Condensing Multivalued Maps and Semilinear Differential Inclusions in Banach Spaces

Mikhail I. Kamenskii, Valeri V. Obukhovskii, Pietro Zecca

• Publisher: De Gruyter
• Year: 2011
• Print ISBN:
  9783110169898
• eBook ISBN:
  9783110870893
• Topic: Mathematics
• Subtopic: Mathematical Analysis
• Index: Mathematics

The theory of set-valued maps and of differential inclusion is developed in recent years both as a field of his own and as an approach to control theory. The book deals with the theory of semilinear differential inclusions in infinite dimensional spaces. In this setting, problems of interest to applications do not suppose neither convexity of the map or compactness of the multi-operators. These assumption implies the development of the theory of measure of noncompactness and the construction of a degree theory for condensing mapping. Of particular interest is the approach to the case when the linear part is a generator of a condensing, strongly continuous semigroup. In this context, the existence of solutions for the Cauchy and periodic problems are proved as well as the topological properties of the solution sets. Examples of applications to the control of transmission line and to hybrid systems are presented.

Table of contents

• Introduction
• 1 Multivalued maps: general properties
• 1.1 Continuity of multimaps
• 1.2 Operations on multimaps. Selections and approximations
• 1.3 Measurable multimaps and the superposition multioperator
• Bibliographic notes
• 2 Measures of noncompactness and condensing multimaps
• 2.1 Measures of noncompactness
• 2.2 Condensing multimaps
• Bibliographic notes
• 3 Topological degree theory for condensing multifields
• 3.1 The relative topological degree and fixed points of compact multimaps
• 3.2 Topological degree for condensing multifields
• 3.3 The evaluation of the degree
• 3.4 The topological degree for condensing nonconvex-valued multimaps
• 3.5 Some properties of the fixed points set
• 3.6 Solvability of systems of inclusions with condensing multioperators
• Bibliographic notes
• 4 Semigroups and measures of noncompactness
• 4.1 Semigroups: general facts
• 4.2 Measures of noncompactness in function spaces: measurability and integrability
• 4.3 Condensing semigroups
• Bibliographic notes
• 5 Semilinear differential inclusions: initial problem
• 5.1 Integral multioperator
• 5.2 Existence and continuous dependence theorems
• 5.3 The structure of the solutions set
• 5.4 The averaging principle
• 5.5 On inclusions with almost lower semicontinuous nonlinearities
• Bibliographic notes
• 6 Semilinear inclusions: periodic problems
• 6.1 The integral multioperator for a periodic problem
• 6.2 The averaging principle for a periodic problem
• 6.3 The translation multioperator
• Bibliographic notes
• Bibliography
• Index