Nonstandard methods in fixed point theory
A unified account of the major new developments inspired by Maurey's application of Banach space ultraproducts to the fixed point theory for non-expansive mappings is given in this text. The first third of the book is devoted to laying a careful foundation for the actual fixed point theoretic results which follow. Set theoretic and Banach space ultraproducts constructions are studied in detail in the second part of the book, while the remainder of the book gives an introduction to the classical fixed point theory in addition to a discussion of normal structure. This is the first book which studies classical fixed point theory for non-expansive maps in the view of non-standard methods.
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Amer assume Banach lattice Banach-Saks property basic sequence bounded sequence canonical basis Chapter compact convex subset completes the proof consider contraction mapping convex set deduce defined Definition denoted diamA dual space equi-integrable equivalent norm finite dimensional subspace finitely representable Fix(T fixed point property fixed point set fixed point theory function Goebel Hausdorff Hence implies inequality integers isometric Let xn lim xn linear M. A. Khamsi Math metric spaces minimal set natural projection nonempty nonexpansive mappings nontrivial ultrafilter obtain Proc proof of Theorem Proposition 2.9 prove R. C. James resp result satisfies Schauder basis sequence of scalars sequence xn spreading model subsequence x'n super-reflexive Theorem 4.1 topological space topology ultrafilter ultranet ultrapower ultraproduct unconditional basis uniform normal structure uniformly convex W. A. Kirk weakly compact convex weakly convergent weakly orthogonal Xi)u Zorn's lemma