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, infx∈K∥x − 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.
