed linear space then by C 1 (I, V ) we denote the space of all continuously differentiable vector-functions φ : I → V . Moreover, for a non-empty subset C ⊆ V , by C 1 (I, C) we denote the set of all functions φ ∈ C 1 (I, V ) such that φ(I) ⊆ C. Now let ||φ||Lip(I,V ) and ||φ||C 1 (I,V ) denote the norms on the space C 1 (I, V ), defined as follows ||φ||Lip(I,V ) := ||φ(0)||V + sup

x1 6=x2

||φ||

C 1 (I,V

)

||φ(x1 ) − φ(x2 )||V , |x1 − x2 |

:= ||φ(0)||V + sup ||φ′ (x)||V ; x∈[0,1]

the second supremum is finite since φ is continuously differentiable and I is a compact set. The first supremum above is also finite; it follows directly from the Mean Value Theorem ||φ(x1 ) − φ(x2 )||V 6 sup ||φ′ (x)||V . (7) |x1 − x2 | x∈[x1 ,x2 ] Moreover, from inequality (7) we get (cf. [13]): ||φ||Lip(I,V ) 6 ||φ||C 1 (I,V ) . Now, let F be a function defined on the interval I with the values in cc(Z). From many definitions of differentiability of set-valued functions we choose the definition due to Banks and Jacobs (cf. [2]); we shall say that F is π-differentiable at x0 ∈ I, if the vector-function π ◦F is differentiable at x0 (for π see Theorem 1.3; the differentiability defined in this way does not depend on the chosen π.) We define the space C 1 (I, cc(Z)) as follows C 1 (I, cc(Z)) := {F ∈ cc(Z)I : π ◦ F ∈ C 1 (I, VZ )}. Let us note that if F belongs to the space C 1 (I, cc(Z)) then F ∈ C(I, cc(Z)), i.e. F is continuous. On the space C 1 (I, cc(Z)) the metric may be defined as follows dC 1 (I,cc(Z)) (F1 , F2 ) := ||π ◦ F1 − π ◦ F2 ||C 1 (I,VZ ) , where F1 , F2 ∈ C 1 (I, cc(Z)).

2. Main results Theorem 2.1. Let Y, Z be normed linear spaces and C be a convex cone in Y . Assume that the Nemytskij operator N generated by G : I × C → cc(Z) satisfies the following conditions 1) N : C 1 (I, C) → C 1 (I, cc(Z)),

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2) there exists L > 0 such that dC 1 (I,cc(Z)) (N φ1 , N φ2 ) 6 L||φ1 − φ2 ||C 1 (I,Y ) ,

φ1 , φ2 ∈ C 1 (I, C).

(8)

Then there exist functions A : I × C → cc(Z), B : I → cc(Z) such that B, Ay belongs to the space C 1 (I, cc(Z)) for every y ∈ C, the function Ax is additive and Lipschitzian for every x ∈ I and G(x, y) = A(x, y) + B(x),

x ∈ I, y ∈ C.

Moreover the function I ∋ x 7→ Ax ∈ L(C, cc(Z)) satisfies the Lipschitz condition with the constant L, i.e. dL (Ax1 , Ax2 ) 6 L|x1 − x2 |

x1 , x2 ∈ I.

(9)

Remark 2.2. Since in the course of the proof of Theorem 2.1 R˚ adstr¨om embedding is applied, it is natural to ask, whether this theorem is a conclusion from its vectorvalued analogue (cf. [17]). The answer, as is easy to see, is “no” – if we formulate the vector-valued analogue and try to prove Theorem 2.1 as a corollary from this analogue, then the crucial step depends on whether the difference of two vectors from the cone is another vector from this cone. Without more detailed information on these vectors we can not generally give a positive answer. Proof. Let us note, that G(·, y) ∈ C 1 (I, cc(Z)). To see that let us fix y ∈ C and set φ1 (x) = y, x ∈ I. By 1) and from the definition of N we get N φ1 = g(·, y) ∈ C 1 (I, cc(Z)). In particular G is continuous with respect to the first variable. Now, let us take x1 , x2 ∈ I such that 0 6 x1 < x2 6 1 and let u, v ∈ C. Consider two functions φ1 , φ2 : I → Y defined by φ1 (x) := y1 + α(x)[y2 − y1 ], φ2 (x) := ye1 + α(x)[e y2 − ye1 ],

(10) (11)

where α is an arbitrary function from the space C 1 (I, I) for which equalities α(x1 ) = 0, α(x2 ) = 1 holds and y1 = ye2 := (u + v)/2, y2 := v, ye1 := u. Note that φ1 , φ2 ∈ C 1 (I, C) and ||φ1 − φ2 ||C 1 (I,Y ) = ||(v − u)/2||Y . Thus, from 2) we get dC 1 (I,cc(Z)) (N φ1 , N φ2 ) 6 L||(v − u)/2||Y . Hence, from the definition of the metric dC 1 (I,cc(Z)) and from inequality (7), there holds |x1 − x2 |−1 ||(π ◦ N φ1 − π ◦ N φ2 )(x1 ) − (π ◦ N φ1 − π ◦ N φ2 )(x2 )||VZ 6 6 L||(v − u)/2||Y . Thus, from the definition of the Nemytskij operator, we get ||π(G(x1 ,

u+v u+v )) + π(G(x2 , )) − [π(G(x1 , u)) + π(G(x2 , v))]||VZ 6 2 2 ||v − u||Y 6L |x1 − x2 |, 2

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and since π is an isometry we infer d(G(x1 ,

u+v u+v ||v − u||Y ) + G(x2 , ), G(x1 , u)) + G(x2 , v)) 6 L |x1 − x2 |. 2 2 2

Now, letting x1 , x2 → x, where x is an arbitrary point of the interval I, we get (since G is continuous with respect to the first variable) u+v 1 ) = [G(x, u) + G(x, v)] 2 2

G(x, and from Lemma 1.2 there is

G(x, y) = A(x, y) + B(x),

(12)

where A : I × C → cc(Z), B : I → cc(Z) and A is additive with respect to the second variable. To prove that B ∈ C 1 (I, cc(Z)), let us note that G(x, 0) = A(x, 0) + B(x) = {0} + B(x) = B(x), and G(·, 0) ∈ C 1 (I, cc(Z)). Now we shall prove that Ay ∈ C 1 (I, cc(Z)) for every y ∈ C. From equalities (2) and (12) and from definition (3) we get π(N φ1 (x)) = π(G(x, y)) = π(A(x, y)) + π(B(x)), where φ1 (x) = y for x ∈ I. Hence π ◦ Ay = π ◦ N φ1 − π ◦ B. Since π ◦ N φ1 , π ◦ B ∈ C 1 (I, VZ ), thus π ◦ Ay ∈ C 1 (I, VZ ), which mplies that Ay ∈ C 1 (I, cc(Z)) for every y ∈ C. Now we shall prove that the inequality d(G(x, y), G(x, ye)) 6 L||y − ye||Y ,

x ∈ I, y, ye ∈ C

(13)

holds. Let us fix x ∈ I, y, ye ∈ C. Define φ1 , φ2 : I → C as follows: φ1 (t) = y, φ2 (t) = ye for t ∈ I. It is obvious that φ1 , φ2 ∈ C 1 (I, C). Let us note that ||φ1 − φ2 ||C 1 (I,Y ) = ||y − ye||Y . According to Lemma 1.1 we get d(G(x, y), G(x, ye)) = d(G(x, y) + G(0, ye), G(x, ye) + G(0, ye)) 6

6 d(G(x, y) + G(0, ye), G(x, ye) + G(0, y)) + d(G(x, ye) + G(0, y), G(x, ye) + G(0, ye)) = = d(G(0, y), G(0, ye)) + d(G(x, y) + G(0, ye), G(0, y) + G(x, ye)). Since π is an isometry, we get d(G(0, y), G(0, ye)) = ||π((G(0, y)) − π(G(0, ye)))||VZ =

= ||(π ◦ N φ1 )(0) − (π ◦ N φ2 )(0)||VZ = ||(π ◦ N φ1 − π ◦ N φ2 )(0)||VZ .

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Moreover from Mean Value Theorem we obtain d(G(x, y) + G(0, ye), G(0, y) + G(x, ye)) =

= ||π((G(x, y)) + π((G(0, ye)) − π((G(0, y)) − π((G(x, ye))||VZ = = ||π((G(x, y)) − π((G(x, ye)) − [π((G(0, y)) − π((G(0, ye))]||VZ =

= ||π(N φ1 (x)) − π(N φ2 (x)) − [π(N φ1 (0)) − π(N φ2 (0))]||VZ = = ||π ◦ N φ1 (x) − π ◦ N φ2 (x) − [π ◦ N φ1 (0) − π ◦ N φ2 (0)]||VZ = = ||(π ◦ N φ1 − π ◦ N φ2 )(x) − (π ◦ N φ1 − π ◦ N φ2 )(0)||VZ 6 6 sup ||(π ◦ N φ1 − π ◦ N φ2 )′ (t)||VZ (x − 0) 6 t∈[0,x]

6 sup ||(π ◦ N φ1 − π ◦ N φ2 )′ (t)||VZ . t∈I

Thus from (8) we get d(G(x, y), G(x, ye)) 6 ||(π◦N φ1 −π◦N φ2 )(0)||VZ +sup ||(π◦N φ1 −π◦N φ2 )′ (t)||VZ = t∈I

= ||(π ◦ N φ1 − π ◦ N φ2 )||C 1 (I,VZ ) = dC 1 (I,cc(Z)) (N φ1 , N φ2 ) 6 L||y − ye||Y ,

which completes the proof of inequality (13). Now from (12) and from Lemma 1.1 we get d(Ax (y), Ax (e y )) = d(A(x, y), A(x, ye)) = d(A(x, y) + B(x), A(x, ye) + B(x)) =

= d(G(x, y), G(x, ye)) 6 L||y − ye||Y ,

and we conclude that Ax is Lipschitzian. Finally, we shall prove that (9) holds. Let z, w ∈ C and let φ1 , φ2 be given by (10) and (11), where y1 = ye2 = z + w, y2 = 2z + w, ye1 = w. Then (φ1 − φ2 )(x) = z for x ∈ I. Hence, from (8) and from Mean Value Theorem we obtain ||π(G(x1 , y1 ) + G(x2 , ye2 )) − π(G(x1 , ye1 ) + G(x2 , y2 ))||VZ 6 L||y1 − y2 ||Y |x1 − x2 |. Thus d(G(x1 , z + w) + G(x2 , z + w), G(x1 , w) + G(x2 , 2z + w)) 6 L||z||Y |x1 − x2 |. From (12) and from additivity of function Ax for x ∈ I we get d(A(x1 , z), A(x2 , z)) 6 L||z||Y |x1 − x2 |. Thus dL (Ax1 , Ax2 ) = which completes the proof.

d(Ax1 (z), Ax2 (z)) 6 L|x1 − x2 |, ||z||Y z∈C\{0} sup

⊔ ⊓

Remark 2.3. It is possible to formulate this theorem in a stronger form, assuming only uniform continuity of N , instead of satisfying the Lipschitz condition – see Theorem 2.6 and its proof.

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Theorem 2.4. Let Y and Z be normed linear spaces, let C be a convex cone with nonempty interior in the space Y and let A, B be given functions such that A : I ×C → cc(Z) and B : I → cc(Z). Assume that Ay , B belong to the space C 1 (I, cc(Z)) for y ∈ C and Ax belongs to the space L(C, cc(Z)) for x ∈ I. Moreover, let the function I ∋ x 7→ Ax ∈ L(C, cc(Z)) satisfies the Lipschitz condition, i.e. there exists a constant L > 0 such that dL (Ax1 , Ax2 ) 6 L|x1 − x2 |, x1 , x2 ∈ I. (14) If we define the function G : I × C → cc(Z) in the following way G(x, y) = A(x, y) + B(x),

x ∈ I, y ∈ C,

then the Nemytski operator N generated by G maps the set C 1 (I, C) into the space C 1 (I, cc(Z)) and satisfies the Lipschitz condition, i.e., there exists a constant L′ > 0 such that dC 1 (I,cc(Z)) (N φ1 , N φ2 ) 6 L′ ||φ1 − φ2 ||C 1 (I,Y ) ,

φ1 , φ2 ∈ C 1 (I, C).

(15)

Proof. Let x1 , x2 ∈ I. Since π ◦ Ax1 , π ◦ Ax2 are additive and continuous, so is π ◦ Ax1 − π ◦ Ax2 . We shall prove now that the following inequality holds ||π ◦ Ax1 − π ◦ Ax2 ||L 6 L|x1 − x2 |.

(16)

Let y ∈ C. From (4) and (14) we get d(Ax1 (y), Ax2 (y)) 6 L||y||Y |x1 − x2 |, whence, as π is an isometry ||[π ◦ Ax1 − π ◦ Ax2 ](y)||VZ 6 L||y||Y |x1 − x2 |. Thus, according to (1) and (5), (16) holds. Now let x ∈ I. By Lemma 1.5 there is exactly one linear, continuous function π ◦ Ax : Y → VZ such that π ◦ Ax (y) = π ◦ Ax (y) for

y ∈ C.

We shall prove now that the function I ∋ x 7→ π ◦ Ax ∈ L(Y, VZ )

(17)

satisfies the Lipschitz condition. Let x1 , x2 ∈ I. Obviously π ◦ Ax1 − π ◦ Ax2 = π ◦ Ax1 − π ◦ Ax2 . Hence, from Lemma 1.5 and (6), we infer ||π ◦ Ax1 − π ◦ Ax2 ||L(Y,VZ ) = ||π ◦ Ax1 − π ◦ Ax2 ||L(Y,VZ ) 6 6 M0 ||π ◦ Ax1 − π ◦ Ax2 ||L 6 M0 L|x1 − x2 |,

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and thus we get ||π ◦ Ax1 − π ◦ Ax2 ||L(Y,VZ ) 6 M0 L|x1 − x2 |.

(18)

Now we shall prove that the following inequality ||(π ◦ Ay1 )′ (x) − (π ◦ Ay2 )′ (x)||VZ 6 M0 L||y1 − y2 ||Y

(19)

holds for x ∈ I and y1 , y2 ∈ C. Let h ∈ R, h 6= 0 and let x + h ∈ I. From inequality (18) we get ||(π ◦ Ay1 )(x + h) − (π ◦ Ay1 )(x) − [(π ◦ Ay2 )(x + h) − (π ◦ Ay2 )(x)]||VZ = = ||π(A(x + h, y1 )) − π(A(x + h, y2 )) − [π(A(x, y1 )) − π(A(x, y2 ))]||VZ = = ||π ◦ Ax+h (y1 ) − π ◦ Ax+h (y2 ) − [π ◦ Ax (y1 ) − π ◦ Ax (y2 )]||VZ = = ||[π ◦ Ax+h − π ◦ Ax ](y1 − y2 )||VZ 6 M0 L|h|||y1 − y2 ||Y , whence (π ◦ A )(x + h) − (π ◦ A )(x) (π ◦ A )(x + h) − (π ◦ A )(x) y1 y1 y2 y2 − 6 h h VZ 6 M0 L||y1 − y2 ||Y . Letting t → 0 we get (19). Now, we shall prove that the derivative of the function π ◦ N φ : I → VZ at any point x0 ∈ I is given by the following formula (π ◦ N φ)′ (x0 ) = (π ◦ B)′ (x0 ) + (π ◦ Aφ(x0 ) )′ (x0 ) + π ◦ Ax0 (φ′ (x0 )).

(20)

Indeed, applaying in turn the definition of the derivative, the triangle inequality and equalities (2) and (3) we get ||(π ◦ N φ)(x0 + h) − (π ◦ N φ)(x0 ) − h(π ◦ B)′ (x0 ) − − h(π ◦ Aφ(x0 ) )′ (x0 ) − h[ π ◦ Ax0 (φ′ (x0 ))]||VZ = = ||π(A(x0 + h, φ(x0 + h)) + B(x0 + h)) − π(A(x0 , φ(x0 )) + B(x0 )) − − h(π ◦ B)′ (x0 ) − h(π ◦ Aφ(x0 ) )′ (x0 ) − h[ π ◦ Ax0 (φ′ (x0 ))]||VZ 6 6 ||(π ◦ B)(x0 + h) − (π ◦ B)(x0 ) − h(π ◦ B)′ (x0 )||VZ + || − h[ π ◦ Ax0 (φ′ (x0 ))] + + π(A(x0 + h, φ(x0 + h))) − π(A(x0 , φ(x0 ))) − h(π ◦ Aφ(x0 ) )′ (x0 )||VZ 6 6 o(h) + ||π(A(x0 + h, φ(x0 ))) − π(A(x0 , φ(x0 ))) − h(π ◦ Aφ(x0 ) )′ (x0 )||VZ + + ||π(A(x0 + h, φ(x0 + h))) − π(A(x0 + h, φ(x0 ))) − h[ π ◦ Ax0 (φ′ (x0 ))]||VZ 6 6 o(h) + ||(π ◦ Aφ(x0 ) )(x0 + h)) − (π ◦ Aφ(x0 ) )(x0 )) − h(π ◦ Aφ(x0 ) )′ (x0 )||VZ + + ||π(A(x0 , φ(x0 + h)) − π(A(x0 , φ(x0 )) − h[ π ◦ Ax0 (φ′ (x0 ))] + + π(A(x0 + h, φ(x0 + h)) − π(A(x0 , φ(x0 + h)) − − π(A(x0 + h, φ(x0 )) + π(A(x0 , φ(x0 ))||VZ 6 6 o(h) + o(h) + ||(π ◦ Ax0 )(φ(x0 + h)) − (π ◦ Ax0 )(φ(x0 )) − h π ◦ Ax0 (φ′ (x0 ))||VZ +

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+ ||(π ◦ Ax0 +h )(φ(x0 + h)) − (π ◦ Ax0 +h )(φ(x0 )) − − (π ◦ Ax0 )(φ(x0 + h)) + (π ◦ Ax0 )(φ(x0 ))||VZ 6 6 o(h) + ||π ◦ Ax0 (φ(x0 + h)) − π ◦ Ax0 (φ(x0 )) − π ◦ Ax0 (φ′ (x0 )(h))||VZ + + ||π ◦ Ax0 +h (φ(x0 + h)) − π ◦ Ax0 +h (φ(x0 )) − − π ◦ Ax0 (φ(x0 + h)) + π ◦ Ax0 (φ(x0 ))||VZ 6 6 o(h) + ||π ◦ Ax0 [φ(x0 + h) − φ(x0 ) − φ′ (x0 )(h)]||VZ + + ||π ◦ Ax0 +h [φ(x0 + h) − φ(x0 )] + π ◦ Ax0 [φ(x0 + h) − φ(x0 )]||VZ 6 6 o(h) + ||π ◦ Ax0 [φ(x0 + h) − φ(x0 ) − φ′ (x0 )(h)]||VZ + + ||[ π ◦ Ax0 +h − π ◦ Ax0 ](φ(x0 + h) − φ(x0 ))||VZ 6 6 o(h) + ||π ◦ Ax0 ||L(Y,VZ ) ||[φ(x0 + h) − φ(x0 ) − φ′ (x0 )(h)]||Y + + ||[ π ◦ Ax0 +h − π ◦ Ax0 ]||L(Y,VZ ) ||(φ(x0 + h) − φ(x0 ))||Y , (21) where o(h) is the Landau symbol. Let us note that there exists a constant M > 0 such that ||π ◦ Ax ||L(Y,VZ ) 6 M for all x ∈ I, since the function (17) is continuous and its domain is a compact set. Now let us write φ(x0 + h) = φ(x0 ) + hφ′ (x0 ) + |h|f (h), where ||f (h)||Y → 0 for h → 0. Thus, from inequality (18), (21) is less than or equal to o(h) + M o(h) + M0 L[φ′ (x0 ) + f (h)] |h|2 = o(h), which completes the proof of (20). Moreover, (20) implies that N maps the set C 1 (I, C) into the space C 1 (I, cc(Z)). Now we shall prove that there exists a constant L′ > 0 such that (15) holds. Let φ1 and φ2 belong to the space C 1 (I, C). From the definition of the metric dC 1 (I,cc(Z)) and from the definition of the norm || · ||C 1 (I,VZ ) we get dC 1 (I,cc(Z)) (N φ1 , N φ2 ) = ||π ◦ N φ1 − π ◦ N φ2 ||C 1 (I,VZ ) = = ||(π ◦ N φ1 )(0) − (π ◦ N φ2 )(0)||VZ + sup ||(π ◦ N φ1 )′ (x) − (π ◦ N φ2 )′ (x)||VZ . x∈I

According to (19) and (20) we infer ||(π ◦ N φ1 )′ (x0 ) − (π ◦ N φ2 )′ (x0 )||VZ 6 6 ||(π ◦ Aφ1 (x0 ) )′ (x0 ) − (π ◦ Aφ2 (x0 ) )′ (x0 )||VZ + + ||π ◦ Ax0 (φ′1 (x0 )) − π ◦ Ax0 (φ′2 (x0 ))||VZ 6 6 M0 L||φ1 (x0 ) − φ2 (x0 )||Y + ||π ◦ Ax0 [φ′1 (x0 ) − φ′2 (x0 )]||VZ . It is easily seen that ||φ1 (x0 ) − φ2 (x0 )||Y 6 ||φ1 − φ2 ||C 1 (I,Y ) , thus sup ||(π ◦ N φ1 )′ (x) − (π ◦ N φ2 )′ (x)||VZ 6 x∈I

6 M0 L||φ1 − φ2 ||C 1 (I,Y ) + M sup ||(φ1 − φ2 )′ (x)||Y . x∈I

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229

Moreover, ||(π ◦ N φ1 )(0) − (π ◦ N φ2 )(0)||VZ = = ||π(A(0, φ1 (0)) + B(0)) − π(A(0, φ2 (0)) + B(0))||VZ = = ||π ◦ A0 (φ1 (0)) − π ◦ A0 (φ2 (0))||VZ 6 ||π ◦ A0 ||L(Y,VZ ) ||(φ1 − φ2 )(0)||Y 6 6 M ||(φ1 − φ2 )(0)||Y . It follows that dC 1 (I,cc(Z)) (N φ1 , N φ2 ) 6 L′ ||φ1 − φ2 ||C 1 (I,Y ) , where L′ := M + M0 L (> 0).

⊔ ⊓

Now lets turn our attention to absolutely continuous functions with values in the space cc(R). For brevity, in the following we shall write K instead of cc(R). Thus K consists of all non-empty, compact intervals (including degenerate ones) in R. For I, J ∈ K and α ∈ R we define I + J := {x + x′ : x ∈ I, x′ ∈ J} and αI := {αx : x ∈ I}. It is clear that if [a, b], [c, d] ∈ K and α > 0, then [a, b] + [c, d] = [a + c, b + d], α[a, b] = [αa, αb]. Of course we have d([a, b], [c, d]) = max{|a − c|, |b − d|}. Now consider the norm || · || in R2 , defined by ||(x, y)|| := max{|x|, |y|},

(x, y) ∈ R2 ,

and the map π : K ∋ [a, b] 7→ (a, b) ∈ R2 .

(22)

It is clear that the following relations π([a, b] + [c, d]) = π([a, b]) + π([c, d]), π(α[a, b]) = απ([a, b])

for α > 0,

d([a, b], [c, d]) = ||π([a, b]) − π([c, d])||, R2 = π(K) − π(K) hold. Thus, K can be embedded into the (of course, complete and reflexive) space R2 , endowed with the maximum norm (cf. [11]). Moreover, a space into which K is embedded in the above fashion is unique up to isometrical isomorphism – see Theorem 1.3. A function φ : I → Y is said to be absolutely continuous (cf. [3], p. 15), if for every ǫ > 0 there exists δ > 0 such that for any positive integer N and aPdisjoint family of N intervals (α1 , β1 ), (α2 , β2 ), . . . , (αN , βN ) in I whose lengths satisfy i=1 (βi − αi ) < δ, PN the inequality i=1 ||φ(βi ) − φ(αi )||Y < ǫ holds true. We define absolute continuity of a set-valued function F : I → K in a similar manner; instead of the distance generated by the norm || · ||Y , we consider the Hausdorff distance in K. We denote the space of

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all absolutely continuous functions defined on the interval I and with the values in the space Y by AC(I, Y ). Moreover, for a subset C ⊆ Y , by AC(I, C) we denote the set of all functions φ ∈ AC(I, Y ) such that φ(I) ⊆ C. Lemma 2.5 ([4], p. 44, Theorem 3.4). Let Y be a reflexive Banach space and let a function φ : I → Y be absolutely continuous. Then φ is differentiable a.e. on the interval I (with respect to the Lebesgue measure), φ′ is integrable in the sense of Bochner and Z φ(x) − φ(a) = φ′ (t)dt for a, x ∈ I. [a,x]

Now, let (Y, ||·||Y ) be a reflexive Banach space and let us define the norm ||·||AC(I,Y ) in the space AC(I, Y ) in the following way Z ||φ||AC(I,Y ) := ||φ(0)||Y + ||φ′ (t)||Y dt; (23) [0,1]

since φ′ is integrable the integral in (23) is finite. It is easy to see, that the following relation holds F ∈ AC(I, K) ⇔ π ◦ F ∈ AC(I, R2 ), where π is the R˚ adstr¨ om embedding (22). Since the function π is invertible, thus the formula dAC(I,K) (F1 , F2 ) := ||π ◦ F1 − π ◦ F2 ||AC(I,R2 ) ,

F1 , F2 ∈ AC(I, K),

(24)

defines a metric in AC(I, K). Now we shall formulate the following set-valued analogue of the J. Matkowski’s result (cf. [17], Theorem 1 and [18]). Theorem 2.6. Let Y be a reflexive Banach space and let C be a convex cone in Y . Assume that the Nemytskij operator N generated by G : I × C → K maps the set AC(I, C) into the space AC(I, K) and is uniformly continuous. Then there exist functions A : I → L(C, K), B : I → K such that B and A(·) y belong to the space AC(I, K) for every y ∈ C, the function A(x)(·) is uniformly continuous for every x ∈ I and G(x, y) = A(x)y + B(x), x ∈ I, y ∈ C. Also, if we assume that there exists L > 0, such that dAC(I,K) (N φ1 , N φ2 ) 6 L||φ1 − φ2 ||AC(I,Y ) ,

φ1 , φ2 ∈ AC(I, C),

(25)

then the function A(x) satisfies the Lipschitz condition with the constant L for every x ∈ I. Proof. Let us note, that G(·, y) ∈ AC(I, K). To see that let us fix y ∈ C and set φ1 (x) = y, x ∈ I. From the definition of N and assumption, that N maps the set AC(I, C) into the space AC(I, K), we get N φ1 = G(·, y) ∈ AC(I, K). In particular G is continuous with respect to the first variable. Now, let us take x1 , x2 , ..., x2n ∈ I such that 0 6 x1 < x2 < ... < x2n 6 1 and let u, v ∈ C. Moreover, let us define y1 = y 2 := (u + v)/2, y2 := v, y 1 := u. Consider a function φ1 : I → Y defined by

Nemytskij operator, R˚ adstr¨ om embedding and set-valued functions

   

y1 x−x2i−1 x2i −x2i−1 [y 1 x−x2i x2i+1 −x2i [y1

y1 + φ1 (x) = y +    1

231

for x ∈ [0, x1 ], for x ∈ [x2i−1 , x2i ], for x ∈ [x2i , x2i+1 ], for x ∈ [x2n , 1],

− y1 ] − y1 ]

y1

(26)

Moreover, let us define a function φ2 : I → Y , by putting y2 , y 2 instead of y1 , y1 , respectively, in definition (34). It is not difficult to check that φ1 , φ2 ∈ AC(I, C) and ||φ1 − φ2 ||AC(I,Y ) = ||(v − u)/2||Y . Now, let us note (cf. [18]), that there exists a function γ : [0, +∞) → [0, +∞), which is continuous at 0, satisfies the condition γ(0) = 0, and, moreover, for which the inequality dAC(I,K) (N φ1 , N φ2 ) 6 γ( ||φ1 − φ2 ||AC(I,Y ) ),

φ1 , φ2 ∈ AC(I, C)

holds; in fact, we can take γ(t) := sup{dAC(I,K)(N φ1 , N φ2 ) : φ1 , φ2 ∈ AC(I, C),

||φ1 − φ2 ||AC(I,Y ) 6 t}.

It is not difficult to show, that uniform continuity of N implies, that the values of γ are finite, γ(0) = 0 and γ is continuous at 0. Thus, from Lemma 2.5, (23) and (24) we get γ( ||(v − u)/2||Y ) > dAC(I,K) (N φ1 , N φ2 ) > >

2n−1 X k=1

=

2n−1 X

||

Z

Z

||(π ◦ N φ1 − π ◦ N φ2 )′ (t)||dt > I

(π ◦ N φ1 − π ◦ N φ2 )′ (t)dt|| =

[xk ,xk+1 ]

||π ◦ N φ1 (xk+1 ) − π ◦ N φ2 (xk+1 ) − π ◦ N φ1 (xk ) + π ◦ N φ2 (xk )|| =

k=1

=

2n−1 X

||π(G(xk+1 , φ1 (xk+1 ))) − π(G(xk+1 , φ2 (xk+1 ))) −

k=1

− π(G(xk , φ1 (xk ))) + π(G(xk , φ2 (xk )))|| = =

n X

||π(G(x2k , y2 )) + π(G(x2k−1 , y1 )) − π(G(x2k , y 1 )) − π(G(x2k−1 , y2 ))|| +

k=1

+

n−1 X

||π(G(x2k , y2 )) + π(G(x2k+1 , y1 )) − π(G(x2k , y1 )) − π(G(x2k+1 , y2 ))||.

k=1

Let x be an arbitrary point of interval I and let xk → x for all k = 1, 2, . . . , 2n − 1. From the continuity of G with respect to the first variable and from the definitions of y1 , y2 , y1 , y2 , we infer that (2n − 1)||π(G(x, (u + v)/2) + π(G(x, (u + v)/2) − π(G(x, u)) − π(G(x, v))|| 6 6 γ( ||(v − u)/2||Y ).

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According to the fact that π is an isometry, we obtain   u + v  u + v   v − u  1 d G x, + G x, , G(x, u) + G(x, v) 6 γ , 2 2 2n − 1 2 Y and the above inequality holds for each positive integer n. Hence we get G(x, (u + v)/2) = [G(x, u) + G(x, v)]/2 for arbitrary x from I and u, v from C, i.e. G satisfies the Jensen equation with respect to the second variable. By virtue of Lemma 1.2 we obtain G(x, y) = A(x)y + B(x),

(27)

where B : I → K and A(x) : C → K, for every x ∈ I. To prove that B ∈ AC(I, K), let us note that G(x, 0) = A(x)0 + B(x) = {0} + B(x) = B(x). Moreover, G(·, 0) = N φ, where the function φ : I → C is given by φ(x) = 0 for x ∈ I. Since N takes its values in the space AC(I, K), B is absolutely continuous. Now we shall prove that A(·)y ∈ AC(I, K) for y ∈ C. Fix y ∈ C. Let us consider a function φ : I → C given by φ(x) = y for x ∈ I. From definition (3), equality (27) and from additivity of π we get π(N φ(x)) = π(G(x, y)) = π(A(x)y) + π(B(x)). Hence π ◦ A(·)y = π ◦ N φ − π ◦ B. Moreover π ◦ N φ, π ◦ B ∈ AC(I, R2 ). Thus π ◦ A(·)y ∈ AC(I, R2 ), which implies that A(·)y ∈ AC(I, K). Now we shall prove that the inequality d(G(x, y1 ), G(x, y2 )) 6 γ( ||y1 − y2 ||Y ),

x ∈ I, y1 , y2 ∈ C

(28)

holds. Let us fix x ∈ I, y1 , y2 ∈ C. Define φ1 , φ2 : I → C as follows: φ1 (t) = y1 , φ2 (t) = y2 for t ∈ I. It is obvious that φ1 , φ2 ∈ AC(I, C). Let us note that ||φ1 − φ2 ||AC(I,Y ) = ||y1 − y2 ||Y . Since π is an isometry, from triangle inequality we get d(G(x, y1 ), G(x, y2 )) = ||π(G(x, y1 )) − π(G(x, y2 ))|| 6 6 ||π(G(0, y1 )) − π(G(0, y2 ))|| + + ||π(G(x, y1 )) − π(G(x, y2 )) − [π(G(0, y1 )) − π(G(0, y2 ))]||. Moreover, since d(G(0, y1 ), G(0, y2 )) = ||(π ◦ N φ1 − π ◦ N φ2 )(0)|| and ||π(G(x, y1 )) − π(G(x, y2 )) − [π(G(0, y1 )) − π(G(0, y2 ))]|| = = ||(π ◦ N φ1 − π ◦ N φ2 )(x) − (π ◦ N φ1 − π ◦ N φ2 )(0)|| = Z Z = (π ◦ N φ1 − π ◦ N φ2 )′ (t)dt 6 ||(π ◦ N φ1 − π ◦ N φ2 )′ (t)||dt, [0,x]

I

Nemytskij operator, R˚ adstr¨ om embedding and set-valued functions

233

we have d(G(x, y1 ), G(x, y2 )) 6 6 ||(π ◦ N φ1 − π ◦ N φ2 )(0)|| +

Z

||(π ◦ N φ1 − π ◦ N φ2 )′ (t)||dt =

I

= ||(π ◦ N φ1 − π ◦ N φ2 )||AC(I,R2 ) = dAC(I,K) (N φ1 , N φ2 ) 6 6 γ( ||φ1 − φ2 ||AC(I,Y ) ) 6 γ( ||y1 − y2 ||Y ), which completes the proof of inequality (28). Now from (28) and from Lemma 1.1 we get d(A(x)y1 , A(x)y2 ) = d(G(x, y1 ), G(x, y2 )) 6 γ( ||y1 − y2 ||Y ). Thus A(x) is uniformly continuous for x ∈ I, and hence A is a map from I into L(C, K). Moreover, if the operator N satisfies (25), then in an analogous way we can show, that the function A(x) satisfies the Lipschitz condition with the constant L for all x from the interval I, which completes the proof. ⊔ ⊓ In the following theorem we shall give sufficient conditions for the Nemytskij operator to be lipschitzian. Let us note, that the reflexivity of the space Y implies the reflexivity of the space L(Y, R2 ) (cf. [8]). Theorem 2.7. Let Y be a reflexive Banach space, and let C be a convex cone with nonempty interior in the space Y . Moreover, let A and B be given functions such that A : I → L(C, K) and B : I → K. Assume, that the functions B and I ∋ x 7→ A(x) ∈ L(C, K) are absolutely continuous (the set L(C, K) is endowed with the metric (4)). If G : I × C → K is of the form G(x, y) = A(x)y + B(x),

x ∈ I, y ∈ C,

then the Nemytskij operator N generated by G maps the set AC(I, C) into the space AC(I, K) and satisfies the Lipschitz condition, i.e., there exists a constant L > 0 such that dAC(I,K) (N φ1 , N φ2 ) 6 L||φ1 − φ2 ||AC(I,Y ) , φ1 , φ2 ∈ AC(I, C). Proof. Let x ∈ I. Since the function π ◦ A(x) : C → R2 is additive and continuous, from Lemma 1.5 we get, that there is exactly one linear and continuous function π ◦ A(x) : Y → R2 such that π ◦ A(x)(y) = [π ◦ A(x)](y)

for y ∈ C.

We divide the proof into five steps. 1. We shall prove that the functions I ∋ x 7→ π ◦ A(x) ∈ L(Y, R2 )

(29)

and A(·)y for y ∈ C are absolutely continuous. Let 0 6 α1 < β1 6 α2 < β2 6 ... 6 αN < βN 6 1. Since π ◦ A(βi ), π ◦ A(αi ) are additive and continuous for i = 1, 2, ..., N , the function π ◦ A(βi ) − π ◦ A(αi ) is also additive and continuous. For y ∈ C \ {0} from (4) we get

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d(A(βi )y, A(αi )y) 6 ||y||Y dL(C,K) (A(βi ), A(αi )), which implies, that the function A(·)y is absolutely continuous for y ∈ C. Moreover, 1 ||[π ◦ A(βi ) − π ◦ A(αi )](y)|| 6 dL(C,K) (A(βi ), A(αi )). ||y||Y Thus ||π ◦ A(βi ) − π ◦ A(αi )||L(C,R2 ) 6 dL(C,K) (A(βi ), A(αi )). Obviously π ◦ A(βi ) − π ◦ A(αi ) = π ◦ A(βi ) − π ◦ A(αi ). Hence, from Lemma 1.5 we infer ||π ◦ A(βi ) − π ◦ A(αi )||L(Y,R2 ) 6 M0 ||π ◦ A(βi ) − π ◦ A(αi )||L(C,K) and thus we get N X

||π ◦ A(βi ) − π ◦ A(αi )||L(Y,R2 ) 6 M0

N X

dL(C,K) (A(βi ), A(αi )).

i=1

i=1

Since the function I ∋ x 7→ A(x) ∈ L(I, K) is absolutely continuous, so is the function I ∋ x 7→ π ◦ A(x) ∈ L(Y, R2 ). 2. Now we shall prove that N maps the set AC(I, C) into the space AC(I, K). Let us note that there exists a constant K such that ||π ◦ A(x)||L(Y,R2 ) 6 K,

x ∈ I.

(30)

In fact the function (29) is continuous (even absolutely continuous) and its domain is a compact set. Moreover, let us note, that inequality (30) implies that the function A(x) satisfies the Lipschitz condition with the constant K. Let φ ∈ AC(I, C) and consider the function I ∋ x 7→ A(x)φ(x) ∈ K. Moreover, let 0 6 α1 < β1 6 α2 < β2 6 ... 6 αN < βN 6 1. Since A(x) satisfies the Lipschitz condition with the constant K for every x from I, we get d(A(αi )φ(αi ), A(βi )φ(βi )) 6 6 d(A(αi )φ(αi ), A(αi )φ(βi )) + d(A(αi )φ(βi ), A(βi )φ(βi )) 6 6 K||φ(αi ) − φ(βi )||Y + ||[π ◦ A(αi )]φ(βi ) − [π ◦ A(βi )]φ(βi )|| 6 6 K||φ(αi ) − φ(βi )||Y + ||[π ◦ A(αi ) − π ◦ A(βi )]φ(βi )|| 6 6 K||φ(αi ) − φ(βi )||Y + max ||φ(t)||Y ||[π ◦ A(αi ) − π ◦ A(βi )]||L(Y,R2 ) , t∈I

Nemytskij operator, R˚ adstr¨ om embedding and set-valued functions

235

whence N X

d(A(αi )φ(αi ), A(βi )φ(βi )) 6

i=1

6K

N X

||φ(αi ) − φ(βi )||Y + max ||φ(t)||Y t∈I

i=1

N X

||[π ◦ A(αi ) − π ◦ A(βi )]||L(Y,R2 ) .

i=1

Since the functions I ∋ x 7→ φ(x) ∈ C and I ∋ x 7→ π ◦ A(x) ∈ L(Y, R2 ) are absolutely continuous, so is the function I ∋ x 7→ A(x)φ(x) ∈ K. Therefore, the function N φ, given by N φ(x) = A(x)φ(x) + B(x), is absolutely continuous. 3. Now, let x ∈ (0, 1) and assume that the derivatives φ′ (x), (π ◦ B)′ (x) and (π ◦ N φ)′ (x) exist (since the functions φ, π ◦ B, π ◦ N φ are absolutely continuous, their derivatives exist a.e. on the interval I). We shall prove that the following formula holds (π ◦ [A(·)φ(x)])′ (x) = (π ◦ N φ)′ (x) − (π ◦ B)′ (x) − π ◦ A(x) (φ′ (x)).

(31)

Since φ is differentiable at x we can write φ(x + h) = φ(x) + hφ′ (x) + hf (h), where f is a function defined on the neighbourhood of 0 in R and with the values in the space Y , such that limh→0 f (h) = 0. For h ∈ R with sufficiently small |h| we have (π ◦ N φ)(x + h) − (π ◦ N φ)(x) = = π(A(x + h)φ(x + h)) + π(B(x + h)) − π(A(x)φ(x)) − π(B(x)) = = π(A(x + h)φ(x)) − π(A(x)φ(x)) + h π ◦ A(x + h)φ′ (x) + + h π ◦ A(x + h)f (h) + π(B(x + h)) − π(B(x)); whence (1/h)[π(A(x + h)φ(x)) − π(A(x)φ(x))] = = (1/h)[(π ◦ N φ)(x + h) − (π ◦ N φ)(x)] − (1/h)[(π ◦ B)(x + h) − (π ◦ B)(x)] − − π ◦ A(x + h)φ′ (x) − π ◦ A(x + h)f (h), which implies that (31) holds. 4. Now, let x ∈ I, y1 , y2 ∈ C, and let us assume that the derivatives of the functions π◦[A(·)y1 ], π◦[A(·)y2 ], π ◦ A(·) exist at the point x. It is easily seen that the inequality   ||(π ◦ [A(·)y1 ])′ (x) − (π ◦ [A(·)y2 ])′ (x)|| 6 || π ◦ A(·) ′ (x)||L(Y,R2 ) ||y1 − y2 ||Y (32) holds for x ∈ I, y1 , y2 ∈ C. 5. Now we shall prove that the operator N is lipschitzian. Let φ1 and φ2 belong to the set AC(I, C). From (23) and (24) we get

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dAC(I,K) (N φ1 , N φ2 ) = ||π ◦ N φ1 − π ◦ N φ2 ||AC(I,R2 ) = Z = ||(π ◦ N φ1 )(0) − (π ◦ N φ2 )(0)|| + ||(π ◦ N φ1 )′ (x) − (π ◦ N φ2 )′ (x)||dx. [0,1]

According to (31) and (32) we infer ||(π ◦ N φ1 )′ (x) − (π ◦ N φ2 )′ (x)|| 6 6 ||(π ◦ [A(·)φ1 (x)])′ (x) − (π ◦ [A(·)φ2 (x)])′ (x)|| + ||π ◦ A(x)φ′1 (x) − π ◦ A(x)φ′2 (x)|| 6   6 || π ◦ A(·) ′ (x)||L(Y,R2 ) ||φ1 (x) − φ2 (x)||Y + + ||π ◦ A(x)||L(Y,R2 ) ||φ′1 (x) − φ′2 (x)||. It is easy to see that ||φ1 (x) − φ2 (x)||Y 6 ||φ1 − φ2 ||AC(I,Y ) for x ∈ I; thus Z

||(π ◦ N φ1 )′ (x) − (π ◦ N φ2 )′ (x)|| dx 6 Z Z   6 ||φ1 − φ2 ||AC(I,Y ) || π ◦ A(·) ′ (x)||L(Y,R2 ) dx + K [0,1]

[0,1]

||(φ1 − φ2 )′ (x)||Y dx.

