Relatively Nonexpansive Mappings in k-Uniformly Convex Banach Spaces. (18th November 2021) - Normohamadi, Z.; Moosaei, M.; Gabeleh, M.
Abstract: Let A and B be two nonempty subsets of a normed linear space X . A mapping T : A ∪ B → A ∪ B is said to be noncyclic if T ( A ) ⊆ A and T ( B ) ⊆ B . In the current paper, we consider the problem of finding the best proximity pair for the noncyclic mapping T, that is, two fixed points of T which achieve the minimum distance between the sets A and B . We do it from some different approaches. The common condition on these results is relatively nonexpansivity of the mapping T . At the first conclusion, we obtain the existence of best proximity pairs in the setting of uniformly convex in every direction Banach spaces where the pair ( A, B ) is nonconvex. Then we conclude a similar result by replacing the geometric property of Opial's property of the Banach space and adding another assumption on the mapping T, called condition ( C ) . We also show that the same result is true when X is a 2-uniformly convex Banach space. In the setting of k -uniformly convex Banach spaces, we prove that every nonempty, and convex pair of subsets has a geometric notion of proximal normal structure and then, we deduce the existence of best proximity pairs for relatively nonexpansive mappings in such spaces.