Bloch-Type Periodic Functions Theory and Applications to Evolution Equations 1122778800__99778899881111225544335522__TTPP..iinndddd 11 99//66//2222 1122::0088 PPMM SERIES ON CONCRETE AND APPLICABLE MATHEMATICS ISSN: 1793-1142 Series Editor: Professor George A. Anastassiou Department of Mathematical Sciences University of Memphis Memphis, TN 38152, USA Published* Vol. 13 Problems in Probability, Second Edition by T. M. Mills Vol. 14 Evolution Equations with a Complex Spatial Variable by Ciprian G. Gal, Sorin G. Gal & Jerome A. Goldstein Vol. 15 An Exponential Function Approach to Parabolic Equations by Chin-Yuan Lin Vol. 16 Frontiers in Approximation Theory by George A. Anastassiou Vol. 17 Frontiers in Time Scales and Inequalities by George A. 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Senthil Kumar Vol. 22 Bloch-Type Periodic Functions: Theory and Applications to Evolution Equations by Yong-Kui Chang, Gaston Mandata N'Guérékata & Rodrigo Ponce *To view the complete list of the published volumes in the series, please visit: http://www.worldscientific/series/scaam SSoouunnddaarraarraajjaann -- 1122778800 -- BBlloocchh--TTyyppee PPeerriiooddiicc FFuunnccttiioonnss..iinndddd 11 55//55//22002222 88::4411::5500 aamm Series on Concrete and Applicable Mathematics – Vol. 22 Bloch-Type Periodic Functions Theory and Applications to Evolution Equations Yong-Kui Chang Xidian University, China Gaston M. N'Guérékata Morgan State University, USA Rodrigo Ponce Universidad de Talca, Chile NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO 1122778800__99778899881111225544335522__TTPP..iinndddd 22 99//66//2222 1122::0088 PPMM Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Names: Chang, Yong-Kui, author. | N'Guérékata, Gaston M., 1953– author. | Ponce, Rodrigo (Professor of mathematics), author. Title: Bloch-type periodic functions : theory and applications to evolution equations / Yong-Kui Chang, Xidian University, China; Gaston M. N'Guérékata, Morgan State University, USA; Rodrigo Ponce, Universidad de Talca, Chile. Description: New Jersey : World Scientific, [2022] | Series: Series on concrete and applicable mathematics, 1793-1142 ; vol. 22 | Includes bibliographical references and index. Identifiers: LCCN 2022006842 | ISBN 9789811254352 (hardcover) | ISBN 9789811254369 (ebook for institutions) | ISBN 9789811254376 (ebook for individuals) Subjects: LCSH: Periodic functions. | Bloch constant. Classification: LCC QA353.P4 .C43 2022 | DDC 515/.94--dc23/eng20220422 LC record available at https://lccn.loc.gov/2022006842 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Cover image: Glaciar Grey, Torres del Paine National Park, Chile, January 25, 2017 Photographed by Rodrigo Ponce Copyright © 2022 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher. For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher. For any available supplementary material, please visit https://www.worldscientific.com/worldscibooks/10.1142/12780#t=suppl Desk Editors: Soundararajan Raghuraman/Lai Fun Kwong Typeset by Stallion Press Email: [email protected] Printed in Singapore SSoouunnddaarraarraajjaann -- 1122778800 -- BBlloocchh--TTyyppee PPeerriiooddiicc FFuunnccttiioonnss..iinndddd 22 55//55//22002222 88::4411::5500 aamm June28,2022 9:18 Bloch-TypePeriodicFunctions:TheoryandApplications-9inx6in b4622-fm FA9 pagev Preface Theaimofthismonographistogiveforthefirsttimeaunifiedandhomoge- nouspresentationofrecentworksonthetheoryofBloch-typefunctionsand their applications to evolution equations. The concept of Bloch functions goes back to the Swiss physicist F´elix Bloch while working on the conduc- tivity of crystalline solids (see for instance Bloch, 1929). Bloch functions generalizebothperiodicandantiperiodicfunctions.Theycanbeexpressed in the form Ψ(r)=eik·rψ(r), where r denotes the position, Ψ the wave function, ψ a periodic function and k the wave vector. Since the paper by Hasler and N’Gu´er´ekata (2014) thestudyofperiodicfunctionsofBloch-typehasarousedgreatinterestdue totheirimportanceinquantumphysicsandotherbranchesofmathematical physics. This book consists of nine chapters and anappendix. Chapter 1 is con- cerned with some preliminary facts. In Chapter 2, we introduce some new notions ofgeneralizedBloch-type periodic functions and presentsome fun- damental properties on spaces of such functions. In Chapter 3, we inves- tigate the existence and uniqueness of generalized Bloch-type periodic solutionstosemilinearintegrodifferentialequationswithmixedkernelviaa uniformlyexponentialstableresolventoperatorfamily.Chapter4isdevoted toestablishsomeexistenceresultsforgeneralizedBloch-typeperiodicsolu- tionstomulti-termfractionalevolutionequationsviaauniformlyintegrable resolventoperatorfamily.Chapter5ismainlyconcernedwiththeexistence and uniqueness of generalized Bloch-type periodic solutions to semilinear fractional evolution equations of degenerate type via the uniform integra- bility of a well-defined resolvent operator family. In Chapter 6, we show the existence and uniqueness of