[0,1]

Moreover, ||(π ◦ N φ1 )(0) − (π ◦ N φ2 )(0)|| = = ||π(A(0)φ1 (0) + B(0)) − π(A(0)φ2 (0) + B(0))|| = = ||π ◦ A(0)(φ1 (0)) − π ◦ A(0)(φ2 (0))|| 6 6 ||π ◦ A(0)||L(Y,R2 ) ||(φ1 − φ2 )(0)||Y 6 K||(φ1 − φ2 )(0)||Y and finally dAC(I,K) (N φ1 , N φ2 ) 6 L||φ1 − φ2 ||AC(I,Y ) ,   where L = K + [0,1] || π ◦ A(·) ′ (x)||L(Y,R2 ) dx (> 0). R

⊔ ⊓

Remark 2.8. It is a natural reaction to try to generalize the method of proofs of Theorems 2.6 and 2.7 to the more general situations. Thus we are led to the question, whether the Hausdorff completion of the R˚ adstr¨om space VZ is reflexive. The positive answer to the simplest case of the space cc(R) is given above. However, even in the case of cc(Rn ) for n > 1, this problem is still unresolved.

Bibliography 1. Appel J., Zabrejko P.P.: Nonlinear Superposition Operators. Cambridge Univ. Press, Cambridge 1990. 2. Banks H.T., Jacobs M.Q.: A differential calculus for multifunctions. J. Math. Anal. Appl. 29 (1970), 246–272.

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3. Barbu V.: Nonlinear Semigroups and Differential Equations in Banach Spaces. Editura Academiei Bucure¸sti Romˆ ania, Noordhoff International Publishing Leyden, The Netherlands 1976. 4. Barbu V., Precupanu Th.: Convexity and Optimization in Banach Spaces. Editura Academiei Bucure¸sti Romˆ ania, Sijthoff and Noordhoff International Publishers 1978. 5. Chistyakov V.V.: Generalized variation of mappings with applications to composition operators and multifunctions. Positivity 5 (2001), 323–358. 6. Chistyakov V.V.: Selections of bounded variation. J. Appl. Anal. 10 (2004), 1–82. 7. Chistyakov V.V.: Lipschitzian Nemytskii operators in the cones of mappings of bounded Wiener φ-variation. Folia Math. 11 (2004), 15–39. 8. Heinrich S.: The reflexivity of the Banach space L(E, F ). Functional Anal. Appl. 8 (1974), 97–98 (in Russian). 9. Ludew J.J.: On Lipschitzian operators of substitution generated by set-valued functions. Opuscula Math. 27 (2007), 13–24. 10. Ludew J.J.: On Nemytskij operator of substitution in the C 1 space of set-valued functions. Demonstratio Math. 41 (2008), 403–414. 11. Ludew J.J.: On Nemytskij operator in the space of absolutely continuous set-valued functions. J. Appl. Anal. 17 (2011), 277–290. 12. Matkowska A.: On a characterization of Lipschitzian operators of substitution in the class of H¨ olders functions. Zeszyty Nauk. Politech. L´ odz. Mat. 17 (1984), 81–85. 13. Matkowska A., Matkowski J., Merentes N.: Remark on globally Lipschitzian composition operators. Demonstratio Math. 28 (1995), 171–175. 14. Matkowski J.: On Lipschitzian solution of a functional equation. Ann. Polon. Math. 28 (1973), 135–139. 15. Matkowski J.: Functional equations and Nemytskij operators. Funkc. Ekvacioj Ser. Int. 25 (1982), 127–132. 16. Matkowski J.: Form of Lipschitz operators of substitution in Banach spaces of differentiable functions. Zeszyty Nauk. Politech. L´ odz. Mat. 17 (1984), 81–85. 17. Matkowski J.: Lipschitzian composition operators in some function spaces. Nonlinear Anal., Theory, Methods Appl. 30 (1997), 719–726. 18. Matkowski J.: Uniformly continuous superposition operators in the spaces of differentiable functions and absolutely continuous functions. Intern. Ser. Numer. Math. 157 (2008), 155–166. 19. Merentes N., Nikodem K.: On Nemytskii operator and set-valued functions of bounded pvariation. Rad. Mat. 8 (1992), 139–145. 20. Nikodem K.: K-convex and K-concave set-valued functions. Zeszyty Nauk. Politech. L´ odz., Mat. 559 (1989), Rozprawy Nauk. 114. 21. R˚ adstr¨ om H.: An embedding theorem for spaces of convex sets. Proc. Amer. Math. Soc. 3 (1952), 165–169. 22. Smajdor A.: On regular multivalued cosine families. Ann. Math. Sil. 13 (1999), 271–280. 23. Smajdor A., Smajdor W.: Jensen equation and Nemytskij operator for set-valued functions. Rad. Mat. 5 (1989), 311–320. 24. Solycheva O.M.: Lipschitzian superposition operators on metric semigroups and abstract convex cones of mappings of finite Λ-variation. Siberian Math. J. 47 (2006), 473–486. 25. Zawadzka G.: On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation. Rad. Mat. 6 (1990), 279–293.

Functional equations and Nemytskij operator Ewelina Mainka-Niemczyk and Jakub Jan Ludew

Abstract. In this chapter we prove that a functional equation of the form φ(x) = g(x, φ[f (x)]) has a unique solution under some assumptions and consider the problem of determining it (for that we use a classical theorem due to J. Matkowski). Furthermore, we prove the set-valued analogue of Matkowski’s result. Keywords: Nemytskij operator, Lipα space, set-valued functions, functional equations, Jensen equation. 2010 Mathematics Subject Classification: 47H04, 47H30, 54C60, 26E25.

Introduction The following chapter consists of two sections. In the first section we consider a functional equation of the form φ(x) = g(x, φ[f (x)]). We apply the Schauder Fixed Point Theorem to show that this equation has a solution under some conditions. Moreover, we consider the problem of determining its solution with the help of the Banach Fixed Point Theorem. We prove that the Banach Principle can be applied only in the linear case. The crucial result on which the proof of this fact is based on, is a classical theorem due to J. Matkowski. In 1982 he showed (cf. [11]), that a Nemytskij operator N mapping the function space Lip(I, R) into itself is Lipschitzian with respect to the Lipschitzian norm if and only if its generator is of the form E. Mainka-Niemczyk, J.J. Ludew Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: {Ewelina.Mainka, Jakub.Ludew}@polsl.pl R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 239–257. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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g(x, y) = a(x)y + b(x),

x ∈ I, y ∈ R,

for some a, b ∈ Lip(I, R). This result was extended to many spaces by J. Matkowski and others (cf. e.g. [12]). Set-valued versions of Matkowski’s results were investigated for instance in papers [3,6–9,15,17,21]. Recently Matkowski has shown (cf. [14]), that if we only assume, that the operator N is uniformly continuous, then the generator g is of the form above. The second part of this chapter contains results from a paper by E. Mainka (cf. [9]). The main goal of it is to prove the set-valued analogue of Matkowski’s result for superposition operators mapping the set Hα (I, C) of all H¨older functions ϕ : I → C into the set Hβ (I, clb(Z)) of all H¨ older set-valued functions φ : I → clb(Z).

1. Fixed Point Theorems and Nemytskij operators In the following we shall write I for a unit interval [0, 1] on the real line. If E, E ′ are nonempty sets, then let us denote by F (E, E ′ ) the set consisting of all maps from E into E ′ . Real bounded functions defined on interval I form a Banach space B(I, R) with the uniform convergence norm || · ||B(I,R) . Real continuous functions defined on I form closed linear subspace C(I, R) of B(I, R), therefore it is a Banach space. The uniform convergence norm in it we denote by || · ||C(I,R) . Moreover, the set Lip(I, R) of all real functions defined on I and satisfying the Lipschitz condition with the norm given by |φ(x1 ) − φ(x2 )| ||φ||Lip(I,R) := |φ(0)| + sup , φ ∈ Lip(I, R) |x1 − x2 | x1 ,x2 ∈I x1 6=x2

is also a Banach space. Now let us consider functional equation φ(x) = g(x, φ[f (x)]),

(1)

where g : I × R → R and f : I → I are given functions. We seek for the solution φ : I → R of equation (1). Theorem 1.1. Assume that a function f : I → I satisfies the Lipschitz condition |f (x1 ) − f (x2 )| 6 s|x1 − x2 |,

x1 , x2 ∈ I

(2)

with the constant 0 < s < 1. Let f (0) = 0 and let g : I × R → R be a function for which the inequality |g(x1 , y1 ) − g(x2 , y2 )| 6 p|x1 − x2 | + q|y1 − y2 |,

x1 , x2 ∈ I, y1 , y2 ∈ R,

(3)

holds for p, q > 0 and let sq < 1. Moreover, let d ∈ R be a fixed point of the function g(0, ·), i.e. d = g(0, d). (4) Then there exists exactly one solution of equation (1) in the class Lip(I, R), for which φ(0) = d.

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Proof. Let Φ0 be a subset of C(I, R) consisting of all the functions φ, for which φ(0) = d and which satisfy the Lipschitz condition with a constant k, where k := p/(1 − sq).

(5)

It is easy to see that Φ0 is a convex and uniformly closed subset of C(I, R). Moreover, functions from the set Φ0 satisfy Lipschitz condition with a fixed constant (equal to k). Hence Φ0 is uniformly bounded and constitutes an equicontinuous family of functions. Thus Arzela-Ascoli theorem implies that Φ0 is a compact subset of the space C(I, R). Now, let us define a map T : Φ0 → F(I, R) as follows: T (φ)(x) := g(x, φ[f (x)]),

φ ∈ Φ0 , x ∈ I.

We are going to show that the range of T is a subset of Φ0 . Let φ ∈ Φ0 and let x1 , x2 ∈ I. From (5) and from inequalities (2) and (3) we get |T (φ)(x1 ) − T (φ)(x2 )| = |g(x1 , φ[f (x1 )]) − g(x2 , φ[f (x2 )]) 6 6 p|x1 − x2 | + q|φ[f (x1 )] − φ[f (x2 )]| 6 p|x1 − x2 | + qks|x1 − x2 | 6 6 (p + qks)|x1 − x2 | = k|x1 − x2 |, thus T (φ) satisfies the Lipschitz condition with the constant k. Moreover, from (4) it follows that T (φ)(0) = g(0, φ[f (0)]) = g(0, φ(0)) = g(0, d) = d, which finishes the proof of the fact, that T is a self-map of the set Φ0 . Now we are going to show that T is continuous. For this, assume that φ1 , φ2 ∈ Φ0 . From inequality (3) we get |T (φ1 )(x) − T (φ2 )(x)| = |g(x, φ1 [f (x)]) − g(x, φ2 [f (x)])| 6 q|φ1 [f (x)] − φ2 [f (x)]| 6 6 q sup |φ1 (x) − φ2 (x)| = q||φ1 − φ2 ||C(I,R) x∈I

and in consequence ||T (φ1 ) − T (φ2 )||C(I,R) 6 q||φ1 − φ2 ||C(I,R) , which implies the continuity of T . From Schauder Theorem we infer that T has a fixed point φ0 , which, on account of the definition of the set Φ0 , is a lipschitzian solution of (1) satisfying the equality φ0 (0) = d. It remains to show that these conditions determine a solution uniquely. Let φ1 and φ2 be solutions of equation (1), for which φ1 (0) = φ2 (0) = d and which satisfy the Lipschitz condition with a constant L. Let us define Q1 , Q2 : I → R as follows:  (1/x)[φi (x) − d] for x ∈ (0, 1], Qi (x) := 0 for x = 0, for i = 1, 2 and let us note, that φi (x) = d + xQi (x), x ∈ I. Since φ1 and φ2 satisfy the Lipschitz condition with the constant L, we obviously have |Qi (x)| 6 L, i = 1, 2, x ∈ I. Thus (1) implies that

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d + Qi (x)x = g(x, d + f (x)Qi [f (x)]). Now define a map h : (0, 1] × R → R in the following way h(x, y) :=

1 [g(x, d + yf (x)) − d] x

and let us note, that for x ∈ (0, 1] we have h(x, Qi [f (x)]) =

1 [g(x, d + f (x)Qi [f (x)]) − d] = Qi (x). x

(6)

Let x ∈ (0, 1], y1 , y2 ∈ R. From (2), (3) and from the inclusion f (I) ⊆ I we get |h(x, y1 ) − h(x, y2 ))| =

1 |g(x, d + y1 f (x)) − g(x, d + y2 f (x))| 6 x

1 1 qf (x)|y1 − y2 | 6 q|f (x) − f (0)||y1 − y2 | = qs|y1 − y2 |. x x Thus, on account of (7) we get for x ∈ (0, 1] 6

(7)

|Q1 (x) − Q2 (x)| = |h(x, Q1 [f (x)]) − h(x, Q2 [f (x)])| 6 6 qs|Q1 [f (x)] − Q2 [f (x)]| 6 qs||Q1 − Q2 ||B(I,R) . Taking the supremum over x ∈ [0, 1] we obtain ||Q1 − Q2 ||B(I,R) 6 qs||Q1 − Q2 ||B(I,R) . Since we assumed that qs < 1, we have Q1 = Q2 and as a consequence we get φ1 = φ2 , which finishes the proof. ⊓ ⊔ Theorem 1.1 with local Lipschitz conditions on the functions f and g and with assumption f (x) > 0 for x > 0 is formulated in J. Matkowski’s work [10] (see also [5, Theorem 5.5.1, p. 205]). In the proof of Theorem 1.1 we obtained the existence of a lipschitzian solution of the equation (1) from the Schauder Fixed Point Theorem. Independently, we proved that such a solution is unique. Thus, in this situation it seems likely, that Theorem 1.1 could be proved with the help of the Banach Fixed Point Theorem. We are going to show, that the properties of the Nemytskij operator enable us to formulate the following conclusion: it is possible to apply the Banach Fixed Point Theorem only for the linear equation of type (1) (cf. [11] and [5, p. 206– 209]). Let E and E ′ be any given non-empty sets and let g : I × E → E ′ be a given function. We shall say, that g generates the Nemytskij operator N : F (I, E) → F(I, E ′ ), defined in the following way (N φ)(x) := g(x, φ(x)),

φ ∈ F(I, E), x ∈ I.

(8)

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Let us again consider (this time in the context of the definition of the Nemytskij operator), the functional equation of the form φ(x) = g(x, φ[f (x)]),

(9)

where g : I × R → R and f : I → I. Let us note that if we define S by S(φ) := φ ◦ f,

φ ∈ F(I, R),

then this definition enables us to write equation (9) in the following form φ = (N ◦ S)φ. Assume now, that f : I → I is a lipschitzian function. It is easy to see, that the composition φ◦f (= S(φ)) is an element of the space Lip(I, R) for every φ ∈ Lip(I, R). Thus S is a self-map of the space Lip(I, R). Moreover, S is a continuous, linear operator on the Banach space Lip(I, R). We shall now prove continuity. Assume that f (0) > 0. First, let us note that |φ(f (0))| 6 |φ(0)| + |φ(f (0)) − φ(0)| = |φ(0)| + Thus |φ(f (0))| 6 |φ(0)| + f (0) sup x1 ,x2 ∈I x1 6=x2

|φ(f (0)) − φ(0)| f (0). |f (0) − 0|

|φ(x1 ) − φ(x2 )| |x1 − x2 |

and the obtained inequality is also true in the case f (0) = 0. Moreover, let us note that for t1 , t2 ∈ I, for which the inequality f (t1 ) 6= f (t2 ) is satisfied, we have |φ(f (t1 )) − φ(f (t2 ))| |φ(f (t1 )) − φ(f (t2 ))| |f (t1 ) − f (t2 )| = 6 |t1 − t2 | |f (t1 ) − f (t2 )| |t1 − t2 | 6 sup x1 ,x2 ∈I x1 6=x2

|φ(x1 ) − φ(x2 )| |f (x1 ) − f (x2 )| sup |x1 − x2 | |x1 − x2 | x1 ,x2 ∈I x1 6=x2

and the inequality (obtained, on account of the assumption t1 , t2 ∈ I, f (t1 ) 6= f (t2 )): |φ(f (t1 )) − φ(f (t2 ))| |φ(x1 ) − φ(x2 )| |f (x1 ) − f (x2 )| 6 sup sup |t1 − t2 | |x1 − x2 | |x1 − x2 | x1 ,x2 ∈I x1 ,x2 ∈I x1 6=x2

x1 6=x2

is obviously true also in the case f (t1 ) = f (t2 ). We conclude, that sup x1 ,x2 ∈I x1 6=x2

|φ(f (x1 )) − φ(f (x2 ))| |φ(x1 ) − φ(x2 )| |f (x1 ) − f (x2 )| 6 sup sup . |x1 − x2 | |x1 − x2 | |x1 − x2 | x1 ,x2 ∈I x1 ,x2 ∈I x1 6=x2

x1 6=x2

Thus ||S(φ)||Lip(I,R) = ||φ ◦ f ||Lip(I,R) = |φ(f (0))| + sup

x1 6=x2 x1 ,x2 ∈I

|φ(f (x1 )) − φ(f (x2 ))| = |x1 − x2 |

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= |φ(0)| + sup x1 6=x2 x1 ,x2 ∈I

|φ(x1 ) − φ(x2 )| h |f (x1 ) − f (x2 )| i f (0) + sup = |x1 − x2 | |x1 − x2 | x1 6=x2

= |φ(0)| + sup x1 6=x2 x1 ,x2 ∈I

x1 ,x2 ∈I

|φ(x1 ) − φ(x2 )| ||f ||Lip(I,R) 6 M ||φ||Lip(I,R) , |x1 − x2 |

where M := max{1, ||f ||Lip(I,R) }, from which we infer the continuity of S. Now we are going to show that f is a surjective self-map of the interval I if and only if S maps the space Lip(I, R) injectively into itself. Assume first, that f (I) = I and consider functions φ, ψ ∈ Lip(I, R) for which S(φ) = S(ψ). The definition of S implies, that φ(f (x)) = ψ(f (x)) for x ∈ I. On account of surjectivity of f , every element y in the interval I is of the form y = f (x) and we infer that φ(y) = ψ(y) for every y ∈ I, which finishes the proof of the injectivity of the map S. Conversely, assume that f (I) ⊆ I, f (I) 6= I. Continuity of f (which is a consequence of the Lipschitz condition) and compactness of its domain imply that f is bounded and f (I) = [m, M ], where m := inf f (x), M := sup f (x). x∈I

x∈I

The assumption f (I) ⊆ I, f (I) 6= I implies that at least one of the inequalities 0 < inf x∈I f (x), supx∈I f (x) < 1 hold. Let us define fe: I → R by  f (m)  for x ∈ [0, m], if 0 < m,  m x e f (x) for x ∈ [m, M ](= f (I)), f (x) =   1−f (M) (x − M ) + f (M ) for x ∈ [M, 1], if M < 1. 1−M Thus fe is an affine function on the interval [0, m] (if 0 < m) and on the interval [M, 1] (if M¡1). Moreover, it satisfies conditions fe(0) = 0, fe(1) = 1 and it is a continuous extension of the function f |f (I) to the interval I. It is also obvious, that fe satisfies the Lipschitz condition. From the definition of the function fe we infer, that for x ∈ I the equality fe(f (x)) = f (f (x)) holds. Thus fe ◦ f = f ◦ f and we obtain equality S(fe) = S(f ). Moreover, let us note that at least one of the inequalities 0 < f (0), f (1) < 1 holds (since f (I) 6= I), which, along with the equalities fe(0) = 0, fe(1) = 1 implies, that fe 6= f . Thus, the equality S(fe) = S(f ) enables us to conclude, that S is not injective. Now, assume that f (I) = I and let S be a surjective map of the space Lip(I, R) onto itself. Thus S is a continuous and linear bijection, for which the inverse map (on account of the Banach Open Mapping Theorem) is also continuous. From surjectivity of S we obtain that for every function φ ∈ Lip(I, R) there exists a function ψ ∈ Lip(I, R), for which φ(x) = ψ(f (x))(= S(ψ)), x ∈ I. In particular, for a function φ given by φ(x) = x for x ∈ I, there exists a function ψ0 ∈ Lip(I, R), for which the equality

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245

ψ0 (f (x)) = x holds for x ∈ I. From this equality we infer that f is injective (and, in consequence, it is a bijection), and that f −1 = ψ0 , which implies that f −1 satisfies the Lipschitz condition. Let us assume once again that φ and ψ belong to the space Lip(I, R) and let the equality S(ψ) = φ hold. Hence φ(x) = ψ(f (x)) for x ∈ I, which implies that ψ(y) = φ(f −1 (y)) for y ∈ I. Thus S −1 (φ) = φ ◦ f −1 and this equality is satisfied for every function φ ∈ Lip(I, R). Conversely, if f is bijective and lipschitzian function on I, for which f −1 is also lipschitzian, then S is a bijective self-map of the space Lip(I, R), the equality S −1 (φ) = φ ◦ f −1 holds and the map S −1 is continuous. Assume that we are trying to apply the Banach Fixed Point Theorem to equation (9). For this, we have to assume that N ◦ S is a contraction (with a constant k < 1) of the space Lip(I, R) (with the lipschitzian norm). Let φ ∈ Lip(I, R). Then S −1 (φ) ∈ Lip(I, R) and (N ◦S)(S −1 (φ)) ∈ Lip(I, R), since we are assuming that N ◦S maps the space Lip(I, R) into itself. Also (N ◦ S)(S −1 (φ)) = N φ and we infer that N maps the space Lip(I, R) into itself. Now, let φ1 , φ2 ∈ Lip(I, R). Since N ◦ S is a contraction, we obtain ||N φ1 − N φ2 ||Lip(I,R) 6 ||(N ◦ S)(S −1 (φ1 )) − (N ◦ S)(S −1 (φ2 ))||Lip(I,R) 6 6 k||S −1 (φ1 − φ2 )||Lip(I,R) 6 k||S −1 ||L(Lip(I,R)) ||φ1 − φ2 ||Lip(I,R) . Thus we infer, that N satisfies the Lipschitz condition. Now we shall quote the following Theorem of J. Matkowski [11] (see also [5, Theorem 5.5.2, p. 207]). Theorem 1.2. Let N be a Nemytskij operator generated by a function g : I × R → R. Conditions 1) N : Lip(I, R) → Lip(I, R), 2) there exists a constant L > 0 such that ||N φ1 − N φ2 ||Lip(I,R) 6 L||φ1 − φ2 ||Lip(I,R) ,

φ1 , φ2 ∈ Lip(I, R)

are simultaneously satisfied if and only if there exist functions a, b ∈ Lip(I, R) for which g(x, y) = a(x)y + b(x), x ∈ I, y ∈ R. This theorem enables us to conclude, that if the operator N ◦ S is a contraction, then (given that f is bijective and lipschitzian along with its inverse) N φ(x) = a(x)φ(x) + b(x), where a, b ∈ Lip(I, R). Thus, in the nonlinear case it is not possible to apply the Banach Fixed Point Theorem to investigate the solvability of equation (1). Theorem 1.2 implies also, that it is possible to apply the Banach Principle to determine the lipschitzian solution of the linear equation of the form φ(x) = a(x)φ[f (x)] + b(x).

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2. Uniformly continuous Nemytskij operators For given intervals I, J ⊂ R and numbers α ∈ (0, 1], x0 ∈ I let us define the set Lipα (I, J) of all functions ϕ : I → J for which the set   |ϕ(x) − ϕ(y)| : x, y ∈ I, x = 6 y |x − y|α is bounded with the functional ||ϕ||Lipα = |ϕ(x0 )| +

sup x,y∈I,x6=y

|ϕ(x) − ϕ(y)| . |x − y|α

(10)

In [14] Matkowski has shown that if a uniformly continuous with respect to the norm (10) superposition operator N of a generator f maps the set Lipα (I, J) into the Banach space Lipα (I, R), then for some a, b ∈ Lipα (I, R) we have f (x, y) = a(x)y + b(x),

x ∈ I, y ∈ J.

Our main goal is to prove a counterpart of Matkowski’s result for Nemytskij operators generated by set-valued functions with values in a set clb(Z) of all nonempty, bounded, closed, convex subsets of a normed linear space Z. Let (Z, || · ||) be a real, normed linear space. For a bounded A ⊂ Z one can define a number ||A|| as follows ||A|| := sup{||z|| : z ∈ A}. ∗

By + we denote a binary operation in clb(Z) defined by the formula ∗

A + B = cl(A + B), where A + B is an algebraic sum of A and B and clA is the closure of A. Note, that for arbitrary A, B ∈ clb(Z) the set A + B does not have to be closed. A corresponding ∗

example can be found e.g. in [20]. The pair (clb(Z), +) is an Abelian semigroup with the set {0} as the zero element. We can multiply elements of clb(Z) by nonnegative numbers and the conditions ∗





1 · A = A, λ(µA) = (λµ)A, λ(A + B) = λA + λB, (λ + µ)A = λA + µA hold for all A, B ∈ clb(Z) and λ, µ > 0. This means that the set clb(Z) with operations ∗

+ and · is an abstract convex cone. The cancellation law, i.e. ∗



A + B = C + B =⇒ A = C in clb(Z) follows e.g. from Theorem II-17 in [2, p. 48]. It is easy to check that (clb(Z), d) is a metric space. It is complete, provided Z is a Banach space (cf. e.g. [2, p. 40]). It is easily seen that the Hausdorff distance is invariant with respect to translation, i.e., ∗



d(A + B, C + B) = d(A + B, C + B) = d(A, C)

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247

(cf. e.g. [4]) and d(λA, λB) = λd(A, B) for all λ > 0 and A, B, C ∈ clb(Z). We say that a subset C of a real linear space Y is a convex cone if λC ⊂ C for all λ > 0 and C + C ⊂ C. A set-valued function F : C → clb(Z) defined on a convex cone C is *additive (*Jensen) if     ∗ ∗ x+y 1 F (x + y) = F (x) + F (y) F = (F (x) + F (y)) 2 2 for all x, y ∈ C. A function F is Q+ -homogeneous if F (λy) = λF (y) for all λ ∈ Q ∩ [0, ∞) and y ∈ C. We shall need the following lemmas. Lemma 2.1 (Corollary 4 in [19]). Let C be a convex cone in a real linear space Y and let Z be a Banach space. A set-valued function F : C → clb(Z) is *Jensen if and only if there exist a *additive set-valued function A : C → clb(Z) and a set B ∈ clb(Z) such that ∗ F (x) = A(x) + B for all x ∈ C. Lemma 2.2 (Lemma 2 in [16]). Let Y, Z be two real, normed linear spaces and let C be a convex cone in Y . Suppose F is a Q+ -homogeneous set-valued function defined on C with nonempty values in Z. The equality lim

y→0, y∈C

||F (y)|| = 0

(11)

holds if and only if there exists a positive constant M such that ||F (y)|| 6 M ||y||

for y ∈ C.

In the set of all Q+ -homogeneous set-valued functions in C with nonempty values in Z, satisfying condition (11) we can introduce the functional ||F || =

||F (x)|| . ||x|| x6=0

sup x∈C,

(12)

By Lemma 2.2, ||F || is finite. We will call this functional a norm. Lemma 2.3 (Theorem 3 in [18], see also Lemma 4 in [16]). Let Y be a Banach space, Z a real, normed linear space and let C be a convex cone in Y . Suppose that (Fj : j ∈ J) is a family of *additive, continuous set-valued functions Fj : C → clb(Z). S If intC 6= ∅ and for each y ∈ C the set j∈J Fj (y) is bounded in Z, then there exists a constant M ∈ (0, ∞) such that sup ||Fj || 6 M. j∈J

We say that a function α : [0, ∞) → [0, ∞) is an α-function, if α(t) > 0 for t ∈ (0, ∞), α(0) = 0 = limt→0+ α(t), α(1) = 1 and both α and α∗ , where

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α∗ (t) =



t α(t)

0

for t ∈ (0, ∞), for t = 0,

are increasing (cf. [1, p. 182]). Observe, that the function α(t) = tp , where p ∈ (0, 1] is an α-function. For two α-functions α and β we write α ≺ β ⇐⇒ α(t) = O(β(t))

as t → 0+ .

Let α be an α-function, I = [0, 1] and let C be a convex cone in a real, normed linear space Y . The set Hα (I, C) consists, by definition, of all functions ϕ : I → C such that ω(ϕ, s) < ∞, s∈(0,1] α(s)

hα (ϕ) := sup

(13)

where ω(ϕ, s) := sup{||ϕ(x1 ) − ϕ(x2 )|| : x1 , x2 ∈ I, |x1 − x2 | 6 s} (cf. [12]). By Hα (I, clb(Z)) we denote the set of all set-valued functions φ : I → clb(Z) such that hα (φ) < ∞, where ω(φ, s) := sup{d(φ(x1 ), φ(x2 )) : x1 , x2 ∈ I, |x1 − x2 | 6 s}. Note, that all functions from Hα (I, C) and from Hα (I, clb(Z)) are continuous. In fact, let us fix x1 , x2 ∈ I and let ϕ ∈ Hα (I, C). We have ||ϕ(x1 ) − ϕ(x2 )|| 6 ω(ϕ, |x1 − x2 |) 6 hα (ϕ)α(|x1 − x2 |).

(14)

Since α is continuous at 0, by (13) and (14) ϕ is uniformly continuous. The same reasoning applies to φ ∈ Hα (I, clb(Z)). We introduce a metric ρα in the set Hα (I, C) putting ρα (ϕ, ϕ) = ||ϕ − ϕ||α , where ||ϕ||α := ||ϕ(0)|| + hα (ϕ). In the set Hα (I, clb(Z)) one can define a metric setting ω(φ, φ, s) , α(s) s∈(0,1]

dα (φ, φ) := d(φ(0), φ(0)) + sup

φ, φ ∈ Hα (I, clb(Z)),

where ω(φ, φ, s) := sup{d(φ(x1 ) + φ(x2 ), φ(x2 ) + φ(x1 )) : x1 , x2 ∈ I, |x1 − x2 | 6 s} (cf. [17]). It may be checked that dα (φ, φ) < ∞ and that dα is a metric in Hα (I, clb(Z)) (cf. [6]). Consider the set L(C, clb(Z)) := {A : C → clb(Z) : A is ∗ additive and continuous}.

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249

Since every *additive set-valued function A : C → clb(Z) is Q+ -homogeneous, for each A ∈ L(C, clb(Z)) we have ||A(y)|| 6 ||A|| ||y||,

y ∈ C,

where ||A|| is defined by (12). Thus, for A, B ∈ L(C, clb(Z)) we have d(A(y), B(y)) 6 ||A(y)|| + ||B(y)|| 6 (||A|| + ||B||)||y|| and dL (A, B) :=

d(A(y), B(y)) ||y|| y6=0

sup y∈C,

is finite. It is easily seen, that dL yields a metric in L(C, clb(Z)). Now let α, β be α-functions. We will prove (in Theorem 2.8) that a uniformly continuous operator of substitution N mapping Hα (I, C) into Hβ (I, clb(Z)) has to be generated by a function F : I × C → clb(Z) of the form ∗

F (x, y) = A(x, y) + B(x), where A(x, ·) is a *additive continuous set-valued function and A(·, y), B belong to Hβ (I, clb(Z)). Theorem 2.4. Let I = [0, 1] and Y be a real normed linear space, Z a Banach space and let C be a convex cone in Y . Assume that γ : [0, ∞) → [0, ∞) is continuous at 0, γ(0) = 0, and the superposition operator N is generated by a set-valued function F : I × C → clb(Z). (a) Suppose that N maps Hα (I, C) into Hβ (I, clb(Z)) and dβ (N ϕ, N ϕ) 6 γ(||ϕ − ϕ||α ),

ϕ, ϕ ∈ Hα (I, C)

(15)

Then there exist functions A : I×C → clb(Z) and B : I → clb(Z) such that A(·, y), B ∈ Hβ (I, clb(Z)) for every y ∈ C, A(x, ·) ∈ L(C, clb(Z)) for every x ∈ I and ∗

F (x, y) = A(x, y) + B(x),

x ∈ I, y ∈ C.

Moreover, the inequality d(A(x, y1 ) + A(x, y2 ), A(x, y1 ) + A(x, y2 )) 6 γ(||y1 − y2 ||)β(|x − x|)

(16)

holds for all x, x ∈ I and y1 , y2 ∈ C. (b) Assume that γ is increasing and the condition β1 ≺ γ( α1 ) does not hold. Then the operator N maps Hα (I, C) into Hβ (I, clb(Z)) and satisfies inequality (15) if and only if the function F is of the form F (x, y) = B(x),

x ∈ I, y ∈ C,

where B ∈ Hβ (I, clb(Z)). In this case N is a constant operator. Proof. (a) Note, that for a given y ∈ C a constant function ϕ(t) = y, t ∈ I, belongs to the space Hα (I, C). Since N maps Hα (I, C) into Hβ (I, clb(Z)), we have N ϕ = F (·, y) ∈ Hβ (I, clb(Z)). Consequently, F (·, y) is continuous.

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For arbitrarily fixed y, y ∈ C, take ϕ, ϕ : I → C defined by ϕ(t) = y,

ϕ(t) = y,

t ∈ I.

Then ϕ, ϕ ∈ Hα (I, C) and, by the assumption, functions N ϕ = F (·, y), N ϕ = F (·, y) belong to Hβ (I, clb(Z)) and ||ϕ − ϕ||α = ||y − y||. From the definition of the metric dβ there is d(N ϕ(0), N ϕ(0)) +

ω(N ϕ, N ϕ, 1) 6 dβ (N ϕ, N ϕ). β(1)

Therefore, by (15), for all x ∈ I: d(F (0, y), F (0, y)) + d(F (x, y) + F (0, y), F (x, y) + F (0, y)) 6 γ(||y − y||).

(17)

Since d(F (x, y), F (x, y)) = d(F (x, y) + F (0, y), F (x, y) + F (0, y)) 6 6 d(F (x, y) + F (0, y), F (x, y) + F (0, y)) + d(F (x, y) + F (0, y), F (x, y) + F (0, y)) = = d(F (0, y), F (0, y)) + d(F (x, y) + F (0, y), F (x, y) + F (0, y)) (17) shows that d(F (x, y), F (x, y)) 6 γ(||y − y||) for x ∈ I. This inequality, the continuity of γ at 0 and the equality γ(0) = 0 imply that F is continuous with respect to the second variable. Let us fix x, x ∈ I, x < x, y1 , y2 , y1 , y 2 ∈ C and define functions  for 0 6 t 6 x,  yi i −yi ϕi (t) := yx−x (t − x) + yi for x < t < x  yi for x 6 t 6 1 for i = 1, 2. Obviously, ϕi (I) ⊆ C. We shall prove that ϕi ∈ Hα (I, C). It is easily seen that ω(ϕi , s) = ||y i − yi || for x − x 6 s 6 1, s ω(ϕi , s) = ||y − yi || for 0 6 s 6 x − x. x−x i Since the function t 7−→

t α(t)

is increasing ω(ϕi , s) ||y − yi || = i . α(s) α(x − x) s∈(0,1] sup

Hence, ϕi ∈ Hα (I, C) and ||ϕi ||α = ||yi || +

||yi −yi || α(x−x) .

||ϕ1 − ϕ2 ||α = ||y1 − y2 || +

In particular

||y1 − y 2 − y1 + y2 || . α(x − x)

(18)

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251

From (15) and the definition of dβ : ω(N ϕ1 , N ϕ2 , x − x) 6 dβ (N ϕ1 , N ϕ2 ) 6 γ(||ϕ1 − ϕ2 ||α ) β(x − x) and since ϕi (x) = yi and ϕi (x) = yi , d(F (x, y1 ) + F (x, y 2 ), F (x, y 1 ) + F (x, y2 )) 6 γ(||ϕ1 − ϕ2 ||α )β(x − x). Taking arbitrary u, v ∈ C and putting y1 = y 2 = ||ϕ1 − ϕ2 ||α =

u+v 2 ,

(19)

y 1 = u, y2 = v we get

||u − v|| 2

and

        u+v u+v ||u − v|| d F x, + F x, , F (x, u) + F (x, v) 6 γ β(x − x). 2 2 2

Letting x tend to x, since limt→0+ β(t) = 0, from the continuity of F with respect to the first variable we obtain     u+v d 2F x, , F (x, u) + F (x, v) = 0, 2 i.e.

  ∗ u+v 1 F x, = [F (x, u) + F (x, v)] 2 2

for all x ∈ I. This shows that F (x, ·) is *Jensen, therefore there exist functions A : I × C → clb(Z) and B : I → clb(Z) such that A(x, ·) is *additive for x ∈ I and ∗

F (x, y) = A(x, y) + B(x),

x ∈ I, y ∈ C

(20)

(cf. Lemma 2.1). To prove that A(x, ·) (x ∈ I) is continuous let us fix y, y ∈ C. We have ∗



d(A(x, y), A(x, y)) = d(A(x, y) + B(x), A(x, y) + B(x)) = d(F (x, y), F (x, y)), therefore, the continuity of F (x, ·) implies the continuity of A(x, ·). From the *additivity of A(x, ·) we get A(x, 0) = {0}, whence ∗

F (x, 0) = A(x, 0) + B(x) = B(x).

(21)

Since F (·, y) ∈ Hβ (I, clb(Z)) for all y ∈ C, (21) shows that B ∈ Hβ (I, clb(Z)). Now we shall prove that A(·, y) ∈ Hβ (I, clb(Z)) for every y ∈ C. Let us fix s ∈ (0, 1], x, x ∈ I such that |x − x| 6 s and y ∈ C. Obviously

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d(A(x, y), A(x, y)) = d(A(x, y) + B(x), A(x, y) + B(x)) 6 6 d(A(x, y) + B(x), A(x, y) + B(x)) + d(A(x, y) + B(x), A(x, y) + B(x)) = = d(F (x, y), F (x, y)) + d(B(x), B(x)), whence d(A(x, y), A(x, y)) 6 ω(F (·, y), s) + ω(B, s) and

ω(A(·, y), s) 6 hβ (F (·, y)) + hβ (B). β(s)

The inequality above shows now, that A(·, y) ∈ Hβ (I, clb(Z)) for every y ∈ C. To show (16) take x, x ∈ I such that x 6 x and y1 , y2 ∈ C. Setting y 1 = y1 , y 2 = y2 in (18) and (19) we obtain d(F (x, y1 ) + F (x, y2 ), F (x, y1 ) + F (x, y2 )) 6 γ(||y1 − y2 ||)β(x − x). Hence d(A(x, y1 ) + B(x) + A(x, y2 ) + B(x), A(x, y1 ) + B(x) + A(x, y2 ) + B(x)) = = d(A(x, y1 ) + A(x, y2 ), A(x, y1 ) + A(x, y2 )) 6 γ(||y1 − y2 ||)β(x − x). The obtained inequality d(A(x, y1 ) + A(x, y2 ), A(x, y1 ) + A(x, y2 )) 6 γ(||y1 − y2 ||)β(x − x) for all y1 , y2 ∈ C and x, x ∈ I, x < x is also true in the case when x > x, which completes the proof of part (a). (b) It is sufficient to prove necessity. Setting y1 = y2 in (18) and (19) we get   ||y1 − y 2 || β(x − x) d(F (x, y 1 ), F (x, y 2 )) 6 γ α(x − x) for all x, x ∈ I such that x < x and for all y 1 , y 2 ∈ C. In the case ||y 1 − y 2 || 6 1 by the monotonicity of γ we have   1 d(F (x, y 1 ), F (x, y 2 )) 6 γ β(x − x). (22) α(x − x) Since the condition tn → 0, such that

1 1 γ( α )

≺ β does not hold, we can find a sequence (tn ), tn ∈ (0, 1], β(tn )γ



1 α(tn )



→0

as n → ∞.

