Developments in Mathematics Yong Zhou Rong-Nian Wang Li Peng Topological Structure of the Solution Set for Evolution Inclusions Developments in Mathematics Volume 51 Series editors Krishnaswami Alladi, Gainesville, USA Hershel M. Farkas, Jerusalem, Israel More information about this series at http://www.springer.com/series/5834 Yong Zhou Rong-Nian Wang (cid:129) Li Peng Topological Structure of the Solution Set for Evolution Inclusions 123 Yong Zhou LiPeng Schoolof Mathematics andComputational Schoolof Mathematics andComputational Science Science Xiangtan University Xiangtan University Xiangtan, Hunan Xiangtan, Hunan China China Rong-Nian Wang Mathematics andScienceCollege ShanghaiNormal University Shanghai China ISSN 1389-2177 ISSN 2197-795X (electronic) Developments inMathematics ISBN978-981-10-6655-9 ISBN978-981-10-6656-6 (eBook) https://doi.org/10.1007/978-981-10-6656-6 LibraryofCongressControlNumber:2017953788 MathematicsSubjectClassification(2010): 34G25,37C70,34K09,35R70,60H15 ©SpringerNatureSingaporePteLtd.2017 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpart of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission orinformationstorageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilar methodologynowknownorhereafterdeveloped. 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Theregisteredcompanyaddressis:152BeachRoad,#21-01/04GatewayEast,Singapore189721,Singapore Preface A lot of phenomena investigated in hybrid systems with dry friction, processes of controlled heat transfer, obstacle problems, and others can be described with the help of various differential inclusions, both linear and nonlinear. The theory of differential inclusions is highly developed and constitutes an important branch of nonlinear analysis. To the best of our knowledge, there were very few monographs concerning the topological theory and dynamics for evolution inclusions. This monographgives a systematicpresentationofthetopologicalstructureofsolutionsetsandattractability fornonlinearevolutioninclusionsanditsrelevantapplicationsincontroltheoryand partial differential equations. The materials in this monograph are based on the researchworkcarriedoutbythe author and other excellent experts during thepast four years. The contents of this book are very new and rich. It provides the nec- essary background material required to go further into the subject and explore the rich research literature. All abstract results are illustrated by examples. This monograph deals with the focused topic with high current interest and complements the existing literature in differential equations and inclusions. It is useful for researchers, graduate or Ph.D., students dealing with differential equa- tions, applied analysis, and related areas of research. We acknowledge with gratitude the support of National Natural Science Foundation of China (11671339, 11471083). Xiangtan, China Yong Zhou Shanghai, China Rong-Nian Wang Xiangtan, China Li Peng v Contents 1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Basic Facts and Notations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Multivalued Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.1 Multivalued Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2.2 Measure of Noncompactness . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Rd-Set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.4 Inverse Limit Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 1.3 Multivalued Semiflows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4 Pullback Attractor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.5 Evolution System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.6 Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 1.6.1 C -Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 0 1.6.2 Analytic Semigroup. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 1.6.3 Integrated Semigroup. . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.7 Weak Compactness of Sets and Operators . . . . . . . . . . . . . . . . . . 31 1.8 Stochastic Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.8.1 Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 1.8.2 Stochastic Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 2 Evolution Inclusions with m-Dissipative Operator. . . . . . . . . . . . . . . 37 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 2.2 The m-Dissipative Operators and C0-Solution . . . . . . . . . . . . . . . 39 2.3 Topological Structure of Solution Set . . . . . . . . . . . . . . . . . . . . . 42 2.3.1 Compact Intervals Case . . . . . . . . . . . . . . . . . . . . . . . . . . 42 2.3.2 Noncompact Intervals Case . . . . . . . . . . . . . . . . . . . . . . . 52 2.4 Nonlocal Cauchy Problem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.5 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 vii viii Contents 3 Evolution Inclusions with Hille–Yosida Operator . . . . . . . . . . . . . . . 65 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.2 Topological Structure of Solution Set . . . . . . . . . . . . . . . . . . . . . 67 3.2.1 Existence of Integral Solution. . . . . . . . . . . . . . . . . . . . . . 68 3.2.2 Structure of Solution Set . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 3.3 Global Attractor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 3.3.1 Existence of Integral Solution. . . . . . . . . . . . . . . . . . . . . . 82 3.3.2 Existence of Global Attractor . . . . . . . . . . . . . . . . . . . . . . 88 3.3.