Table Of ContentBloch-Type Periodic
Functions
Theory and Applications
to Evolution Equations
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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
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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
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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
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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
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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