(23)

Take x ∈ [0, 1) and xn := x + tn . Then xn ∈ [0, 1] for a large enough n and xn → x. Since F (·, y), y ∈ C is continuous, from (22) and (23) we deduce that F (x, y 1 ) = F (x, y 2 ), x ∈ [0, 1] and y 1 , y 2 ∈ C. In the case ||y 1 − y 2 || > 1, fix n large enough to have n1 ||y 1 − y2 || 6 1. Setting i y = y 1 + ni (y 2 −y1 ), i = 0, 1, ..., n−1, we obtain ||y i+1 −y i || 6 1. By the above, we get

Functional equations and Nemytskij operator

253

F (x, y i ) = F (x, y i+1 ) for all x ∈ I and i = 0, 1, ..., n − 1, whence F (x, y 1 ) = F (x, y 2 ) for all x ∈ I and y1 , y2 ∈ C. In consequence, F (x, y) = F (x, 0) =: B(x) for x ∈ I and x ∈ C, which completes the proof. ⊔ ⊓ Remark 2.5. We denote by A the set of all functions ϕ ∈ Hα (I, C) of the form  for 0 6 t 6 x, y y−y (t − x) + y for x < t < x, ϕ(t) := x−x  y for x 6 t 6 1 for some x, x ∈ I, x < x, y, y ∈ C. Theorem 2.4 remains true if inequality (15) is assumed only for all ϕ, ϕ ∈ A. Remark 2.6. Assuming, that γ in Theorem 2.4 is increasing does not cause any loss of generality. For a given γ : [0, ∞) → [0, ∞) we can take γ ∗ : [0, ∞) → [0, ∞) defined by γ ∗ (t) = sups∈[0,t] γ(s). Remark 2.7. If in Theorem 2.4, γ(t) = Lt (for some constant L > 0) and the function F maps I × C into the space cc(Z) of all nonempty, convex and compact ∗

subsets of Z, we can replace + by the usual algebraic sum of two sets and we get the result obtained by J.J. Ludew in [6]. Condition (15) in Theorem 2.4 can be replaced by the uniform continuity of N . Theorem 2.8. Let Y be a real normed linear space, Z a Banach space and C a convex cone in Y . Suppose that the superposition operator N of the generator F : I × C → clb(Z) maps Hα (I, C) into Hβ (I, clb(Z)) and that N is uniformly continuous. Then there exist functions A : I × C → clb(Z) and B : I → clb(Z) such that A(·, y), B ∈ Hβ (I, clb(Z)) for every y ∈ C, A(x, ·) ∈ L(C, clb(Z)) for every x ∈ I and ∗

F (x, y) = A(x, y) + B(x),

x ∈ I, y ∈ C.

Proof. Suppose that N is uniformly continuous. Then for every ε > 0 there is δ > 0 such that for all ϕ, ϕ ∈ Hα (I, C) ||ϕ − ϕ||α 6 δ =⇒ dβ (N ϕ, N ϕ) 6 ε. Let γ : [0, ∞) → [0, ∞) be defined by γ(t) := sup{dβ (N ϕ, N ϕ) : ||ϕ − ϕ||α 6 t},

t > 0.

The function γ is well defined. Indeed, fix δ > 0 such that for all ϕ, ϕ ∈ Hα (I, C) ||ϕ − ϕ||α 6 δ =⇒ dβ (N ϕ, N ϕ) 6 1.

(24)

Therefore, we have γ(t) 6 1 for all t ∈ [0, δ]. Take t > 0, s > 0, t + s > 0 and t s ϕ, ϕ ∈ Hα (I, C) such that ||ϕ − ϕ||α 6 t + s. The function ψ = t+s ϕ + t+s ϕ also belongs to Hα (I, C) and

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||ϕ − ψ||α =

s ||ϕ − ϕ||α 6 s, t+s

||ψ − ϕ||α =

t ||ϕ − ϕ||α 6 t. t+s

Thus, by the definition of γ dβ (N ϕ, N ϕ) 6 dβ (N ϕ, N ψ) + dβ (N ψ, N ϕ) 6 γ(s) + γ(t) and consequently γ(s + t) 6 γ(s) + γ(t). In particular, γ(2t) 6 2γ(t), whence by induction we obtain γ(nt) 6 nγ(t)

(25)

for all n ∈ N and t > 0. For a given t > 0 there exists a positive integer n such that t n < δ. From (24) and (25) it follows that     t t 6 nγ 6 n < ∞. γ(t) = γ n n n Since N is uniformly continuous, γ is continuous at 0, γ(0) = 0 and obviously dβ (N ϕ, N ϕ) 6 γ(||ϕ − ϕ||α ),

ϕ, ϕ ∈ Hα (I, C),

the result is a consequence of Theorem 2.4.

⊔ ⊓

The following result may be proved in the same way as Lemma 5 in [16]. Lemma 2.9. Let Y and Z be two real, normed linear spaces and C a convex cone in Y with nonempty interior. Then there exists a positive constant M0 such that for every continuous, ∗ additive, set-valued function F : C → clb(Z) the inequality d(F (x), F (y)) 6 M0 ||F || ||x − y||,

x, y ∈ C

holds. The following theorem is a converse of part (a) of Theorem 2.4. Theorem 2.10. Let Y be a Banach space, Z a real normed linear space, C a convex cone in Y with nonempty interior and let α, β be two α-functions such that α ≺ β. Assume that A(·, y), B ∈ Hβ (I, clb(Z)) for y ∈ C and A(x, ·) ∈ L(C, clb(Z)) for x ∈ I. Moreover, assume that for some increasing, continuous at 0 function γ : [0, ∞) → [0, ∞), such that γ(0) = 0, the inequality d(A(x, y1 ) + A(x, y2 ), A(x, y1 ) + A(x, y2 )) 6 γ(||y1 − y2 ||)β(|x − x|)

(26)

holds for all x, x ∈ I and y1 , y2 ∈ C. If a set-valued function F : I × C → clb(Z) is of the form ∗

F (x, y) = A(x, y) + B(x),

x ∈ I, y ∈ C,

then the operator of substitution N generated by F maps the set Hα (I, C) into the set Hβ (I, clb(Z)) and satisfies inequality (15) with a function γ1 , where γ1 (t) = c(t+γ(t)), t > 0 and c is a constant.

Functional equations and Nemytskij operator

Proof. First, we will prove that the set Let x ∈ I, y ∈ C. We have

255

S

x∈I

A(x, y) is bounded for an arbitrary y ∈ C.

||A(x, y)|| = d(A(x, y), {0}) 6 d(A(x, y), A(0, y)) + d(A(0, y), {0}) = = d(A(x, y), A(0, y)) + ||A(0, y)||. Moreover, since A(·, y) ∈ Hβ (I, clb(Z)), d(A(x, y), A(0, y)) 6 ω(A(·, y), 1) =

ω(A(·, y), 1) 6 hβ (A(·, y)) < ∞. β(1)

Hence ||A(x, y)|| 6 hβ (A(·, y)) + ||A(0, y)||,

x ∈ I.

Since {A(x, ·)}x∈I is a family of ∗ additive and continuous functions, by Lemma 2.3 there exists a constant M > 0 such that sup ||A(x, y)|| 6 M ||y||, x∈I

y ∈ C.

Hence, and by Lemma 2.9, we deduce that d(A(x, y), A(x, y)) 6 M0 M ||y − y||

(27)

for all x ∈ I and y, y ∈ C. We shall prove now that N maps Hα (I, C) into Hβ (I, clb(Z)). Let ϕ ∈ Hα (I, C) and x, x ∈ I. The inequality d(A(x, y), A(x, y)) 6 γ(||y||)β(|x − x|)

(28)

is a consequence of (26). From (27) and (28) we obtain d(N ϕ(x), N ϕ(x)) = d(A(x, ϕ(x)) + B(x), A(x, ϕ(x)) + B(x)) 6 6 d(A(x, ϕ(x)), A(x, ϕ(x))) + d(B(x), B(x)) 6 6 d(A(x, ϕ(x)), A(x, ϕ(x))) + d(A(x, ϕ(x)), A(x, ϕ(x))) + d(B(x), B(x)) 6 6 γ(||ϕ(x)||)β(|x − x|) + M0 M ||ϕ(x) − ϕ(x)|| + d(B(x), B(x)) for all x, x ∈ I. Since ||ϕ(x)|| 6 ||ϕ(0)|| +

||ϕ(x) − ϕ(0)|| α(x − 0), α(x − 0)

x ∈ (0, 1]

we have ||ϕ(x)|| 6 ||ϕ||α for every x ∈ I. Now take s ∈ (0, 1] and x, x ∈ I such that |x − x| 6 s. The monotonicity of γ and β implies, that d(N ϕ(x), N ϕ(x)) 6 γ(||ϕ||α )β(s) + M0 M ω(ϕ, s) + ω(B, s).

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Therefore, for every s ∈ (0, 1] we obtain ω(N ϕ, s) ω(ϕ, s) α(s) ω(B, s) 6 γ(||ϕ||α ) + M0 M + 6 β(s) α(s) β(s) β(s) 6 γ(||ϕ||α ) + LM0 M hα (ϕ) + hβ (B), where L > 1 is a constant such that α(s) β(s) 6 L, s ∈ (0, 1] (by the assumption α ≺ β). Thus, hβ (N ϕ) < ∞ and N ϕ ∈ Hβ (I, clb(Z)). What remains to show is the fact, that N satisfies (15). Let ϕ, ϕ ∈ Hα (I, C), s ∈ (0, 1] and take x, x ∈ I such that |x − x| 6 s. Inequalities (27) and (26) imply that d(N ϕ(x) + N ϕ(x), N ϕ(x) + N ϕ(x)) = = d(A(x, ϕ(x)) + A(x, ϕ(x)), A(x, ϕ(x)) + A(x, ϕ(x))) =  = d A(x, ϕ(x) + ϕ(x)) + A(x, ϕ(x)) + A(x, ϕ(x)),  A(x, ϕ(x) + ϕ(x)) + A(x, ϕ(x)) + A(x, ϕ(x)) 6 6 d(A(x, ϕ(x) + ϕ(x)), A(x, ϕ(x) + ϕ(x))) + + d(A(x, ϕ(x)) + A(x, ϕ(x)), A(x, ϕ(x)) + A(x, ϕ(x))) 6 6 M0 M ||(ϕ − ϕ)(x) − (ϕ − ϕ)(x)|| + γ(||(ϕ − ϕ)(x)||)β(|x − x|) 6 6 M0 M ω(ϕ − ϕ, s) + γ(||ϕ − ϕ||α )β(s). Hence

ω(N ϕ, N ϕ, s) ω(ϕ − ϕ, s) α(s) 6 M0 M + γ(||ϕ − ϕ||α ) β(s) α(s) β(s)

and therefore sup s∈(0,1]

ω(N ϕ, N ϕ, s) 6 M0 M Lhα (ϕ − ϕ) + γ(||ϕ − ϕ||α ). β(s)

In consequence ω(N ϕ, N ϕ, s) 6 β(s) s∈(0,1]

dβ (N ϕ, N ϕ) = d(N ϕ(0), N ϕ(0)) + sup

6 M0 M ||(ϕ − ϕ)(0)|| + M0 M Lhα (ϕ − ϕ) + γ(||ϕ − ϕ||α ) 6 6 M0 M L||ϕ − ϕ||α + γ(||ϕ − ϕ||α ). Setting c = max{1, M0 M L} and γ1 (t) = c(t + γ(t)) we get dβ (N ϕ, N ϕ) 6 γ1 (||ϕ − ϕ||α ). This finishes the proof.

⊔ ⊓

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257

Bibliography 1. Appel J., Zabrejko P.P.: Nonlinear Superposition Operators. Cambridge University Press, Cambridge 1990. 2. Castaing C., Valadier M.: Convex Analysis and Measurable Multifunctions. Lecture Notes in Math. 580, Springer-Verlag, Berlin 1977. 3. Chistyakov V.V.: Generalized variation of mappings with applications to composition operators and multifunctions. Positivity 5, no. 4 (2001), 323–358. 4. De Blasi F.S.: On differentiability of multifunctions. Pacific J. Math. 66 (1976), 67–81. 5. Kuczma M., Choczewski B., Ger R.: Iterative Functional Equations. Encyclopedia of Mathematics and its Applications 32, Cambridge University Press, Cambridge 1990. 6. Ludew J.J.: On Lipschitzian operators of substitution generated by set-valued functions. Opuscula Math. 27, no. 1 (2007), 13–24. 7. Ludew J.J.: On Nemytskij operator of substitution in the C 1 space of set-valued functions. Demonstratio Math. 41, no. 2 (2008), 403–414. 8. Ludew J.J.: On Nemytskij operator in the space of absolutely continuous set-valued functions. J. Appl. Anal. 17, no. 2 (2011), 277–290. 9. Mainka E.: On uniformly continuous Nemytskij operators generated by set-valued functions. Aequat. Math. 79 (2010), 293–306. 10. Matkowski J.: On Lipschitzian solution of a functional equation. Ann. Polon. Math. 28 (1973), 135–139. 11. Matkowski J.: Functional equations and Nemytskij operators. Funkc. Ekvacioj Ser. Int. 25 (1982), 127–132. 12. Matkowski J.: Lipschitzian composition operators in some function spaces. Nonlinear Anal. Theory Methods Appl. 30, no. 2 (1997), 719–726. 13. Matkowki J.: Uniformly continuous superposition operators in the spaces of differentiable functions and absolutely continuous functions. Intern. Ser. Numer. Math. 157 (2008), 155–166. 14. Matkowski J.: Uniformly continuous superposition operators in the space of H¨ older functions. J. Math. Anal. Appl. 359 (2009), 56–61. 15. Merentes N., Nikodem K.: On Nemytskii operatorand set-valued functions of bounded pvariation. Rad. Mat. 8, no. 1 (1992), 139–145. 16. Smajdor A.: On regular multivalued cosine families. Ann. Math. Sil. 13 (1999), 271–280. 17. Smajdor A., Smajdor W.: Jensen equation and Nemytskii operator for set-valued functions. Rad. Math. 5 (1989), 311–320. 18. Smajdor W.: Superadditive set-valued functions and Banach-Steinhaus theorem. Rad. Mat. 3 (1987), 203–214. 19. Smajdor W.: Note on Jensen and Pexider functional equations. Demonstratio Math. 32, no. 2 (1999), 363–376. 20. Smajdor W.: Note on a Jensen type functional equation. Publ. Math. Debrecen 63, no. 4 (2003), 703–714. 21. Zawadzka G.: On Lipschitzian operators of substitution in the space of set-valued functions of bounded variation. Rad. Mat. 6 (1990), 279–293.

Fun with cascades Andrzej Starosolski

Abstract. This paper is, in a certain sense, a supplement of S. Dolecki’s paper Multisequences [Quaest. Math. 29 (2006), 239–277]. We describe some operations on cascades, together with their influence on a contour and sometimes on other notions connected with cascades. We illustrate these operations by a sketch of their use in some proofs of the results published last years concerning (ultra)filters. The paper contains also some new results on subsequential filters, answering a question from [Garcia-Ferreira S., Uzc´ ateui C.: Subsequential filters, Toppology. Appl. 156 (2009), 2949–2959]. To make the paper self-contained, we repeat all necessary definitions and re-describe some properties. Keywords: Monotone sequential cascade, contour, subsequential filters. 2010 Mathematics Subject Classification: 03E04, 03E05.

1. Preliminaries Monotone sequential cascades were introduced by S. Dolecki and F. Mynard in [5] to describe topological sequential spaces. Recently they were also used by S. GarciaFerreira and C. Uzc´ ateui in [10, 11] as a tool to investigate subsequential spaces. The reader interested in topological/convergence use of cascades is invited to look also at [6, 7]. Here we focus on pure set-theoretical aspects of cascades and, in fact, on infinite combinatorics used to this end. For brevity we omit proofs of quoted theorems and present only sketches of those of them, which show the typical way how to work with cascades. A cascade is a tree V , ordered by “⊑”, without infinite branches and with the least element ∅V . A cascade is sequential if for each non-maximal element v of V (v ∈ V \ max V ) the set v +V of immediate successors of v (in V ) is countably infinite. A. Starosolski Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 259–271. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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We say that v is a predecessor of v ′ (we write v = pred (v ′ ) or v = v ′−V ) if v ′ ∈ v +V . We write v + (v − ) instead of v +V (v −V ) if it is known in which cascade the successors (predecessor) of v is considered. As a convention, since max V is a countably infinite set, we think about max V as of a copy (or of a subset) of ω. The rank of v ∈ V (rV (v) or r(v)) is defined inductively as follows: r(v) = 0 if v ∈ max V , otherwise r(v) is the least ordinal greater than the ranks of all immediate successor of v. The rank r(V ) of the cascade V is, by definition, the rank of ∅V . Note that for each countable ordinal α there is a (monotone – see next paragraph) sequential cascade of rank α. The cascade V is said to be monotone if it is possible to order all sets v + (for v ∈ V \ max V ) in ω-sequences (vn )n∞ un of ultrafilters and take functions fm : ω → ω, each witnessing that um−1 >∞ um . Since formally levels in a cascade cannot intersect, we may assume that domain of f1 and ranges of fm are subsets of pairwise disjoint copies of ω.

Fun with cascades

269

We will build a monotone sequential cascade V which corresponds to the sequence above with respect to some U ∈ u. Simply start with a monotone sequential cascade of rank 1 and on each of n steps take an inversed image of the cascade (by fn−k+1 for step k). Those inverse images may not be sequentional, but a set of elements of infinite inversed images belongs to the un−k+1 (for step k), so we can restrict our cascade and make the next step on this resricted cascade. The number of restrictions is finite and at the end we obtain the cascade we were looking for. R Now take any monotone sequential cascade V of finite rank, with V ⊂ u. Without loss of generality we may assume that all branches of V have the same length n. For each v ∈ V let vˆ be an arbitrary element of max v ↑ . Consider functions fi : ω → ω such that fi (v1 ) = vˆ for each v1 ∈ max v ↑ , where r(v) = i. Thus u >∞ f1 (u) >∞ f2 (u) >∞ . . . >∞ fn (u).

2.4. Decreasing the rank, i.e., destroying nodes We say that a cascade V is built by destruction of nodes of rank 1 in a cascade W of rank r(W ) > 2 if and only if V = W \ R, where R = {w ∈ W : r(w) = 1, r(w− ) = 2}. Observe that if W is a monotone sequential cascade, then V is also a monotone sequential cascade. Moreover, if r(W ) is finite, then r(V ) = r(W ) − 1 and if r(W ) is infinite, then r(V ) = r(W ). Assume that there is a given cascade of rank α and an ordinal 1 6 β 6 α. We shall describe the operation of decreasing the rank of a cascade W to β. The construction is inductive: For a finite α, we can decrease the rank of W from α to β by applying α − β times the operation of destroying nodes of rank 1 (i.e., if α = β, then the cascade is unchanged). For infinite α, suppose that for each pair (δ, γ), where 1 < δ 6 γ < α, and for each cascade W of rank γ the operation of decreasing the rank of W from γ to δ is defined. Let W be a monotone sequential cascade of rank α, let (βn ) be a nondecreasing sequence of ordinals such that βn = 0 if and only if r(Wn ) = 0, βn 6 r(Wn ) and limn→∞ (βn + 1) = β. For each n < ω let Vn be the cascade obtained by decreasing the rank of Wn to βn . Finally, let V = (n) " Vn . Clearly, for infinite α the operation of decreasing the rank is not defined uniquely. Observe also that the described above decrease of the rank of a cascade W does not change max W . If a cascade V is obtained from W by decreasing R R the rank, then we write V ⊳ W . Trivially V ⊳ V and inductively V ⊳ W ⇒ V ⊂ W . This operation has been used for example in the proof of Lemma 2.19. The idea is to take (by contradiction) a sequence of cascades and a cascade of limit rank which contour is in the sum of contours of sequence of cascade. By [6] for each cascade from the sequence there is a set in limit contour which does not belong to the contour of the cascade in sequence. The point is to control “shrinking” of these sets, to be fixed on some parts of limit cascades. To do it we apply the operation described above. Lemma 2.19 ([21, Lemma 6.3]). Let α < ω1 be a limit ordinal and let (V n : n < ω) be a sequence of monotone sequential contours such that r(V n ) < r(V n+1 ) < α for S every n and such that n 0 such that if a mapping h : G → G1 satisfies the inequality



d h(xy), h(x)h(y) < δ

for all x, y ∈ G,

M. Adam Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 273–289. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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then there exists a homomorphism H : G → G1 with



d h(x), H(x) < ε

for all x ∈ G?

In the next year, D.H. Hyers [18] gave a partial affirmative answer to the question of Ulam in the context of Banach spaces. That was the first significant breakthrough and a step toward more solutions in this area. Since then, a large number of papers have been published in connection with various generalizations of Ulam’s problem and Hyers’ theorem. Th.M. Rassias in [27] (see also [2]), G.L. Forti in [14] and Z. Gajda in [16] considered the stability problem with unbounded Cauchy differences. The above results can be partially summarized in the following theorem: Theorem 1.1. Let X and Y be a real normed space and a real Banach space, respectively, and let p 6= 1 be a nonnegative constant. Suppose that a function f : X → Y satisfies the inequality



f (x + y) − f (x) − f (y) 6 ε kxkp + kykp , x, y ∈ X for some ε > 0. Then there exists a unique additive function A : X → Y such that

f (x) − A(x) 6

2ε kxkp , |2 − 2p |

x ∈ X.

This phenomenon is called Hyers-Ulam-Rassias stability. The function A : X → Y can be explicitly constructed, started from the given function f , by the formulae 1 f (2n x), n→∞ 2n

A(x) = lim

p < 1,

and A(x) = lim 2n f n→∞

x , 2n

p > 1.

This method is called the direct method or Hyers’ method. It is often used to construct a solution of a given functional equation and is a powerful tool for studying the stability of many functional equations. The second most often considered equation is the quadratic functional equation. The Hyers-Ulam stability of this equation was first proved by F. Skof [30] and generalized by P.W. Cholewa [5]. Thereafter, S. Czerwik [8] proved the Hyers-Ulam-Rassias stability of the quadratic functional equation and his result reads as follows: Theorem 1.2. Let X and Y be a real normed space and a real Banach space, respectively, and let p 6= 2 be a nonnegative constant. If a function f : X → Y satisfies the inequality



f (x + y) + f (x − y) − 2f (x) − 2f (y) 6 ε kxkp + kykp , x, y ∈ X for some ε > 0, then there exists a unique quadratic function Q : X → Y such that

f (x) − Q(x) 6

2ε kxkp , |4 − 2p |

x ∈ X.

The stability problems of several functional equations have been extensively investigated by many authors. For more information and primary references, the reader should refer to the monographs [10, 11, 19, 22], and papers, e.g. [9, 15, 20, 28].

A fixed point approach to the stability of some functional equation. . .

275

The Hyers’ method is the most popular technique of proving the Hyers-Ulam stability of functional equations. Nevertheless, there are also known several different approaches proving the Hyers-Ulam stability, for example the method of invariant means (see [17, 31]), the method based on sandwich theorems (see [25]) and on the concept of shadowing (see [32]). L. Cˇadariu and V. Radu applied the fixed point method to the investigation of Jensen and Cauchy functional equations (see [3] and [4], respectively). Now it is the second most popular technique of proving the Hyers-Ulam stability of functional equations. An extensive source of information on applications of fixed point theorems to the Hyers-Ulam stability of functional equations is Ciepli´ nski’s survey paper [6]. In this paper, we will apply the fixed point method to prove the Hyers-Ulam-Rassias stability of the following functional equation f (x + 3y) + 3f (x − y) = f (x − 3y) + 3f (x + y),

(1)

which is connected with additive and quadratic mappings. We will consider a class of functions between a linear space and a complete β-normed space. The above functional equation is interesting because of its connection with a single variable functional equation (see [1, 13, 29]) which is also closely associated with additive and quadratic mappings. It is well known (see [21]) that the general solution of (1) in the class of functions between real or complex linear spaces is of the form f = Q + A + c, where Q is a quadratic mapping, A is an additive one and c = f (0). It is worth to notice that the equation (1) is equivalent to the functional equation ∆32y f (x − 3y) = 0, where ∆ is the difference operator defined by ∆h f (x) = f (x + h) − f (x) and ∆3 denotes its third iterate. Therefore a solution of the above equation is a polynomial of degree at most two (see, e.g., [24]). Standard symbols R, C denote the sets of real and complex numbers, respectively, and N0 := N ∪ {0}, where N denotes the set of positive integers.

2. Preliminaries In this section we present some definitions and auxiliary result which will be needed in the sequel. Definition 2.1 ([12]). Let X be a nonempty set. A function d : X × X → [0, ∞] is called a generalized metric on X if and only if d satisfies the following conditions: (i) d(x, y) = 0 if and only if x = y, (ii) d(x, y) = d(y, x) for all x, y ∈ X, (iii) d(x, z) 6 d(x, y) + d(y, z) for all x, y, z ∈ X. Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity. We now introduce one of fundamental results of fixed point theory. For the proof, refer to [12].

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Theorem 2.2 ([12]). Let (X, d) be a generalized metric space. Assume that Λ : X → X is a strictly contractive operator with the Lipschitz constant L < 1. Then, for each given element x ∈ X, either (a) d(Λn+1 x, Λn x) = ∞ for all n ∈ N0 , or (b) there exists k ∈ N0 such that d(Λn+1 x, Λn x) < ∞ for all n > k, n ∈ N0 . Actually, if (b) holds and the respective k ∈ N0 is fixed, then (i) the sequence (Λn x)n∈N0 converges to the fixed point x∗ of Λ, (ii) x∗ is the unique fixed point of Λ in the space  X ∗ := y ∈ X : d(Λk x, y) < ∞ , (iii) if y ∈ X ∗ , then d(y, x∗ ) 6

1 d(Λy, y). 1−L

Remark 2.3. It is known that the fixed point x∗ , if it exists, is not necessarily unique in the whole space X, it may depend on the starting point x. Moreover, in the case (b), the pair (X ∗ , d) is a complete metric space and Λ(X ∗ ) ⊂ X ∗ . Therefore the properties (i)–(iii) follow from Banach’s Contraction Principle (cf. [23]). Let E be a vector space over the field K = {R, C}. Moreover, from now, let α ∈ (0, ∞) and 0 < β 6 1. Definition 2.4 (cf. [4]). A mapping k · kα : E → [0, ∞) is called a sub-homogeneous functional of order α if and only if kλxkα 6 |λ|α · kxkα ,

λ ∈ K, x ∈ E.

(2)

Similarly, we can formulate the following definition. Definition 2.5. A mapping k·kα : E → [0, ∞) is called a sub-homogeneous functional of order 2α if and only if kλxkα 6 |λ|2α · kxkα ,

λ ∈ K, x ∈ E.

(3)

Actually, a sub-homogeneous functional of order α (or 2α) is a homogeneous functional of order α (or 2α). Indeed, it suffices to substitute λx and λ1 in the place of x and λ in the above conditions, respectively, to obtain the converse inequalities. As usually, E is identified with E × {0} in E × E. Hence kxkα = k(x, 0)kα for all x ∈ E and for each sub-homogeneous functional of order α (or 2α) on E × E. Definition 2.6 ([4]). A mapping k · kβ : E → [0, ∞) is called a β-norm if and only if it has the following properties: (i) kxkβ = 0 if and only if x = 0, (ii) kλxkβ = |λ|β · kxkβ , (iii) kx + ykβ 6 kxkβ + kykβ for all λ ∈ K and x, y ∈ E.

A fixed point approach to the stability of some functional equation. . .

277

3. Main results Throughout this section let E1 , E2 be two linear spaces over the same field K = {R, C}. Moreover, assume that E2 is a complete β-normed space for some 0 < β 6 1. Let us define a number ai , i = 0, 1, by the formula: ( 2, i = 0, ai = 1 i = 1. 2, To shorten some considerations and for the sake of brevity we shall use in the following two theorems the same notations for the case i = 0 and i = 1. We investigate the stability problem of (1) by decomposing the unknown function f into its even and odd part. Next, we will connect these two cases receiving the main Theorem 3.3. Theorem 3.1. Suppose ϕ : E1 × E1 → [0, ∞) is a given function and there exists a constant L, 0 < L < 1, such that the mapping x x , , x ∈ E1 , x → ψ(x) = ϕ 2 2 has the property ψ(x) 6 L · a2β i ψ



x ai



,

x ∈ E1 , i = 0, 1,

(4)

x, y ∈ E1 , i = 0, 1.

(5)

and the mapping ϕ satisfies the condition lim

n→∞

ϕ(ani x, ani y) a2nβ i

= 0,

If an even function f : E1 → E2 satisfies the inequality

f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ϕ(x, y), β

x, y ∈ E1 ,

then there exists a unique quadratic mapping Q : E1 → E2 such that i

f (x) − f (0) − Q(x) 6 1 L ψ(x), β β 4 1−L

x ∈ E1 , i = 0, 1.

(6)

Proof. Let f1 (x) := f (x) − f (0) for all x ∈ E1 . Then f1 (0) = 0 and the function f1 satisfies the inequality

f1 (x + 3y) + 3f1 (x − y) − f1 (x − 3y) − 3f1 (x + y) 6 ϕ(x, y), x, y ∈ E1 . (7) β Consider the set

 X := h : E1 → E2 : h(0) = 0 ,

and introduce the generalized metric on X:

 d(g, h) := inf C ∈ [0, ∞] : g(x) − h(x) β 6 Cψ(x) ,

x ∈ E1 .

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As usually, inf ∅ := ∞. First, we will verify that (X, d) is a complete space. Let (gn )n∈N be a Cauchy sequence in (X, d), i.e. ^ _ ^ d(gm , gn ) 6 ε. ε>0 N0 ∈N m,n>N0

By considering the definition of the generalized metric d, we see that ^ _ ^ ^

gm (x) − gn (x) 6 εψ(x). β

(8)

ε>0 N0 ∈N m,n>N0 x∈E1

 For fixed x ∈ E1 , condition (8) implies that gn (x) n∈N is a Cauchy sequence in     E2 , k · kβ . Since E2 , k · kβ is complete, gn (x) n∈N converges in E2 , k · kβ for each x ∈ E1 . Hence we can define a function g : E1 → E2 by g(x) := lim gn (x), n→∞

Letting m → ∞ in (8), we get ^ _ ^

x ∈ E1 .

^

g(x) − gn (x) 6 εψ(x), β

ε>0 N0 ∈N n>N0 x∈E1

i.e.

^

_

^

d(g, gn ) 6 ε.

ε>0 N0 ∈N n>N0

This fact leads us to the conclusion that the sequence (gn )n∈N converges in (X, d). Hence (X, d) is a complete space. We define an operator Λ : X → X by the formula (Λh)(x) :=

1 h(ai x), a2i

x ∈ E1 , i = 0, 1.

We show that Λ is strictly contractive on X. Given g, h ∈ X and let C ∈ [0, ∞] be an arbitrary constant with d(g, h) 6 C, i.e.

g(x) − h(x) 6 Cψ(x), x ∈ E1 , i = 0, 1. β Substituting ai x instead of x in the above inequality and dividing both sides of the resulting expression by a2β i , we get

g(ai x) h(ai x) 1

a2 − a2 6 2β Cψ(ai x), x ∈ E1 , i = 0, 1. ai i i β Therefore in view of (4) and the definition of Λ we see that

(Λg)(x) − (Λh)(x) 6 LCψ(x), x ∈ E1 , β i.e. d(Λg, Λh) 6 LC. Hence we conclude that d(Λg, Λh) 6 Ld(g, h) for any g, h ∈ X.

A fixed point approach to the stability of some functional equation. . .

279

Next, we assert that d(Λf1 , f1 ) < ∞. Consider the case where i = 0. Substituting instead of x and y in (7) and dividing both sides of the resulting inequality by 4β , we obtain

(Λf1 )(x) − f1 (x) 6 1 ψ(x), x ∈ E1 , β 4β i.e. 1 d(Λf1 , f1 ) 6 β < ∞. 4 Let us now consider the second case where i = 1. Replacing x and y by x4 in (7) and applying (4) gives x 2



(Λf1 )(x) − f1 (x) 6 1 Lψ(x), β 4β i.e. d(Λf1 , f1 ) 6

x ∈ E1 ,

1 L < ∞. 4β

Therefore we have

1 i L < ∞, i = 0, 1. (9) 4β By Theorem 2.2 (i) there exists a mapping Q : E1 → E2 with Q(0) = 0, which is a fixed point of Λ, i.e. (ΛQ)(x) = Q(x) for all x ∈ E1 . Hence Q(2x) = 4Q(x) for all x ∈ E1 and Λn f1 → Q, i.e. d(Λf1 , f1 ) 6

lim

n→∞

1 f1 (ani x) = Q(x), a2n i

x ∈ E1 , i = 0, 1.

Since k = 0 (see (9)) and f1 ∈ X ∗ in Theorem 2.2, by Theorem 2.2 (iii) and (9) we obtain 1 1 Li d(f1 , Q) 6 d(Λf1 , f1 ) 6 β , 1−L 4 1−L i.e. i

f1 (x) − Q(x) 6 1 L ψ(x), x ∈ E1 , i = 0, 1, β 4β 1 − L which means that the inequality (6) is true. We verify that a function Q is quadratic. Substituting ani x and ani y instead of x and y in (7), respectively, and dividing both sides of the resulting inequality by a2nβ , i we get

f an (x + 3y) 3f an (x − y) f an (x − 3y) 3f an (x + y) 1 i 1 i 1 i

1 i

+ − −



a2n a2n a2n a2n i i i i

β

6

ϕ(ani x, ani y) a2nβ i

,

x, y ∈ E1 ,

where i = 0, 1. Taking the limit in the above expression as n → ∞ and applying (5) we conclude that

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Q(x + 3y) + 3Q(x − y) = Q(x − 3y) + 3Q(x + y),

x, y ∈ E1 .

This implies that Q is a quadratic function (since Q is even, cf. [21]). To prove the uniqueness of the solution assume that there exists another quadratic function Q1 : E1 → E2 satisfying the condition (6). Therefore Q1 (x) = a12 Q1 (ai x) = i (ΛQ1 )(x) for all x ∈ E1 , i.e. Q1 is a fixed point of Λ. In view of (6) with Q1 and the definition of d, we know that i

f1 (x) − Q1 (x) 6 1 L ψ(x), β β 4 1−L

x ∈ E1 , i = 0, 1,

i.e.

1 Li < ∞, i = 0, 1. 4β 1 − L  Thus Q1 ∈ X ∗ = y ∈ X : d(Λf1 , y) < ∞ and Theorem 2.2 (ii) implies that Q = Q1 , which proves the uniqueness of Q. The proof is completed. ⊔ ⊓ d(f1 , Q1 ) 6

Theorem 3.2. Suppose ϕ : E1 × E1 → [0, ∞) is a given function and there exists a constant L, 0 < L < 1, such that the mapping x x x → ψ(x) = ϕ , , x ∈ E1 2 2 has the property ψ(x) 6 L ·

aβi ψ



x ai



,

x ∈ E1 , i = 0, 1,

(10)

x, y ∈ E1 , i = 0, 1.

(11)

and the mapping ϕ satisfies the condition lim

n→∞

ϕ(ani x, ani y) anβ i

= 0,

If an odd function f : E1 → E2 satisfies the inequality

f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ϕ(x, y), β

x, y ∈ E1 ,

(12)

then there exists a unique additive mapping A : E1 → E2 such that i

f (x) − A(x) 6 1 L ψ(x), β 2β 1 − L

x ∈ E1 , i = 0, 1.

(13)

Proof. Since a function f is odd then obviously f (0) = 0. Similarly as in the proof of Theorem 3.1 we define the set X and the generalized metric d. Then (X, d) is a complete space. We define an operator Λ : X → X by the formula (Λh)(x) :=

1 h(ai x), ai

x ∈ E1 , i = 0, 1.

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We assert that Λ is strictly contractive on X. For given g, h ∈ X, let C ∈ [0, ∞] be an arbitrary constant such that d(g, h) 6 C, i.e.

g(x) − h(x) 6 Cψ(x), x ∈ E1 . β Substituting ai x instead of x in the above inequality and dividing both sides of the resulting expression by aβi , we obtain

g(ai x) h(ai x)

6 1 Cψ(ai x), x ∈ E1 , i = 0, 1. −

ai ai β aβi Therefore in view of (10) and the definition of Λ we see that

(Λg)(x) − (Λh)(x) 6 LCψ(x), x ∈ E1 , β i.e. d(Λg, Λh) 6 LC. Thus d(Λg, Λh) 6 Ld(g, h) for any g, h ∈ X. We show that d(Λf, f ) < ∞. Consider the case where i = 0. Substituting x2 instead of x and y in (12) and dividing both sides of the resulting inequality by 2β , we obtain

(Λf )(x) − f (x) 6 1 ψ(x), β 2β

x ∈ E1 ,

i.e.

1 < ∞. 2β Let us now consider the second case where i = 1. Replacing x and y by applying (10) gives d(Λf, f ) 6



(Λf )(x) − f (x) 6 1 Lψ(x), β 2β i.e. d(Λf, f ) 6

x 4

in (12) and

x ∈ E1 ,

1 L < ∞. 2β

Therefore we have

1 i L < ∞, i = 0, 1. (14) 2β Then, it follows from Theorem 2.2 (i) that there exists a function A : E1 → E2 with A(0) = 0, which is a fixed point of Λ, i.e. (ΛA)(x) = A(x) for all x ∈ E1 . Thus A(2x) = 2A(x) for all x ∈ E1 and Λn f → A, i.e. d(Λf, f ) 6

lim

n→∞

1 f (ani x) = A(x), ani

x ∈ E1 , i = 0, 1.

Since k = 0 (see (14)) and f ∈ X ∗ in Theorem 2.2, by Theorem 2.2 (iii) and (14) we obtain 1 1 Li d(f, A) 6 d(Λf, f ) 6 β , 1−L 2 1−L i.e.

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f (x) − A(x) 6 1 L ψ(x), β β 2 1−L

x ∈ E1 , i = 0, 1,

which means that the inequality (13) holds true. We show that a function A is additive. Substituting ani x i ani y instead of x and y in (12), respectively, and dividing both sides of the resulting inequality by anβ i , we get

f an (x + 3y) 3f an (x − y) f an (x − 3y) 3f an (x + y)

i i i i + − −



ani ani ani ani

β

6

ϕ(ani x, ani y) anβ i

,

x, y ∈ E1 ,

where i = 0, 1. Letting n → ∞ in the above inequality and applying (11) we have A(x + 3y) + 3A(x − y) = A(x − 3y) + 3A(x + y),

x, y ∈ E1 .

This implies that A is a additive function (since A is odd, cf. [21]). Assume that inequality (13) is also satisfied with another additive function A1 : E1 → E2 besides A. Therefore A1 (x) = a1i A1 (ai x) = (ΛA1 )(x) for all x ∈ E1 , i.e. A1 is a fixed point of Λ. In view of (13) with Q1 and the definition of d, we know that i

f (x) − A1 (x) 6 1 L ψ(x), β β 2 1−L

x ∈ E1 , i = 0, 1,

i.e.

1 Li < ∞, i = 0, 1. 2β 1 − L  Thus A1 ∈ X ∗ = y ∈ X : d(Λf, y) < ∞ and Theorem 2.2 (ii) implies that A = A1 , which proves the uniqueness of A. This completes the proof. ⊔ ⊓ d(f, A1 ) 6

Theorem 3.3. Suppose ϕ : E1 × E1 → [0, ∞) is a given function and there exists a constant L, 0 < L < 1, such that the mapping x x x → ψ(x) = ϕ , , x ∈ E1 2 2 has the properties: i

ψ(x) 6 L · a2i β ψ



x ai



,

x ∈ E1 , i = 0, 1,

and the mapping ϕ satisfies the conditions lim

n→∞

ϕ(ani x, ani y) i nβ

a2i

= 0,

x, y ∈ E1 , i = 0, 1.

If a function f : E1 → E2 satisfies the inequality

A fixed point approach to the stability of some functional equation. . .

283



f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ϕ(x, y), β

x, y ∈ E1 ,

(15)

then there exist a unique quadratic mapping Q : E1 → E2 and a unique additive mapping A : E1 → E2 such that β i 



f (x)−f (0)−Q(x)−A(x) 6 2 + 1 L ψ(x)+ψ(−x) , x ∈ E1 , i = 0, 1. (16) β β 8 1−L

Proof. We define functions f1 , f2 : E1 → E2 by f1 (x) :=

f (x) + f (−x) , 2

f2 (x) :=

f (x) − f (−x) , 2

x ∈ E1 .

Then f = f1 + f2 . Since a function f safisfies the condition (15), so the following inequalities hold true

 

f1 (x + 3y) + 3f1 (x − y) − f1 (x − 3y) − 3f1 (x + y) 6 1 ϕ(x, y) + ϕ(−x, −y) , β β 2

 

f2 (x + 3y) + 3f2 (x − y) − f2 (x − 3y) − 3f2 (x + y) 6 1 ϕ(x, y) + ϕ(−x, −y) β 2β for all x, y ∈ E1 . It follows from Theorems 3.1 and 3.2 that there exist a unique quadratic function Q : E1 → E2 and aunique additive function A : E1 → E2 such that i 



f1 (x) − f1 (0) − Q(x) 6 1 L ψ(x) + ψ(−x) , β β 8 1−L i 



f2 (x) − A(x) 6 1 L ψ(x) + ψ(−x) β 4β 1 − L

for all x ∈ E1 , i = 0, 1, respectively. From the above inequalities we easily obtain the condition (16). ⊔ ⊓ Corollary 3.4. Suppose that we have given a sub-homogeneous functional of order 2α on E1 × E1 , α 6= β. Then for each ε > 0 there exists δ(ε) > 0 such that for every even function f : E1 → E2 which satisfies



f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 δ(ε) · (x, y) , x, y ∈ E1 , β α there exists a unique quadratic mapping Q : E1 → E2 such that



f (x) − f (0) − Q(x) 6 ε · (x, x) , x ∈ E1 . β α Proof. Define

ϕ(x, y) := δ(ε) · (x, y) α ,

(17)

x, y ∈ E1 .

