Abstract
AbstractWe discuss Panayanak’s results on the existence of an endpoint of a multivalued nonexpansive mapping. We show that all of his results can be extended and some can be established in a wider class of mappings. Out of his three open questions, two of them are solved in affirmative.
Highlights
The distance from an element x ∈ X to a nonempty subset E ⊂ X is defined by d(x, E) := inf d(x, y) : y ∈ E
For a nonempty subset E ⊂ X, we denote by BC(E) (K(E), respectively) the family of nonempty bounded and closed subsets of E
The set of all endpoints of T is denoted by End(T) (Fix(T), respectively)
Summary
Let us recall the following simple example of Ko [ ] It suggests that fixed point results in the single-valued case should be extended to endpoint results in the corresponding multivalued one. Theorem P Suppose that E is a nonempty bounded closed convex subset of a uniformly convex Banach space X. Question P Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X, and T : E → K(X) be a nonexpansive mapping. Question P Let E be a nonempty closed convex subset of a uniformly convex Banach space X, and T : E → BC(X) be a nonexpansive mapping. Uv Theorem Let E be a nonempty closed convex subset of a strictly convex Banach space X. Lemma Let E be a nonempty bounded closed convex subset of a uniformly convex Banach space X.
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