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Fixed point theorems for point-to-set mappings Ko, Hwei-Mei


Let f be a point-to-set mapping from a topological X space X into the family 2(X) of nonempty closed subsets of X . K. Fan [13] proved that if X is a Hausdorff locally convex linear topological space and K is a nonempty compact convex subset of X , then an upper semicontinuous mapping (abbreviated by u.s.c.) f from K into k(K), the family of nonempty closed convex subsets of K, has a fixed point in K . Our main object in this work is to weaken "compactness" of K to "weak compactness" and prove a fixed point theorem for a mapping f on K into certain subfamily of 2(K). The definition of convex function has been extended to point-to-set mappings in Chapter I. Let I denote the identity mapping on a Banach space X. Assume that I-f is a convex mapping on a weakly compact closed convex subset K of X. Then any of the following conditions implies the existence of the fixed point of f on K: (1) f : K → 2(K) is u.s.c. and [formula omitted] d(x,f(x)) = 0. (2) f : K → 2(K) is u.s.c. and is asymptotically regular (see definition 1.3) at some point in K . (3) f : K → cc(K) is nonexpansive and the Banach space X has a strictly convex norm. Moreover, it has been shown that if f : K → cpt(K) (see definition 0.3) is nonexpansive and I-f is strictly convex (see definition 1.5) on K, then K has a fixed point on K . Finally, an effort has been made to investigate the properties of the set of fixed points of a point-to-set mappings. In Chapter II, we have confined ourselves to a reflexive Banach space X which has a weakly continuous duality map J (see definition 2.3) and X has a strictly convex norm. On such a special space we are able to prove that a nonexpansive mapping f : X → cc(X) such that f(x)ʗ K, for any x in a closed convex bounded subset K of X , has a fixed point. As an application of this result we prove a fixed point theorem for semicontractive mappings (see definition 2.7). F : X → cc(X) such that F(x)ʗK for any x ε K , where K and X are the same as above. . In the last Chapter, we have proved that if f is strictly nonexpansive on a.Banach space X into cpt(X) and if there is x(o) ε X such that [formula omitted] has a subsequence convergent to a set A ε cpt(X) under the Hausdorff metric D on cpt(X), then f has a fixed point in A . Furthermore we prove that a nonexpansive mapping f : K → cpt(K), where K is a weakly compact convex subset of a metrizable locally convex linear topological space X, has a fixed point in K, provided that a constant k > 0 exists such that the set E(x) = {y ε K ; d(x,y) ≥ kd(y,f(y))} is nonempty and convex and the mapping E : K → k(K), with E(x) defined above, is weakly locally closed (see definition 3.1). Finally the comparisons of the continuities of a point-to-set mapping have been made.

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