For a0 = 2 and α − β < 0 we have ϕ(ani x, ani y) a2nβ i

=

δ(ε) 2n(α−β) · (ani x, ani y) α 6 δ(ε) · ai · (x, y) α −−−−→ 0, 2nβ n→∞ ai

x, y ∈ E1 .

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We obtain the same condition for a1 = 12 and α − β > 0. Hence (5) is true. Moreover, for a0 = 2 and α − β < 0 we get ψ(x) = ϕ

 x x 

 x x x x 



, = δ(ε) · ,

= δ(ε) · 2 · , 2 ·

6 2 2 2 2 α  4 4 α x x

x x 6 4α · δ(ε) · , , =

= 4α · ϕ 4 4 α x 4 4 x = 4α · ψ = L · 4β · ψ , 2 2

x ∈ E1 ,

where L = 4α−β < 1. For a1 = 12 and α − β > 0 we have ψ(x) = ϕ

 x x  x x

1

, = δ(ε) · ,

6 α · δ(ε) · (x, x) α = 2 2 2 2 α 4 1 1 = α · ϕ(x, x) = L · β · ψ(2x), 4 4

x ∈ E1 ,

where L = 4β−α < 1. Therefore the inequality (4) is satisfied. So, in view of Theorem 3.1 there exists a unique quadratic mapping Q : E1 → E2 such that

f (x) − f (0) − Q(x) 6 1 1 ψ(x), β 4β 1 − L

x ∈ E1

holds, with L = 4α−β , or

f (x) − f (0) − Q(x) 6 1 L ψ(x), β 4β 1 − L

x ∈ E1

holds, with L = 4β−α . Thus, the inequality (17) holds true for δ(ε) = ε · 4α(4β − 4α) and δ(ε) = ε · 4α (4α − β 4 ), respectively. ⊔ ⊓ Corollary 3.5. Suppose that we have given a sub-homogeneous functional of order α on E1 × E1 , α 6= β. Then for each ε > 0 there exists δ(ε) > 0 such that for every odd function f : E1 → E2 which satisfies



f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 δ(ε) · (x, y) , x, y ∈ E1 , β α there exists a unique additive mapping A : E1 → E2 such that



f (x) − A(x) 6 ε · (x, x) , x ∈ E1 . α β Proof. Define

ϕ(x, y) := δ(ε) · (x, y) α ,

(18)

x, y ∈ E1 .

For a0 = 2 and α − β < 0 we have ϕ(ani x, ani y) ainβ

=

δ(ε) n(α−β) · (ani x, ani y) α 6 δ(ε) · ai · (x, y) α −−−−→ 0, nβ n→∞ ai

x, y ∈ E1 .

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285

We obtain the same condition for a1 = 12 and α − β > 0. Hence (11) is true. Moreover, for a0 = 2 and α − β < 0 we get ψ(x) = ϕ

 x x 

 x x x x 



, = δ(ε) · ,

= δ(ε) · 2 · , 2 ·

6 2 2 2 2 α  4 4 α   x x

x x α α 6 2 · δ(ε) · , , =

=2 ·ϕ 4 4 α x 4 4 x = 2α · ψ = L · 2β · ψ , 2 2

x ∈ E1 ,

where L = 2α−β < 1. For a1 = 12 and α − β > 0 we have ψ(x) = ϕ

 x x  x x

1

, = δ(ε) · ,

6 α · δ(ε) · (x, x) α = 2 2 2 2 α 2 1 1 = α · ϕ(x, x) = L · β · ψ(2x), 2 2

x ∈ E1 ,

where L = 2β−α < 1. Therefore the inequality (10) is satisfied. So, in view of Theorem 3.2 there exists a unique additive mapping A : E1 → E2 such that

f (x) − A(x) 6 1 1 ψ(x), β 2β 1 − L

x ∈ E1

holds, with L = 2α−β , or

f (x) − A(x) 6 1 L ψ(x), β 2β 1 − L

x ∈ E1 ,

holds, with L = 2β−α . Thus, the inequality (18) holds true for δ(ε) = ε · 2α(2β − 2α) and δ(ε) = ε · 2α (2α − β 2 ), respectively. ⊔ ⊓ Corollary 3.6. Let E1 be a normed space over K. Let us fix p ∈ (0, 2), p2 < β 6 1 in the case i = 0, and p ∈ (2, ∞), 0 < β 6 1 in the case i = 1. If an even function f : E1 → E2 satisfies the inequality



f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ε kxkp + kykp , x, y ∈ E1 (19) β and ε > 0, then there exists a unique quadratic mapping Q : E1 → E2 such that

f (x) − f (0) − Q(x) 6 β

2ε kxkp , 2p · |4β − 2p |

x ∈ E1 .

(20)

Proof. Without loss of generality we can assume that f (0) = 0. Define ϕ(x, y) :=  ap i , where i = 0 when ε kxkp + kykp for all x, y ∈ E1 . Moreover, let L := Li = a2β p < 2β, and i = 1 when p > 2β. Hence 0 < L < 1 and

i

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ψ(x) = ϕ



p

p

x p x x

x

p x

, = 2ε = 2ε a · 2εa = = i i

2 2 2 2ai 2ai       x x x x = api ψ = L · a2β ψ , , = api ϕ i 2ai 2ai ai ai

Furthermore, ϕ(ani x, ani y) ai2nβ

= api

=

ε kani xkp + kani ykp !n

a2β i



a2nβ i

=

ε · anp kxkp + kykp i



a2nβ i

x ∈ E1 .

=

  · ε kxkp + kykp = Ln · ε kxkp + kykp −−−−→ 0, n→∞

x, y ∈ E1 .

Therefore, in view of Theorem 3.1 there exists a unique quadratic mapping Q : E1 → E2 which satisfies (20). The proof is completed. ⊔ ⊓ Corollary 3.7. Let E1 be a normed space over K. Let us fix p ∈ (0, 1), p < β 6 1 in the case i = 0, and p ∈ (1, ∞), 0 < β 6 1 in the case i = 1. If an odd function f : E1 → E2 satisfies the inequality



f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ε kxkp + kykp , x, y ∈ E1 (21) β and ε > 0, then there exists a unique additive mapping A : E1 → E2 such that

f (x) − A(x) 6 β

2ε kxkp , 2p · |2β − 2p |

x ∈ E1 .

(22)

 Proof. Clearly, f (0) = 0. Define ϕ(x, y) := ε kxkp + kykp for all x, y ∈ E1 . Moreover, let L := Li = and ψ(x) = ϕ

ap i , aβ i

where i = 0 when p < β, and i = 1 when p > β. Hence 0 < L < 1

p

p

x p x x

x

p x

= 2ε = 2ε , a · = 2εa = i i

2 2 2 2ai 2ai       x x x x = api ϕ , = api ψ = L · aβi ψ , 2ai 2ai ai ai

Furthermore, ϕ(ani x, ani y) anβ i =

= api aβi

ε kani xkp + kani ykp !n

anβ i



=

ε · anp kxkp + kykp i anβ i



x ∈ E1 .

=

  · ε kxkp + kykp = Ln · ε kxkp + kykp −−−−→ 0, n→∞

x, y ∈ E1 .

Therefore, in view of Theorem 3.2 there exists a unique additive mapping A : E1 → E2 which satisfies (22). This completes the proof. ⊔ ⊓ Corollary 3.8. Let E1 be a normed space over K. Let p 6= 1, p 6= 2 and let us fix β > p with p ∈ (0, 1), β > p2 with p ∈ (1, 2) and β > 0 with p ∈ (2, ∞), respectively.

A fixed point approach to the stability of some functional equation. . .

287

If a function f : E1 → E2 satisfies the inequality



f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ε kxkp + kykp , β

x, y ∈ E1

and ε > 0, then there exist a unique quadratic mapping Q : E1 → E2 and a unique additive mapping A : E1 → E2 such that  

1 1

f (x) − f (0) − Q(x) − A(x) 6 4ε + kxkp , x ∈ E1 . (23) β 2p+β |4β − 2p | |2β − 2p | Proof. Similarly as in the proof of Theorem 3.3 one can obtain the inequality (23). ⊓ ⊔ In the following two corollaries, we deal with the inequalities (19) and (21) for the case p = 0. We need only to set L = 41β and L = 21β and apply Theorems 3.1 and 3.2 for their proofs, respectively. It is worth to note that the above constants L are the smallest ones satisfying conditions (4) and (10), respectively. Corollary 3.9. If an even function f : E1 → E2 satisfies the inequality

f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ε, x, y ∈ E1 β for some ε > 0, then there exists a unique quadratic mapping Q : E1 → E2 such that

f (x) − f (0) − Q(x) 6 β



ε , −1

x ∈ E1 .

Corollary 3.10. If an odd function f : E1 → E2 satisfies the inequality

f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ε, x, y ∈ E1 β for some ε > 0, then there exists a unique additive mapping A : E1 → E2 such that

f (x) − A(x) 6 β

ε , 2β − 1

x ∈ E1 .

(24)

Corollary 3.11. If a function f : E1 → E2 satisfies the inequality

f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ε, x, y ∈ E1 β for some ε > 0, then there exist a unique quadratic mapping Q : E1 → E2 and a unique additive mapping A : E1 → E2 such that 2−β

+2

f (x) − f (0) − Q(x) − A(x) 6 2 ε, β 4β − 1

x ∈ E1 .

It is worth noticing that we can prove Theorem 3.2 with another definition of the number ai . Namely, let us define a number ai , i = 0, 1, by the formula: ( 3, i = 0, ai = 1 i = 1. 3,

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Theorem 3.12. Suppose ϕ : E1 × E1 → [0, ∞) is a given function and there exists a constant L, 0 < L < 1, such that the mapping x → ψ(x) = ϕ(0, x), has the property ψ(x) 6 L · aβi ψ



x ai



,

x ∈ E1

x ∈ E1 , i = 0, 1,

and the mapping ϕ satisfies the condition lim

n→∞

ϕ(ani x, ani y) anβ i

= 0,

x, y ∈ E1 , i = 0, 1.

If an odd function f : E1 → E2 satisfies the inequality

f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ϕ(x, y), β

x, y ∈ E1 ,

then there exists a unique additive mapping A : E1 → E2 such that i

f (x) − A(x) 6 1 L ψ(x), β 6β 1 − L

x ∈ E1 , i = 0, 1.

Corollary 3.13. If an odd function f : E1 → E2 satisfies the inequality

f (x + 3y) + 3f (x − y) − f (x − 3y) − 3f (x + y) 6 ε, x, y ∈ E1 β for some ε > 0, then there exists a unique additive mapping A : E1 → E2 such that

f (x) − A(x) 6 β

ε , 6β − 2β

x ∈ E1 .

It can be easily checked that the above approximation constant is better than that one obtained in (24). Recently, B. Przebieracz [26] presented an application of the Markov-Kakutani common fixed point theorem to the theory of stability of functional equations by proving some version of the Hyers theorem concerning approximate homomorphisms. It seems to be interesting to consider applications of another fixed point theorems to the theory of the Hyers-Ulam stability of functional equations (see, e.g., fixed point theorems investigated in [7]).

Bibliography 1. Adam M.: On the Hyers-Ulam-Rassias stability of a generalized quadratic and additive functional equation. Nonlinear Funct. Anal. Appl. 14, no. 5 (2009), 699–705. 2. Aoki T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Japan 2 (1950), 64–66. 3. Cˇ adariu L., Radu V.: Fixed points and the stability of Jensen’s functional equation. J. Inequal. Pure and Appl. Math. 4, no. 1 (2003), art. 4.

A fixed point approach to the stability of some functional equation. . .

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4. Cˇ adariu L., Radu V.: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Math. Ber. 346 (2004), 43–52. 5. Cholewa P.W.: Remarks on the stability of functional equations. Aequationes Math. 27 (1984), 76–86. 6. Ciepli´ nski K.: Applications of fixed point theorems to the Hyers-Ulam stability of functional equations – a survey. Ann. Funct. Anal. 3 (2012), 151–164. 7. Czerwik S.: Fixed Point Theorems and Special Solutions of Functional Equations. Prace Nauk. ´ ask. 428, Katowice 1980. Uniw. Sl¸ 8. Czerwik S.: On the stability of the quadratic mapping in normed spaces. Abh. Math. Sem. Univ. Hamburg 62 (1992), 59–64. 9. Czerwik S.: The stability of the quadratic functional equation. In: Stability of Mappings of HyersUlam Type, Rassias Th.M., Tabor J. (eds.), Hadronic Press, Palm Harbor 1994, 81–91. 10. Czerwik S.: Functional Equations and Inequalities in Several Variables. World Scientific, New Jersey 2002. 11. Czerwik S.: Stability of Functional Equations of Ulam-Hyers-Rassias Type. Hadronic Press, Palm Harbor 2003. 12. Diaz J., Margolis B.: A fixed point theorem of the alternative for contractions on a generalized complete metric space. Bull. Amer. Math. Soc. 74 (1968), 305–309. 13. Fechner W.: On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings. J. Math. Anal. Appl. 322 (2006), 774–786. 14. Forti G.L.: An existence and stability theorem for a class of functional equations. Stochastica 4 (1980), 23–30. 15. Forti G.L.: Hyers-Ulam stability of functional equations in several variables. Aequationes Math. 50 (1995), 143–190. 16. Gajda Z.: On stability of additive mappings. Internat. J. Math. & Math. Sci. 14 (1991), 431–434. 17. Gajda Z.: Invariant Means and Representations of Semigroups in the Theory of Functional ´ ask. 1273, Katowice 1992. Equations. Prace Nauk. Uniw. Sl¸ 18. Hyers D.H.: On the stability of the linear functional equation. Proc. Nat. Acad. Sci. USA 27 (1941), 222–224. 19. Hyers D.H., Isac G., Rassias Th.M.: Stability of Functional Equations in Several Variables. Birkh¨ auser Verlag, Basel 1998. 20. Hyers D.H., Rassias Th.M.: Approximate homomorphisms. Aequationes Math. 44 (1992), 125– 153. 21. Jun K.-W., Kim H.-M., Lee D.O: On the Hyers-Ulam-Rassias stability of a modified additive and quadratic functional equation. J. Korea Soc. Math. Educ. Ser. B: Pure Appl. Math. 11, no. 4 (2004), 323–335. 22. Jung S.-M.: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor 2000. 23. Kirk W.A., Sims B. (eds.): Handbook of Metric Fixed Point Theory. Kluwer Academic Publishers, Dordrecht 2001. 24. Kuczma M.: An Introduction to the Theory of Functional Equations and Inequalities. Cauchy’s Equation and Jensen’s Inequality. Second edition (edited by A. Gil´ anyi), Birkh¨ auser Verlag, Basel 2009. 25. P´ ales Z.: Generalized stability of the Cauchy functional equation. Aequationes Math. 56 (1998), 222–232. 26. Przebieracz B.: The Hyers theorem via the Markov-Kakutani fixed point theorem. J. Fixed Point Theory Appl. 12 (2012), 35–39. 27. Rassias Th.M.: On the stability of the linear mapping in Banach spaces. Proc. Amer. Math. Soc. 72 (1978), 297–300. 28. R¨ atz J.: On approximately additive mappings. In: General Inequalities 2, Beckenbach E.F. (ed.), Birkh¨ auser Verlag, Basel 1980, 233–251. 29. Sikorska J.: On a direct method for proving the Hyers-Ulam stability of functional equations. J. Math. Anal. Appl. 372 (2010), 99–109. 30. Skof F.: Propriet´ a locali e approssimazione di operatori. Rend. Sem. Mat. Fis. Milano 53 (1983), 113–129. 31. Szek´ elyhidi L.: Note on a stability theorem. Canad. Math. Bull. 25 (1982), 500–501. 32. Tabor J., Tabor J.: General stability of functional equations of linear type. J. Math. Anal. Appl. 328 (2007), 192–200. 33. Ulam S.M.: Problems in Modern Mathematics. Wiley, New York 1960.

Selected issues in the theory of nonlinear oscillations Viktor Kulyk and Dariusz P¸aczko

Abstract. The paper presents results concerning the theory of oscillations in the field of linear extensions of dynamical systems. An overview of the basic results was done, the direction of research was outlined and the results obtained were given in this regard. Keywords: dynamical systems, oscillation. 2010 Mathematics Subject Classification: 34D10, 34D35, 37L25.

1. Introduction In the theory of non-linear multi-frequency oscillations several questions related to the research of invariant tori of autonomous systems of differential equations arise. One of the important issues is to maintain invariant tori for small disturbances, as well as the behaviour of the solutions of an equation on the same tori and within their neighbourhood. Together with deep research in this direction (see [2, 10, 11]), there is a number of problems that currently can not be fully resolved. This article is a review of some of the problems that arise in the use of Lyapunov functions in the theory of linear extensions of dynamical systems on the torus. Similar research can be found in [1, 9, 14, 12, 15]. Let us consider the system of differential equations dϕ = a(ϕ), dt

dx = A(ϕ)x + f (ϕ), dt

(1)

V. Kulyk Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] D. P¸ aczko Opole University of Technology, Pr´ oszkowska 76, 45-758 Opole, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 291–303. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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where ϕ = (ϕ1, ϕ2 , . . . , ϕm ), x = (x1 , x2 , . . . , xn ), the vector function a(ϕ) = (a1 (ϕ) , a2 (ϕ), . . ., am (ϕ)) defined for all ϕ ∈ Rm is real, continuous and periodic with respect to each variable ϕj with the period 2π. Usually, it is said that the function a(ϕ) is defined on the m-dimensional torus Tm , thus it belongs to the space of continuous functions C(Tm ). It is assumed that the Cauchy problem dϕ dt = a(ϕ), ϕ|t=0 = ϕ has a unique solution, denoted by ϕt (ϕ) (see [2, 10]). Since the periodic function a(ϕ) is bounded, the solutions ϕt (ϕ) will always be defined on the whole real axis R. The matrix A(ϕ) in the system (1) is a square matrix whose elements are real, continuous and 2π-periodic functions, i.e. A(ϕ) ∈ C(Tm ), the vector function f (ϕ) ∈ C(Tm ). The system (1) is used to be called a linear extension of a dynamical system on the torus. Together with the system (1), we will consider the corresponding homogeneous system in the following form dx dϕ = a(ϕ), = A(ϕ)x. (2) dt dt Recall the definition of space C ′ (Tm ; a) of an invariant torus of the system (1) and the definition of the Green-Samoilenko function for the problem with an invariant torus G0 (τ, ϕ) for the system (2). Definition 1.1. C ′ (Tm ; a) is a subspace C(Tm ) of continuous functions Φ(ϕ) such that the superposition Φ(ϕt (ϕ)) is continuously differentiable with respect to the varidf ˙ able t, t ∈ R, whereas dΦ(ϕt (ϕ)) |t=0 = Φ(ϕ). dt

Definition 1.2 (see [5]). We say that the system (1) has an invariant torus, defined by the equality x = u(ϕ), if u(ϕ) ∈ C ′ (Tm ; a) and an identity: u(ϕ) ˙ ≡ A(ϕ)u(ϕ) + f (ϕ) for every ϕ ∈ Tm , is satisfied. Denoting a fundamental matrix of solutions normalised for t = τ , i.e. Ωτt |t=τ = In , where In is an n-dimensional identity matrix of the linear system dx = A(ϕt (ϕ))x, dt

(3)

with Ωτt (ϕ) (Ωτt (ϕ) = Ωτt (ϕ; A)), let us recall the definition of the Green-Samoilenko function [2]. Definition 1.3. If there exists an n-dimensional square matrix C(ϕ) ∈ C(Tm ), such that the function ( Ωτ0 (ϕ)C(ϕτ (ϕ)), τ 6 0, G0 (τ, ϕ) = (4) 0 Ωτ (ϕ) [C(ϕτ (ϕ)) − In ] , τ > 0, satisfies the estimate kG0 (τ, ϕ)k 6 Ke−γ|τ |

∀τ ∈ R

∀ϕ ∈ Tm ,

(5)

where K, γ are positive constants independent of τ and ϕ, then the function (4) is called the Green-Samoilenko function for the problem with an invariant torus of the system (2).

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Remark 1.4. If there exists a Green-Samoilenko function (4) satisfying the estimate (5), then the system (1) has an invariant torus in the form of +∞ Z x = u(ϕ) = G0 (τ, ϕ)f (ϕτ (ϕ))dτ ,

(6)

−∞

for all fixed functions f (ϕ) ∈ C(Tm ). Remark 1.5. (See [10]). To establish the Green-Samoilenko function, the estimate (5) can be weakened – it is enough to require that the integral +∞ Z kG0 (τ, ϕ)kdτ

(7)

−∞

converges uniformly with respect to variables ϕ ∈ Tm , and then the equation (6) determines the invariant torus (1), whereas a problem with the study of the function smoothness (6) with respect to ϕ appears. There are examples of systems (2) that have a function in the form of (4) for which the estimate (5) is not satisfied, and the integral (7) is uniformly convergent with respect to the variables ϕ. However, in the examples considered there was another Green-Samoilenko function that satisfied the estimate (5). The question arises whether or not, if there is a function in the form of (4), for which the integral (7) is uniformly convergent, there always exists a function (possibly different) in the form of (4) for which the estimate (5) is satisfied. This question still remains unanswered. Remark 1.6. Since ϕt (ϕz (ϕ)) ≡ ϕt+z (ϕ), the fulfilment of the inequality (5) for the function (4) is equivalent to the fulfilment of the following estimate kGt (0, ϕ)k 6 Ke−γ|t|

∀ t ∈ R ∀ϕ ∈ Tm

for the auxiliary function Gt (0, ϕ) =

(

Ω0t (ϕ)C(ϕ), t > 0, Ω0t (ϕ) [C(ϕ) − In ] , t < 0,

(8)

with the same positive constants K, γ. Remark 1.7. If there exists a Green-Samoilenko function (4) with an estimate (5), the following function ( Ωτt (ϕ)C(ϕτ (ϕ)), τ 6 t, Gt (τ, ϕ) = (9) t Ωτ (ϕ) [C(ϕτ (ϕ)) − In ] , τ > t, is a Green’s function for a problem with bounded solutions of the system (3), i.e. the heterogeneous system

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dx = A(ϕt (ϕ))x + f (t) (10) dt +∞ R has the bounded solution x = Gt (τ, ϕ)f (τ )dτ for each function f (t) being contin−∞

uous and bounded on R. Corollary 1.8. If a heterogeneous linear system (10) has no bounded solution on R for a certain parameter value ϕ and a certain function f (t) continuous and bounded on R, then a Green-Samoilenko function (4) does not exist.

2. Overview of the main results The system (2) that has a unique Green-Samoilenko function with an estimate (5) is used to be called regular, and if it is known that there is at least one such function (4), then the system (2) is weakly regular, and in the case where there is infinitely many different of such functions, the system (2) is strictly weakly regular. n p P Let us recall that hy, y¯i = yj y¯j represents the inner product in Rn , kyk = hy, yi j=1

is a norm of the vector y in Rn , hSy, yi is a quadratic form associated with the symmetric matrix S. In the book [10] on page 124 the following theorem is formulated. Theorem 2.1. Suppose that the following quadratic form exists W = hS(ϕ)y, yi,

y ∈ Rn ,

(11)

associated with a symmetric matrix S(ϕ) ∈ C ′ (Tm ; a), whose derivative with respect to the system of equations dϕ = a(ϕ), dt

dy = −AT (ϕ)y, dt

(12)

is positive definite, thus Dh i E ˙ = S(ϕ) ˙ W − S(ϕ)AT (ϕ) − A(ϕ)S(ϕ) y, y > kyk2 ,

y ∈ Rn .

(13)

Under the above assumptions the system (2) is weakly regular. If in addition we assume that det S(ϕ) 6= 0 ∀ϕ ∈ Tm , (14) then the system (2) is regular and in the structure of the Green-Samoilenko function (4) the matrix C(ϕ) is a projection matrix for which the following identities are satisfied C 2 (ϕ) ≡ C(ϕ),

C(ϕt (ϕ)) ≡ Ω0t (ϕ)C(ϕ)Ωt0 (ϕ)

∀ϕ ∈ Tm , t ∈ R.

(15)

Selected issues in the theory of nonlinear oscillations

295

If the condition (14) is not satisfied and there exist such values ϕ = ϕ0 ∈ Tm for which the following the equality is satisfied det S(ϕ0 ) = 0 then the system (2) is strictly weakly regular and each of the matrices C(ϕ) does not satisfy any of the identities (15). In the book [11] on page 130 identities are examined in detail. It is easy to see (by making the change of variables y = S −1 (ϕ)x) that the following remark is true. Remark 2.2. If the symmetric matrix S(ϕ) ∈ C ′ (Tm ; a) is non-degenerated in the ¯ inequality (13), then for the matrix S(ϕ) = −S −1 (ϕ) the following inequality is satisfied h i ¯˙ ¯ ¯ h S(ϕ) + S(ϕ)A(ϕ) + AT (ϕ)S(ϕ) x, xi > γkxk2 , x ∈ Rn , γ = const > 0, and which means that the derivative of the non-degenerated quadratic form ¯ −hS −1 (ϕ)x, xi = hS(ϕ)x, xi with respect to the system (2) is positive definite. A converse theorem is established (see [10] on page 123). Theorem 2.3. If the system (2) has the Green-Samoilenko function (4) with the estimate (5), then there exist symmetric matrices S(ϕ) that satisfy the inequality (13). Some of these matrices can be expressed in the following form S(ϕ) = 2[S1 (ϕ) − S2 (ϕ)], Z 0  T S1 (ϕ) = Ωτ0 (ϕ)C(ϕτ (ϕ)) Ωτ0 (ϕ)C(ϕτ (ϕ)) dτ, −∞ +∞

S2 (ϕ) =

Z

0

 T Ωτ0 (ϕ) [C(ϕτ (ϕ)) − In ] Ωτ0 (ϕ) [C(ϕτ (ϕ)) − In ] dτ.

Remark 2.4. The form of matrices S(ϕ) was generalized (the proof can be found in [7]): Z 0  T Ωτ0 (ϕ)C(ϕτ (ϕ))H1 (ϕτ (ϕ)) Ωτ0 (ϕ)C(ϕτ (ϕ)) dτ − S(ϕ) = −

Z 0

−∞ +∞

 T Ωτ0 (ϕ) [C(ϕτ (ϕ)) − In ] H2 (ϕτ (ϕ)) Ωτ0 (ϕ) [C(ϕτ (ϕ)) − In ] dτ, (16)

where Hi (ϕ) ∈ C(Tm ) are any symmetric matrices, satisfying the inequality hHi (ϕ)y, yi > 2kyk2,

i = 1, 2.

Simultaneously, if the system (2) is regular, then the inequality (14) is satisfied for any symmetric matrix S(ϕ) ∈ C ′ (Tm ; a) that satisfies the condition (13). Regarding to the norm of a matrix A defined as kAk = max kAxk, it has been kxk=1

proven in [13] (pp. 1685–1686) that from the inequality kΩ0∆ (ϕ)k < 1 ∀ϕ ∈ Tm

(17)

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for any fixed value ∆ > 0 we obtain an estimation kΩτt (ϕ)k 6 Ke−γ(t−τ ),

t, τ ∈ R,

τ 6 t,

(18)

where K and γ are certain positive constants. Hence it follows the theorem. Theorem 2.5. Let the constant ∆ > 0 such that kΩ0∆ (ϕ)k < 1 for every ϕ ∈ Tm exist. Then the system (2) is regular and the Green–Samoilenko function has the form ( Ωτ0 (ϕ), τ 6 0, G0 (τ, ϕ) = 0, τ > 0. Another proof of the fact that the inequality (17) implies the estimation (18) has been proposed in [4] (pp. 92–93). Namely, a symmetric matrix was considered in the form ∆+t Z T (Ωtσ (ϕ)) Ωtσ (ϕ) dσ = S(ϕt (ϕ)). (19) t

Whereas the quadratic form hS(ϕt (ϕ)x, x) is positive definite and its derivative with respect to the system (3) is negative definite. Example 2.6. Let us examine the regularity of the system dϕ = 1 + ε cos ϕ, dt

dx = (cos ϕ)x. dt

0 < ε < 1,

The regularity of this system is equivalent to the regularity of the following system dϕ = 1, dt

dx cos ϕ = x, dt 1 + ε cos ϕ

for this system the value of Ω02π (ϕ) will be less than 1, namely Ω02π (ϕ) = exp

Z



0



=

 Z 2π cos(t + ϕ) cos(t + ϕ) dt + dt = 1 + ε cos(t + ϕ) 1 + ε cos(t + ϕ) π Z0 π    cos(t + ϕ) cos(t + ϕ) = exp − dt = 1 + ε cos(t + ϕ) 1 − ε cos(t + ϕ) 0 Z π  −2ε cos2 (t + ϕ) = exp dt < 1. 2 2 0 1 − ε cos (t + ϕ) = exp

Z

cos(t + ϕ) dt 1 + ε cos(t + ϕ)

π

In this way the system considered is regular for the parameter value ε ∈ (0, 1). If ε = 0, the system is not regular. If we now consider the matrix Z T2 (ϕ) T (Ω0σ (ϕ)) Ω0σ (ϕ) dσ = S(ϕ), T1 (ϕ)

Selected issues in the theory of nonlinear oscillations

297

which is a generalization of (19), then calculating the derivative of a quadratic form hS(ϕ)x, xi with respect to the system (2) we obtain the following theorem. Theorem 2.7. Let the two scalar functions T1 (ϕ), T2 (ϕ) ∈ C ′ (Tm ; a) exist such that the quadratic form T (ϕ)

Φ(x) = (1 + T˙2 (ϕ))kΩ0 2

T (ϕ)

(ϕ)xk2 −(1 + T˙1 (ϕ))kΩ0 1

(ϕ)xk2

is negative definite: Φ(x) 6 −γkxk2 , then the system (12) is weakly regular. In addition, if the inequality T1 (ϕ) < T2 (ϕ) is satisfied, the systems (2) and (12) are regular. Let us recall kAk0 = max kA(ϕ)k for matrix A = A(ϕ). Then the generalization ϕ∈Tm

of the theorem 2.5 is the following theorem. Theorem 2.8. Suppose that there exist ∆i ∈ R, i = 1, . . . , k, satisfying the inequality k·

k X

kΩ0∆i (ϕ) · Pi (ϕ)k20 < 1

∀ϕ ∈ Tm ,

(20)

i=1

where matrices Pi (ϕ) ∈ C(Tm ), i = 1, . . . , k, fulfil the condition k X

Pi (ϕ) ≡ In ,

(21)

i=1

then the system (2) is regular. Proof. Let us consider a symmetric matrix in the form S(ϕ) =

k X

Si (ϕ) =

i=1

k Z0 X

 T Ωτ0 (ϕ)Pi (ϕτ (ϕ)) · Ωτ0 (ϕ)Pi (ϕτ (ϕ)) dτ.

(22)

i=1 −∆

i

We show that if the inequality (20) is satisfied, the derivative of a quadratic form hS(ϕ)x, xi with respect to the system (12) is positive definite. Let us examine one of the element of the series (22): Z0

Si (ϕ) =

 T Ωτ0 (ϕ)Pi (ϕτ (ϕ)) · Ωτ0 (ϕ)Pi (ϕτ (ϕ)) dτ,

−∆i

and write down a composition Si (ϕt (ϕ)) =

Zt −∆i +t

 T Ωτt (ϕ)Pi (ϕτ (ϕ)) · Ωτt (ϕ)Pi (ϕτ (ϕ)) dτ.

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Differentiating with respect to variable t, we obtain d Si (ϕt (ϕ)) = dt  t T t = Pi (ϕt (ϕ)) · [Pi (ϕt (ϕ))]T − Ωt−∆ (ϕ)Pi (ϕt−∆i (ϕ)) · Ωt−∆ (ϕ)Pi (ϕt−∆i (ϕ)) + i i + A(ϕt (ϕ))Si (ϕt (ϕ)) + Si (ϕt (ϕ))AT (ϕt (ϕ)), and hence for t = 0 we get S˙ i (ϕ) − Si (ϕ) AT (ϕ) − A (ϕ) Si (ϕ) =

 0 T T 0 (ϕ) Pi (ϕ−∆i (ϕ)) . = Pi (ϕ) · [Pi (ϕ)] − Ω−∆ (ϕ) Pi (ϕ−∆i (ϕ)) · Ω−∆ i i

In this way we obtain equality for the matrix (22): ˙ S(ϕ) − S(ϕ)AT (ϕ) − A(ϕ)S(ϕ) = =

k X

Pi (ϕ) · [Pi (ϕ)]T −

i=1

k X

0 0 Ω−∆ (ϕ)Pi (ϕ−∆i (ϕ)) · [Ω−∆ (ϕ)Pi (ϕ−∆i (ϕ))]T , i i

i=1

hence the corresponding quadratic form is as follows Dh i E S˙ (ϕ) − S (ϕ) AT (ϕ) − A (ϕ) S (ϕ) x, x = =

k X

T

[Pi (ϕ)] i=1

k

2 X T

 0 x −

Ω−∆i (ϕ) Pi (ϕ−∆i (ϕ)) i=1

2

x . (23)

Taking into account the identity (21), the first component of the right-hand side of the equation (23) is estimated from below in the following manner k X

Indeed, since

k P

i=1

T

k [Pi (ϕ)] xk2 >

i=1

Pi (ϕ) ≡ In , then

k

X T kxk = [Pi (ϕ)]

i=1

1 kxk2 . k

k

X T x 6 k [Pi (ϕ)] xk.

i=1

Hence, based on the Cauchy–Schwarz inequality X 2  X  X  k k k 2 2 ai · bi , ai b i 6 i=1

i=1

i=1

T

by substituting ai = 1, bi = k [Pi (ϕ)] xk, we obtain X 2 k k X T T 2 kxk 6 k [Pi (ϕ)] xk 6 k · k [Pi (ϕ)] xk2 , i=1

which implies the fulfilment of the inequality (24).

i=1

(24)

Selected issues in the theory of nonlinear oscillations

299

Now let us estimate the second component on the right-hand side of (23). To do this, let us consider the following inequality



T T

0

 0

(ϕ) P (ϕ (ϕ))

Ω−∆i (ϕ) Pi (ϕ−∆i (ϕ)) x 6 Ω−∆

· kxk = i −∆i i 0 0 = kΩ−∆ (ϕ) Pi (ϕ−∆i (ϕ))k·kxk 6 max kΩ−∆ (ϕ) Pi (ϕ−∆i (ϕ))k · kxk = i i ϕ∈Tm

= max kΩ0∆i (ϕ) Pi (ϕ)k · kxk = kΩ0∆i (ϕ) Pi (ϕ)k0 · kxk. ϕ∈Tm

Thus, the second term on the right-hand side of (23) is estimated as follows k k X T

2 X ∆i

 0 kΩ0 (ϕ) Pi (ϕ)k20 kxk2 .

Ω−∆i (ϕ) Pi (ϕ−∆i (ϕ)) x 6

(25)

i=1

i=1

Taking into account inequalities (24), (25), from equation (23) follows Dh i E S˙ (ϕ) − S (ϕ) AT (ϕ) − A (ϕ) S (ϕ) x, x >

! k 1 X ∆i 2 − kΩ0 (ϕ) Pi (ϕ) k0 ·kxk2 . k i=1

This shows that if the inequality (20) is satisfied, then the derivative of the quadratic form hS (ϕ) x, xi with respect to the system (12) is positive definite, which means that the system (2) is weakly regular. ⊔ ⊓ Theorem 2.9 ([6]). Let the system (2) be weakly regular, then the expanded system dϕ = a(ϕ), dt

dx = A(ϕ)x, dt

dy = x − AT (ϕ)y, dt

(26)

is regular. Whereas the derivative of the following non-degenerated quadratic form Vp = phx, yi + hS(ϕ)y, yi,

(27)

with respect to the system (26) is positive definite for sufficiently large values of the parameter p (here the matrix S(ϕ) ∈ C ′ (Tm ; a) satisfies the inequality (13)). dx Example 2.10. The system of equations dϕ dt = sin ϕ, dt = (cos ϕ)x is strictly weakly regular, because the derivative of the function V = −(cos ϕ)y 2 with respect to the dy adjoint system: dϕ dt = sin ϕ, dt = −(cos ϕ)y is positive definite

V˙ = (sin ϕ)ϕy ˙ 2 − (cos ϕ)2y y˙ = (sin2 ϕ + 2 cos2 ϕ)y 2 > y 2 whereas cos ϕ = 0, ϕ =

π 2

+ πn. The following expanded system

dϕ = sin ϕ, dt

dx = (cos ϕ)x, dt

dy = x − (cos ϕ)y dt

(28)

is regular, because the derivative of the non-degenerated quadratic form V = 2xy − (cos ϕ)y 2

(29)

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with respect to the system (28) is positive definite V˙ = 2x2 − 2xy cos ϕ + (sin2 ϕ + 2 cos2 ϕ)y 2 > x2 + y 2 . Remark 2.11. If the system (2) is weakly regular, then the expanded system dϕ = a(ϕ), dt

dx = A(ϕ)x, dt

dy = B(ϕ)x − AT (ϕ)y, dt

(30)

is regular for each n × n-dimensional matrix B(ϕ) ∈ C(Tm ) that satisfies one of the inequalities hB(ϕ)x, xi > βkxk2 ,

hB(ϕ)x, xi 6 −βkxk2 ,

β = const > 0.

(31)

Whereas the Green-Samoilenko function of the system (30) will be 2n-dimensional  ! !  Ωτ0 (ϕ) 0 C11 (ϕτ (ϕ)) C12 (ϕτ (ϕ))   , τ 6 0,   ω(0, τ, ϕ) (Ω τ (ϕ))T C21 (ϕτ (ϕ)) C22 (ϕτ (ϕ)) 0 ¯ ! ! G0 (τ, ϕ) =  Ωτ0 (ϕ) 0 C11 (ϕτ (ϕ)) − In C12 (ϕτ (ϕ))   , τ >0   ω(0, τ, ϕ) (Ω τ (ϕ))T C (ϕ (ϕ)) C (ϕ (ϕ)) − I 0

21

τ

22

τ

n

and will be changing with the change of the matrix B(ϕ) ∈ C(Tm ). The n-dimensional ¯ block G11 0 (τ, ϕ) of the matrix G0 (τ, ϕ) is the Green-Samoilenko function of the system (2). Still, there is no answer to the question whether all the Green-Samoilenko functions G11 0 (τ, ϕ) of the system (2) can be obtained from the system (30) by changing the matrix B(ϕ) ∈ C(Tm ). Remark 2.12. If we resign from the conditions (31), the system (30) may not be regular. Whereas the question what necessary and sufficient conditions need to be imposed on the matrix B(ϕ) ∈ C(Tm ) so that the system (30) is regular under the condition of strictly weakly regularity of the system (2) remains open. Continuing the study of the example (28), we will consider the following system dϕ = sin ϕ, dt

dx = (cos ϕ)x, dt

dy = (sin ϕ)x − (cos ϕ)y. dt

(32)

Although the derivative of the quadratic form V = pxy − (cos ϕ)y 2 with respect to the system (32) is not positive definite independently of the parameter p ∈ R, a quadratic form can be written in another form Φ = x2 cos ϕ + 2xy sin ϕ − y 2 cos ϕ, whose derivative with respect to the system (32) will be positive definite. It can be concluded that under the condition of strictly weakly regularity of the system (2), the expanded system (30) may be regular with some matrices B(ϕ) ∈ C(Tm ) that do not fulfil any of the inequalities (31). One of the generalizations of the regular system of equations (32) is known. Consider the case n = 1, A(ϕ) = λ(ϕ) is a continuous, 2π-periodic scalar function with respect to each variable ϕj , j = 1, . . . , m. Let us denote

Selected issues in the theory of nonlinear oscillations

301

ψ = k1 ϕ1 + . . . + km ϕm + θ = hk, ϕi + θ, where kj are certain integers, k = (k1 , . . . , km ) is a vector with integer coordinates, |k| = |k1 | + . . . + |km |, θ is a constant. The following theorem is true. Theorem 2.13 ([6]). Let the inequality σ = hk, a(ϕ)i sin ψ + 2λ(ϕ) cos ψ > 0 for some integer vector k, |k| > 0, for all ϕj , j = 1, . . . , m, and for some constant θ be satisfied. Then the system of equations dϕ = a(ϕ), dt

dx = λ(ϕ)x, dt

dy = hk, a(ϕ)i − λ(ϕ)y, dt

(33)

is regular whereas the derivative of the quadratic form x2 cos ψ + 2xy sin ψ − y 2 cos ψ with respect to the system (33) is positive definite. Let us consider the following example (see [6]). Example 2.14. dϕ1 dϕ2 = 3 sin ϕ1 cos ϕ2 , = 2 cos ϕ1 sin ϕ2 , dt dt dx = x [n cos(ϕ1 − ϕ2 ) + ε sin(ϕ1 + ϕ2 )] , dt where n = 1, 2, . . ., |ε| < 0, 5. Denoting ψ = ϕ1 − ϕ2 , k = (1, −1), we have hk, ai sin ψ = 2 sin2 ψ + sin ϕ1 cos ϕ2 sin ψ, 2λ(ϕ) cos ψ = 2n cos2 ψ + 2ε cos ψ sin(ϕ1 + ϕ2 ). It is clear that the conditions of Theorem 2.13 given above are met. In this way, the expanded system of equations dϕ1 dϕ2 = 3 sin ϕ1 cos ϕ2 , = 2 cos ϕ1 sin ϕ2 , dt dt dx = [n cos(ϕ1 − ϕ2 ) + ε sin(ϕ1 + ϕ2 )] x, dt dy = [3 sin ϕ1 cos ϕ2 − 2 cos ϕ1 sin ϕ2 ] x − [n cos(ϕ1 − ϕ2 ) + ε sin(ϕ1 + ϕ2 )] y, dt is regular for any natural value n and the fulfilment of the inequality |ε| < 0, 5. Let us consider the system of differential equations dϕ = a cos ϕ + b sin ϕ, dt n  X dx  = a0 + (aj cos jϕ + bj sin jϕ) x, dt j=1

(34)

with some real coefficients a, b, aj , bi , j = 0, . . . , n, i = 1, . . . , n. The problem is to find such conditions for these coefficients that the system (34) has the GreenSamoilenko function.

