Fixed point theorems in reflexive Banach spaces

Abstract

Abstract.In this paper fixed point theorems are established first for mappings T, mapping a closed bounded convex subset K of a reflexive Banach space into itself and satisfying || Tx -Ty\\ g i{\\x -Tx\\ + \\y -Ty\\}, x,yeK, and then an analogous result is obtained for nonexpansive mappings giving rise to a question regarding the unification of these theorems.Let X be a reflexive Banach space and let K be a nonempty bounded closed and convex subset of X. In [10] Kirk proved the following theorem: If F be a nonexpansive mapping of K into itself i.e., \\Tx -Fj||_||x-_y||, x, y e K, and if K has normal structure, then F has a fixed point in K. This result was also proved in a uniformly convex space X by Browder [2], Göhde [4] and Goebel [5], the reflexivity of the space and the normal structure of K being consequences of the uniform convexity of X.In this paper first we establish some fixed point theorems for mappings T of K into itself which satisfy || Tx -Ty\\ ^ «I* -Tx\\ + \\y -Ty\\}, x, y e K.Mappings T of this type will be referred to as having property A over K. Such mappings have been used to study fixed point and other allied problems in [6], [7], [8], [9]. Then we obtain a theorem, analogous to the one proved for a mapping T having property A over K, for nonexpansive mappings. We conclude the paper with some observations on this last theorem.Before going to the theorems, we first recollect the following definitions. Definition 1 (Normal Structure [10]). A bounded convex set K in a Banach space X is said to have normal structure if for each convex subset S of K which contains more than one point, there exists x e S such that supvesllx-j|| <à(S), ô(S) being the diameter of S.Definition 2 [9]. A mapping F of a bounded subset A" of a Banach space X into itself is said to have property B on K [9] if for every closed

Fixed points of asymptotically regular multivalued mappings

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