Topological Structure of the Solution Set for Evolution Inclusions

278 Pages·2017·3.567 MB·English

by  Yong Zhou,Rong-Nian Wang,Li Peng (auth.)

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
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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