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It should be noted that if a = b = 0 in the system (34), it may have only one n P Green-Samoilenko function under the condition a0 + (aj cos jϕ + bj sin jϕ) 6= 0 for j=1

every ϕ ∈ R. Suppose that a2 + b2 6= 0 and denote M1 = a1 cos θ − b1 sin θ + a3 cos 3θ − b3 sin 3θ + . . . + a2l−1 cos(2l − 1)θ − b2l−1 sin(2l − 1)θ, M2 = a0 + a2 cos 2θ − b2 sin 2θ + a4 cos 4θ − b4 sin 4θ + . . . + a2m cos 2mθ − b2m sin 2mθ, where max{2l − 1, 2m} = n,

a sin θ = √ , 2 a + b2

b cos θ = √ . 2 a + b2

The following theorem is true. Theorem 2.15 ([3]). The system (34) has a unique Green-Samoilenko function whenever the inequality |M1 | < |M2 | is satisfied. If the inequality M1 > |M2 | holds, then the system (34) has infinitely many different Green-Samoilenko functions. In the case of |M1 | = |M2 |, M1 < −|M2 |, the system (34) has no Green-Samoilenko function. Theorem 2.16 ([8]). Let the two systems ( dϕ ϕ ∈ Tm , dt = ω1 (ϕ), dx = A (ϕ)x, x ∈ Rn , 1 dt

(

dψ dt dx dt

= ω2 (ψ), = A2 (ψ)x,

ψ ∈ Tk , x ∈ Rn ,

(35)

 dϕ  dt = ω1 (ϕ),   dψ     dt = ω2 (ψ),   dx1   (ψ) + 12 (A1 (ϕ) + AT1 (ϕ)) − In x1 +  dt = A 2  + A1 (ϕ) + AT2 (ψ) x2 , xj ∈ Rn ,     dx2 1 T  = −A2 (ψ) + 2 (A1 (ϕ) − A1 (ϕ)) + In x1 − AT2 (ψ)x2  dt      dx3  = A2 (ψ) + 12 (AT1 (ϕ) − A1 (ϕ)) + In x1 −      dt − A1 (ϕ) + AT2 (ψ) x2 − AT1 (ϕ)x3 ,

(36)

be weakly regular, then the following system

is regular. In addition, the derivative of the quadratic form Vp = p2 (hx1 , x2 i + hx1 , x3 i + hx2 , x3 i) + phS2 (ψ)x2 , x2 i + hS1 (ϕ)x3 , x3 i, with respect to the system (28) for sufficiently large values of the parameter p > 1 is positive definite.

Selected issues in the theory of nonlinear oscillations

303

Corollary 2.17. Let the system (2) be weakly regular, then the following system  dϕ  dt = a(ϕ),   dx  3 1 T T 1 xj ∈ Rn , dt = ( 2 A(ϕ) + 2 A (ϕ) − In )x1 + (A(ϕ) + A (ϕ))x2 , dx 1 1 T T 2   dt = (− 2 A(ϕ) + 2 A (ϕ) + In )x1 − A (ϕ)x2 ,   dx 3 1 T T T 3 dt = ( 2 A(ϕ) − 2 A (ϕ) + In )x1 − (A(ϕ) + A (ϕ))x2 − A (ϕ)x3 ,

(37)

is regular whereas the derivative of the quadratic form Vp = p2 (hx1 , x2 i + hx1 , x3 i + hx2 , x3 i) + phS(ϕ)x2 , x2 i + hS(ϕ)x3 , x3 i, with respect to the system (37) for sufficiently large values of the parameter p > 1 is positive definite. Remark 2.18. In the systems (36) and (37) the identity matrix can be replaced with the matrix B(ϕ), which is positive definite and in this case the systems (36) and (37) remain regular.

Bibliography 1. Boichuk A.A.: A condition for the existence of a unique Green-Samoilenko function for the problem of invariant torus. Ukrainian Math. J. 53 (2001), 637–641. 2. Kulyk V.L., Mitropolski Y.A., Samoilenko A.M.: Dichotomies and Stability in Nonautonomous Linear Systems. Taylor & Francis, London 2003. 3. Kulyk H.M., Kulyk V.L.: Existence of Green-Samoilenko Functions of Some Linear Extensions of Dynamical Systems. Nonlinear Oscillations 7 (2004), 454-460. ´ Gliwice 4. Kulyk V.L., P¸ aczko D.: Selected Issues of Mathematical Control Theory. Wyd. Pol. Sl., 2008 (in Polish). 5. Kulyk V.L., P¸ aczko D.: Selected Issues of Qualitative Theory of Differential Equations. Wyd. ´ Gliwice 2012 (in Polish). Pol. Sl., 6. Kulyk V.L., P¸ aczko D.: Some methods of complement of weak regular linear extensions of dynamical systems to regular. Nonlinear Oscillations 16 (2013), 65–74. 7. Kulyk V.L., Stepanenko N.: Alternating-sign Lyapunov functions in the theory of linear extensions of dynamical systems on a torus. Ukrainian Math. J. 59 (2007), 546–562. 8. Kulyk V.L., Wojtowicz B.: Linear extensions of dynamical systems on a torus that possess Green-Samoilenko functions. Ukrainian Math. J. 50 (1998), 203–215. 9. Melnikov V.K.: On the stability of the center for time periodic perturbations. Proc. Moscow Math. Soc. 2 (1963), 3–52. 10. Mitropolsky Y.A., Samoilenko A.M., Kulik V.L.: Investigation of Dichotomy of Linear Systems of differential Equations Using Lyapunov Functions. Naukova Dumka, Kiev 1990. 11. Samoilenko A.M.: Elements of the Mathematical Theory of Multi-Frequency Oscillations. Mathematics and its Applications. Springer Netherlands, 1991. 12. Samoilenko A.M., Timchishin O.Ya., Prikarpatskii A.K.: Poincar´ e-Mel’nikov geometric analysis of the transversal splitting of manifolds of slowly perturbed nonlinear dynamical systems. Ukrainian Math. J. 45 (1993), 1878–1892. 13. Samoilenko A.M.: On certain problems of perturbation theory of smooth invariant tori of dynamical systems. Ukrainian Math. J. 46 (1994), 1665–1699. 14. Samoilenko A.M.: On the existence of a unique Green function for the linear extension of a dynamical system on a torus. Ukrainian Math. J. 53 (2001), 584–594. 15. Samoilenko A.M., Prykarpats’kyi A.K., Samoilenko V.H.: Lyapunov–Schmidt approach to studying homoclinic splitting in weakly perturbed lagrangian and hamiltonian systems. Ukrainian Math. J. 55 (2003), 82–92.

On identities satisfied by cancellative semigroups and their groups of fractions Olga Macedo´ nska and Piotr Slanina

Abstract. A subsemigroup S of a group G is called generating if elements of S generate G as a group. If S satisfies an identity, then G = SS −1 = S −1 S is a group of fractions of S. We recall the results concerning the problem which was open for more then 20 years, whether each identity satisfied in S must be satisfied in its group of fractions. The groups where the answer for this question is positive for every generating semigroup we call S-R-groups. We show that varieties of S-R-groups form a sublattice in the lattice of all group varieties. Keywords: semigroup identities, groups of fractions. 2010 Mathematics Subject Classification: 20E10.

1. Preliminaries Let F be the free group on the set X = {x1 , x2 , x3 , . . .}, and let F ⊆ F be the cancellative free semigroup with the unity, generated by X. • A group G satisfies an identity u(x1 , . . . , xm ) ≡ v(x1 , . . . , xm ) if for every elements g1 , . . . , gm in G the equality u(g1 , . . . , gm ) = v(g1 , . . . , gm ) holds. • An identity in a semigroup has a form u(x1 , . . . , xn ) = v(x1 , . . . , xn ) where the words u and v are written without inverses of variables, u, v ∈ F. Such an identity in a group is called a semigroup identity. It is clear that abelian groups and groups of finite exponent satisfy semigroup identities. • A semigroup S satisfies left (right) Ore condition if for arbitrary a, b ∈ S there are a′ , b′ ∈ S such that a′ a = b′ b (resp. aa′ = bb′ ). • A subsemigroup S of a group G is called a generating semigroup if elements of S generate G as a group.

O. Macedo´ nska, P. Slanina Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected]; [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 305–312. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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• If a generating semigroup S ⊆ G satisfies a nontrivial identity, then S satisfies Ore conditions (see e.g. [12, Proposition 1]). • If a generating semigroup S ⊆ G satisfies a nontrivial identity, then G is the group of fractions of S, that is G = SS −1 = S −1 S [5]. • Each congruence ρ ⊆ F × F defines a subset Aρ ⊆ F F −1 , where Aρ := { ab−1 | (a, b) ∈ ρ }, AF ρ denotes the normal closure of Aρ in F. • Let G be a group with a generating semigroup S. If S satisfies a nontrivial identity then there exists a congruence ρ on F such that S ∼ = F /ρ, G ∼ = F/AF ρ and −1 F Aρ ∩ F F = Aρ [6, Construction 12.3 and Corollary 12.8]. X ւ ↓ ց S∼ = F /ρ ←− F −→ F ց ւ G∼ = F/AF ρ So we have the following commutative diagram F ֒→ F ↓ ↓ ∼ S∼ = F /ρ ֒→ F/AF ρ =G

2. History of the topic Since the 2-generator free semigroup sgp(x, y) contains a free semigroup of infinite rank sgp(x1 , x2 , . . .), where each xi is a word on x, y, an identity u(x1 , . . . , xn ) = v(x1 , . . . , xn ) implies a binary identity. So if a generating semigroup S ⊆ G satisfies a nontrivial identity then S satisfies a 2-variable semigroup identity which, by the cancellation property, may be assumed as xa(x, y) = yb(x, y) or u(x, y)x = v(x, y)y. It follows that S satisfies left and right Ore conditions. Then elements in S satisfy the identities y −1 x = b(x, y)a−1 (x, y)

and

xy −1 = u−1 (x, y)v(x, y),

which implies that G = SS −1 = S −1 S, that is G is the group of fractions for S [13, 15, 9] and [6, Theorem 1.23]. It is clear that if S is abelian then the group G is necessary abelian. However, if S is a free semigroup, then G is not necessary free, because by result of A.I. Mal’tsev [13], F/F ′′ has a free generating semigroup. The problem how far properties of a generating semigroup S define the properties of the group G attracted attention of many authors. A.I. Mal’tsev considered semigroup identities of the form un = vn , where un , vn are words on letters x, y, z1 , . . . , zn , defined inductively as follows

On identities satisfied by cancellative semigroups and their groups of fractions

u0 = x,

v0 = y,

u1 = u0 z1 v0 ,

un+1 = un zn+1 vn ,

v1 = v0 z1 u0 ,

and for

307

n>0

vn+1 = vn zn+1 un .

In [13] A.I. Mal’tsev proved that a group is nilpotent of class at most n if and only if it satisfies the identity un = vn . Moreover, if the identity un = vn is satisfied in a generating semigroup S ⊆ G then it is satisfied in G which must be nilpotent of class at most n. We shall call the laws with such a property transferable.

2.1. Transferable identities The following well known problems are due to G. Bergman [1, 2]. GB-Problem. Let G be a group with a generating semigroup S. Must each identity satisfied in S be satisfied in G? Another formulation of this problem is whether every proper variety of semigroups is closed with respect to groups of fractions [17, Question 11.1]. The following question was posed in [10, page 95]. Let a semigroup identity a = b imply a semigroup identity u = v in groups. Does the same implication hold in semigroups? The equivalence of this question with the GB-Problem was proved in [11]. Definition 2.1. Let S be a generating semigroup in a group G. We call an identity u = v transferable if being satisfied in S, it must be satisfied in G. For example, the nilpotent identities un = vn found by A.I. Mal’tsev [13] are transferable. In this terminology the GB-Problem asks: Is every semigroup identity transferable? Another weaker problem posed by G. Bergman was: Must the group G satisfy some group identity if its generating semigroup S satisfies a nontrivial semigroup identity? In 2005 S.V. Ivanov and A.M. Storozhev [7] gave negative answer to both questions. Their counterexample-group G (in fact a family of them) with a generating semigroup S contains a free subgroup (hence satisfies no group identity) while S satisfies a semigroup identity similar to that introduced by A.Yu. Ol’shanskii in [16]. Since the problems have negative answers in general, the questions arise: 1. Which semigroup identities are transferable? 2. In which groups all semigroup identities are transferable? In 1986 when the problems were discussed in G. Bergman’s “Problem Seminar” in Berkeley, it was shown that the identity x2 y 2 = y 2 x2 is transferable. In 1992 it was shown that the identities xn y n = y n xn are transferable for all natural n [8]. Besides these identities, Mal’tsev identities and the identity xn ≡ 1 no other examples are described. As to the second of the above question, Theorem C in [3] says that Bergman’s question has an affirmative answer for soluble groups: if G is a soluble group (or, slightly more generally, an extension of a soluble group by a locally finite group of finite exponent), and S ⊆ G is any generating subsemigroup satisfying a positive law, then that law holds in G.

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2.2. Semigroup respecting groups We define now so called S-R groups, the groups where all semigroup identities are transferable. Definition 2.2 ([12]). We call a group G semigroup respecting (S-R group) if all of the identities holding in any generating semigroup of G, hold in G. For example, torsion groups are S-R groups, since S −1 = S = G. In 2008 the following properties of S-R groups were proved [12]: • The property of a group to be S-R group is a “local property” in the sense of Mal’tsev, that is if every finitely generated subgroup of a group G is the S-R group, then so is G. • The class of locally residually finite groups consists of S-R groups. • Every linear group over a field is the S-R group. • The class of locally graded groups without free noncyclic subsemigroups consists of S-R groups. We recall that a group G is called locally graded if every nontrivial finitely generated subgroup of G has a proper subgroup of finite index. The class ˇ of locally graded groups was introduced in 1970 by Cernikov to avoid groups such as the infinite Burnside groups or Ol’shanskii-Tarski monsters. We note that all locally or residually soluble groups and all locally or residually finite groups are locally graded. The class of locally graded groups is closed for taking subgroups, extensions and cartesian products.

3. End+ invariance and S-R-property Let F be the free group on the set X = {x1 , x2 , x3 , . . .}, and let F ⊆ F be the cancellative free semigroup with the unity, generated by X. By End F we denote the set of all endomorphisms in F . + By End we denote the set of so called positive endomorphisms in F , which map + X → F . The set End can be identified with the set End F of all endomorphisms + of the semigroup F . The inclusion End ⊆ End F is clear. We recall that End F invariant subgroup is called fully invariant or verbal. Every group variety is uniquely defined by a verbal subgroup V ⊆ F [14]. In view of the construction [6, Theorem 1.23] we have the following: • A semigroup S ∼ = F /ρ is relatively free if and only if the set Aρ and hence the + normal subgroup AF ρ are End -invariant. • The group of fractions of S, G = F/AF ρ is relatively free if and only if the normal subgroup AF is End-invariant. ρ • Each normal End+ -invariant subgroup N in F defines a cancellative congruence ρ on F by Aρ = N ∩ FF −1 . If Aρ 6= 1, then AF ρ = N. Hence the following two questions concerning S-R-groups are equivalent: Must a group of fractions of a relatively free semigroup be the relatively free group?

On identities satisfied by cancellative semigroups and their groups of fractions

309

Must each End+ -invariant normal subgroup in F be fully invariant? We show that being invariant with respect to a special automorphism α gives a criterion for a normal End+ -invariant subgroup N ⊳ F to be fully invariant. We start with two Lemmas. Lemma 3.1. If a normal subgroup N ⊳ F is End+-invariant and N ∩ F F −1 6= 1 then F = F F −1 N = F −1 F N. Proof. By assumption there are two different words a′ , b′ ∈ F such that a′ b′−1 ∈ N hence a′ ≡ b′ modulo N . Then by cancellation there are a, b ∈ F such that xa(x, y) ≡ yb(x, y) modulo N . That is x−1 y ∈ a(x, y)b−1 (x, y)N . Then since N is normal and End+ -invariant, we get F = FF −1 N. The second equality follows by conjugation. ⊓ ⊔ Lemma 3.2. If F = FF −1 N = F −1 F N and g1 , g2 , . . . , gn are in F , then there are s1 , s2 , . . . , sn , and r in F such that modulo N gi = si r−1 , i = 1, 2, . . . , n. Proof. All calculations in the proof are assumed modulo N . If n = 1, the statement is clear. To proceed by induction, let gi = ti q −1 for i 6 n − 1 and gn = ab−1 for some ti , q, a, b ∈ F. By Ore conditions, which are satisfied in F modulo N , there exist q ′ , b′ ∈ F such that modulo N qq ′ = bb′ . We denote r := qq ′ = bb′ , si := ti q ′ for i 6 n − 1 and sn := ab′ . Then gi = ti q −1 = ti (q ′ q ′

−1

gn = ab−1 = a(b′ b′

)q −1 = (ti q ′ )(q ′

−1

)b−1 = (ab′ )(b′

−1 −1

q

−1 −1

b

which finishes the proof.

) = si r−1 ,

) = sn r−1 ⊔ ⊓

We recall that F is the free group on the set X = {x1 , x2 , x3 , . . .}. Let α ∈ Aut F fix x1 and map xi → xi x−1 1 , i 6= 1. Lemma 3.3. If N ⊳ F is End+ -invariant and α-invariant then N is fully invariant. −1

Proof. First we note that α−1 ∈ End+ and hence N α invariant subgroup N in F satisfies the inclusion N ⊆ N α.

⊆ N. Hence every End+ (1)

Since by assumption N α ⊆ N , we have N α = N . To show that N is fully invariant it suffices to check that if w(x1 , . . . , xn−1 ) is a word in N then w(g2 , . . . , gn ) also is in N for every g2 , g3 , . . . , gn in F . So let w(x1 , . . . , xn−1 ) be in N . Since the map xi → xi+1 is in End+ , we have that −1 w = w(x2 , . . . , xn ) also is a word in N . Then N contains wα = w(x2 x−1 1 , . . . , xn x1 ) −1 In view of Lemma 3.2 we can find s2 , . . . , sn , r ∈ F such that gi = si r modN. Then we map x1 → r and xi → si . Since N , being End+ invariant, is invariant to this mapping, we get w(g2 , . . . gn ) ∈ N and hence N is fully invariant as required. ⊔ ⊓ Corollary 3.4. Let N be a normal End+ -invariant subgroup in F . Then N is fully invariant if and only if N = N α .

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4. Varieties of S-R groups The varieties of groups form a set partially ordered by inclusion, which is a complete lattice by means of the following definitions of greatest lower and least upper bound: M1 ∧ M2 is the variety with the set of laws defined by the verbal subgroup M1 M2 , while M1 ∨ M2 has the set of laws defined by the verbal subgroup M1 ∩ M2 [14]. The one-to-one correspondence between verbal subgroups in F and varieties reverses the inclusion relations. In this section we consider group varieties consisting of S-R-groups. These varieties are defined by specific verbal subgroups in F which, as we show, form a sublattice in the lattice of all verbal subgroups in F . Because of the duality the same follows for the S-R-varieties. Definition 4.1. We say that a variety M is semigroup respecting (S-R-variety) if it consists of S-R-groups. By VN we denote the fully invariant closure of a subgroup N ⊆ F . Corollary 4.2. A verbal subgroup M ⊆ F defines an S-R-variety if and only if every End+ -invariant normal subgroup N in F is fully invariant modulo M , that is N M = VN M, which by Corollary 3.4 is equivalent to N M = N α M = (N M )α . For example, since by result of P. Hall [4] finitely generated nilpotent groups are residually finite, it follows (see Section 2.2) that nilpotent varieties are semigroup respecting. We recall here the following well known result Lemma 4.3. Let A, B, C ⊆ G and A ⊆ C then AB ∩ C = A(B ∩ C). Proof. The inclusion “⊇” is clear. The opposite inclusion follows since ab = c ∈ AB∩C implies b = a−1 c ∈ B ∩ C, hence ab = c ∈ A(B ∩ C). ⊔ ⊓ Now we can prove Theorem 4.4. Semigroup respecting varieties form a modular lattice. Proof. Let M1 and M2 be two S-R-varieties defined by verbal subgroups M1 , M2 ∈ F . By Corollary 4.2 this is equivalent to the fact that for any normal End+ -invariant subgroup N the equalities hold: N M1 = N α M1 ,

N M2 = N α M2 .

(2)

Thus we have to prove that for any normal End+ -invariant subgroup N , N M1 M2 = N α M1 M2 ,

N (M1 ∩ M2 ) = N α (M1 ∩ M2 ).

The first equality is clear by assumptions on M1 . For any fixed N we denote D = N (M1 ∩ M2 ), then Dα = N α (M1 ∩ M2 ). So, the only we have to prove is the equality D = Dα . Now we note the following properties of the subgroup D.

On identities satisfied by cancellative semigroups and their groups of fractions

311

• If a subgroup M is fully invariant then Dα ∩ M = (D ∩ M )α . • Since the subgroup D is End+ -invariant, D ⊆ Dα and by Lemma 4.3, for A := D the following equality holds DM1 ∩ Dα = D(M1 ∩ Dα ).

(3)

• Since D is End+ -invariant and M1 defines an S-R-variety, so by (2) DM1 = Dα M1 .

(4)

• Since D ∩ M1 is End+ -invariant and M2 defines an S-R-variety, also by (2) (D ∩ M1 )M2 = (D ∩ M1 )α M2 ⊇ (D ∩ M1 )α .

(5)

• By Lemma 4.3, for A := (D ∩ M1 ) ⊆ M1 the following equality holds (D ∩ M1 )M2 ∩ M1 = (D ∩ M1 )(M2 ∩ M1 ) = (D ∩ M1 ).

(6)

We intersect (5) with M1 then (D ∩ M1 )M2 ∩ M1 ⊇ (D ∩ M1 )α ∩ M1 = (D ∩ M1 )α , which, in view of (6), gives D ∩ M1 ⊇ (D ∩ M1 )α . Since (D ∩ M1 ) is End+ -invariant, D ∩ M1 ⊆ (D ∩ M1 )α , which implies the equalities D ∩ M1 = (D ∩ M1 )α = Dα ∩ M1 .

(7)

Now we can obtain required Dα = D, because (4)

(3)

(7)

Dα = Dα M1 ∩ Dα = DM1 ∩ Dα = D(Dα ∩ M1 ) = D(D ∩ M1 ) = D. So, by Corollary 4.2, D is fully invariant, which finishes the proof.

⊔ ⊓

Bibliography 1. Bergman G.: Hyperidentities of groups and semigroups. Aequat. Math. 23 (1981), 55–65. 2. Bergman G.: Questions in Algebra. Preprint, Berkeley, U.D. 1986. 3. Burns R.G., Macedo´ nska O., Medvedev Y.: Groups satisfying semigroup laws, and nilpotent-byburnside varieties. J. Algebra 195 (1997), 510–525. 4. Hall P.: On the finiteness of certain soluble groups. Proc. London Math. Soc. 9 (1959), 595–622. 5. Clifford A.H., Preston G.B.: The Algebraic Theory of Semigroups, vol. I. Math. Surveys, Amer. Math. Soc. Providence, R.I. 1964. 6. Clifford A.H., Preston G.B.: The Algebraic Theory of Semigroups, vol. II. Math. Surveys, Amer. Math. Soc. Providence, R.I. 1967. 7. Ivanov S.V., Storozhev A.M.: On identities in groups of fractions of cancellative semigroups. Proc. Amer. Math. Soc. 133 (2005), 1873–1879. 8. Krempa J., Macedo´ nska O.: On identities of cancellative semigroups. Dontemporary Math. 131 (1992), 125–133. 9. Lewin J., Lewin T.: Semigroup laws in varieties of solvable groups. Proc. Dambr. Phil. Soc. 65, no. 1 (1969), 1–9.

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10. McCune W., Padmanabhan R.: Automated Deduction in Equational Logic and Cubic Curves. Lect. Notes Artificial Intelligence 1095, Springer, Berlin 1996. 11. Macedo´ nska O.: Two questions on semigroup laws. Bull. Austral. Math. Soc. 65 (2002), 431–437. 12. Macedo´ nska O., Slanina P.: GB-problem in the class of locally graded groups. Domm. Algebra 36, no. 3 (2008), 842–850. 13. Mal’tsev A.I.: Nilpotent semigroups. Ivanov. Gos. Ped. Inst. Uc. Zap. 4 (1953), 107–111 (Russian). 14. Neumann H.: Varieties of Groups. Springer-Verlag, Berlin 1967. 15. Neumann B.H., Taylor T.: Subsemigroups of nilpotent groups, Proc. Roy. Soc. Ser. A 274 (1963), 1–4. 16. Ol’shanskii A.Yu., Storozhev A.: A group variety defined by a semigroup law, J. Austral. Math. Soc. Series A 60 (1996), 255–259. 17. Shevrin L.N., Sukhanov E.V.: Structural aspects of the theory of varieties of semigroups. Izv. Vyssh. Uchebn. Zaved. Mat. 6 (1989), 3–39 (in Russian, English transl. Soviet Math. (Iz. VUZ.) 6, no. 33 (1989), 1–34).

On some M¨ obius transformations generating free semigroup Piotr Slanina

Abstract. A complex  number λ is called  a “semigroup free” if the semigroup gener12 10 ated by matrices and λ = is free. In the paper we give a short survey of 01 λ1 the results about domains of semigroup free points on the complex plane. We make also a graphical visualisation of this domain. Keywords: free semigroups, semigroups of matrices. 2010 Mathematics Subject Classification: 20M05, 20E05.

1. Introduction Let λ be any complex number and let         12 10 1λ 01 A= , Bλ = , Cλ = ,J = 01 λ1 01 10 are the matrices from GL(2, C). A number λ is called a ”free point” or ”free” if gp(A, Bλ ) – the subgroup generated by A and Bλ is free (otherwise it is called a ”nonfree point”). If sgp(A, Bλ ) – the semigroup generated by A and Bλ is a free semigroup then λ is called a ”semigroup free point” or ”semigroup free” (otherwise it is called a ”semigroup nonfree point”). The natural question is which complex numbers are free (or semigroup free). There are two main approaches to that question:

P. Slanina Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 313–323. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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(a) enlarging the set of known free points which form an open set (see Figure 1 – the set outside the black area) – the most important results are presented in [23, 3, 5, 20, 10, 16, 17, 26]; (b) determining new families of nonfree points (for example [25, 2, 8, 13, 21, 18, 14, 15, 22, 6, 12]). We recall some facts proved in [27], Proposition 2.1. Property 1.1. (i) Let A1 , A2 , ... be any square matrices of the same order over the same ring. If the group gp(A1 , A2 , ...) is free, then the semigroup sgp(A1 , A2 , ...) is free. (ii) Let 2λ = νµ. Then the semigroup sgp(A, Bλ ) is free if and only if sgp(Bµ , Cν ) is free. The same fact holds for the groups gp(A, Bλ ) and gp(Bµ , Cν ) (iii) The semigroup sgp(Bµ , Cµ ) is free if and only if sgp(Bµ , J) is a free product of cyclic semigroups generated by Bµ and J. The same property holds for groups gp(Bµ , Cµ ) and gp(Bµ , J). (iv) Every transcendental λ is free (and hence semigroup free) [7]. Note that there is an isomorphism between P SL(2, C) and the group of homographic functions namely     az + b ab ϕ = z→ . cd cz + d Let

 α = (x 7→ x + 2) , βλ = x 7→

x λx + 1



  1 , γλ = (x 7→ x + λ) , ι = x 7→ . x

Then ϕ(A) = α, ϕ(Bλ ) = βλ , ϕ(Cλ ) = γλ , ϕ(J) = ι, βλ = ιγλ ι. Then sgp(A, Bλ ) is free if and only if sgp(α, βλ ) is free. Further we will consider homographic functions instead of matrices. Let G be any subgroup of P SL(2, C). By [24], Definition 2.1.1, an elements of a group G form a normal family in a domain Ω(G) ⊂ C if every sequence of elements {gn } ⊂ G contains either a subsequence which converges to a limit element g 6≡ ∞ uniformly on each compact subset of Ω(G), or a subsequence which converges uniformly to ∞ on each compact subset. If G is discrete then the set Ω(G) on which the elements form a normal family is called the regular set of G. For these groups, the set of λ’s, for which Ω(gp(α, βλ ))/gp(α, βλ ) is a four times punctured sphere is called The Riley slice of Schottky space and consists of free points [19]. In the Figure 1 (due to David Wright, see for example [19]), the Riley slice of Schottky space is the set of points lying outside the black area. Thus the complement of the Riley slice is the black quasi-rhombus which has vertices at the points 2, −2, i, −i. This Figure includes all information about free points we can get from publications mentioned in (a). However, there are no papers describing the algorithm of this fractal creation.

On some M¨ obius transformations generating free semigroup

315

Fig. 1

Our aim is to collect all known facts about semigroup free points. In contrast to free points, semigroup free points are not so recognized and to prove that some complex numbers are semigroup free or not, we need often different and more complicated methods. For example, for gp(A, Bλ ) to be nonfree we only need to find any nontrivial word W (A, Bλ ) which is equal to identity while sgp(A, Bλ ) is nonfree if and only if either there exists any nontrivial word W (A, Bλ ) with positive exponents which is equal to identity matrix or if two different words with positive exponents are equal.

2. Semigroup free points The first result was Theorem 2.6 in [4] and it can be described by Figure 2, where points outside the grey area are semigroup free.

Fig. 2

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Bamberg used a computer program to find some free points and showed his result in the Figure 4 in [1]. In this figure, white points outside the black area represent free points, white points inside the black area represent nonfree points. The border of the black area consists of some arcs and lines based on [5, 20, 9, 11]. Thanks to the picture with the Riley slice of Schottky space (which consists of free points) from [19] and Proposition 1.1 (i), the border in the Figure 4 in [1] can be improved. In [27], Corollary 3.6, the author presents another picture (see Figure 3) based on Figure 2, Figure 4 in [1] and Corollary 3.2 in [27]:

Fig. 3

Property 2.1. In the Figure 3: (i) White points outside the grey and black areas are free. (ii) White points inside the black area mark area where the set of nonfree points is “almost dense” 1 and these points are semigroup free at the same time. (iii) All points outside the grey area are semigroup free. We will show that the set of known semigroup free points can be enlarged. First we recall some known results. Property 2.2. ([27], Proposition 2.2) Let a1 , a2 , b1 , b2 ∈ R, b1 6= 0 and b2 6= 0. If z = a1 + a2 i + (b1 + b2 i)t, t ∈ R is the parametric equation of a line in the complex plane C, not including the origin, then the transformation ι : z −→ z1 maps this line into a circumference (which includes origin) defined as b2 + b1 i z + = |b1 + b2 i| . 2(a2 b1 − a1 b2 ) 2|a2 b1 − a1 b2 | 1 The area of nonfree points is called “almost dense” if every pixel in this area contains a nonfree point [1].

On some M¨ obius transformations generating free semigroup

317

By Lemma 2.5 and Corollary 2.6 in [27], we have: Lemma 2.3. Let G be a group which acts on the set X and let H1 , H2 be infinite cyclic semigroups of the group G. Let X1 , X2 be two nonempty disjoint subsets of the set X such that (i) for every h1 ∈ H1 , h1 (X1 ∪ X2 ) ⊂ X2 , (ii) for every h2 ∈ H2 , h2 (X2 ) ⊂ X1 . Then the semigroup generated by H1 and H2 is free. Let λ = λ1 + λ2 i, where λ1 , λ2 ∈ R and let X1 = {z ∈ C : λ2 Re(z) < −λ1 Im(z) < λ2 Re(z − 2) ∧ Im(z) > 0}, X2 = {z ∈ C : λ2 Re(z − 2) < −λ1 Im(z) ∧ Im(z) > 0}, H1 = gp hαλ i ,

H2 = gp hβλ i .

Then for every natural m, n: αm (X1 ) = = {z : λ2 Re(z − 2m) < −λ1 Im(z − 2m) < λ2 Re(z − 2m − 2) ∧ Im(z) > 0} ⊂ X2 and αm (X2 ) ⊂ X2 . Without loss of generality, we assume that λ2 < 0. The equality λ2 Re(z − 2) = −λ1 Im(z) defines a line z =1+

λ2 i + (λ2 − λ1 i)t, t ∈ R. λ1

Hence by Property 2.2, ι(X2 ) is an intersection of the circle p 2 2 z − 1 + λ1 i < λ1 + λ2 4 4λ2 −4λ2 and the half-plane {z ∈ C : Im(z) < 0}. To simplify the notation, let p λ21 + λ22 r(ι(X2 )) = . −4λ2 The sets ιγλn (X2 ) are tangent to the line λ2 Re(z) = −λ1 Im(z). Because the vector [λ1 , λ2 ] is parallel to the line λ2 Re(z) = −λ1 Im(z), the sets ιγλn (X2 ) are tangent to this line for any natural n and z ∈ ιγλn (X2 ) implies Im(z) < 0. We will find the conditions implying ιγλ(X2 ) ∩ ι(X2 ) = ∅

(1)

(i) Let λ2 > λ1 (see Figure 4). Then (1) holds if 2r(ι(X2 )) 6 |λ1 + λ2 i| and hence − 21 > λ2 (observe that − 21 > λ2 > λ1 imply 8λ1 > 1 − 16λ22 ). (ii) Let λ1 > 0 (see Figure 5). Then (1) holds if r 1 |λ2 | > r(ι(X2 )) − r2 (ι(X2 )) − . 16

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(iii) Let λ2 6 λ1 6 0 (see Figure 5). Then (1) holds if r |λ2 | > r(ι(X2 )) +

Fig. 4

Fig. 5

r2 (ι(X2 )) −

1 . 16

On some M¨ obius transformations generating free semigroup

Each of the two last inequalities is equivalent to q 4λ22 + λ1 > λ21 + λ22 .

319

(2)

If 4λ22 + λ1 < 0 then inequality (2) is false so let 4λ22 + λ1 > 0. Then the inequality (2) can be transformed to 8λ1 > 1 − 16λ22 (3) (observe that 8λ1 > 1−16λ22 implies 4λ22 +λ1 > 0). Hence (3) implies ιγλn (X2 )∩ι(X2 ) = ∅ and βλn (X2 ) = ιγλn ιn (X2 ) ⊂ X1 for any natural n and hence by Lemma 2.3, λ is semigroup free. If we assume that λ2 = Im(λ) > 0 then we also get (3). All the above considerations give us the following Theorem 2.4. If 8λ1 > 1 − 16λ22 and λ2 6= 0 then λ = λ1 + λ2 i is a semigroup free point. We present results from Theorem 2.4, Figure 1 and [4] (Theorems 2.5 and 2.6) at the Figure 6.

Fig. 6

Theorem 2.5. Points outside the grey area at the Figure 6 are semigroup free points.

3. Semigroup nonfree points To show that sgp(A, Bλ ) is nonfree it suffices to find two different words W1 (A, Bλ ) and W2 (A, Bλ ) (one of them can be empty) such that W1 (A, Bλ ) = W2 (A, Bλ ). Note that the shortest nontrivial relations are of the form Ak1 Bλk2 Ak3 = Bλk4 Ak5 Bλk6

320

and if

P. Slanina

1 λ= 2



1 1 1 − − k3 k6 k4 k5 k5 k6



,

k1 =

k5 k6 , k2

k2 =

k4 k5 k3

(4)

for any natural k1 , · · · , k6 then λ is semigroup nonfree point (compare [4] Theorem 3.04). Property 3.1. The set of λ satisfying (4) has limit points −1, 12 , 1 1 2a − 2b for any positive integer a and b. Proof. The fact that the numbers −1, Theorem 3.04. If k6 = k4 = 1 and k3 = a then lim

k5 →∞

1 2

1 2a ,

1 − 2a −

1 2b

and

are limits of nonfree points follows from [4],

 1 −1 1 a − k5−1 − k5−1 = . 2 2a

If k5 = 1, k4 = a and k6 = b then  1 1 1 (k3 b)−1 − a−1 − b−1 = − − . k3 →∞ 2 2a 2b lim

If k6 = 1, k3 = a and k5 = b then  1 −1 1 1 a − (k4 b)−1 − b−1 = − . k4 →∞ 2 2a 2b lim

⊔ ⊓ In [4], Brenner and Charnov investigated mainly in the semigroup sgp(Cµ , Bµ ); we reformulate Theorems 4.1–4.6, 5.1, 5.2 and 6.1 from [4] to the semigroup sgp(A, Bλ ); the reason is to collect finally all results concerning semigroup free and nonfree points in the common picture. Theorem 3.2. (1) The nonfree points λ are dense on [−2, 0] and are arbitrarily close to 12 . (2) If the word W (A, Bλ ) has finite order and length not greater than 4 then λ is real and negative. (3) Let n be a nonzero integer. Then sgp(A, Bλ ) has torsion element for λ ∈ {− 2n1 2 , − n12 , − 2n3 2 }. (4) If λ is real and positive then sgp(A, Bλ ) is torsion free. p2 (5) If λ = − 2q 2 , p and r are nonzero integers and p 6= ±1, gcd(p, q) = 1 then gp(A, Bλ ) (and hence sgp(A, Bλ )) is torsion free. (6) Let n be an integer. Then sgp(A, Bλ ) is not free if λ = ± n82 for n > 4. √ (7) Let 2λ be a primitive r-th root of 1. Then sgp(A, Bλ ) is free if and only if r 6= 4. Fact given in Theorem 3.2 (7) was proved in [4], Theorem 6.1, for all the cases except q ∈ {3, 6}. Observe that (cos(2π/3) + i sin(2π/3))/2 and (cos(4π/3) + i sin(4π/3))/2 are free points by Theorem 2.4, because the parabola 8λ1 = 1 − 16λ22 includes such λ’s.

On some M¨ obius transformations generating free semigroup

321

The following equality Ak1 Bλk2 Ak3 Bλk4 Ak5 Bλk6 = Bλk6 Ak5 Bλk4 Ak3 Bλk2 Ak1 holds if and only if 4k1 k2 k3 k4 k5 k6 λ2 +

 + 2 k1 k2 k3 k4 + k3 k4 k5 k6 + k1 k2 k5 k6 + k1 k4 k5 k6 + k1 k2 k3 k6 − k2 k3 k4 k5 λ + + k3 k4 + k1 k2 + k1 k4 + k5 k6 + k1 k6 + k3 k6 − k2 k5 − k4 k5 − k2 k3 = 0

(5)

and the equality Ak1 Bλk2 Ak3 Bλk4 Ak5 Bλk6 Ak7 Bλk8 = Bλk8 Ak7 Bλk6 Ak5 Bλk4 Ak3 Bλk2 Ak1 holds if and only if 8k1 k2 k3 k4 k5 k6 k7 k8 λ3 + + 4 k5 k6 k7 k8 k1 k4 + k5 k6 k7 k8 k3 k4 + k5 k6 k7 k8 k1 k2 + k5 k6 k1 k2 k3 k4 +  + k7 k8 k1 k2 k3 k4 + k5 k8 k1 k2 k3 k4 + k6 k7 k8 k1 k2 k3 − k5 k6 k7 k2 k3 k4 λ2 + + 2 k5 k6 k7 k8 + k5 k6 k1 k2 + k5 k6 k3 k4 + k5 k6 k1 k4 + k7 k8 k1 k2 + k7 k8 k3 k4 + + k7 k8 k1 k4 + k1 k2 k3 k4 + k5 k8 k1 k2 + k5 k8 k3 k4 + k5 k8 k1 k4 + k8 k1 k2 k3 + + k6 k1 k2 k3 + k6 k7 k8 k1 + k6 k7 k8 k3 − k7 k3 k4 k2 − k5 k2 k3 k4 − k5 k6 k7 k2 −  − k5 k6 k7 k4 − k6 k7 k3 k2 λ + k8 k1 + k8 k3 + k6 k1 + k6 k3 + k1 k2 + k3 k4 + k1 k4 + + k5 k8 + k7 k8 + k5 k6 − k7 k2 − k7 k4 − k5 k2 − k5 k4 − k6 k7 − k2 k3 = 0. (6) We have used a computer program to make a graphical visualisation of nonfree points based on Figure 6 and then we get Figure 7. Light grey points are roots of the polynomial (5), evaluated for k1 , k4 , k5 , k6 ∈ {1, . . . , 20} and k2 , k3 ∈ {1, . . . , 200} and roots of the polynomial (6) for k1 , . . . , k8 ∈ {1, . . . , 13} and for k1 , k4 , k5 , k8 ∈ {1, . . . , 4} and k2 , k3 , k6 , k7 ∈ {1, . . . , 50}. Property 3.3. In the Figure 7: (1) Light grey points mark area where the set of semigroup nonfree points is “almost dense”. (2) White points inside the quasi-rhombus mark area where the set of nonfree points is “almost dense”. All this points are semigroup free. (3) Points outside the grey area are semigroup free. (4) White points outside the quasi-rhombus are free. There are still much of complex numbers we don’t know if they are semigroup free or not – for example, some algebraic numbers from the quasi-rhombus in the Figure 7.

