Vol. 42, No. 2, 1972

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Fixed point theorems for point-to-set mappings and the set of fixed points

H. M. Ko

Vol. 42 (1972), No. 2, 369–379
Abstract

Let X be a Banach space and K be a nonempty convex weakly compact subset of X. Belluce and Kirk proved that (1) If f : K K is continuous, inf xKx f(x)= 0 and I f is a convex mapping, then f has a fixed point in K. (2) If f : K K is nonexpansive and I f is a convex mapping on K, then f has a fixed point in K. In this paper the concept of convex mapping has been extended to point-to-set mappings. Theorems 1 and 2 in §2 extend the above fixed point theorems by Belluce and Kirk.

Let W stand for the set of fixed points of f : K cc(K). The set W is called a singleton in a generalized sense if there is x0 W such that W f(x0). In §3 two examples are given to show that W is not necessarily a singleton in a generalized sense if f is strictly nonexpansive or if I f is convex. But one can be sure that W is a convex set if I f is a convex or a semiconvex mapping.

Mathematical Subject Classification 2000
Primary: 47H10
Secondary: 54H25
Milestones
Received: 12 April 1971
Revised: 15 September 1971
Published: 1 August 1972
Authors
H. M. Ko