generalized Bloch-type periodic solutions v June28,2022 9:18 Bloch-TypePeriodicFunctions:TheoryandApplications-9inx6in b4622-fm FA9 pagevi vi Bloch-Type Periodic Functions: Theory and Applications to semilinear fractional integrodifferential equations via a uniformly inte- grable growth of a corresponding resolvent operator family. In Chapter 7, weestablishsomeexistenceofpseudoS-asymptoticallyBlochtypeperiodic solutionstodampedevolutionequationswithlocalornonlocalinitialcondi- tionsonthenonnegativerealaxisviasuitableregularizedfamilies.Chapter 8isfocuseduponsomeexistenceresultsforpseudoS-asymptoticallyBloch- type periodic solutionsto partial integrodifferentialequationswith localor nonlocal initial conditions via a uniformly exponentially stable resolvent operator family. In Chapter 9, we present some existence results of gener- alized Bloch-type solutions to semilinear integral equations via asymptotic decayofanintegralresolventfamily.Thefinalisanappendix,whichmainly includesnormcontinuityandcharacterizationofcompactnessforfractional resolventoperatorfamiliesappearinginChapters4and5,andapplications toasymptoticbehaviorofsolutionstoabstractfractionalCauchyproblems in the Caputo and Riemann–Liouville fractional derivatives, respectively. The content of this monograph includes some new and unpublished results.Itisuseful forgraduatestudents andresearchersasseminartopics, graduate courses and reference book in Pure and Applied Mathematics, Physics and Engineering. June28,2022 9:18 Bloch-TypePeriodicFunctions:TheoryandApplications-9inx6in b4622-fm FA9 pagevii About the Authors Dr Yong-KuiChangisnowworkingasafullpro- fessor in the School of Mathematics and Statistics, Xidian University, Xi’an, China. His main research interests include Bloch periodicity, almost period- icity and almost automorphy with applications to abstract evolution equations, functional differential equationsandinclusionwithapplications,evolution systems and controls. Dr Gaston Mandata N’Gu´er´ekata is a Univer- sity Distinguished Professorand Associate Dean at Morgan State University in Baltimore, Maryland, USA. He received his college education from the University of Montreal in Canada. He is an American Mathematical Society (AMS) Fellow, a The World Academy of Sciences (TWAS) Fellow, an African Academy of Sciences (AAS) Fellow and author of over 280 publications including 11 books at the graduate/research level, some of them are cornerstones on the subjects. His contributions range from abstract har- monic analysis to almost periodicity, almost automorphy, fractional calcu- lusandevolutionequations.DrN’Gu´er´ekataisalsoontheEditorialBoards of over 20 international journals. vii June28,2022 9:18 Bloch-TypePeriodicFunctions:TheoryandApplications-9inx6in b4622-fm FA9 pageviii viii Bloch-Type Periodic Functions: Theory and Applications Professor Rodrigo Ponce received his PhD in MathematicsattheUniversityofSantiagodeChile, Chile, in 2011. He has published more than 40 sci- entific articles in the area of functional analysis, mainlyonevolutionaryequations,maximalregular- ity on UMD spaces, theory of semigroups of linear operators, theory of operators, resolventfamilies in Banach spaces, differential and integral equations, and fractional differential equations. Currently, he works as an Assistant Professor at the Institute of Mathematics of the University of Talca, Chile, where he teaches math- ematics in undergraduate and graduate courses, and collaborates in the Mathematics Olympiadand in the training of secondaryschool teachersin his country. June28,2022 9:18 Bloch-TypePeriodicFunctions:TheoryandApplications-9inx6in b4622-fm FA9 pageix Acknowledgments Yong-Kui Chang would like to thank his graduate students Zhi-Han Zhao, Rui Zhang, Jian-Qiong Zhao, Xiao-Xia Luo, Zhuan-Xia Cheng, Yan-Tao Bian, Chao Tang, Xue-Yan Wei, Mei-Juan Zhang, Shan Zheng, Yanyan Wei,SiqiChen,fortheirhelpandcollaboration.Healsoacknowledgeswith gratitude the support of NSF of China, the Key Project of Chinese Min- istry ofEducation,ChinaPostdoctoralScience Foundationfunded project, Program for New Century Excellent Talents in University, Program for LongyuanYouth Innovative Talents of Gansu Provinceof China, and NSF of Gansu Province of China, NSF of Shaanxi Province of China during his working. G. M. N’Gu´er´ekata would like to thank professor Maximillian Hasler andhisstudentsDarinBrindleandRogerEnockOueama-Guengaifortheir collaboration. R.PoncewouldliketothankhismentorprofessorC.Lizama,andtohis graduatestudentsA.PereiraandS.Ruedafortheirhelpandcollaboration. He also thanks the support of Conicyt-ANID of Chile (11130619). We are thankful to professors B. de Andrade, E. Cuesta, C. Cuevas, H. R. Henr´ıquez, E. Herna´ndez, C. Lizama, M. Pierri for their excep- tionalcontributiontotheoriesofresolventoperatorfamilies,dampedevolu- tion equations and S-asymptotic periodicity. We also tender our thanks to anonymousrefereesforcarefullyreadingthemanuscriptandgivingvaluable commentstoimprovethisbook.Meanwhile,weexpressourthankstoseries ix