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Fig. 7

Bibliography 1. Bamberg J.: Non-free points for groups generated by a pair of 2 × 2 matrices. Proc. London Math. Soc. II 62 (2000), 795–801. 2. Beardon A.-F.: Pell’s equation and two generator free Moebius groups. London Math. Soc. 25 (1993), 527–532. 3. Brenner J.-L.: Quelques groupes libres de matrices. C. R. Acad. Sci. Paris 241 (1955), 1689–1691. 4. Brenner J.-L., Charnow A.: Free semigroups of 2 × 2 matrices. Pacific J. Math. 77 (1978), 57–69. 5. Chang B., Jennings S.-A., Ree R.: On certain matrices which generate free groups. Canadian J. Math. 10 (1958), 279–284. 6. Evans R.-J.: Non-free groups generated by two parabolic matrices. J. Res. Nat. Bur. Stand. 84 (1979), 179–180. 7. Fouxe-Rabinowitch D.-I.: On a certain representation of a free group. Leningrad State Univ. Annals (Uchennye Zapiski). Math. Ser. 10 (1940), 154–157. 8. Grytczuk A., Wojtowicz M.: The problem of freeness for Euler monoids and Moebius groups. Semigroup Forum 61 (2000), 277–282. 9. Ignatov Yu.-A.: Free groups generated by two parabolic-fractional linear transformations. Modern Algebra 4 (1976), 87–90. 10. Ignatov Yu.-A.: Free and nonfree subgroups of PSL2 (C) that are generated by two parabolic elements. Mat. Sb. (N.S.) 106 (148) (1978), 372–379 (in Russian). 11. Ignatov Yu.-A.: Free and nonfree subgroups of PSL2 (C) that are generated by two parabolic elements. Mat. Sb. 35 (1979), 49–55 (in Russian). 12. Ignatov Yu.-A.: Roots of unity as nonfree points of the complex plane. Mat. Zametki 27 (1980), 825–827. 13. Ignatov Yu.-A.: Rational nonfree points of the complex plane. Tulsk. Gos. Ped. Inst., Tula 1986, 72–80. 14. Ignatov Yu.-A.: Rational nonfree points of the complex plane. Tulsk. Gos. Ped. Inst., Tula 1990, 53–59. 15. Ignatov Yu.-A.: Rational nonfree points of the complex plane, part III. Tulsk. Gos. Ped. Inst., Tula 1995, 78–84. 16. Ignatov Yu.-A., Gruzdeva T.-N., Sviridova I.-A.: Free groups of linear-fractional transformations. Tulsk. Gos. Univ. Ser. Mat. Mekh. Inform. 5 (1999), 116–120 (in Russian). 17. Ignatov Yu.-A., Evtikhova A.V.: Free groups of linear-fractional transformations. Chebyshevskii Sb. 3 (2002), 78–81 (in Russian). 18. Kabieniuk M.-I.: Groups Generated by Two Transitive Matrices. Kemerowskij Gos. Univ., Kemerovo 1988.

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19. Keen L., Series C.: The Riley slice of Schottky space. Proc. London Math. Soc. II 69 (1994), 72–90. 20. Lyndon R.-C., Ullman J.-L.: Groups generated by two parabolic linear fractional transformations. Canadian J. Math. 21 (1969), 1388–1403. 21. Lysenkov S.-W.: Groups generated by two 2 × 2 matrices. Kemerowskij Gos. Univ. (1983), 27–31. 22. Newman M.: A conjecture on a matrix group with two genetators. J. Res. Nat. Bur. Standards. 78B (1974), 69–70. 23. Sanov I.-N.: A property of a representation of a free group. DAN CCCP 57 (1947), 657–659 (in Russian). 24. Schiff J.-L.: Normal Families. Springer-Verlag, New York 1993. 25. Skuratskii A.-I.: The problem of the generation of free groups by two unitriangular matrices. Mat. Zametki 24 (1978), 411–414. 26. Slanina P.: On some free groups, generated by matrices. Demonstratio Math. 37 (2004), 55–61. 27. Slanina P.: On some free semigroups, generated by matrices. Czech. Math. J. (in press).

Periodic orbits of boundary logistic map and new kind of modified Chebyshev polynomials Edyta Hetmaniok, Piotr Lorenc, Damian Slota and Roman Witula

Abstract. In the paper we discuss the periodic orbits of maps connected with the boundary logistic map. In consequence some new kind of the modified Chebyshev polynomials is defined and intensively studied. Many fundamental relations for these polynomials are presented and discussed. Concepts of the Chebyshev functions of any real order are also introduced and compared with other parallel concepts. Keywords: periodic orbits, logistic map, modified Chebyshev polynomials. 2010 Mathematics Subject Classification: 37C25, 12E10.

Organization of the paper The paper is divided into three main sections completed by references and three tables. The sections are: 1. Introduction – where besides the ideas and notations used in the paper also the investigated polynomials are introduced. Background of the paper subject-matter is presented as well. 2. Boundary logistic map (with coefficient 4) – in this section the periodic orbits of two classes of maps gw (x) =

1 x (4 w − x), w

and

hw (x) =

1 (2 w − x)2 w

E. Hetmaniok, D. Slota, R. Witula Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: {edyta.hetmaniok, damian.slota, roman.witula}@polsl.pl P. Lorenc is the MSc graduate student in Mathematics Faculty of Applied Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 325–343. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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are discussed. In the sequel we prove that both gw and hw possess 3-periodic orbits for every w ∈ C, w 6= 0, which implies that gw and hw are chaotic in the Li-Yorke sense. 3. New types of modified Chebyshev polynomials – in this section on the grounds of discussion on the periodic orbits of polynomials gw and hw the new modified Chebyshev polynomials (perhaps the name “Chebyshev polynomials of the fifth kind” would be the most appropriate here) are defined Wn (c2 (x)) := c2 (n x) = 4 Tn2

 c(x)  2

and 2 Vn (s2 (x)) := s2 (n x) = s2 (x) Un−1

 c(x) 

, 2 for every n ∈ N, where c(x) := 2 cos(x), s(x) := 2 sin(x), Tn (x) and Un (x) denote the n-th Chebyshev polynomials of the first and second kind, respectively (see [17, 21, 22, 29] for fundamental information about the Chebyshev polynomials). A number of basic properties of these polynomials are also presented, including the recurrence relations for Wn and Vn . Moreover, in Remark 3.2 we discuss some analytical generalizations of the Chebyshev polynomials which we intend to use for further investigations in a separate paper. The paper constitutes an essential supplement for discussion started by the authors in papers [23, 24, 31].

1. Introduction In the paper we intend to discuss the periodic orbits of boundary logistic map. Main rˆole in discussion will be played by the following polynomials (see [30, 33, 35, 37]): p(X) = X3 − 3 X + 1 =

2  Y

 X − c(2k β) ,

k=0

q(X) = X3 + X2 − 2 X − 1 =

2   Y X − c(2k α) , k=0

pc2 (X) := p(X − 2) = X3 − 6 X2 + 9 X − 1 =

2  Y

 X − c2 (2k β) ,

k=0 2 Y

  X − s2 (2k β) ,

ps2 (X) := −p(2 − X) = X3 − 6 X2 + 9 X − 3 = qc2 (X) := q(X − 2) = X3 − 5 X2 + 6 X − 1 =

k=0 2 Y

 X − c2 (2k α) ,

k=0

qs2 (X) := −q(2 − X) = X3 − 7 X2 + 14 X − 7 =

2   Y X − s2 (2k α) . k=0

Periodic orbits of boundary logistic map. . .

where α :=

2π 7 ,

β :=

2π 9 .

327

Furthermore γ :=

c(x) := 2 cos(x)

2π 11 ,

δ :=

2π 13 ,

ε :=

2π 15

and

s(x) := 2 sin(x).

and

The above notation will be applied throughout the entire paper. In this connection we obtain the following form of the known trigonometric identities c(2 x) = c2 (x) − 2 = 2 − s2 (x), s(2 x) = s(x) c(x), s(x)

N −1 Y

c(2k x) = s(2N x).

k=0

We also use the notation −A := {−a : a ∈ A} for every nonempty A ⊂ R. Certainly, the problem of analytic description of such orbits in specific cases is still interesting for us. It turned out that there exist several particular quadratic and cubic polynomials, in case of which the analysis of their orbits is especially interesting because it is connected, among others, with determination of some new sequences of integers (associated with the length of appropriate orbits of given polynomials) and new sequences of “modified” Chebyshev polynomials possessing the intriguing properties. It seems that the obtained results are worth to be popularized for the sake of their accessibility and creativity. This paper is a substantial suplement of paper [32]. We say that n-periodic real orbit {x1 , x2 , . . . , xn } of polynomial p ∈ R[x] possesses the trigonometric form if there exist α1 , α2 , . . . , αn ∈ [0, 2π) such that xi = c(αi ) for every i = 1, 2, . . . , n. In particular, if there exist α ∈ (0, 2π) and k ∈ N such that xi = c2 (k i α) or xi = s2 (k i α), respectively, for every i = 1, 2, . . . , n, then orbit {x1 , x2 , . . . , xn } will be called the n-periodic square trigonometric orbit. Remark 1.1. All the above four decompositions can be deduced from respective connections with p(X) and q(X) (see also [29, 30]). Remark 1.2. Polynomial qs2 (X) is called the Johannes Kepler polynomial (see [29] volume III). We have X qs2 (X) = 2 T7 (X/2) where T7 (X) is the seventh Chebyshev polynomial of the first kind. Remark 1.3. Let us also notice the unusual similarity between forms of coefficients of polynomial qs2 (X) and the following one X3 − 7 X2 + 7 X + 7 = =

2   Y √ X − 7 cot(2k α) = k=0 2  Y

 X − 3 − 2 c(2k α) .

k=0

Sums of the powers of roots of the above polynomial is described by sequence A215575 in Sloane’s OEIS.

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One more polynomial “similar” to polynomial qs2 (X) will appear in this paper. On the occasion of discussing the 4-elements orbits rescaling the limit logistic polynomial gw (x) we will deal with polynomial (see [29] volume I): X4 − 7 X3 + 14 X2 − 8 X + 1 = =

Y 16k,l62



3   π  Y X − s2 2 k = 15

k=0

√ 1 X − (7 + (−1)k 5 + (−1)k+l 4

q √  30 + (−1)k 6 5) .

Remark 1.4. In paper [9] the authors have proven that a real analytic function, ∞ P whose Maclaurin series expansion has the form an xn , behaves chaotically whenn=0

ever the following condition holds a22 − a1 a3 > 0.

(1)

It can be easily observed that all the polynomials presented in this section, and simultaneously the all ones discussed in this paper, satisfy condition (1), i.e. they are chaotic.

2. Boundary logistic map (with coefficient 4) Maps, discussed in this section (functions gw and hw defined in theorem given below), have a close connection with the boundary logistic map x 7→ 4 x(1 − x) [7]. Let us notice that if we replace coefficient 4 by number a > 4 then the obtained logistic map is chaotic [11]. Furthermore, this boundary logistic map has no attracting periodic points. In the following theorem we present all the n−periodic orbits of functions gw and hw for n = 1, 2, 3, 4, 5, 6. We note that since both functions gw and hw possess the 3periodic orbits, these functions possess also the n−periodic orbits for every n ∈ N (by Sharkovsky’s Theorem). Unfortunately, we do not know the description of all these orbits for other n ∈ N. We only suppose that for every positive integer n ≥ 7 both gw and hw possess exclusively the n−periodic square trigonometric orbits of the form {s2 (2k x) :

k = 0, 1, ..., n − 1},

where x = x(n) is a rational number (probably x = {c2 (2k y) :

r 2n ±1 ,

(r, 2n ± 1) = 1), and

k = 0, 1, ..., n − 1},

where y = y(n) is also a rational number (probably y = respectively.

t 2n ±1 ,

(t, 2n ± 1) = 1),

Theorem 2.1. Let gw (x) := w1 x(4w−x) and hw (x) := w1 (2w−x)2 , where w ∈ C\{0}. Then we have gw ([0, 4w]) = [0, 4w] and hw ([0, 4w]) = [0, 4w], for every w ∈ C, and the following identities hold

Periodic orbits of boundary logistic map. . .

329

gw (w s2 (x)) = w s2 (2x), 2

2

hw (w c (x)) = w c (2x).

(2) (3)

In the sequel, from these relations we deduce that sets {w s2 (α), w s2 (2α), w s2 (4α)}

and

{w s2 (β), w s2 (2β), w s2 (4β)}

are the 3-periodic orbits of gw and that sets {w c2 (α), w c2 (2α), w c2 (4α)}

and

{w c2 (β), w c2 (2β), w c2 (4β)}

are the 3-periodic orbits of hw . In both cases these are the unique 3−periodic orbits of these functions. Hence, from Sharkovsky’s Theorem [3,5,6,15,19] we obtain that gw and hw possess the periodic orbits of any finite cardinality. We note that the given above sets are the only possible 3-periodic orbits of gw and hw , respectively. Moreover, numbers 0 and 3w are the only fixed points of gw (x), whereas numbers w and 4w are the only fixed points of hw (x), sets √ ! √ !  π 5− 5 5+ 5 2 k {w s 2 : k = 0, 1} = {w ,w } 5 2 2 and

!2 √ 5+1 {w c 2 : k = 0, 1} = {w } 5 2 √  are the only 2-periodic orbits of gw and hw , respectively (we note that 5 c 2k π5 =  s2 2k+1 π5 for every k = 0, 1), sets 2





!2 √ 5−1 ,w 2



{w s2 (2k γ) : k = 0, 1, . . . , 4}

and

{w c2 (2k γ) : k = 0, 1, . . . , 4}

are the 5-periodic orbits of gw and hw , respectively (remaining 5-periodic orbits of hw are presented in Remark 2.4), sets {w s2 (2k δ) : k = 0, 1, . . . , 5}

and

{w c2 (2k δ) : k = 0, 1, . . . , 5}

are the 6-periodic orbits of gw and hw , respectively, and at last, sets q q √ √  w √ √  w {w s2 (2k ε) : k = 0, 1, 2, 3} = 7 + 5 ± 6(5 + 5) , 7 − 5 ± 6(5 − 5) 4 4 and

q q √ √  w √ √  {w c (2 ε) : k = 0, 1, 2, 3} = 9 + 5 ± 6(5 − 5) , 9 − 5 ± 6(5 + 5) 4 4 2

k

w

are the 4-periodic orbits of gw and hw , respectively.

330

E. Hetmaniok, P. Lorenc, D. Slota and R. Witula

Function gw possesses additionally two other 4-periodic orbits: n  π o n  3π  o w s2 2 k : k = 1, 2, 3, 4 and w s2 2 k : k = 0, 1, 2, 3 17 17 (all these eight numbers for w = 1 are roots of the following polynomial x8 − 17 x7 + 119 x6 − 442 x5 + 935 x4 − 1122 x3 + 714 x2 − 204 x + 17). Similarly, function hw possesses as well two other 4-periodic orbits: n

o  π : k = 1, 2, 3, 4 and w c2 2 k 17

n

 3π  o w c2 2 k : k = 0, 1, 2, 3 17

(all these eight numbers for w = 1 form the set of zeros of polynomial x8 − 15 x7 + 91 x6 − 286 x5 + 495 x4 − 462 x3 + 210 x2 − 36 x + 1). Proof. Proof of the second part of theorem results from equalities s2 (8α) = s2 (α)

and s2 (8β) = s2 (β)

(we note that gw

gw

gw

gw

gw

s2 (γ) → s2 (2γ) → s2 (4γ) → s2 (3γ) → s2 (5γ) → s2 (γ), h

h

h

h

w 2 w 2 w 2 w 2 c2 (ε) → c (2ε) → c (4ε) → c (7ε) → c (ε).

etc.) and from the following decompositions g1 ◦ g1 ◦ g1 (x) − x = −x (x − 3) ps2 (x) qs2 (x), h1 ◦ h1 ◦ h1 (x) − x = (x − 1) (x − 4) pc2 (x) qc2 (x), since we have gw ◦ gw ◦ gw (w x) = w g1 ◦ g1 ◦ g1 (x) and hw ◦ hw ◦ hw (w x) = w h1 ◦ h1 ◦ h1 (x). ⊔ ⊓ Corollary 2.2. For every positive integer n ≥ 3 both gw and hw possess the n−periodic square trigonometric orbits of the form   π : k = 0, 1, ..., n − 1}, {s2 2k n 2 +1   π {s2 2k n : k = 0, 1, ..., n − 1} 2 −1 and

 {c2 2k

π n 2 +1



: k = 0, 1, ..., n − 1},

Periodic orbits of boundary logistic map. . .

 {c2 2k

π 2n − 1

331



: k = 0, 1, ..., n − 1},

respectively. Certainly, these functions may possess also the other n-periodic square trigonometric orbits which is especially interestingly exhibited in Remark 2.4. Remark 2.3. Both maps gw [0,4w] and hw [0,4w] are transitive. It follows easily from   ∞   ∞ the facts that both sequences w s2 2n x π n=0 and w c2 2n x π n=0 are dense in [0, for almost all x ∈ R. It is a consequence of uniform distribution of the sequence  n4w] ∞ 2 x n=0 for almost all x ∈ R [12]. Remark 2.4. Polynomial h1 (x) (and in consequence every hw (x)) possesses additionally five other 5-periodic orbits: – the first one:  2π   4π  π h1 h1 x1 = c2 = 3.96386 −→ x2 = c2 = 3.85674 −→ x3 = c2 = 3.447483 33 33 33  8π  π h1 h1 −→ x4 = c2 = 2.09516 −→ x5 = s2 = 0.00905615, 33 66 – the second one:  5π   10π   13π  h1 h1 x1 = c2 = 3.16011 −→ x2 = c2 = 1.34586 −→ x3 = c2 = 0.427894 33 33 33  7π   5π  h1 h1 −→ x4 = c2 = 2.47152 −→ x5 = s2 = 0.222329, 33 66 – the third one: π  2π   4π  h1 h1 x1 = c2 = 3.95906 −→ x2 = c2 = 3.83792 −→ x3 = c2 = 3.37793 31 31 31  8π  π h1 h1 −→ x4 = c2 = 1.8987 −→ x5 = s2 = 0.0102614, 31 62 – the fourth one:  3π   6π   7π  h1 h1 = 3.64153 −→ x2 = c2 = 2.69461 −→ x3 = s2 = 0.482484 x1 = c2 31 31 62  7π   3π  h1 h1 −→ x4 = c2 = 2.30286 −→ x5 = s2 = 0.0917215, 31 62 – the fifth one:  5π   10π   9π  h1 h1 x1 = c2 = 3.05793 −→ x2 = c2 = 1.11921 −→ x3 = s2 = 0.775788 31 31 62  9π   5π  h1 h1 −→ x4 = c2 = 1.49869 −→ x5 = s2 = 0.251307. 31 62 Elements of the first and second orbit are zeros of polynomial x10 − 21 x9 + 188 x8 − 934 x7 + 2806 x6 − 5202 x5 + + 5809 x4 − 3629 x3 + 1090 x2 − 120 x + 1,

332

E. Hetmaniok, P. Lorenc, D. Slota and R. Witula Table 1  π Orbits of g1 including sin2 n for successive odd n > 3, where l := min{k ∈   π N : sin2 2k n = sin2 π } denotes the length of every orbit and N := ⌊ nl ⌋ n  denotes the number of remaining orbits of g1 generated by sin2 rπ , (r, n) = 1 n

3 1 3 5 2 2



9 3 3

15 4 3



π 11



sin2

π 13



19 9 2

21 6 3 23 11 2

25 10 2

27 9 3

29 14 2



π 19

sin2

sin2

π 21

sin2

π 23



 

sin2



sin2



sin2

35 12 2



sin2

sin2

sin2

5π 38

, sin2

2π 21

, sin2

2π 23



, cos2

5π 22

, cos2

3π 26

, cos2

 7π

, cos2

π 22

, sin2

3π 13

, cos2

, cos2

π 30

, cos2

π 34



, sin2

3π 19









30

 4π 17



, cos2

π 38

, sin2

4π 21

, cos2

5π 42

, sin2

5π 21

, sin2

4π 23



, cos2

7π 46

, cos2

9π 46

, cos2

3π 46

, sin2

3π 23

2π 25



, sin2

4π 25



, cos2



, sin2

4π 27

, cos2

cos2

7π 54



, cos2



, sin2

4π 29

, cos2

, sin2

5π 29







3π 50 2π 27











2π 31



, sin2

2π 33

2π 35

, sin2



 11π 70





7π 29





13π 54

9π 50



9π 58

, cos2

, sin2

π 54





15π 62





, sin2

8π 33

8π 35

, cos2

, cos2

 9π



70



π 26



, cos2

7π 38



, cos2

π 42

, cos2

5π 46











π 46

, cos2

,

11π 50



,

,

5π 54



, sin2

5π 27



3π 58



, sin2

3π 29

, cos2

7π 58





, cos2

π 62 π 66

3π 70

, sin2



 17π 70



, ,



, cos2

, cos2

,







11π 58

, cos2

, cos2

4π 35

π 50

, cos2



4π 31



7π 50





, cos2

, cos2

π 58

4π 33

70







, cos2

 13π

11π 46

, cos2 13π 58



, cos2



11π 54

, sin2





6π 25

, cos2





, cos2

, sin2

, sin2



, cos2



3π 25

, sin2

sin2

, sin2



, sin2

, cos2





9π 38

5π 58

π 33

π 18

, cos2

, cos2

sin2

, cos2





3π 38

2π 29

π 31

π 14

, sin2

17

cos2

, cos2



, cos2

, sin2

35

, sin2

o

5 8

+





cos2



 2π

4π 19





15

, sin2



 6π







2π 19

, sin2

π 35



π 17





3π 22

, sin2

, sin2

sin2

33 5 6

, cos2

π 15

sin2

, sin2



31 5 6



 2π





2π 9

5π 26

π 25

6π 29

sin2

, sin2

, cos2

5π 23

π 29



2π 13

sin2

π 27

3π 14







, cos2

π 9



, sin2







2π 11





17 4 4

sin2

4 √ 5 5 ,8 8

5 8

, sin2

sin2

11 5 2



3

n

π 7

sin2

7 3 2

13 6 2

π n

the orbit of g1 including sin2

n l N



3π 35

, cos2



π 70

,



whereas the elements of the other three orbits are zeros of polynomial x15 − 29 x14 + 378 x13 − 2925 x12 + 14950 x11 − 53130 x10 + + 134596 x9 − 245157 x8 + 319770 x7 − 293930 x6 + 184756 x5 − − 75582 x4 + 18564 x3 − 2380 x2 + 120 x − 1.

Periodic orbits of boundary logistic map. . .

333

Properties (2) and (3) for the cubic polynomials We give here a description of all the cubic polynomials satisfying equalities (2) and (3) for given x = α and separately for given x = β. Theorem 2.5. Let r(x) = x3 + a x2 + b x + c, a, b, c ∈ C. a) If

 r w s2 (2k α) = w s2 (2k+1 α),

(4)

for every k = 0, 1, 2, where w ∈ C \ {0}, then  1 2 x r(x) = x3 − 7 w + x + (14 w2 + 4) x − 7 w3 = w3 qs2 + gw (x). w w Polynomial r(x) is the only one which satisfies condition (4). Set {w s2 (α), w s2 (2α), w s2 (4α)} is the 3-periodic orbit of r(x) and r(x) possesses the n-periodic orbits for every n ∈ N. b) If

 r w c2 (2k α) = w c2 (2k+1 α),

(5)

for every k = 0, 1, 2, where w ∈ C \ {0}, then r(x) = x3 +



−5w +

1 2 x x + (6 w2 − 4) x + 4 w − w3 = w3 qc2 + hw (x). w w

Polynomial r(x) is the only one which satisfies condition (5). Set {w c2 (α), w c2 (2α), w c2 (4α)} is the 3-periodic orbit of r(x) and r(x) possesses the n-periodic orbits for every n ∈ N. c) If

 r w s2 (2k β) = w s2 (2k+1 β),

(6)

for every k = 0, 1, 2, where w ∈ C \ {0}, then  4 2 x r(x) = x3 − 6 w + x + (4 + 9 w2 ) x − 3 w3 = w3 ps2 + gw (x). w w Moreover, r(x) is the only polynomial which satisfies condition (6). Set {w s2 (β), w s2 (2β), w s2 (4β)} is the 3-periodic orbit of r(x) and r(x) possesses the periodic orbits of every finite cardinality.

334

E. Hetmaniok, P. Lorenc, D. Slota and R. Witula

d) If

 r w c2 (2k β) = w c2 (2k+1 β),

(7)

for every k = 0, 1, 2, where w ∈ C \ {0}, then r(x) = x3 +

1

 x − 6 w x2 + (9 w2 − 4) x + 4 w − w3 = w3 pc2 + hw (x). w w

Polynomial r(x) is the only polynomial which satisfies condition (7). Set {w c2 (β), w c2 (2β), w c2 (4β)} is the 3-periodic orbit of r(x) and r(x) possesses the periodic orbits of every finite cardinality. Remark 2.6. In this case we may put a question whether the reduction of conditions (4)–(7) influences the description of polynomials r(x)?

3. New types of modified Chebyshev polynomials Properties (2) and (3) of polynomials gw and hw lead in natural way to generating ∞ two new types of the sequences of polynomials {Wn (x)}∞ n=0 and {Vn (x)}n=0 defined by conditions  c(x)  (8) Wn (c2 (x)) = c2 (n x) = 4 Tn2 2 and  c(x)  2 Vn (s2 (x)) = s2 (n x) = s2 (x) Un−1 , (9) 2 for every n ∈ N∪{0}, where Tn (x) and Un (x) denote the n-th Chebyshev polynomials of the first and second kind, respectively. Hence we get x Wn (x2 ) = 4 Tn2 , 2 for every x ∈ R, and 2 Vn (x) = x Un−1

r x 1− , 4

for every x ∈ (−∞, 4]. It turns out that these polynomials  are closely connected with the modified Chebyshev polynomials Ωn (x) := 2Tn x2 (see [32, 38]): Wn (x2 ) = Ωn2 (x) = Ω2n (x) + 2

(10)

Vn (x2 ) = (−1)n−1 Ω2n (x) + 2,

(11)

and from which the following identities result

Periodic orbits of boundary logistic map. . .

335

W2n−1 (t) = V2n−1 (t), W2n (t) + V2n (t) = 4 and 2 2 2 W2n (x2 ) + V2n (x2 ) = 2 Ω4n (x) + 8 = 2 Ω8n (x) + 12

for every t, x ∈ R. Hence we deduce   π   π = c2 n , Wn (1) = Wn c2   π3  π3 Vn (1) = Vn s2 = s2 n , 6 6 which implies Vn (1) =



Wn (1) = c2 n π3 s2 n2 · π3



for odd n, for even n.

Obviously one can prove inductively that (10)⇒(8) and (11)⇒(9). Let us present the proof of implication (11)⇒(9). Proof. From (11) we obtain (−1)n−1 Vn (s2 (x)) = Ω2n (s(x)) + 2(−1)n−1 = 2 T2n (sin x) + 2(−1)n−1 =   π = 2 cos 2 n − x + 2(−1)n−1 = 2(−1)n cos(2 n x) + 2(−1)n−1 = 2 = 2(−1)n (1 − 2 sin2 (n x)) + 2(−1)n−1 = (−1)n−1 s2 (n x) which implies (9).

⊔ ⊓

We know that [32]: Ωn (θ + θ−1 ) = θn + θ−n , for every θ ∈ C \ {0}, which implies two interesting relations  (10) n = (θ + θ−n )2

(12)

 (11) = (−1)n−1 (θn − (−θ)−n )2 .

(13)

Wn (θ + θ−1 )2 and Vn (θ + θ−1 )2

Moreover, the following decompositions are proven in paper [32]: Ω2n−1 (x) − Ω2n−1 (θ + θ−1 ) =

2n−2 Y

 x − θ ξ 2k − θ−1 ξ −2k ,

(14)

k=0

where ξ := exp(i π/(2n − 1)), and n

(−1) Ω2n (i x) + Ω2n (θ + θ

−1

)=

2n−1 Y

 x − θ ζ 2k+1 + θ−1 ζ −2k−1 ,

k=0

where ζ := exp(i π/(2n)). It implies, among others, that

(15)

336

E. Hetmaniok, P. Lorenc, D. Slota and R. Witula

    (−1)n Wn (−x2 ) − Wn (θ + θ−1 )2 = (−1)n Ω2n (i x) − θ2n − θ−2n = = (−1)n Ω2n (i x) − (i θ)2n − (i θ)−2n = (−1)n Ω2n (i x) + (i ζθ)2n + (i ζθ)−2n = 2n−1 Y   = x − i θζ 2k+2 + θ−1 ζ −2k−2 , (16) k=0

or in “positive” version (x 7→ i x):  Y   2n−1 Wn (x2 ) − Wn (θ + θ−1 )2 = x − θζ 2k+2 − θ−1 ζ −2k−2 .

(17)

k=0

Similarly we determine  − Vn (−x2 ) + Vn (θ + θ−1 )2 = (−1)n Ω2n (i x) − (−1)n Ω2n (θ + θ−1 ) = = (−1)n Ω2n (i x) − (−1)n (θ2n + θ−2n ) = (−1)n Ω2n (i x) + (i ζθ)2n + (i ζθ)−2n = 2n−1 Y   = x − i θζ 2k+2 − i θ−1 ζ −2k−2 . (18) k=0

Recurrence relations for polynomials Wn and Vn The following recurrence relation for Ωn (x) holds (see [30, 32]): Ωn+2 (x) = x Ωn+1 (x) − Ωn (x) which implies the two steps recurrence relation [14]: Ωn+4 (x) = (x2 − 2) Ωn+2 (x) − Ωn (x).

(19)

Hence we deduce that Ω2(n+2) (x) + 2 = (x2 − 2)(Ω2(n+1) (x) + 2) − (Ω2n (x) + 2) − 2 x2 + 8, i.e., by (10), Wn+2 (t) = (t − 2) Wn+1 (t) − Wn (t) − 2 t + 8

(20)

which is the recurrence relation for Wn (t). The first eleven polynomials Wn are presented in Table 2. Similarly, from (19) one can conclude the relation Ω2(n+2) + 2(−1)n+1 = (x2 − 2)(Ω2(n+1) (x) + 2(−1)n ) − − (Ω2n (x) + 2(−1)n−1 ) − 2(−1)n x2 , which generates the recurrence relation for Vn (t): (−1)n+1 Vn+2 (t) = (t − 2)(−1)n Vn+1 (t) − (−1)n−1 Vn (t) − 2 t (−1)n ,

Periodic orbits of boundary logistic map. . .

337 Table 2

Polynomials Wn n

Wn (t)

0

4

1

t

2

4 − 4t + t2

3

9t − 6t2 + t3

4

4 − 16t + 20t2 − 8t3 + t4

5

25t − 50t2 + 35t3 − 10t4 + t5

6

4 − 36t + 105t2 − 112t3 + 54t4 − 12t5 + t6

7

49t − 196t2 + 294t3 − 210t4 + 77t5 − 14t6 + t7

8

4 − 64t + 336t2 − 672t3 + 660t4 − 352t5 + 104t6 − 16t7 + t8

9

81t − 540t2 + 1386t3 − 1782t4 + 1287t5 − 546t6 + 135t7 − 18t8 + t9

10

4 − 100t + 825t2 − 2640t3 + 4290t4 − 4004t5 + 2275t6 − 800t7 + 170t8 − 20t9 + t10

Table 3 Polynomials Vn n

Vn (t)

0

0

1

t

2

4t − t2

3

9t − 6t2 + t3

4

16t − 20t2 + 8t3 − t4

5

25t − 50t2 + 35t3 − 10t4 + t5

6

36t − 105t2 + 112t3 − 54t4 + 12t5 − t6

7

49t − 196t2 + 294t3 − 210t4 + 77t5 − 14t6 + t7

8

64t − 336t2 + 672t3 − 660t4 + 352t5 − 104t6 + 16t7 − t8

9

81t − 540t2 + 1386t3 − 1782t4 + 1287t5 − 546t6 + 135t7 − 18t8 + t9

10

100t − 825t2 + 2640t3 − 4290t4 + 4004t5 − 2275t6 + 800t7 − 170t8 + 20t9 − t10

i.e., Vn+2 (t) = (2 − t) Vn+1 (t) − Vn (t) + 2 t.

(21)

The first eleven polynomials Vn are given in Table 3. Let us also notice that polynomials Wn (t) and Vn (t) satisfy many identities of trigonometric nature compatible with the respective identities of standard trigonometry. For example, the following one’s hold

338

E. Hetmaniok, P. Lorenc, D. Slota and R. Witula

Wn (c2 (x)) + Vn (s2 (x)) = 4, Wn (c2 (x)) Vn (s2 (x)) = V2n (s2 (x)), (Wn (c2 (x)))2 + (Vn (s2 (x)))2 = 1 − 2V2n (s2 (x)), N Y

Vn (s2 (x))

W2k n (c2 (x)) = V2N +1 n (s2 (x)).

k=0

In particular, we obtain from this the following interesting numerical relations

x→0+

N Y V2N +1 n (x) = W2k n (1), Vn (x)

Vn (2)

N Y

lim

k=0

W2k n (2) = V2N +1 n (2),

k=0

Wn (2) + Vn (2) = 4, 2V2n (2) = 1 − (Wn (2))2 − (Vn (2))2 , Wn (0) + Vn (1) = Wn (1) + Vn (0) = 4, which implies π Vn (0) = 4 − Wn (1) = s2 n ,  2 3 π s n3 Wn (0) = 4 − Vn (1) = c2 n π6

for odd n, for even n, N Y V N +1 n (x) π  π π 2 lim 2 = c2 2 k n = sin 2N +1 n / sin n . Vn (x) 3 3 3 x→0+ k=0

Moreover, we get   π   π = c2 n = 2 + ((−1)n + 1)(−1)⌈n/2⌉ , Wn (2) = Wn c2  π4   π4 = s2 n = 2 − ((−1)n + 1)(−1)⌈n/2⌉ , Vn (2) = Vn s2 4 4   π   π Wn (3) = Wn c2 = c2 n ,   π6  π6 2 2 Vn (3) = Vn s =s n , 3 3 which implies Wn (3) = 1 − Vn (1), Vn (3) = 1 − Wn (1),  2 Wn (4) = Wn c (0) = 4,  π   0 for even n,   π  2 2 =s n Vn (4) = Vn s = 4 for odd n, 2 2 and so on. Simultaneously, from (12) we can deduce that

Periodic orbits of boundary logistic map. . .

339

Wn (5) = L2n + 2, where Ln denote the n-th Lucas number, since √ √ 5+1 −1 θ + θ = 5 ⇐⇒ θ1 = 2

√ 5−1 or θ2 = 2

and by Binet’s formula for Ln we have θi2n + θi−2n = L2n , for every i = 1, 2, and n ∈ N. Generally we get √ √ √ √ 2n −2n k+ k−1 +2+ k− k−1 Wn (4k) = for k ∈ C, which implies  c2   c + a 2n  c − a −2n Wn 4 2 = +2+ b b b for a, b, c ∈ C, b 6= 0, and a2 + b2 = c2 . For example, we obtain Wn and Wn

 13 2 

= 5n + 5−n

6

 26 2  5

=

 3 n 2

+

whenever a = 5, b = 12, c = 13. At last, let us observe that if √  √5 + 1 2k ak + b k 5 = , 2 2 then

2

 2 n 2 3

,

ak , bk , k ∈ N,

 Wn a2k = L2k n + 2.

For example, Wn (9) = L4n + 2, Wn (49) = L8n + 2, Wn (2209) = L16n + 2. k 5+1 2 2 a2k − 5 b2k =





ak +bk 2 2

√ 5

then ak , bk ∈ 2N − 1 and

k 5−1 2 2

Proof. If

=

implies

4 and θ − ak θ + 1 = 0 which is equivalent to θ =

We note also that from (17) we obtain Wn (0) − Wn (θ + θ

−1 2

)



=

2n Y k=1

 θζ 2k + θ−1 ζ −2k .

√ ak −bk 5 . 2 √ ak ±bk 5 . 2

=

It ⊔ ⊓

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Remark 3.1. From relations (8) and (9) it follows that polynomials Vn (x), n ∈ N ∪ {0}, and separately Wn (x), n ∈ N ∪ {0}, are not orthogonal on any nontrivial interval I ⊂ R. However, from (10) and (11) we deduce the following orthogonal relations Z2

  Wm (x2 ) − 2 Wn (x2 ) − 2 (4 − x2 )−1/2 dx = Nm δm,n

−2

with N0 = 4π and Nm = 2π if m 6= 0, and Z2

  Vm (x2 ) − 2 Vn (x2 ) − 2 (4 − x2 )−1/2 dx = (−1)m+n Nm δm,n .

−2

More properties of these polynomials, as well as their new applications, we intend to discuss in a separate paper. Remark 3.2. (of conceptional nature) Next generalizations of the Chebyshev polynomials, from among many known ones (see [2]) together with the one discussed in the current paper, led us to one more kind of questions concerning the generalizations of these polynomials, this time to the real indices (and even the complex indices). For example Tξ (cos x) = cos(ξ x), x ∈ R, ξ > 0. We note that after simple algebra we get then the following differential equation (1 − t2 ) Tξ′′ (t) − t Tξ′ (t) + ξ 2 Tξ (t) = 0. The idea of proposing that kind of generalization can arise from the connection of polynomial Tn (x) with the hypergeometric functions and next with their integral representation (the latter is not necessary, however it enables to analyse better this generalization). Thus we have (see [1, 18]):  1 1 − x Tn (x) =2 F1 − n, n; ; , 2 2 for every x ∈ C, and Γ (γ) 2 F1 (α, β; γ; x) = Γ (β)Γ (γ − β)

Z1

tβ−1 (1 − t)γ−β−1 (1 − t x)−α dt,

0

for x, α, β, γ ∈ C, | arg(1 − x)| < π and Re(γ) > Re(β) > 0. By this facts we may take 1 1 − x − ξ, ξ; ; , 2 2 for every x ∈ C satisfying condition 1−x 2 < 1, and even Tξ (x) :=

2 F1



Periodic orbits of boundary logistic map. . .

1 1 − x − ξ, ζ; ; , 2 2   > Re(ζ) > 0, arg 1+x < π. 2

Tξ,ζ (x) := for ξ, ζ ∈ C, Re(ξ) > 0,

1 2

341 2 F1



Another source of possible generalization of the Chebyshev polynomials can be found in the inspiring paper [27]. For example, starting from identity Tn (θ + θ−1 ) = θn + θ−n , for θ ∈ C \ {0}, we can take either the first hyperbolic cosine formula n ∞   X (x2 − 1)n Y 2 (ξ − (2k − 1)2 ) , Tξ (x) = x 1 + (2n)! n=1 k=1

or the second hyperbolic cosine formula ∞ n   X (2(x − 1))n Y 2 Tξ (x) = 1 + ξ 2 (x2 − 1) 1 + 2 (ξ − k 2 ) . (2n + 2)! n=1 k=1

Both formulae are compatible with our hypergeometric ones. Some other type of generalization of the Chebyshev polynomials is discussed in [2] (see also [8]). In the similar manner, i.e. by applying the hypergeometric function, the other classical orthogonal polynomials can also be generalized, for example the Legendre polynomials (see [16], page 56): Pξ (x) :=

2 F1



− ξ, ξ + 1; 1;

1 1  − x , 2 2

for −1 6 x 6 1, which generates the excellent connections with the classical Legendre polynomials Pξ (x) =

∞ h 1 i sin ξπ X 1 (−1)n − Pn (x), π n=0 ξ−n ξ+n+1

for ξ ∈ R \ Z, x ∈ (−1, 1], and Pξ (cos θ)Pξ (cos θ′ ) =

∞ h 1 i sin ξπ X 1 (−1)n − Pn (cos θ)Pn (cos θ′ ). π n=0 ξ−n ξ+n+1

Bibliography 1. Beals R., Wong R.: Special Functions. Cambridge University Press, Cambridge 2010. 2. Bojanov B.D., Rahman Q.J.: On certain extremal problems for polynomials. J. Math. Anal. Appl. 189 (1995), 781–800. 3. Ciesielski K, Pogoda Z.: On ordering the natural numbers or the Sharkovski Theorem. Amer. Math. Monthly 115 (2008), 159–165.

