Topological fixed point theory for singlevalued and multivalued mappings and applications [electronic resource] / Afif Ben Amar, Donal O'Regan.
2016
QA329.9
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Title
Topological fixed point theory for singlevalued and multivalued mappings and applications [electronic resource] / Afif Ben Amar, Donal O'Regan.
Author
Ben Amar, Afif, author.
ISBN
9783319319483 (electronic book)
3319319485 (electronic book)
9783319319476
3319319485 (electronic book)
9783319319476
Published
Switzerland : Springer, 2016.
Language
English
Description
1 online resource (x, 194 pages)
Item Number
10.1007/978-3-319-31948-3 doi
9783319319476
9783319319476
Call Number
QA329.9
Dewey Decimal Classification
515/.73
Summary
This is a monograph covering topological fixed point theory for several classes of single and multivalued maps. The authors begin by presenting basic notions in locally convex topological vector spaces. Special attention is then devoted to weak compactness, in particular to the theorems of Eberlein-Šmulian, Grothendick and Dunford-Pettis. Leray-Schauder alternatives and eigenvalue problems for decomposable single-valued nonlinear weakly compact operators in Dunford-Pettis spaces are considered, in addition to some variants of Schauder, Krasnoselskii, Sadovskii, and Leray-Schauder type fixed point theorems for different classes of weakly sequentially continuous operators on general Banach spaces. The authors then proceed with an examination of Sadovskii, Furi-Pera, and Krasnoselskii fixed point theorems and nonlinear Leray-Schauder alternatives in the framework of weak topologies and involving multivalued mappings with weakly sequentially closed graph. These results are formulated in terms of axiomatic measures of weak noncompactness. The authors continue to present some fixed point theorems in a nonempty closed convex of any Banach algebras or Banach algebras satisfying a sequential condition (P) for the sum and the product of nonlinear weakly sequentially continuous operators, and illustrate the theory by considering functional integral and partial differential equations. The existence of fixed points, nonlinear Leray-Schauder alternatives for different classes of nonlinear (ws)-compact operators (weakly condensing, 1-set weakly contractive, strictly quasi-bounded) defined on an unbounded closed convex subset of a Banach space are also discussed. The authors also examine the existence of nonlinear eigenvalues and eigenvectors, as well as the surjectivity of quasibounded operators. Finally, some approximate fixed point theorems for multivalued mappings defined on Banach spaces. Weak and strong topologies play a role here and both bounded and unbounded regions are considered. The authors explicate a method developed to indicate how to use approximate fixed point theorems to prove the existence of approximate Nash equilibria for non-cooperative games. Fixed point theory is a powerful and fruitful tool in modern mathematics and may be considered as a core subject in nonlinear analysis. In the last 50 years, fixed point theory has been a flourishing area of research. As such, the monograph begins with an overview of these developments before gravitating towards topics selected to reflect the particular interests of the authors. .
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Online resource; title from PDF title page (SpringerLink, viewed May 11, 2016).
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O'Regan, Donal, author.
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Table of Contents
Basic Concepts
Nonlinear Eigenvalue Problems in Dunford-Pettis Spaces
Fixed Point Theory in Locally Convex Spaces
Fixed Points for Maps with Weakly Sequentially-Closed
Fixed Point Theory in Banach Algebras
Fixed Point Theory for (ws)-Compact Operators
Approximate Fixed Point Theorems in Banach Spaces. .
Nonlinear Eigenvalue Problems in Dunford-Pettis Spaces
Fixed Point Theory in Locally Convex Spaces
Fixed Points for Maps with Weakly Sequentially-Closed
Fixed Point Theory in Banach Algebras
Fixed Point Theory for (ws)-Compact Operators
Approximate Fixed Point Theorems in Banach Spaces. .