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4 Quasi-autonomous Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . 97 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.2 Generalized Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2.1 Limit Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.2.2 Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 4.2.3 One-Sided Perron Condition. . . . . . . . . . . . . . . . . . . . . . . 110 4.3 Relaxation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.3.1 Relaxation Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117 4.3.2 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 4.4 Pullback Attractor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.4.1 Solvability. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 4.4.2 Existence of Pullback Attractor. . . . . . . . . . . . . . . . . . . . . 133 4.4.3 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 5 Non-autonomous Evolution Inclusions and Control System . . . . . . . 143 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 5.2 Nonhomogeneous Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . 146 5.3 Topological Structure of Solution Set . . . . . . . . . . . . . . . . . . . . . 148 5.3.1 Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148 5.3.2 Control Problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 5.4 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 5.4.1 An Existence Result. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 5.4.2 Invariance of Reachability Set . . . . . . . . . . . . . . . . . . . . . 160 5.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 6 Neutral Functional Evolution Inclusions. . . . . . . . . . . . . . . . . . . . . . 169 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.2 Topological Structure of Solution Set . . . . . . . . . . . . . . . . . . . . . 170 6.2.1 Compact Semigroup Case . . . . . . . . . . . . . . . . . . . . . . . . 172 6.2.2 Noncompact Semigroup Case. . . . . . . . . . . . . . . . . . . . . . 185 7 Impulsive Evolution Inclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197 7.2 Existence and Weak Compactness. . . . . . . . . . . . . . . . . . . . . . . . 199 Contents ix 7.2.1 Compact Operator Case . . . . . . . . . . . . . . . . . . . . . . . . . . 202 7.2.2 Noncompact Operator Case . . . . . . . . . . . . . . . . . . . . . . . 206 7.3 Topological Structure of Solution Set . . . . . . . . . . . . . . . . . . . . . 209 7.3.1 Compact Interval Case. . . . . . . . . . . . . . . . . . . . . . . . . . . 211 7.3.2 Noncompact Intervals Case . . . . . . . . . . . . . . . . . . . . . . . 228 8 Stochastic Evolution Inclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231 8.2 Statement of the Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232 8.3 Existence via Weak Topology. . . . . . . . . . . . . . . . . . . . . . . . . . . 235 8.4 Topological Structure of Solution Set . . . . . . . . . . . . . . . . . . . . . 240 8.4.1 Compact Operator Case . . . . . . . . . . . . . . . . . . . . . . . . . . 241 8.4.2 Noncompact Operator Case . . . . . . . . . . . . . . . . . . . . . . . 248 References.... .... .... .... ..... .... .... .... .... .... ..... .... 257 Index .... .... .... .... .... ..... .... .... .... .... .... ..... .... 267 Introduction Since the dynamics of nonlinear and hybrid systems is multivalued, differential inclusions serve as natural models in many dynamical processes. In addition, dif- ferential inclusions also provide a powerful tool for various branches of mathe- maticalanalysis.Inthepasttwentyyears,theoryofdifferentialinclusionshasbeen developed very rapidly. The several excellent monographs by Aubin and Cellina [20],BenchohraandAbbas[34],Borisovichetal.[42],Bothe[46],Deimling[80], Djebali et al. [89, 90], Dragoni et al. [96], Górniewicz [113], Graef [116], Hu and Papageorgiou [125], Kamenskii et al. [130], Kisielewicz [135, 136], Mahmudov [141],Smirnov[176],Tolstonogov[185],Vrabie[189],andZgurovskyetal.[207] summarize a lot of important works in this area. Sinceadifferentialinclusionusuallyhasmanysolutionsstartingatagivenpoint, new issues appear, such as investigation of topological properties of solution sets. In the study of the topological structure of solution sets for integral/differential equations and inclusions, an important aspect is the Rd-property. Recall that a subsetofametricspaceiscalledanRd-setifitcanberepresentedastheintersection ofadecreasingsequenceofcompactandcontractiblesets.ItisknownthatanRd-set isacyclicand,inparticular,nonempty,compact,andconnected.From thepointof viewofalgebraictopology,anRd-setisequivalenttoapoint,inthesensethatithas the same homology group as one-point space. For the Cauchy problems of ordinary differential equations having no unique- ness,Kneser[137]provedin1923thatthesetsoftheirsolutionsareateveryfixed timecontinua,andthen,Hukuhara[127]showedthatthesolutionset(onacompact interval) itself is a continuum (i.e., closed and connected). Later, Yorke [203] improved this result inthe sense that solution sets are Rd-sets. Let us also mention that byusing topologicaldegree arguments, Górniewiczand Pruszko [115] proved that the solution set (on a compact interval) of a Darboux problem for hyperbolic equation is an Rd-set; an analogous result was also established by De Blasi and Myjak [79] by using a different approach and recently, by means of the theory of condensing mappings and multivalued analysis tools, Ke et al. [133] investigated theRd-structureofthesolutionsetforanabstractVolterraintegralequationwithout uniqueness on a compact interval. xi