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4. Darst R.B., Palagallo J.A., Price T.E.: Curious Curves. World Scientific, New Jersey 2010. 5. Devaney R.L.: An Introduction to Chaotic Dynamical Systems. Addison-Wesley, Boston 1989. 6. Du B.S.: A simple proof of Sharkovsky’s Theorem revisited. Amer. Math. Monthly 114 (2007), 152–155. 7. Elaydi S.N.: An Introduction to Difference Equations. Springer, New York 2005. 8. Garc´ıa Ravelo J., Cuevas R., Queijeiro A., Pe˜ na J.J., Morales J.: Chebyshev functions of halfinteger order. Integral Transforms Spec. Funct. 18 (2007), 743–749. 9. Hacibekiro˜ glu G., Ca˜ glar M., Polato˜ glu Y.: The higher order schwarzian derivative: its applications for chaotic behavior and new invariant sufficient condition of chaos. Nonlinear Anal. Real World Appl. 10 (2009), 1270–1275. 10. Hsu C.-H., Li M.-C.: Transitivity implies period six: a simple proof. Amer. Math. Monthly 109 (2002), 840–843. 11. Kraft R.L.: Chaos, Cantor sets and hyperbolicity for the logistic maps. Amer. Math. Monthly 106 (1999), 400–408. 12. Kuipers L., Niederreiter H.: Uniform Distribution of Sequences. Wiley, New York 1974. 13. Kwietniak D., Oprocha P.: Chaos theory from the mathematical viewpoint. Mat. Stosow. 9 (2008), 1–45 (in Polish). 14. Latushkin Y., Ushakov V.: A representation of regular subsequences of recurrent sequences. Fibonacci Quart. 43.1 (2005), 70–84. 15. Li T.Y., Yorke J.A.: Period three implies chaos. Amer. Math. Monthly 82 (1975), 985–992. 16. Magnus W., Oberhettinger F.: Formeln und S¨ atze f¨ ur die Speziellen Funktionen der Mathematischen Physik. Springer, Berlin 1943. 17. Mason J.C., Handscomb D.C.: Chebyshev Polynomials. Chapman & Hall/CRC, Boca Raton 2003. 18. Mathai A.M., Haubold H.J.: Special Functions for Applied Scientists. Springer, New York 2008. 19. Misiurewicz M.: Remarks on Sharkovsky’s Theorem. Amer. Math. Monthly 104 (1997), 846–847. 20. Peitgen H.-O., J¨ urgens H., Saupe D.: Chaos and Fractals, New Frontiers of Science. Springer, New York 1992. 21. Rivlin T.J.: Chebyshev Polynomials from Approximation Theory to Algebra and Number Theory. Wiley, New York 1990. 22. Paszkowski S.: Numerical Applications of Chebyshev Polynomials and Series. PWN, Warsaw 1975 (in Polish). 23. Trawi´ nski T., Kolton W., Hetmaniok E., Slota D., Witula R.: Analysis of chaotic phenomena occurring in chosen branched kinematic chains of robot manipulators. In: Wybrane zagadnienia elektrotechniki i elektroniki, WZEE 2013, Czarna (2013), 1–8. 24. Trawi´ nski T., Kolton W., Hetmaniok E., Slota D., Witula R.: Chaotic phenomena occurring in chosen kinematic chains of robot manipulators (in preparation). 25. Triˇ ckovi´ c S.B., Stankovi´ c M.S.: On periodic solutions of a certain difference equation. Fibonacci Quart. 42.4 (2004), 300–305. 26. Vellekoop M., Berglund R.: On intervals, transitivity = chaos. Amer. Math. Monthly 101 (1994), 353–355. 27. Verde-Star L., Srivastava H.M.: Some binomial formulas of the generalized Appel form. J. Math. Anal. Appl. 274 (2002), 755–771. 28. Wang Z.X., Guo D.R.: Special Functions. World Scientific, New Jersey 2010. 29. Witula R.: Complex Numbers, Polynomials and Partial Fraction Decomposition, vol. I–III. Wyd. ´ Gliwice 2010 (in Polish). Pol. Sl., 30. Witula R.: Formulae for Sums of Unimodular Complex Numbers. WPKJS, Gliwice 2011 (in Polish). 31. Witula R., Hetmaniok E., Slota D., Trawi´ nski T., Kolton W.: On the Three, Five and Other Periodic Orbits of Some Polynomials. In: Analysis and Simulation of Electrical and Computer Systems, L. Gol¸ebiowski and D. Mazur (eds.), Lecture Notes in Electrical Engineering 324 (2015), doi 10.1007/978-3-319-11248-0 8. 32. Witula R., Slota D.: On modified Chebyshev polynomials. J. Math. Anal. Appl. 324 (2006), 321–343. 33. Witula R., Slota D.: New Ramanujan-type formulas and quasi-Fibonacci numbers of order 7. J. Integer Seq. 10 (2007), article 07.5.6.

Periodic orbits of boundary logistic map. . .

343

34. Witula R., Slota D.: Quasi-Fibonacci numbers of order 11. J. Integer Seq. 10 (2007), article 07.8.5. 35. Witula R., Slota D.: Decomposition of certain symmetric functions of powers of cosine and sine functions. Int. J. Pure Appl. Math. 50 (2009), 1–12. 36. Witula R., Slota D.: Fixed and periodic points of polynomials generated by minimal polynomials of 2 cos(π/n). Internat. J. Bifur. Chaos Appl. Sci. Engrg. 19 (2009), 3005–3016. 37. Witula R., Slota D.: Quasi-Fibonacci numbers of order 13 on the occasion of the Thirteenth International Conference on Fibonacci Numbers and Their Applications. Congr. Numer. 201 (2010), 89–107. 38. Witula R., Slota D.: Some identities for the moments of random variables having a linear rescaling the Student’s t distribution. J. Math. Anal. Appl. 361 (2010), 276–279.

Continua and dimension Lukasz Przontka and Alicja Samulewicz

Abstract. We briefly discuss the relationship between chainability and dimension and describe a non-metric chainable continuum of inductive dimensions 2. We also give topological characterizations of selected hyperspaces of infinite dimensional compacta in the Hilbert cube. Keywords: continuum, dimension, absorber, Hurewicz set. 2010 Mathematics Subject Classification: 54F15, 54F45.

1. Introduction Although the development of both dimension theory and continuum theory began in the early XXth century, even nowadays new results are obtained. We would like to present a few facts from the border of this two fields. In the paper all spaces are Hausdorff and a continuum is a non-empty, compact connected space. Let A, B be disjoint closed subsets of a space X. Suppose that C is a closed subset of X and C ⊂ X \ (A ∪ B). Then C is called a separator between A and B if there are disjoint open subsets U , V of X such that A ⊂ U , B ⊂ V and X \ C = U ∪ V ; and C is called a cut between A and B provided that every continuum that meets both A and B meets C. Notice that every separator is a cut. If X is a compact, metric and locally connected space, then a closed subset C ⊂ X cuts X between A and B if and only if C is a separator in X between A and B [13, Theorem 1, p. 238].

L. Przontka Institute of Mathematics, University of Silesia, Bankowa 14, 40-007 Katowice, Poland, e-mail: [email protected] A. Samulewicz Institute of Mathematics, Silesian University of Technology, Kaszubska 23, 44-100 Gliwice, Poland, e-mail: [email protected] R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 345–354. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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The definitions of the inductive dimension functions Ind and Dg are similar: Ind X or Dg X equals −1 iff X = ∅. For a non-empty normal space X, Ind X (Dg X, respectively) is the smallest non-negative integer n such that between any pair of disjoint closed subsets A and B of X, there is a separator (cut, respectively) C with Ind C (Dg C, respectively) 6 n − 1, provided that such a number n exists. If there is no such n then Ind X (Dg X, respectively) = ∞. Assuming that the set A in the above definition of Ind is a singleton we obtain the definition of the dimension function ind. Transfinite extensions of the above dimension functions are obtained in the usual manner and denoted by trDg, trind and trInd. It is known that if X is a separable metric space then ind X = Ind X = dim X. In particular, for metric continua and subspaces of I ∞ , where I = [0, 1], all dimension functions coincide. A space X is strongly infinite dimensional if there exists a sequence (An , Bn )n of closed disjoint subsets of X suchTthat for each sequence (Cn )n of closed separators of X between An and Bn we have n Cn 6= ∅. A space is weakly infinite dimensional if it is not strongly infinite-dimensional. A space is strongly countable dimensional if it is a countable union of closed finite dimensional subsets. By a Hilbert cube we mean a homeomorphic copy of I ∞ . The Hilbert cube is strongly infinite dimensional and not strongly countable dimensional.

2. Chainable continua and dimension A chain is a finite collection of sets U1 , . . . , Un such that Ui ∩ Uj 6= ∅ iff |i − j| 6 1. A non-empty normal space X is said to be chainable if every open cover of X can be refined by an open (or equivalently, closed) chain. Is is easy to see that any chainable space X is a continuum, and dim X = 1 unless X is a single point. We say that a point x of a chainable space X is an end point if every open cover of X can be refined by an open (or equivalently closed) chain {U1 , . . . , Un } such that x ∈ U1 . In a sense, chainable continua are the simplest ones among continua of covering dimension 1. It was an open question if every chainable continuum is one dimensional in the sense of dimensions ind, Ind and Dg. In 1959 Mardeˇsiˇc [16] constructed the first chainable continuum X with ind X = Ind X = 2 and thus answered this question in the negative.1 Later in 1963 Pasynkov [17] strengthened Lokucievski˘ı’s [14] counterexample to a sum theorem for Ind. He has constructed a chainable continuum X with Ind X = 2 such that X is the union of two chainable subcontinua one dimensional in all sense. Bobkov [1] constructed in 1979 the first known first-countable chainable continuum X with ind X = 2, and later Chatyrko [4] in 1990 gave, for every n ∈ N, examples of first-countable chainable continua Xn such that ind Xn = n. The spaces Xn+1 and Xn are linked by natural projections fn : Xn+1 → Xn which lead to an inverse sequence whose limit space X∞ is a chainable continuum such that every proper subcontinuum of X∞ has infinite inductive dimension ind. Recently Charalambous 1

Lunc [15] constructed the first known example of a continuum with non-coinciding dimensions.

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347

and Krzempek [3], for each pair of ordinals α,β with 1 6 α 6 β 6 ω(c+ ), where ω(c+ ) is the first ordinal of cardinality c+ , have presented first-countable chainable continua Sα,β such that trDg Sα,β = α and trind Sα,β = trInd Sα,β = β. In this chapter we construct a chainable continuum K(A) with Ind K(A) = 2 and then we describe how to obtain chainable continua Q(A, B) such that Ind Q(A, B) = 2 and Q(A, B) is the union of two continua which are chainable and one-dimensional with respect to ind and Ind. The construction described below is a modification of the ones of Chatyrko [5] and Pasynkov [17]. Krzempek noticed that one can combine techniques from [17] and [5] in order to obtain a new class of examples. He suggested this method to the first named author. Denote by ω(c) the first ordinal of cardinality c, by W (W 0 ) the set of all ordinals < ω(c) (6 ω(c), respectively) and by L the long segment (cf. [9, Example 2.2.13]) of length ω(c). If x ∈ L then x = α + t where α < ω(c) and t ∈ [0, 1) (for convention if t = 0 then we will simply write α instead α + 0) or x = ω(c). Denote by Ω the product space L × [0, 1]. For α ∈ W 0 put Iα = {α} × I ⊂ Ω. For each α ∈ W fix a homeomorphism hα : [α, α + 1] → [0, 1] such that hα (α) = 0 and hα (α + 1) = 1. For any 0 < a 6 b < 1 denote by [a, b]α the subspace h−1 α ([a, b]). Fix a non-empty set A ⊂ (0, 1) and write SA for the family of all sequences {xn }∞ n=0 such that: (1) (2) (3) (4)

xn ∈ (0, 1) for all n ∈ N, limn→∞ xn = x0 ∈ A, subsequence {x2k−1 }∞ k=1 is strictly monotonically increasing and subsequence {x2k }∞ k=1 is strictly monotonically decreasing.

Consider any function φ : W → SA such that card φ−1 x = c for every x ∈ SA . For α ∈ W denote by Fα (A) the subspace of [α, α + 1] × [0, 1] which is the union of – segments [0, 1/3]α × {0} and [2/3, 1]α × {1}, – segments [1/3, 2/3]α × φ(α), – all segments which connect points (α + 1/3, 0) and (α + 2/3, x1 ), (α + 1/3, x1 ) and (α + 2/3, x3 ),. . . and all segments which connect points (α + 2/3, 1) and (α + 1/3, x2 ), (α+2/3, x2 ) and (α+1/3, x4 ), . . . , where limn→∞ xn = x0 and {xn }∞ n=0 = φ(α). From definition of Fα (A) we get the following proposition. Proposition 2.1. Fα (A) is a chainable continuum for each α ∈ W . In order to show the next statement we will need the following trivial proposition. Proposition 2.2. Suppose that X = X1 ∪ X2 , where X1 and X2 are chainable continua, x1 , x2 ∈ X1 are endpoints of X1 , x2 , x3 ∈ X2 are endpoints of X2 and X1 ∩ X2 = {x2 }. Then X is a chainable continuum with endpoints x1 , x3 . Define K0 (A) = {(0, 0)}, Kα+1 (A) = Kα (A) ∪ Fα (A) ∪ Iα+1 , [ Kα (A) = Iα ∪ Kβ (A), if α is a limit ordinal, and β β such that {xin : n ∈ N and i = 1, 2} ⊂ φ(α). Of course [1/3, 2/3]α × {xin : n ∈ N} ⊂ Ui for i = 1, 2 and, by the definition of Fα , we have [1/3, 2/3]α × {p} ⊂ cl U1 ∩ cl U2 ⊂ K(A) \ (U1 ∪ U2 ). ⊔ ⊓

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Theorem 2.7. ind K(A) = Ind K(A) = 2 if and only if int A 6= ∅. Proof. Suppose that int A 6= ∅. Thus there exist a, b ∈ A such that [a, b] ⊂ A. Put x1 = (ω(c), a) and x2 = (ω(c), b). Consider a separator P in the space K(A) between the points x1 , x2 and disjoint open sets U ′ ,V ′ of K(A) such that K(A) \ P = U ′ ∪ V ′ and x1 ∈ U ′ , x2 ∈ V ′ . There exist disjoint open sets U, V ⊂ Ω such that K(A) \ P = K(A)∩(U ∪V ) and x1 ∈ U , x2 ∈ V . If int π[0,1] [P ∩Iω(c) ] 6= ∅ then ind P > 1. Suppose that int π[0,1] [P ∩ Iω(c) ] = ∅. Thus there exists a point p such that p ∈ [a, b] ∩ cl π[0,1] [U ∩ Iω(c) ] ∩ cl π[0,1] [V ∩ Iω(c) ]. By Lemma 2.6 there exists an ordinal number α < ω(c) such that [1/3, 2/3]α × {p} ⊂ cl U ∩ cl V ⊂ K(A) \ (U ∪ V ) = P. Thus ind P > 1 and ind K(A) = Ind K(A) = 2. Now suppose that int A = ∅. It is enough to prove that ind K(A) 6 1. Using rectangular open sets it is easy to see that the statement holds. ⊔ ⊓ S S Remark 2.8. Notice that X(A) = α∈W 0 Iα ∪ α∈W [1/3, 2/3]α × φ(α) is a closed subspace of K(A). In order to prove Theorem 2.7 it is enough to show that ind X(A) = Ind X(A) = 2 if and only if int A 6= ∅, but it follows from [5, Proposition 4.1]. Example 2.9. Let A = (0, 1), then ,by Corollary 2.4, K(A) is a chainable continuum and, by Theorem 2.7, we have ind K(A) = Ind K(A) = 2. Consider product {0, 1} × L with the linear order 6 such that (a, α) 6 (b, β) if and only if (1) a < b, (2) a = b = 0 and α 6 β or (3) a = b = 1 and β 6 α where (a, α), (b, β) ∈ {0, 1} × L. Denote by E the upper semi-continuous decomposition of {0, 1} × L into the set X = {(0, ω(c)), (1, ω(c))} and singletons of {0, 1} × L \ X. The quotient space M = ({0, 1} × L)/E is a linearly ordered continuum. For convention, if q : {0, 1} × L → M is the quotient mapping then we will denote points q(0, α), q(1, α) by α0 , α1 respectively. Notice that L0 = q[{0} × L] and L1 = q[{1} × L] are homeomorphic to L. There exists a continuous symmetry σ : M → M such that (1) σ(00 ) = 01 and σ(01 ) = 00 , (2) σ ◦ σ = id and (3) if x, y ∈ M and x 6 y, then σ(y) 6 σ(x). Consider product space M × [0, 1] and homeomorphism ρ : M × [0, 1] → M × [0, 1] given by the formula ρ(x, t) = (σ(x), 1 − t) for (x, t) ∈ M × [0, 1]. Then the sets Li × [0, 1] for i = 0, 1 are homeomorphic to Ω. Thus previously constructed chainable continuum K(A) can be considered as a subset of L0 × [0, 1]. For a set B ⊂ (0, 1) denote by K ′ (B) the chainable continuum ρ[K(1 − B)] where 1 − B = {1 − b : b ∈ B} ⊂ (0, 1). Put Q(A, B) = K(A)∪K ′ (B). The following theorem is clear. Theorem 2.10. Q(A, B) is a chainable continuum and ind Q(A, B) = Ind Q(A, B) = 2 if and only if int(A ∪ B) 6= ∅.

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Example 2.11. Let A = Q ∩ (0, 1) and B = (0, 1) \ A. Then, by Theorem 2.10, Q(A, B) is a chainable continuum and ind Q(A, B) = Ind Q(A, B) = 2. Notice that Q(A, B) is the union of chainable continua K(A) and K ′ (B) (cf. Corollary 2.4) which are one-dimensional with respect to ind and Ind (cf. Theorem 2.7).

3. Infinite dimensional compact subsets of the Hilbert cube In this chapter we would like to recommend a few recent results, obtained with the method of absorbers and concerning hyperspaces whose definitions refer to the dimension of the elements. Theory of absorbers gives a topological characterization of some incomplete spaces. As an example of its application we present the proof of the theorem stating that strongly countable dimensional compacta of positive dimension form a space homeomorphic to the Hurewicz set. Let (X, d) be a metric space. Denote by 2X the space of all nonempty compact subsets of X equipped with the Hausdorff metric dH (K, L) = inf{ǫ > 0 : K ⊂ B(L; ǫ) and L ⊂ B(K; ǫ)}, where B(A; ǫ) stands for the open ǫ-ball about the subset A in X. Denote by C(X) the subspace of 2X consisting of all nonempty continua in X. By a hyperspace of X we mean a subspace of 2X . If X is a locally connected nondegenerate continuum without free arcs, then 2X ∞ and C(X) are Hilbert cubes [6]. In particular, 2I and C(I ∞ ) are homeomorphic to I ∞. Let X be a Hilbert cube with a metric d. Recall that a closed subset B of X is a Z-set in X if B satisfies the following condition: for any ǫ > 0 there exists a continuous mapping f : X → X such that e idX ) = sup{d(f (x), x) : x ∈ X} < ǫ. f (X) ∩ B = ∅ and d(f,

(Z)

A subset B ⊂ X is called a σZ-set in X if B is the countable union of Z-sets in X. Observe that B is a σZ-set in X if and only if B is an Fσ -set in X and condition (Z) holds. Let M be a class of spaces which is topological (i.e., if M ∈ M then each homeomorphic image of M belongs to M) and closed hereditary (i.e., each closed subset of M ∈ M is in M). A subset A of a Hilbert cube X is a M-absorber in X provided that 1. A ∈ M; 2. A is contained in a σZ-set in X; 3. A is strongly M-universal, i.e., for each subset M ∈ M of I ∞ and for each compact set K ⊂ I ∞ , any embedding f : I ∞ → X such that f (K) is a Z-set in X e by an embedding can be approximated arbitrarily closely (in the “sup” metric d) ∞ ∞ g : I → X such that g(I ) is a Z-set in X, g|K = f |K and g −1 (A)\ K = M \ K. If M-absorbers in a Hilbert cube exist then they are unique up to homeomorphisms.

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Lemma 3.1 ([8]). If A ⊂ X and B ⊂ Y are M-absorbers in Hilbert cubes X and Y then there exists a homeomorphism h : X → Y such that h(A) = B. Moreover, if X = Y then h can be chosen arbitrarily close to the identity. Verifying strong M-universality of a subset of a Hilbert cube is usually difficult. The following lemma, using techniques from [8], may simplify this task. n

Lemma 3.2 ([18]). Assume that a family A ⊂ 2I , n ∈ N ∪ {∞} is topological and n A ∈ M. Let n ∈ N ∪ {∞}, n ≥ 2, S ∈ {C(I n ), 2I }. Suppose A is a subset of S such that 1. A ∈ M, 2. for an arbitrary set M ⊂ I ∞ , M ∈ M, there exists a continuous mapping ξ : I ∞ → S such that ξ −1 (A) = M and 3. there exists an embedding θ : I ∞ → C([−1, 1]∞ ) such that for every ǫ ∈ (0, 12 ], for each graph Γ ∈ S, Γ ⊂ [ǫ, 1 − ǫ]n with straight line edges, for each nonempty subset T of vertices of Γ and for each continuum C ∈ θ(I ∞ ), the union [ [ (v + ǫC) ∪ (v + ǫξ(x)) Γ∪ v∈T

v∈T

belongs to A if and only if x ∈ M . Then A is strongly M-universal in S. Different Borel or desciptive classes satisfy the conditions imposed on a class M in the definition of M-absorbers. These classes play an important role in investigating topological structures of incomplete spaces. It is known that for all Borel classes, with exception of Gδ ’s and lower classes, absorbers in the Hilbert cube do exist. Moreover, absorbers belonging to different descriptive classes are pairwise not homeomorphic. Notice that Fσ -absorbers are, in particular, σZ-sets. A standard example of an Fσ -absorber in I ∞ is its pseudoboundary B(Q) = I ∞ \ (0, 1)∞ . Theorem 3.3 ([7]). 1. If n ≥ 1 then the hyperspace D≥n (I ∞ ) of all compacta of ∞ dimension ≥ n is an Fσ -absorber in 2I . 2. For n ≥ 2 the hyperspace D≥n ∩ C(I ∞ ) of all continua of dimension ≥ n is an Fσ -absorber in C(I ∞ ). ∞ 3. All infinite dimensional compacta in I ∞ form an Fσδ -absorber in 2I . The other example of an Fσ -absorber is the hyperspace of all decomposable subcontinua of the cube I k , k ∈ N ∪ {∞}, considered as a subset of the Hilbert cube C(I k ) [19]. Recall that a continuum is decomposable provided that it is the union of two proper subcontinua. Let Π11 denote the class of coanalytic sets. The Π11 -absorbers are also called coanalytic aborbers. A standard example of a Π11 -absorber in the Hilbert cube 2I is the Hurewicz set H consisting of all nonempty countable closed subsets of the unit interval I [2]. By lemma 3.1 every Π11 -absorber in a Hilbert cube is homeomorphic to H and its complement is homoeomorphic to 2I \ H. Theorem 3.4 ([12]). Denote by SCDk (I ∞ ) and SCDk ∩ C(I ∞ ) the hyperspaces of all strongly countable dimensional compacta and continua in I ∞ of dimension at least ∞ k, considered as subspaces of 2I and C(I ∞ ), respectively. For every positive number

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k the hyperspaces SCDk (I ∞ ) and SCDk+1 ∩ C(I ∞ ) are Π11 -absorbers in the Hilbert ∞ cubes 2I and C(I ∞ ), respectively. Proof. The families SCD k (I ∞ ) and SCDk+1 ∩ C(I ∞ ) are coanalytic [10]. Obviously SCDk (I ∞ ) ⊂ D≥1 (I ∞ ) and SCDk+1 ∩ C(I ∞ ) ⊂ D≥2 ∩ C(I ∞ ) and thus they are contained in appropriate σZ-sets (Theorem 3.3). To verify Π11 -universality we use Lemma 3.2. Let M be a coanalytic set in I ∞ . First we have to construct a continuous mapping ξ : I ∞ → C(I ∞ ) such that ξ −1 (SCD k (I ∞ )) = M . Let NN be the Baire space of all infinite sequences of natural numbers. The set N 0). Then the asymptotic center is a singleton. The proof of this result can be found in [5] and [15] (see also [6, Corollary 3.7]). A very particular example of geodesic spaces are metric trees which have a lot of applications in different fields. This class of spaces can be treated as CAT(κ) spaces for infinite parameter κ (κ = −∞). Definition 2.7. A metric tree is a geodesic metric space M such that: (1) for all x, y ∈ M there is unique metric segment [x, y] joining them; (2) if x, y and z ∈ M are such that [y, x] ∩ [x, z] = {x}, then [y, x] ∪ [x, z] = [y, z]. This definition implies some basic properties which are not shared with other subclasses of CAT(0) spaces. We present them in the following proposition. Proposition 2.8. Let M be a metric tree. Then (1) for all x, y, z ∈ M there is the unique w ∈ M such that [x, z] ∩ [y, z] = [w, z]; (2) if C is a closed and convex subset of M , then for all x ∈ C and all y ∈ M we have ρ(x, y) = ρ(x, PC (y)) + ρ(PC (y), y), so the projection PC (y) is a gate (see [1]).

3. Fixed points on bounded sets We begin with a basic fixed point theorem for nonexpansive mappings defined on nonempty closed convex and bounded subsets of CAT(κ) spaces. Since this theorem will be a fundamental result for the rest of this section we add a short proof. It is worth to emphasize that the same type of proof also works for uniformly convex Banach spaces and, more general, for uniformly convex metric spaces with a modulus of convexity that is monotone or lower–semicontinuous from the right. First we consider subsets of CAT(0) spaces. Theorem 3.1. Let C be a nonempty convex closed and bounded subset of a complete CAT(0) space X and T : C → C be a nonexpansive mapping. Then the set of fixed points of T (denoted by F ix T ) is nonempty closed and convex. Proof. Let us fix x0 and consider an orbit {T n (x0 ) : n ∈ N}, where T 1 (x) := T (x) and T n+1 (x) := T (T n(x)) for all x ∈ C. This orbit forms a bounded sequence since C is bounded, so its asymptotic center is a singleton. Let us denote A := A((T n (x0 ))) ∈ X. It is easy to see that it must be that A ∈ C. Otherwise, considering the projection of A onto C the nonexpansivity of the projection mapping implies that

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ρ(T n (x0 ), PC (A)) = ρ(PC (T n (x0 )), PC (A)) 6 ρ(T n (x0 ), A) and r(PC (A), (xn )) 6 r(A, (xn )) which contradicts the fact that A is an asymptotic center of T n (x0 ). Now repeating our consideration for T (A) instead of PC (A) the uniqueness of asymptotic center implies that A = T (A), i.e., the set of fixed points is nonempty. The closedness follows immediately from the fact that being nonexpansive the mapping T must be continuous. Finally, we want to show that F ix T is convex. Let x, y ∈ F ix T and z ∈ [x, y]. Then ρ(x, T (z)) = ρ(T (x), T (z)) 6 ρ(x, z)

and ρ(y, T (z)) = ρ(T (y), T (z)) 6 ρ(y, z),

and from the uniqueness of metric segments it follows that T (z) ∈ [x, y]. Moreover, if one assume that ρ(x, z) = αρ(x, y), then ρ(x, T (z)) 6 αρ(x, y) and ρ(y, T (z)) 6 (1 − α)ρ(x, y), which completes the proof that T (z) = z. ⊔ ⊓ In a similar way one may obtain the same result for asymptotic nonexpansive and asymptotic pointwise nonexpansive mappings, i.e., the mappings for which ρ(T n (x), T n (y)) 6 kn ρ(x, y)

and lim sup kn 6 1

or ρ(T n (x), T n (y)) 6 kn (x)ρ(x, y)

and lim sup kn (x) 6 1 for each x ∈ C

holds, respectively (cf. [11, Theorem 5.1] or [6, Theorem 3.11]). Now let us consider a more general situation in which T (C) does not have to be a subset of C. Theorem 3.2. Let C be a nonempty convex closed and bounded subset of a complete CAT(0) space X and T : C → X be a nonexpansive mapping. Then there is a best proximity point, i.e., a point x0 ∈ C such that ρ(x0 , T (x0 )) = inf ρ(x, T (x0 )). x∈C

Remark 3.3. 1. Let us note that the point x0 from the previous theorem is the unique projection of T (x0 ) onto C. Clearly, we do not claim that x0 is the unique point satisfying the above condition. 2. Since the domain C of T is a closed convex subset of a complete CAT(0) space X, C is a retract of the whole space X and the projection map PC : X → C is nonexpansive (see Proposition 2.3), so one may define a new mapping T˜ = PC ◦ T which, as a composition of nonexpansive mappings, is also nonexpansive. On account of Theorem 3.1 there is a fixed point x0 of T˜. Hence T (x0 ) = x0 or x0 is a projection of T (x0 ) onto C. In both cases the claim of the theorem holds. Theorem 3.1 may be generalized for subsets of a CAT(κ) space X for all κ ∈ R. In case of negative κ, in the virtue of Proposition 2.2, one may repeat the above proof. If κ > 0, then we may obtain the following result assuming a boundedness condition on the radius of C (cf. [5, Theorem 3.9]):

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Theorem 3.4. Let C be a nonempty convex and closed subset of a complete CAT(κ) space X, κ > 0, and T : C → C be a nonexpansive mapping. If the radius radC (C) =  π inf x∈C supy∈C ρ(x, y) of C is smaller than √ , then the set of fixed points of T 2 κ is nonempty closed and convex. The proof follows the same patterns as in Theorem 3.1. We have repeated the result for positive κ to emphasize differences in behaviour between mappings T : C → X defined on subsets of CAT(0) spaces and CAT(κ) spaces for κ > 0. In both cases if C is bounded (for κ > 0 additionally radC (C) cannot be too large) and T : C → C, then there is a fixed point of T , i.e., a point x0 which is the best proximity one, because ρ(x0 , T (x0 )) = 0 = inf x∈C ρ(x, T (x0 )). Now let us focus on mappings for which T (C) is not contained in C. Theorem 3.1 and the properties of the projection mapping guarantee that T has a best proximity point for κ 6 0. In the case of positive κ, the situation may be completely different as the following example shows. Example 3.5. Let us consider the unit sphere S2 . Clearly, we may view it as a CAT(1) space. Take C a closed ball of S2 centered at the north pole and with radius less π than . Let T : C → S2 be defined by T (x) = −x, i.e., for x = (x1 , x2 , x3 ) we 2 have T (x) = (−x1 , −x2 , −x3 ). Obviously, being an isometry T is also a nonexpansive mapping. Now let us consider any point of T (C). If y ∈ T (C) satisfies T −1 (y) = PC (y), then y must belong to the boundary of C. In that case PC (y) = (y1 , y2 , −y3 ),

for y = (y1 , y2 , y3 )

while T −1 (y) = (−y1 , −y2 , −y3 ) 6= PC (y). So there is no best proximity point in C. In [10] the fixed point property for set-valued mappings has been considered. Before we pass to this result we have to recall some definitions. Let a set-valued mapping T : X → 2X take as values nonempty subsets of X. In this case x will be called a fixed point of F if and only if x ∈ F (x). Obviously, this is not the unique way to define fixed points for set-valued mappings, but we use this definition in the sequel. In case of a metric space (X, ρ) first we define an ε–hull of a nonempty set C ⊂ X. Namely, Nε (C) is defined by Nε (C) = {x ∈ X | ρ(x, C) = inf ρ(x, c) < ε}. c∈C

Next we consider the family of nonempty and closed subsets of X which we denote by cl(X). Then (cl(X), h) is a metric space, where h(C1 , C2 ) = inf {Ci ⊂ Nε (Cj ), i, j ∈ {1, 2}} . ε>0

If additionally X is complete, then so is cl(X). Moreover, instead of closed subsets one can take only compact subsets cc(X) and then the metric space (cc(X), h) is still complete (cf. [4, Theorems II-2, II-3 and II-5,pp. 38–41]). In this case we say that T : X → cc(X) is a nonexpansive mapping if

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h(T (x), T (y)) 6 ρ(x, y),

x, y ∈ X.

The following counterpart of Theorem 15.3 from [10] holds in the setting of CAT(0) spaces (cf. [6, Theorem 4.2]): Theorem 3.6. Let X be a complete CAT(0) space and C a nonempty closed convex and bounded subset of X. Then each nonexpansive set-valued mapping T : C → cc(C) has at least one fixed point, i.e., there exists a point x ∈ C such that x ∈ T (x). Under an additional assumption on the radius of C the same result is true for complete CAT(κ) spaces with a positive κ (see [6, Theorem 4.3]).

4. Fixed points on unbounded sets In 1980 William O’Ray proved that a closed and convex subset C of a Hilbert space has the following property: Theorem 4.1 (O’Ray, 1980). Let C be defined as above. Then C has the fixed point property for nonexpansive mappings if and only if C is bounded. Four year later the counterpart of this result was proved by Kazimierz Goebel and Simeon Reich for spaces of constant negative curvature: Theorem 4.2 (Goebel, Reich, 1984). Let X be a real Hilbert ball with the hyperbolic metric and C ⊂ X a closed and convex subset. Then C has the fixed point property for nonexpansive mappings if and only if C is geodesically bounded. Since the real Hilbert ball with the hyperbolic metric is a space with constant curvature equal to −1, we can extend this result for all separable spaces of constant negative curvature. Moreover, one can get the same equivalence for complete metric trees, which can be treated as spaces of curvature equal to −∞ (cf. [13]). So the natural question one may raise is whether there are any geometric conditions which are equivalent to the fixed point property in CAT(0) spaces. This problem was considered for the first time by Rafa Esp´ınola and the author in [8]. In the same paper the reader can find many examples of subclasses of CAT(0) spaces for which the counterpart of Theorem 4.2 holds. Moreover, very recently the author gave in [21] the final answer for the question on the behaviour of spaces with strictly negative curvature. Theorem 4.3 (Pi¸atek, 2014). Let X be a complete CAT(κ) space, κ < 0 and C ⊂ X a closed and convex subset of X. Then C has the fixed point property for nonexpansive mappings if and only if C is geodesically bounded.

5. Continuous mappings In the previous section we mentioned the result due to William A. Kirk on metric trees. But we considered it only from the viewpoint of a very particular class of nonexpansive mappings. However, this result can be extended in the following way for all continuous mappings.

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Theorem 5.1 (Kirk, 2004). Let X be a complete metric tree. Then each continuous mapping F : X → X has at least one fixed point if and only if X is geodesically bounded. Next this result was explored for a wide class of continuous set-valued mappings. Definition 5.2. Let X be a metric tree and F : X → 2X take nonempty values. Then we say that F is upper–semicontinuous at x0 ∈ X if for each open V ⊂ X such that F (x0 ) ⊂ V there is an open U ∋ x0 with F (x) ⊂ V,

x ∈ U.

We say that F is lower–semicontinuous at x0 ∈ X if for each open V ⊂ X such that F (x0 ) ∩ V 6= ∅ there is an open U ∋ x0 with F (x) ∩ V 6= ∅,

x ∈ U.

Clearly, these definitions can be used for a much wider class of spaces not only for metric trees. But in this particular case we obtain the following result (cf. [14] and [16]): Theorem 5.3. Let X be a complete and geodesically bounded metric tree. Then each upper– (lower–) semicontinuous set-valued mapping F : X → 2X with nonempty closed convex and bounded values has at least one fixed point. Next the author introduced in [19] a weaker notion of semi-continuity. In case of metric spaces both upper and lower semi-continuous mappings belong to this new class. Definition 5.4. Let X be a metric tree and F : X → 2X take nonempty values. Then we say that F is ε–semicontinuous at x0 ∈ X if for each positive ε > 0 there is an open U ∋ x0 such that F (x) ∩ Nε (F (x0 )) 6= ∅,

x ∈ U.

For these mappings it can be also shown that: Theorem 5.5 (Pi¸atek, 2008). Let X be a complete and geodesically bounded metric tree. Then each ε–semicontinuous set-valued mapping F : X → 2X with nonempty closed convex and bounded values has at least one fixed point. Clearly, one cannot expect that the similar result is still true in each complete CAT(0) space. However, in this setting the counterpart of the Schauder Fixed Point Theorem holds (cf. [2, Corollary 18], also [17]): Theorem 5.6 (Ariza–Ruiz, Li, L´ opez-Acedo, 2014). Let X be a CAT(0) space and K a nonempty closed convex and bounded subset of X. Then, any continuous T : K → K with compact range T (K) has at least one fixed point in K.

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Bibliography 1. Aksoy A.G., Khamsi M.A.: A selection theorem in metric trees. Proc. Amer. Math. Soc. 134 (2006), 2957–2966. 2. Ariza–Ruiz D., Li C., L´ opez-Acedo G.: The Schauder fixed point theorem in geodesic spaces. J. Math. Anal. Appl. 417 (2014), 345–360. 3. Bridson M., Haefliger A.: Metric Spaces of Non-positive Curvature. Springer-Verlag, Berlin 1999. 4. Castaing C., Valadier M.: Convex Analysis and Measurable Multifunctions. Springer-Verlag, Berlin 1977. 5. Esp´ınola R., Fern´ andez–Le´ on A.: CAT(κ)–spaces, weak convergence and fixed points. J. Math. Anal. Appl. 353 (2009), 410–427. 6. Esp´ınola R, Fern´ andez-Le´ on A., Pi¸ atek B.: Fixed points of single and set-valued mappings in uniformly convex metric spaces with no metric convexity. Fixed Point Theory Appl. 2010 (2010), article ID 169837. 7. Esp´ınola R., Kirk W.A.: Fixed point theorems in R-trees with applications to graph theory. Topology Appl. 153 (2006), 1046–1055. 8. Esp´ınola R., Pi¸ atek B.: The fixed point property and unbounded sets in CAT(0) spaces. J. Math. Anal. Appl. 408 (2013), 638–654. 9. Goebel K., Reich S.: Uniform convexity, hyperbolic geometry and nonexpansive mappings. Pure Appl. Math., Marcel Dekker, New York 1984. 10. Goebel K., Kirk W.A.: Topics in Metric Fixed Point Theory. Cambridge Studies in Advanced Mathematics 28, Cambridge Univ. Press, Cambridge 1990. 11. Hussain N., Khamsi M.A.: On asymptotic pointwise contractions in metric spaces. Nonlinear Anal. 71 (2009), 4423–4429. 12. Kirk W.A.: Geodesic geometry and fixed point theory. In: Seminar of Mathematical Analysis, Proceedings, Universities of Malaga and Seville (Spain) September 2002 – February 2003. Girela´ Alvarez D., L´ opez-Acedo G., Villa-Caro R. (eds.), Univ. Sevilla Secr. Publ., Seville 2003, 195–225. 13. Kirk W.A.: Fixed points theorem in CAT(0) spaces and R–tress. Fixed Point Theory Appl. 2004, no. 4 (2004), 309–316. 14. Kirk W.A., Panyanak B.: Best approximation in R-trees. Numer. Funct. Anal. Optim. 28 (2007), 681–690. 15. Leu¸stean L.: Nonexpansive iterations in uniformly convex W-hyperbolic spaces. Contemp. Math. 513 (2008), 193–210. 16. Markin J.T.: Fixed points, selections and best approximation for multivalued mappings in Rtrees. Nonlinear Anal. 67 (2007), 2712–2716. 17. Niculescu C.P., Roventa I.: Schauder fixed point theorem in spaces with global nonpositive curvature. Fixed Point Theory Appl. 2009 (2009), article ID 906727. 18. O’Ray W.: The fixed point property and unbounded sets in Hilbert space. Trans. Amer. Math. Soc. 258 (1980), 531–537. 19. Pi¸ atek B.: A best approximation of coincide points in metric trees. Annales UMCS Sec. A 62 (2008), 113–121. 20. Pi¸ atek B.: Viscosity interation in CAT(κ) spaces. Numer. Funct. Anal. Optim. 34 (2013), 1245– 1264. 21. Pi¸ atek B.: The fixed point property and unbounded sets in spaces of negative curvature. Israel J. Math. (to appear). 22. Shafrir I.: The approximate fixed point property in Banach and hyperbolic spaces. Israel J. Math. 71 (1990), 211–223. 23. Witula R.: Divergent vector sequences {yn } with ∆yn → 0. Colloquium Math. 107 (2007), 263–266.

