Abstract
Every non-convex pair (C, D) may not have proximal normal structure even in a Hilbert space. In this article, we use cyclic relatively nonexpansive maps with respect to orbits to show the presence of best proximity points in Ccup D, where Ccup D is a cyclic T-regular set and (C, D) is a non-empty, non-convex proximal pair in a real Hilbert space. Moreover, we show the presence of best proximity points and fixed points for non-cyclic relatively nonexpansive maps with respect to orbits defined on Ccup D, where C and D are T-regular sets in a uniformly convex Banach space satisfying T(C)subseteq C, T(D)subseteq D wherein the convergence of Kranoselskii’s iteration process is also discussed.
Highlights
Introduction and preliminariesLet (X, · ) be a normed linear space and let C and D be non-empty subsets of X
A point w0 ∈ C ∪ D is known as a best proximity point for T when d(w0, Tw0) = dist(C, D) holds true
2 Main results We prove the following proposition
Summary
Introduction and preliminariesLet (X, · ) be a normed linear space and let C and D be non-empty subsets of X. Remark 1 Let (L, M) be a non-empty, convex proximal pair in a Banach space X and T : L ∪ M → L ∪ M be a relatively nonexpansive map. Proposition 2 Let (L, M) be a non-empty, non-convex weakly compact proximal pair in a real Hilbert space satisfying dist(conv(L), conv(M)) = dist(L, M).
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