chapter  3
60 Pages

Approximation and Selection Methods for Set-Valued Maps and Fixed Point Theory

Department of Mathematics, Brock University, Saint Catharines, Ontario, Canada

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.2 Approximative Neighborhood Retracts, Extensors, and Space

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.1 Approximative Neighborhood Retracts and Extensors . 80 3.2.2 Contractibility and Connectedness . . . . . . . . . . . . . . . . . . . . . . 84

3.2.2.1 Contractible Spaces . . . . . . . . . . . . . . . . . . . . . . . . 84 3.2.2.2 Proximal Connectedness . . . . . . . . . . . . . . . . . . . 85

3.2.3 Convexity Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 3.2.4 Space Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

3.2.4.1 The Property A(K;P) for Spaces . . . . . . . . . 90 3.2.4.2 Domination of Domain . . . . . . . . . . . . . . . . . . . . . 92 3.2.4.3 Domination, Extension, and

Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 3.3 Set-Valued Maps, Continuous Selections, and Approximations . 97

3.3.1 Semicontinuity Concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.3.2 USC Approachable Maps and Their Properties . . . . . . . . . 99

3.3.2.1 Conservation of Approachability . . . . . . . . . . . 100 3.3.2.2 Homotopy Approximation, Domination of

Domain, and Approachability . . . . . . . . . . . . . 106 3.3.3 Examples of A−Maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 3.3.4 Continuous Selections for LSC Maps . . . . . . . . . . . . . . . . . . . . 113

3.3.4.1 Michael Selections . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.3.4.2 A Hybrid Continuous

Approximation-Selection Property . . . . . . . . 116 3.3.4.3 More on Continuous Selections for

Non-Convex Maps . . . . . . . . . . . . . . . . . . . . . . . . . 116 3.3.4.4 Non-Expansive Selections . . . . . . . . . . . . . . . . . . 121

3.4 Fixed Point and Coincidence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . 122

3.4.1 Generalizations of the Himmelberg Theorem to the Non-Convex Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 3.4.1.1 Preservation of the FPP from P to

A(K;P) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.4.1.2 A Leray-Schauder Alternative for

Approachable Maps . . . . . . . . . . . . . . . . . . . . . . . . 126 3.4.2 Coincidence Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

A short version of this paper was presented at the International Workshop on Nonlinear Analysis and Optimization held at the University of Tabuk, Saudi Arabia, on March 18-19, 2013. The Chapter was completed while the author was visiting the Canadian University of Dubai, UAE. The kind hospitality of both institutions is gratefully acknowledged.