Przykład łuku prostego w R3 z indykatrysą nigdzie niespójną (zerowymiarową)∗ Zygmunt Zahorski

1. Wstęp Indykatrysą Ik krzywej K : x = f1 (t), y = f2 (t), z = f3 (t), 0 ¬ t ¬ 1, mającej wszędzie oś styczną, nazywam zbiór punktów S(t) powierzchni kuli o promieniu 1 i środku w początku układu współrzędnych, odpowiadających wszystkim wartościom t ∈ [0, 1], przy czym punkt S(t) określam w ten sposób: prowadzę wektor długości 1, o początku w środku układu współrzędnych, równoległy do osi stycznej L(t) krzywej K w punkcie P (t) (tj. x = f1 (t), y = f2 (t), z = f3 (t)) i mający zwrot tej osi. Koniec tego wektora jest punktem S(t). Oś L(t) określam jako wspólną granicę osi −−−−→ −−−−→ P (t), P (t + h) i P (t−h), P (t) przy h > 0, h → 0 (siecznych, przechodzących przez punkty P (t), P (t ± h) krzywej K). Oś ta wg założenia istnieje dla każdego t ∈ [0, 1], −−−→ przy czym dla t = 0 lub t = 1 określam ją jednostronnie, tj. jako granicę osi P (0), P (h) przy h > 0, h → 0. Gdy krzywa K jest płaska, f3 (t) ≡ 0, indykatrysa Ik jest również płaska, mianowicie składa się z punktów obwodu koła w płaszczyźnie XY , o promieniu 1. W szczególności, krzywa y = f (x), gdzie f jest funkcją ciągłą, mającą wszędzie pochodną skończoną lub nieskończoną, x ∈ [a, b]. Wobec tego, że f ′ (x) ma własność Darboux (przechodzi przez wartości pośrednie) ma indykatrysę spójną, tj. łuk koła o promieniu 1. P.M. Wojdysławski1 wykazał, że indykatrysę spójną ma każdy łuk prosty w R2 , mający wszędzie oś styczną. Twierdzenie to nie daje się uogólnić na R3 , łatwo zbudować łuk prosty w R3 z osią styczną w każdym punkcie, o indykatrysie niespójnej. Indykatrysą linii śrubowej na walcu kołowym prostym jest oczywiście koło, dotyczy to również linii śrubowych na stożku ew. innych powierzchniach, jeżeli określić śrubowe jako krzywe, których styczna tworzy stały kąt ze stałym wektorem. ∗

Praca zredagowana na podstawie zachowanego rękopisu profesora Zygmunta Zahorskiego.

1

Na seminarium prof. Mazura z geometrii różniczkowej, na jesieni 1940 r. – jako uogólnienie jednego z twierdzeń Ostrowskiego. P. Wojdysławski podał dwa własne dowody tego twierdzenia: 1) elementarny, ale skomplikowany, 2) prostszy, oparty na tw. topologii. M. Wojdysławski, wybitny topolog, ur. 1918 r. nie daje znaku życia od końca wojny, prawdopodobnie zamordowany przez zbrodniarzy hitlerowskich na tle obłędu rasowego podczas likwidacji ludności getta w Częstochowie, gdzie przebywał ostatnio. R. Witula, D. Slota, W. Holubowski (eds.), Monograph on the Occasion of 100th Birthday Anniver´ askiej, Gliwice 2015, pp. 367–379. sary of Zygmunt Zahorski. Wydawnictwo Politechniki Sl¸

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Rys. 1

Istnienie śrubowych na powierzchniach dostatecznie regularnych wynika z twierdzeń o istnieniu rozwiązań równań różniczkowych. W szczególności, na powierzchni obrotowej, utworzonej przez obrót łuku koła L dookoła osi AB, stycznej do niego w punkcie P istnieje śrubowa K, której styczna tworzy kąt α < π2 z osią AB, wyjątek stanowi punkt P , w którym osią styczną jest oś AB. Indykatrysa Ik składa się z koła, odpowiadającego zwojom spirali i punktu C, leżącego w jednakowej odległości od wszystkich punktów tego koła, odpowiadającego osi stycznej w P (rys. 1a). (Punkt o tej własności, leżący w mniejszym z obszarów, na które koło dzieli kulę, nazywam środkiem koła na kuli). Krzywą K tak zbudowaną nazywam elementem śrubowym I rodzaju, indykatrysa jej, jako złożona z dwóch zbiorów zamkniętych rozłącznych, jest niespójna. Podobnie, choć nieco trudniej, można zbudować łuk prosty, którego indykatrysa składa się z łuku koła i punktu izolowanego (element śrubowy II rodzaju, rys. 1b). Biorę mianowicie dwie powierzchnie, podobne jak w 1a), tylko odpowiednio wygięte, jak na rysunku 1b, T1 i T2 , złączone wierzchołkami w punkcie P , w którym mają wspólną styczną, oś AB. Krzywa K składa się z odcinków śrubowych na T1 i T2 , których styczne tworzą z AB kąt α i odcinków prostych l1 , m1 , l2 , m2 , . . . stycznych do tych śrubowych. Zaczynając krzywą K odcinkiem śrubowej na przedniej powierzchni T1 przechodzę prostą l1 na tylną powierzchnię T2 , następnie odcinkiem śrubowej na przednią powierzchnię T2 , stamtąd prostą m1 na tylną powierzchnię T1 , odcinkiem śrubowej na przednią powierzchnię T2 i prostą l2 na tylną powierzchnię T2 itd. „ósemkami”. [Terminów „przednia” i „tylna” strona powierzchni nie precyzuję bliżej, choć można by to tu zrobić, ponieważ intuicyjne stosowanie tych określeń nie prowadzi do nieporozumień]. Odcinkom śrubowej na T1 odpowiada w indykatrysie Ik łuk koła M L, odcinkom śrubowej na T2 – ten sam łuk o zwrocie LM , prostym l1 , l2 , . . ., punkt L, prostym m1 , m2 , . . ., punkt M , punktowi P punkt C (środek koła M L na kuli). Twierdzenie P. Wojdysławskiego wymaga również koniecznego spełnienia pozostałych dwóch założeń. Choć właściwie, przy nieistnieniu osi stycznej, nie można mówić o indykatrysie, to jednak, w wypadku gdy istnieje prosta styczna, można ostrzom przyporządkować dwa punkty indykatrysy, odpowiadające dwóm zwrotom prostej stycznej. Wtedy jednak, jak widać z rys. 2a, indykatrysa może być niespójna (dwa łuki koła, grubszą linią). Podobnie krzywa z rysunku 2b, mająca wszędzie oś styczną, ale niebędąca łukiem prostym, ma indykatrysę niespójną (łuk koła i punkt izolowany C). Elementy śrubowe I i II rodzaju z rys. 1 o indykatrysie niespójnej mają jednak własność, że indykatrysa zawiera część spójną (koło, ew. łuk koła). Można jednak zagęścić tę osobliwość i otrzymać łuk prosty w R3 z osią styczną w każdym punkcie, którego indykatrysa nie zawiera części spójnej, mianowicie jest

Przykład łuku prostego w R3 z indykatrysą nigdzie niespójną (zerowymiarową)

369

zbiorem zerowymiarowym w sensie Mengera, a nawet podzbiorem zbioru zamkniętego 0-wymiarowego. Celem tej pracy jest skonstruowanie takiego łuku. Posługuję się przy tym elementem śrubowym II rodzaju, zbudowanym nieco inaczej niż na rys. 1b, przy tym słowa „II rodzaju” skreślam, ponieważ elementy śrubowe prostsze (I rodzaju) nie są nigdzie użyte.

Rys. 2

2. Budowa elementu śrubowego Przy dowolnie danych kątach ψ i α, 0 < ψ < π2 , 0 < α < π2 , przesuwam przez oś Z układu współrzędnych (prostokątnego) dwie płaszczyzny Γ , ∆, tworzące z płaszczyzną XZ (ich dwusieczną) kąty ψ (rys. 3).

Rys. 3

Oś Z dzieli te płaszczyzny na cztery półpłaszczyzny. W tej półpłaszczyźnie, która przechodzi przez pierwszą ćwiartkę układu XY , prowadzę przez początek układu O półprostą n, tworzącą z osią Z kąt α, w tej, która przechodzi przez IV ćwiartkę, półprostą m przez punkt A osi Z, tworzącą z osią Z kąt π − α. Oznaczam |OA| = a.

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Przez punkt S osi X prowadzę oś walca stycznego do obu płaszczyzn, oczywiście równoległą do osi Z, oznaczam |OS| = l. Przy danym a obieram l tak, że spełnia warunek następujący: niech C i B oznaczają, odpowiednio, punkty styczności prostych n i m z walcem. Mają one dać się połączyć na powierzchni walca łukiem śrubowej lewoskrętnej odpowiadającym łukowi koła o długości mniejszej niż 2πr, której styczna tworzy z osią Z kąt α. W szczególności, proste n i m będą styczne do śrubowej w C i B. Oznaczając przez h > 0 różnicę współrzędnych z-owych punktów B i C, mam: h = a − 2|OC| cos α,

(1)

1 , (2) sin α promień walca r = l sin ψ, długość rzutu łuku śrubowej CB na płaszczyznę XY jest równa r(π + 2ψ), więc: |OC| = l cos ψ

h = r(π + 2ψ) ctg α = l sin ψ ctg α(π + 2ψ) i warunek dla l jest wg (1) i (2): a − 2l cos ψ ctg α = l sin ψ(π + 2ψ) ctg α, l=

a , (π + 2ψ) sin ψ ctg α + 2 cos ψ ctg α

(3)

czyli walec spełniający wymienione warunki istnieje. Stąd długość łuku: ⌢

|CB|= =

h r(π + 2ψ) = = cos α sin α

(π + 2ψ)l sin ψ a(π + 2ψ) sin ψ = . sin α (π + 2ψ) sin ψ cos α + 2 cos ψ cos α

Odległość dowolnego punktu łuku OCBA od osi Z jest mniejsza od długości tego łuku równej: ⌢ h a a−h + = . (4) 2|OC|+ |CB| = 2 2 cos α cos α cos α Rzut takiego punktu na oś Z leży, oczywiście, na odcinku OA. Oznaczając przez b > 0 liczbę daną, obieram ciąg punktów w odległościach 2b , b b b b b 2 + 4 , 2 + 4 + 8 itd. od 0 na dodatniej osi Z i dzielę nowymi punktami odcinki między kolejnymi dwoma z tych punktów, między pierwszym, tj. 0, a drugim na 10 równych części, między drugim a trzecim na 102 równych części, między n-tym a n+1szym na 10n części. Oznaczając kolejno tak otrzymane punkty przez O, A1 , A2 , A3 , . . . oraz OA1 = a1 , A1 A2 = a2 , . . . , An−1 An = an , . . ., mamy: a1 + a2 + . . . + an + . . . = b,

an ∞ P

¬ ai

1 , 10k

i=n+1

jeżeli tylko punkty An−1 , An leżą w k-tej z dzielonych części. Zauważmy, że przy n → ∞ jest k → ∞, stąd:

Przykład łuku prostego w R3 z indykatrysą nigdzie niespójną (zerowymiarową)

lim

n→∞

an ∞ P

= 0.

371

(5)

ai

i=n+1

Przesuwając przez punkty O, A1 , A2 , . . . w płaszczyznach Γ, ∆ proste n1 , m1 , n2 , m2 , . . . równoległe do n i m i łącząc w sposób opisany ni z mi śrubową lewoskrętną, mi z ni+1 po drugiej stronie osi Z – śrubową prawoskrętną otrzymuję element śrubowy (rys. 4), którego indykatrysą jest łuk koła łączący punkty N i M , odpowiadające kierunkom n i m i punkt P , środek tego koła na kuli, odpowiadający kierunkowi osi Z.

Rys. 4

Istnienie stycznej i jej kierunek (punkt indykatrysy) są łatwo widoczne na odcinkach prostoliniowych i śrubowych elementu śrubowego, a że styczną w wierzchołku, odległym o b od O, jest oś Z, wynika z oszacowania tg ϕ, gdzie ϕ jest kątem, jaki z osią Z tworzy sieczna przechodząca przez wierzchołek i punkt elementu śrubowego, leżący na części śrubowej lub prostoliniowej, odpowiadającej odcinkowi an . Jest wg (4): tg ϕ < cos α



an ∞ P

i=n+1

ai

,

tj. wg (5) tg ϕ → 0, gdy punkt siecznej dąży do wierzchołka. Długość całego łuku elementu śrubowego jest wg (4): a1 a2 a3 l + + + ... = . cos α cos α cos α cos α

(6)

Jednocześnie widać, że przez odpowiednią zmianę kierunku prostych n, m, Z można zbudować element śrubowy, którego indykatrysa składa się z dowolnego punktu na kuli i dowolnego łuku koła o „środku” w tym punkcie, byleby większego od połowy obwodu (ψ > 0). Oś Z nazywam osią elementu śrubowego.

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3. Zastępowanie łuków śrubowej elementami śrubowymi Rozpatruję łuk AB śrubowej na walcu o promieniu r, np. prawoskrętnej, krótszy od całego zwoju i odpowiadający mu w sposób homeomorficzny łuk indykatrysy CD (rys. 5). Połączę końce A i B łuku śrubowej krzywą (łukiem prostym) złożoną z dwóch elementów śrubowych, mających jeden odcinek prostoliniowy wspólny. Odcinek ten ma być równoległy do stycznej do śrubowej w środku E łuku AB, której odpowiada punkt F indykatrysy na środku łuku CD. Osią jednego elementu ma być styczna do śrubowej w A, drugiego – styczna w B. Kierunki pozostałych odcinków prostoliniowych obieram dowolnie, ale tak, aby łuki indykatrysy elementów były równe, niedużo dłuższe od połowy obwodu i skierowane wypukłością w przeciwne strony „S”, co można zrobić na dwa sposoby (rys. 5 I i 5 II).

Rys. 5

Odkładam w tym celu od A na stycznej do śrubowej w A wektor długości c o zwrocie osi stycznej, od B na stycznej do śrubowej w B wektor długości c o zwrocie przeciwnym do stycznej i łączę końce tych wektorów, przy czym wyznaczam c z warunku, że odcinek łączący ma być równoległy do stycznej w E. Jeżeli z równań otrzymam c > 0, to znaczy że konstrukcja jest możliwa. Oznaczam przez α kąt, jaki styczne śrubowej tworzą z osią walca, przez α1 – analogiczny kąt dla odcinków śrubowych elementów śrubowych, z ich osiami, oczywiście α1 jest to kąt między stycznymi do śrubowej w A i E, bo styczna w A jest osią elementu, zaś styczna w E jest równoległa do jednego z odcinków prostoliniowych elementu. Układ współrzędnych obieram tak, że oś Z jest osią walca, oś X przechodzi przez A i oznaczam przez ϕ wartość parametru punktu B w reprezentacji parametrycznej łuku śrubowej AB: x = r cos ϕ, y = r sin ϕ, z = rϕ ctg α. Wreszcie, oznaczając przez p długość odcinka równoległego do stycznej w E, łączącego końce wektorów stycznych o początkach w A i B, otrzymuję wg (6) długość L = p + 2 coscα1 elementu śrubowego zastępującego AB. Z równania śrubowej: x = r cos ϕ, y = r sin ϕ, z = rϕ ctg α, otrzymuję składowe jednostkowego wektora stycznego: − sin ϕ sin α, cos ϕ sin α, cos α i szukany warunek równoległości odcinka łączącego punkty na stycznych śrubowej w A i B do stycznej śrubowej w E jest:

Przykład łuku prostego w R3 z indykatrysą nigdzie niespójną (zerowymiarową)

373

−r + r cos ϕ + c sin ϕ sin α r sin ϕ − c sin α − c cos ϕ cos α = = ϕ − sin 2 sin α cos ϕ2 sin α rϕ ctg α − 2c cos α = p. cos α Pierwsze równanie jest tożsamością, drugie daje: =

2r

rϕ 1 ϕ ϕ sin − 2c cos = − 2c, sin α 2 2 sin α

stąd:

ϕ

c=

ϕ

r 2 − sin 2 > 0, sin α 2 sin2 ϕ4

rϕ r p= − 2c = sin α sin α

ϕ−

ϕ − 2 sin ϕ2 2 sin2

ϕ 4

!

,

skąd: " #   ϕ − 2 sin ϕ2 1 r L= ϕ+ −1 , sin α cos α1 2 sin2 ϕ4 cos α1 = cos ostatecznie:

ϕ sin2 α + cos2 α, 2

" # ϕ − 2 sin ϕ2 2 sin2 α sin2 ϕ4 r L= ϕ+ , sin α cos2 α + sin2 α cos ϕ2 2 sin2 ϕ4

rπ zaś długość połowy zwoju śrubowej jest równa sin α , stosunek v tych miar jest: " #  1 sin2 α ϕ v= ϕ+ ϕ − 2 sin , π 2 cos2 α + sin2 α cos ϕ2

(7)

czyli może być v < 1, przy dostatecznie małym ϕ, dla każdego α i r.

4. Konstrukcja Łuk prosty K o własnościach wymienionych w tytule zbuduję podając jego równanie: x = f (z), y = g(z), gdzie f i g – funkcje wszędzie różniczkowalne w przedziale z ∈ [0, 1], |f ′ (z)| ¬ m, |g ′ (z)| ¬ m. Łuk ten będzie mianowicie granicą łuków x = fn (z), y = gn (z), gdzie fn i gn to funkcje wszędzie różniczkowalne, |fn′ (z)| ¬ m, |gn′ (z)| ¬ m, istnieją lim oraz lim gn′ (z), przy tym granice pochodnych istnieją jedn→∞

n→∞

nostajnie, granice lim fn (z0 ), lim gn (z0 ) istnieją w pewnym punkcie z0 . Jak wiadon→∞

n→∞

mo, f (z) = lim fn (z) i g(z) = lim gn (z) istnieją wtedy jednostajnie, są różniczkon→∞

n→∞

walne i f ′ (z) = lim fn′ (z), g ′ (z) = lim gn′ (z). Istnienie f ′ (z) i g ′ (z) pociąga za sobą n→∞

n→∞

istnienie osi stycznej, a więc i indykatrysy Ik . Funkcje fn (z) i gn (z) określam przez ich

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Z. Zahorski

wykresy: wykresy te są to rzuty łuku Kn łączącego środek O układu współrzędnych z punktem (0, 0, 1) osi Z na płaszczyzny XZ i Y Z odpowiednio. Aby rzuty te były wykresami funkcji jednoznacznej, wystarczy, aby przy przebieganiu łuku Kn w jednym kierunku współrzędna z stale rosła, co uzyskuję przez warunek, aby styczna do Kn tworzyła małe kąty z osią Z, np. mniejsze od π4 . Wtedy styczna nie jest nigdzie prostopadła do płaszczyzny ZX względnie ZY i rzuty łuku Kn na te płaszczyzny  mają również wszędzie styczne, tworzące z osią Z kąty mniejsze od π4 α < π2 , tj. fn′ (z) i gn′ (z) istnieją i są bezwzględnie mniejsze od m (tg α = m). Warunek ten jest równoważny żądaniu, aby indykatrysa Kn leżała wewnątrz pewnego koła na kuli. Konstrukcja stanowić będzie dowód twierdzenia (ze względów rachunkowych przyjmuję m = 1, co jednak nie zmniejsza ogólności i dowód dla m dowolnego jest analogiczny): Twierdzenie 4.1. Dla dowolnej liczby m > 0 istnieją funkcje ciągłe skończone, wszędzie różniczkowalne f (z) i g(z), z ∈ [0, 1], takie że f ′ (z) i g ′ (z) mają wszędziegęste zbiory przedziałów stałości i najwyżej przeliczalne zbiory punktów nieciągłości, |f ′ (z)| < m, |g ′ (z)| < m dla każdego z, takie że łuk prosty K : x = f (z), y = g(z) ma indykatrysę Ik różną od punktu i Ik jest zbiorem 0-wymiarowym w sensie Mengera. Łuki Kn mające wszędzie oś styczną, określę indukcyjnie. Mianowicie K1 określam jako element śrubowy łączący początek układu O i punkt (0, 0, 1) (b = 1), którego π . osią jest oś Z i np. α = π6 , ψ = 12 Punkt (0, 0, 1) nazywam punktem osobliwym elementu, odpowiada mu izolowany punkt P1 indykatrysy IK1 . Jednocześnie określam indukcyjnie ciąg zbiorów domkniętych Fn na kuli jednostkowej, związanych z indykatrysami IKn , kładąc F0 równe całej powierzchni kuli. Przypuśćmy, że jest już określony łuk Kn , mający indykatrysę IKn , i zbiór Fn−1 , tak że spełniają warunki: 1. Łuk Kn składa się z przeliczalnej ilości odcinków prostoliniowych i łuków śrubowych rozłącznych krótszych od jednego zwoju, a dłuższych od połowy zwoju oraz najwyżej przeliczalnej ilości punktów osobliwych. Punkty osobliwe są punktami skupienia punktów leżących na łukach różnych śrubowych; koniec łuku śrubowej nie jest punktem osobliwym. 2. Indykatrysa IKn składa się ze skończonej ilości punktów izolowanych P1 , P2 , . . ., Pmn odpowiadających punktom osobliwym łuku Kn i skończonej ilości łuków kół lub „S-ów” rozłącznych i leży wewnątrz koła K odp. kątom mniejszym od π 4 z osią Z (gdzie „S-em” nazywam sumę dwóch łuków kół nieco większych od połowy obwodu zwróconych wypukłościami w przeciwną stronę i mających punkt wspólny). 3. Fn−1 składa się z punktów P1 , P2 , . . ., Pmn i obszarów zamkniętych zawierających wewnątrz S-y lub łuki indykatrysy IKn (a więc Fn−1 ⊃ IKn ), przy tym rozłącznym S-om lub łukom odpowiadają rozłączne obszary je zawierające. Jak widać, K1 , IK1 i F0 spełniają warunki 1, 2, 3. Określam łuk Kn+1 w sposób następujący: odcinki prostoliniowe i punkty osobliwe łuku Kn należą do Kn+1 . Łuki śrubowych w Kn dzielę na pewną skończoną ilość łuków każdy i zastępuję te części elementami śrubowymi wg rozdziału 2, przy tym łuki indykatrysy IKn zostają zastąpione leżącymi na nich punktami izolowanymi i łukami innych kół, tak zmieniony łuk Kn przyjmuję za Kn+1 . Pozostaje określić bliżej sposób zastępowania łuków śrubowej (rys. 6a). Wystarczy w tym celu podać sposób przejścia

Przykład łuku prostego w R3 z indykatrysą nigdzie niespójną (zerowymiarową)

375

od indykatrysy IKn do IKn+1 , bo choć każdemu łukowi koła w IKn odpowiada nieskończenie wiele łuków różnych śrubowych, to jednak między łukiem jednej śrubowej a odpowiadającym mu łukiem IKn zachodzi homeomorfizm, bo łuk ten jest krótszy od całego zwoju (odpowiadające sobie długości łuku śrubowej i indykatrysy są nawet wprost proporcjonalne).

Rys. 6

Rozważam w IKn łuk lub „S” złożony z dwóch równych łuków. Dzielę ten łuk lub każdy z dwóch łuków na 3k łuków równych o wnętrzach rozłącznych, stykających się końcami i pewne z nich łączę po dwa w łuki dłuższe, mianowicie pierwszy z drugim, czwarty z piątym, siódmy z ósmym, . . . , (3m + 1)-szy z (3m + 2)-gim. W ten sposób każdy łuk jest sumą 2k łuków o wnętrzach rozłącznych i końcach parami wspólnych, takich że z każdych dwóch łuków sąsiednich jeden jest dwa razy dłuższy od drugiego. Liczba k, zależna od n (i ewentualnie od „S”-a) jest przy tym tak duża, że gdy tak otrzymane łuki zastąpimy „S”-ami IKn+1 (tj. odpowiednie łuki śrubowej Kn dwoma elementami śrubowymi Kn+1 ), to: 1) „S”-y zastępujące te łuki leżą w domknięciu Fn−1 , każdy wewnątrz tej samej składowej domknięcia Fn−1 co zastępowany łuk. 2) Średnica zbioru złożonego z „S”-a i zastępowanego przez „S” łuku jest mniejsza od 21n (średnica jest równa kresowi górnemu odległości dwóch punktów mierzonej łukiem koła wielkiego). 3) Element śrubowy Kn+1 zastępujący część łuku śrubowej Kn jest krótszy od całego tego łuku. (Ponieważ cały łuk śrubowej w Kn zawiera więcej niż połowę zwoju, wystarczy, wg wzoru (7) z rozdziału 2, przyjąć v < 1, co da się osiągnąć przy dostatecznie małym ϕ, tj. przy dostatecznie dużej ilości części, 2k). Jak wiadomo, zastąpić łuk „S”-em można na dwa sposoby, więc istnieje 24k sposobów ułożenia „S”-ów IKn+1 na „S”-ie IKn . Obieram jeden z dwóch sposobów wyznaczonych w zupełności przez położenie pierwszego „S”-a (dowolne), zaznaczone dostatecznie jasno na rys. 6a) (I „S” zastępuje AB, II – BC, III – CD itd.). Przy tym sposobie ułożenia, „S”-y IKn+1 zastępujące poszczególne łuki AB, BC, . . . są rozłączne. W ten sposób łuki lub „S”-y IKn są zastąpione przez skończoną ilość rozłącznych „S”-ów IKn+1 i punkty izolowane Pmn +1 , Pmn +2 , . . ., Pmn+1 leżące na łukach lub na „S”-ach IKn . (Wynika z tego, choć jest to bez znaczenia, że dla n ­ 2, IKn

376

Z. Zahorski

składają się tylko z „S”-ów i punktów izolowanych, zaś nie zawierają składowej z łuku pierwszego koła jak IK1 ). Rysunek 6b podaje szkic IK3 . Określam IKn+1 jako sumę wszystkich tych „S”-ów i punktów P1 , P2 , . . ., Pmn+1 i wykażę, że jest to istotnie indykatrysa Kn+1 oraz Kn+1 jest łukiem prostym. Wreszcie określam Fn w ten sposób: obieram liczbę δn > 0 taką małą, że δn otoczenia domknięte S-ów rozłącznych IKn+1 są rozłączne i otoczenia te leżą wewnątrz domknięcia Fn−1 . Zbiór Fn jest sumą tych otoczeń oraz punktów P1 , . . ., Pmn+1 . [δ-otoczeniem punktu P na kuli nazywam czaszę kulistą otwartą o środku P , „promieniu” δ mierzonym łukiem koła wielkiego, zaś δ-otoczeniem zamkniętym punktu P – takąż czaszę zamkniętą. Nazywam δ-otoczeniem zbioru Q punktów sumę δ-otoczeń wszystkich punktów zbioru Q. Gdy Q jest S-em lub łukiem koła, to domknięcie δotoczenia zbioru Q jest sumą δ-otoczeń domkniętych wszystkich punktów zbioru Q i nazywam je δ-otoczeniem domkniętym zbioru Q]. Udowodnię, że tak określone zbiory Kn+1 , IKn+1 , Fn spełniają warunki 1), 2), 3). Przede wszystkim widać, że w tych punktach Kn+1 , które nie należą do ex definitione do Kn styczna istnieje i odpowiada jej punkt indykatrysy IKn+1 , bo w punktach tych przejście od Kn do Kn+1 polega na zastąpieniu łuku śrubowej skończoną ilością elementów śrubowych, których styczne w punktach osobliwych wspólnych są wspólne. Również oczywiste jest, że wewnątrz odcinków prostoliniowych, które wg definicji należą do Kn+1 , styczna istnieje, mianowicie jest nią prosta zawierająca ten odcinek. Punkty osobliwe Kn są, jak widać, również punktami osobliwymi Kn+1 (tj. punktami skupienia różnych śrubowych Kn+1 ), a że ponadto przybywają przy zastępowaniu łuków śrubowych Kn przez elementy śrubowe punkty osobliwe w ilości skończonej dla każdego takiego łuku, więc wobec przeliczalności tych łuków, Kn+1 ma również przeliczalną ilość punktów osobliwych. (Widać stąd, że dla n ­ 2, Kn ma dokładnie przeliczalną ilość punktów osobliwych). Każdy element śrubowy zawiera ℵ0 łuków śrubowych i odcinków prostoliniowych, więc Kn+1 zawiera przeliczalnie wiele łuków śrubowych i odcinków prostoliniowych, że w nowych punktach osobliwych styczne istnieją i odpowiadają im punkty Pnm +1 , . . ., Pnm+1 indykatrysy IKn+1 oraz końce śrubowych Kn+1 nie są punktami osobliwymi, jest widoczne z konstrukcji. Pozostaje więc dowieść, że w tych punktach osobliwych Kn+1 , które są punktami osobliwymi Kn styczna istnieje i to ta sama co dla Kn , a więc tym punktom osobliwym odpowiadają punkty P1 , . . ., Pmn indykatrysy oraz że istnieje styczna w końcach odcinków prostoliniowych wspólnych dla Kn i Kn+1 . Ponieważ koniec odcinka prostoliniowego w Kn jest albo końcem łuku śrubowej w Kn , albo punktem osobliwym w Kn , więc w przypadku pierwszym istnienie w nim stycznej do Kn+1 jest widoczne – jest nią jedna z osi elementów śrubowych Kn+1 , a w przypadku drugim kwestia istnienia stycznej sprowadza się do przypadku stycznej w punktach osobliwych. Aby wykazać, że styczna do Kn+1 w punktach osobliwych Kn istnieje, udowodnię indukcyjnie, że: d lim = 0, (8) h→0 h gdzie d – długość dowolnego łuku śrubowej Kn+1 , h – odległość tego łuku od dowolnego (ustalonego) punktu osobliwego Kn+1 . Dla K1 wynika to z mocniejszego warunku:

Przykład łuku prostego w R3 z indykatrysą nigdzie niespójną (zerowymiarową)

lim

n→∞

cos α



an ∞ P

i=n+1

ai

377

 = 0,

który wynika ze wzoru (5); K1 ma tylko jeden punkt osobliwy. Przypuśćmy, że (8) zachodzi dla Ki . Punkty osobliwe Ki+1 , które nie są punktami osobliwymi Ki , leżą w ilości skończonej na każdej części Ki+1 zastępującej łuk śrubowej Ki , są więc izolowane. W dostatecznie małej odległości od takiego punktu osobliwego, Ki+1 jest elementem śrubowym analogicznym do K1 lub do dwóch łuków takich jak K1 o wspólnym wierzchołku i osi. Wobec tego (8) wynika dla tych punktów, podobnie jak dla K1 z (5). Weźmy pod uwagę punkt T osobliwy dla Ki oraz Ki+1 . Każdy łuk śrubowy Ki+1 leży w pewnym elemencie śrubowym, zastępującym jakąś część łuku śrubowej w Kn , długość d rozważanego łuku Kn+1 jest oczywiście mniejsza od długości całego elementu śrubowego, a ta znów jest wg (7) mniejsza od długości d1 , odpowiedniego łuku śrubowej w Ki . Rozważany łuk śrubowej Ki+1 leży w odległości h od punktu T nie mniejszej niż zawierający go element śrubowy, który wg konstrukcji ma wierzchołki wspólne z łukiem śrubowej w Ki . Odległość elementu śrubowego od T jest niemniejsza od odległości wierzchołka minus długość elementu, a zastępując odległość wierzchołka (punktu śrubowej Ki ) przez niewiększą od niej odległość h1 , odp. łuku śrubowej Ki od T , zaś długość elementu przez większą od niej wg (7) długość d1 , tego łuku otrzymuję, 1 , jeśli tylko h1 − d1 > 0. Ale ponieważ (8) zachodzi dla że h > h1 − d1 , stąd hd < h1d−d 1 d1 1 =0 Ki , więc lim h1 = 0 i d1 < h1 dla dostatecznie małych h1 , więc wobec lim h1d−d 1 h1 →0 d h1 →0 h

mamy lim

h1 →0

= 0. Pozostaje więc wykazać, że lim h1 = 0. W tym celu korzystam h→0

z tego, że indykatrysa IKi (i ¬ n) leży w Fn ⊂ K, więc styczne do Ki tworzą kąt mniejszy od π4 z osią Z, wobec tego można łuk Ki przedstawić parametrycznie przez x = fi (z), y = gi (z), gdzie z rośnie od 0 do 1. Ponieważ koniec łuku śrubowej Ki nie jest punktem osobliwym T , ani T nie leży na tym łuku, więc T ma współrzędną z większą (lub mniejszą) od współrzędnej z wszystkich punktów łuku, przypuśćmy, że większą (w drugim przypadku rozumowanie jest analogiczne). Przypuśćmy, że istnieje ciąg łuków Ki taki, że lim h1 = a > 0, zaś dla pewnych łuków Ki+1 związanych z nimi w sposób wyżej podany mamy lim h = 0. Można więc obrać taki łuk, że h1 > a2 , h < a8 . Ponieważ różnica współrzędnych z jest niewiększa od odległości, która jest √ niewiększa od 2 razy różnica współrzędnych z, więc punkty rozważanego łuku mają a współrzędną z mniejszą co najmniej o 2√ od współrzędnej z punktu T . Ponieważ 2 jednak pewien punkt Ki+1 w części zastępującej rozważany łuk ma współrzędną z różną od współrzędnej z punktu T co najwyżej o a8 , więc współrzędna z osiąga w tej części wartości większe niż na odpowiednim łuku Ki , zaś końce rozważanej części Ki+1 , złożonej ze skończonej ilości elementów śrubowych są wspólne z tym łukiem Ki . Ale ponieważ elementy śrubowe mają styczną, więc rozważana część Ki+1 ma indykatrysę złożoną z „S”-ów i punktów izolowanych, leżącą wg (I) i konstrukcji, w K, wobec tego współrzędna z punktów tej części Ki+1 zmienia się monotonicznie i nie może być większa niż w końcach, co stanowi sprzeczność. Wobec tego (8) jest udowodniona indukcyjnie dla 1 ¬ i ¬ n+1. Weźmy teraz pod uwagę punkt T osobliwy dla Kn i Kn+1 oraz dowolną sieczną przechodzącą przez T i punkt P ∈ Kn+1 zmienny.

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Gdy P należy do odcinka prostoliniowego wspólnego dla Kn i Kn+1 lub jest punktem osobliwym Kn i Kn+1 , sieczna ta jest również sieczną Kn , dąży więc do stycznej do Kn odpowiadającej jednemu z punktów P1 , . . . , Pmn indykatrysy. Pozostaje rozpatrzeć przypadek, gdy P jest punktem elementu śrubowego, zastępującego jakiś łuk śrubowy Kn . Prowadzę wtedy drugą sieczną przez T i P1 , gdzie P1 jest końcem elementu śrubowego zawierającego P , a więc punktem łuku Kn , wobec tego ta druga sieczna dąży do tej samej stycznej. Oszacuję kąt β między tymi siecznymi, wystarczy wykazać, że gdy P → P1 , to β → 0. Oznaczam P T = l, P1 T = l1 , zaś odległość łuku śrubowej Kn (zawierającego P1 ) od T przez h, jest więc h ¬ l1 . Ponieważ P P1 jest niewiększe od długości elementu śrubowego, która jest mniejsza od długości d łuku Kn zawierającego d P1 , wg (7), więc l ­ l1 −P P1 > h−d, l1 > h−d, sin β2 < 2(h−d) , jeśli h−d > 0, ale z (8) d wynika, że dla h → 0, h > d oraz lim h−d = 0, więc lim β = 0 i wystarczy wykazać, h→0

h→0

że lim h = 0. Ponieważ l oznacza odległość punktu P na części Kn+1 zastępującej łuk l→0

śrubowej Kn o wspólnych z nią końcach, dowód jest analogiczny do podanego wyżej dowodu lim h = 0. Wobec tego Kn i Kn+1 mają w T nie tylko wspólny punkt, ale h1 →0

i wspólną oś styczną, co było do dowiedzenia. Z tego wynika, że IKn+1 jest indykatrysą Kn+1 i wobec IKn+1 ⊂ K współrzędna z łuku Kn+1 stale rośnie i może być przyjęta za parametr. Mając wszędzie styczną, łuk Kn jest ciągły, w ten sposób jego rzuty na płaszczyzny ZX i ZY , wykresy fn (x) i gn (x) są określone, z inkluzji IKn+1 ⊂ K wynika też, że |fn′ (x)| ¬ 1, |gn′ (x)| ¬ 1 dla każdego x, a więc fn (x) i gn (x) są wszędzie różniczkowalne i spełniają warunek Lipschitza. To, że IKn+1 spełnia warunek 2) z n + 1 zamiast n, jest widoczne z konstrukcji IKn+1 , podobnie z definicji liczby δn i Fn wynika, że Fn spełnia warunek 3). Wobec tego ciąg zbiorów Kn , IKn , Fn spełniających warunki 1), 2), 3) jest określony. Zbiory zamknięte Fn tworzą ciąg zstępujący. Istotnie, Fn−1 składa się z punktów P1 , . . . , Pmn i obszarów zamkniętych Qi,n zawierających S-y IKn , zaś Fn z punktów P1 , . . . , Pmn , Pmn +1 , . . . , Pmn+1 i obszarów Qj,n+1 zawierających S-y IKn+1 . Ale obszary Qj,n+1 leżą wg definicji w obszarach Qi,n , zaś Pmn , . . ., Pmn+1 – na S-ach IKn , a więc w Qi,n , stąd Fn ⊂ Fn−1 . Wobec tego, gdy punkt P jest granicą ciągu punktów Pn , to: ∞ Y P ∈ Fn , jeśli Pn ∈ Fn . n=1

′ ′ Oszacuję różnicę pochodnych fn′ (z) − fn+1 (z), gn′ (z) − gn+1 (z). Jeśli punktowi Z odpowiada punkt osobliwy lub punkt odcinka prostoliniowego Kn , to styczne Kn i Kn+1 są wspólne i różnice te są równe zero. Wobec tego niech punktowi Z odpowiada punkt A łuku śrubowej Kn , oraz punkt B elementu śrubowego Kn+1 zastępującego ten łuk. (Nie może to być inny element śrubowy, bo przebiegając Kn lub Kn+1 , z stale rośnie, zaś końce łuku i elementu są wg definicji wspólne). Wobec tego stycznym w A i w B odpowiadają punkty α i β, z których α leży na części łuku lub S-a IKn , zaś β na S-ie lub w punkcie izolowanym zastępującym tę część, więc wg (??)2 odległość (geodezyjna na kuli) α od β jest mniejsza od 21n . Styczne w A i w B tworzą więc kąt ϕ < 21n , a że ramiona tego kąta tworzą z płaszczyznami XZ oraz Y Z

2

W rękopisie Profesora brakuje w tym miejscu numeru wzoru (red.).

Przykład łuku prostego w R3 z indykatrysą nigdzie niespójną (zerowymiarową)

379

kąty γ, δ < π4 , więc ramiona jego rzutu prostokątnego na te płaszczyzny tworzą kąt ϕ1 < 2 21n , wg łatwego rachunku. Jest więc: 1 ′ arc tg fn′ (z) − arc tg fn+1 (z) < n−1 , 2 stąd: ′ ′ ′ fn (z) − fn+1 (z) = (1 + ξ 2 ) arc tg fn′ (z) − arc tg fn+1 (z) ¬

1 + L2 ′ ¬ (1 + L2 ) arc tg fn′ (z) − arc tg fn+1 (z) < n−1 . 2 Z tego wynika, że ciąg pochodnych jest jednostajnie zbieżny. Ponieważ punkt (0, 0, 0) jako leżący na odcinku prostoliniowym K1 , jest wspólny dla wszystkich Kn , czyli fn (0) = gn (0) = 0 dla każdego n, ciągi fn (z), gn (z) są zbieżne w punkcie 0, a więc jednostajnie zbieżne w [0, 1]. Kładąc f (z) = lim fn (z), g(z) = lim gn (z), mamy: n→∞

f ′ (z) = lim fn′ (z), n→∞

n→∞

g ′ (z) = lim gn′ (z), n→∞

czyli funkcje f (z) i g(z) są wszędzie różniczkowalne i spełniają warunek Lipschitza, a styczna do łuku K : (x = f (z), y = g(z)) w punkcie z jest granicą stycznych łuków Kn w punkcie z, zaś punkt P indykatrysy Ik jej odpowiadający – granicą punktów Pn ∈ IKn . Według (??)3 jest więc: Y IK ⊂ Fn = F. Zbiór F jest zamknięty, bo Fn są niepuste i ograniczone, jako leżące na powierzchni kuli, wobec tego F ⊃ IK . Zbiór F jest 0-wymiarowy. Wystarczy na to, aby każdy punkt P leżał w obszarze wewnętrznym topologicznego pierścienia kołowego o dowolnie małej średnicy, rozłącznego z F . Ale P ∈ Fn , F ⊂ Fn , dla każdego n, zaś każda składowa Fn da się oddzielić od pozostałych pierścieniem kołowym o średnicy mniejszej od ε plus średnica rozważanej składowej. Wystarczy więc, aby średnice składowych Fn dążyły do zera przy n → ∞. Wykażę to dla tych składowych, które nie są punktami. Ponieważ wg (??)4 S-y IKn mają średnicę mniejszą od 21n , zaś składowe Fn je zawierające są ich δn -otoczeniami, δn < 21n , więc średnice tych składowych są mniejsze od 23n , c.b.d.d. Wobec tego IKn nie zawiera części spójnej. Na odcinkach prostoliniowych Kn i na łukach śrubowych styczna do Kn jest ciągła, wobec tego nieciągłość może mieć miejsce tylko w punktach izolowanych. Wobec tego fn′ (z), gn′ (z) mają najwyżej przeliczalne zbiory punktów nieciągłości. Ponieważ granica jednostajnie zbieżnego ciągu jest ciągła w punkcie, w którym są ciągłe wszystkie składniki, f ′ (z) i g ′ (z) są również ciągłe niemal5 wszędzie. Z konstrukcji wynika, że łuk K ma wszędziegęsty zbiór odcinków prostoliniowych, na których f ′ (z), g ′ (z) są stałe. Biorąc jakiekolwiek dwa odcinki prostoliniowe K należące do K1 o różnych kierunkach, widać, że IK nie jest jednym punktem. Wobec tego twierdzenie jest całkowicie dowiedzione. 3 4 5

W rękopisie Profesora brakuje w tym miejscu numeru wzoru (red.). W rękopisie Profesora brakuje w tym miejscu numeru wzoru (red.). Uwaga: to nie znaczy „prawie”.