Abstract
In this article, we prove a best proximity point theorem for generalized cyclic contractions in convex metric spaces. Then we investigate the structure of minimal sets of cyclic relatively nonexpansive mappings in the setting of convex metric spaces. In this way, we obtain an extension of the Goebel-Karlovitz lemma, which is a key lemma in fixed point theory.
Highlights
Let (X, d) be a metric space, and let A, B be subsets of X
We study the structure of minimal sets for cyclic relatively nonexpansive mappings
We obtain an extension of the Goebel-Karlovitz lemma which plays an important role in fixed point theory
Summary
Let (X, d) be a metric space, and let A, B be subsets of X. The theorem guarantees the existence, uniqueness, and convergence of a best proximity point for cyclic contractions in metric spaces by using the notion of the property UC. We establish a best proximity point theorem for generalized cyclic contraction mappings in convex metric spaces.
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