Non-convex proximal pair and relatively nonexpansive maps with respect to orbits

Abstract

Abstract Every non-convex pair ( C , D ) $(C, D)$ 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 C ∪ D $C\cup D$ , where C ∪ D $C\cup D$ is a cyclic T-regular set and ( C , D ) $(C, D)$ 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 C ∪ D $C\cup D$ , where C and D are T-regular sets in a uniformly convex Banach space satisfying T ( C ) ⊆ C $T(C)\subseteq C$ , T ( D ) ⊆ D $T(D)\subseteq D$ wherein the convergence of Kranoselskii’s iteration process is also discussed.

Keywords

• Proximal parallel pair
• Cyclic T-regular set
• Kranoselskii’s iteration
• Fixed point
• Best proximity point