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2017 | Buch

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Topological Structure of the Solution Set for Evolution Inclusions

verfasst von: Yong Zhou, Rong-Nian Wang, Li Peng

Verlag: Springer Singapore

Buchreihe : Developments in Mathematics

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

This book systematically presents the topological structure of solution sets and attractability for nonlinear evolution inclusions, together with its relevant applications in control problems and partial differential equations. It provides readers the background material needed to delve deeper into the subject and explore the rich research literature.

In addition, the book addresses many of the basic techniques and results recently developed in connection with this theory, including the structure of solution sets for evolution inclusions with m-dissipative operators; quasi-autonomous and non-autonomous evolution inclusions and control systems; evolution inclusions with the Hille-Yosida operator; functional evolution inclusions; impulsive evolution inclusions; and stochastic evolution inclusions. Several applications of evolution inclusions and control systems are also discussed in detail.

Based on extensive research work conducted by the authors and other experts over the past four years, the information presented is cutting-edge and comprehensive. As such, the book fills an important gap in the body of literature on the structure of evolution inclusions and its applications.

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Inhaltsverzeichnis

Frontmatter

Chapter 1. Preliminaries

Abstract

In this chapter, we introduce some basic facts on multivalued analysis, evolution system, semigroups, weak compactness of sets and operators, and stochastic processwhich are needed throughout this monograph.

Yong Zhou, Rong-Nian Wang, Li Peng

Chapter 2. Evolution Inclusions with m-Dissipative Operator

Abstract

This chapter deals with a nonlinear delay differential inclusion of evolution type involving m-dissipative operator and source term of multivalued type in a Banach space. Under rather mild conditions, the \(R_\delta \)-structure of \(C^0\)-solution set is studied on compact intervals, which is then used to obtain the \(R_\delta \) -property on noncompact intervals. Secondly, the result about the structure is furthermore employed to show the existence of \(C^0\)-solutions for the inclusion (mentioned above) subject to nonlocal condition defined on right half-line. No nonexpansive condition on nonlocal function is needed. As samples of applications, we consider a partial differential inclusion with time delay and then with nonlocal condition at the end of the chapter.

Yong Zhou, Rong-Nian Wang, Li Peng

Chapter 3. Evolution Inclusions with Hille–Yosida Operator

Abstract

This chapter deals with a parabolic differential inclusion of evolution type involving a nondensely defined closed linear operator satisfying the Hille–Yosida condition and source term of multivalued type in Banach space. The topological structure of the solution set is investigated in the cases that the semigroup is noncompact. It is shown that the solution set is nonempty, compact and an \(R_\delta \)-set. It is proved on compact intervals and then, using the inverse limit method, obtained on noncompact intervals. Secondly, the existing solvability and the existence of a compact global attractor for the m-semiflow generated by the system are studied by using measures of noncompactness. As samples of applications, we apply the abstract results to some classes of partial differential inclusions.

Yong Zhou, Rong-Nian Wang, Li Peng

Chapter 4. Quasi-autonomous Evolution Inclusions

Abstract

This chapter deals with a kind of semilinear differential inclusions in general Banach spaces. Firstly, we study different types of generalized solutions including limit and weak solutions. Under appropriate assumptions, we show that the set of the limit solutions is a compact \(R_\delta \) -set. When the right-hand side satisfies the one-sided Perron condition, a variant of the well-known lemma of Filippov-Pliś, as well as a relaxation theorem, are proved. Secondly, we study a kind of semilinear evolution inclusions. If the nonlinearity is one-sided Perron with sublinear growth, then we establish the relation between the solutions of the considered differential inclusion and the solutions of the relaxed one. A variant of the well known Filippov-Pliś lemma is also proved. Finally, we analyze the existence of pullback attractor for non-autonomous differential inclusions with infinite delays by using measures of noncompactness. As samples of applications, we apply the abstract results to control systems driven by semilinear partial differential equations and multivalued feedbacks.

Yong Zhou, Rong-Nian Wang, Li Peng

Chapter 5. Non-autonomous Evolution InclusionsEvolution inclusion and Control System

Abstract

In this chapter we consider the topological structure of the solution set of non-autonomous parabolic evolution inclusions with time delay, defined on noncompact intervals. The result restricted to compact intervals is then extended to non-autonomous parabolic control problems with time delay. Moreover, as the applications of the information about the structure, we establish the existence result of global mild solutions for non-autonomous Cauchy problems subject to nonlocal condition, and prove the invariance of a reachability set for non-autonomous control problems under single-valued nonlinear perturbations. Finally, some illustrating examples are supplied.

Yong Zhou, Rong-Nian Wang, Li Peng

Chapter 6. Neutral Functional Evolution InclusionsEvolution inclusion

Abstract

This chapter deals with functional evolution inclusions of neutral type in Banach space when the semigroup is compact as well as noncompact. The topological properties of the solution set is investigated. It is shown that the solution set is nonempty, compact and an \(R_\delta \) -set which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As a sample of application, we consider a partial differential inclusion.

Yong Zhou, Rong-Nian Wang, Li Peng

Chapter 7. Impulsive Evolution InclusionsEvolution inclusion

Abstract

In this chapter, the existence of mild solutions for impulsive differential inclusions in a reflexive Banach space is obtained. Weakly compact valued nonlinear terms are considered, combined with strongly continuous evolution operators generated by the linear part. A continuation principle or a fixed point theorem are used, according to the various regularity and growth conditions assumed. Secondly, a topological structure of the set of solutions to impulsive functional differential inclusions on the half-line is investigated. It is shown that the solution set is nonempty, compact and, moreover, an \(R_\delta \) -set. It is proved on compact intervals and then, using the inverse limit method, obtained on the half-line.

Yong Zhou, Rong-Nian Wang, Li Peng

Chapter 8. Stochastic Evolution Inclusions

Abstract

In this chapter, we investigate the topological structure of solution sets for stochastic evolution inclusions in Hilbert spaces in cases that semigroup is compact and noncompact, respectively. It is shown that the solution set is nonempty, compact and \(R_\delta \) -set which means that the solution set may not be a singleton but, from the point of view of algebraic topology, it is equivalent to a point, in the sense that it has the same homology group as one-point space. As applications of the obtained results, an example is given.

Yong Zhou, Rong-Nian Wang, Li Peng

Backmatter

Titel: Topological Structure of the Solution Set for Evolution Inclusions
verfasst von: Yong Zhou
Rong-Nian Wang
Li Peng
Copyright-Jahr: 2017
Verlag: Springer Singapore
Electronic ISBN: 978-981-10-6656-6
Print ISBN: 978-981-10-6655-9
DOI: https://doi.org/10.1007/978-981-10-6656-6

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