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Condensing multivalued maps and semilinear differential inclusions in Banach spaces. (English) Zbl 0988.34001
de Gruyter Series in Nonlinear Analysis and Applications. 7. Berlin: de Gruyter. xi, 231 p. (2001).
The book is devoted to some aspects of multivalued analysis. It is organized into six chapters. Chapter 1 recalls general definitions and properties of multivalued maps and focuses special attention to measurable multimaps and to the superposition multioperator. Chapter 2 describes the main results of the theory of measures of noncompactness in Banach spaces and specifies the notion of condensing multimap relative to a measure of noncompactness. Chapter 3 investigates the topological degree for different types of condensing multifields in Banach spaces. A study of the existence of solutions to a system of inclusions with condensing multioperators is developed and applied to obtain optimal control for systems governed by a neutral functional-differential equation. Chapter 4 summarizes some results dealing with strongly continuous semigroups that are necessary to the study of semilinear inclusions. This is done in the last two chapters. Chapter 5 presents results concerning the existence of local and global solutions, the topological structure of the solution set and the dependence of the solutions on parameters and initial data. In Chapter 6, the authors develop methods for justifying the averaging principle in periodic problems and for proving the existence of a periodic solution and the existence of a global attractor of semilinear differential inclusions satisfying a dissipativity condition.
The presentation is self-contained, and the subject is addressed to graduate students as well as to researchers in applied functional analysis.

##### MSC:
 34-02 Research exposition (monographs, survey articles) pertaining to ordinary differential equations 34G25 Evolution inclusions 47-02 Research exposition (monographs, survey articles) pertaining to operator theory 47H04 Set-valued operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 47H11 Degree theory for nonlinear operators 34A60 Ordinary differential inclusions 34C25 Periodic solutions to ordinary differential equations 34C29 Averaging method for ordinary differential equations 49-02 Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control 49J24 Optimal control problems with differential inclusions (existence) (MSC2000) 49J27 Existence theories for problems in abstract spaces 49J53 Set-valued and variational analysis 54C60 Set-valued maps in general topology 54C65 Selections in general topology 54H25 Fixed-point and coincidence theorems (topological aspects) 55M20 Fixed points and coincidences in algebraic topology 55M25 Degree, winding number 34H05 Control problems involving ordinary differential equations 34D45 Attractors of solutions to ordinary differential equations