Some Results on Iterative Proximal Convergence and Chebyshev Center

Abstract

In this paper, we prove a sufficient condition that every nonempty closed convex bounded pair M,N in a reflexive Banach space B satisfying Opial’s condition has proximal normal structure. We analyze the relatively nonexpansive self-mapping T on M∪N satisfying TM⊆M and TN⊆N, to show that Ishikawa’s and Halpern’s iteration converges to the best proximity point. Also, we prove that under relatively isometry self-mapping T on M∪N satisfying TN⊆N and TM⊆M, Ishikawa’s iteration converges to the best proximity point in the collection of all Chebyshev centers of N relative to M. Some illustrative examples are provided to support our results.