Best Proximity Point Theorems for Cyclic Relatively ρ-Nonexpansive Mappings in Modular Spaces

Abstract

In this paper we introduce the notion of proximal ρ-normal structure of pair of ρ-admissible sets in modular spaces. We prove some results of best proximity points in this setting without recourse to Zorn’s lemma. We provide some examples to support our conclusions.

On best proximity point theorems for relatively nonexpansive mappings in locally convex spaces

“…As indicated in [1], a proof establishes that in the case where H is a normed linear space, the equivalence between strict convexity of H and the d-property of H holds.…”

“…An influential best proximity point theorem, attributed to Fan [7], asserts that within a Hausdorff locally convex topological vector space H, if K represents a compact and convex nonempty subset, and T stands as a continuous self-mapping over K, then the existence of an element υ within K is guaranteed, satisfying p(υ, Tυ) = p(Tυ, K). This theorem has been extended by several authors, including Chaira et al [1], Chi et al [3], El harmouchi et al [5], Espìnola [6], Prolla [9], Reich [11], Saadaoui et al [12] and Sehgal and Singh [13,14], across a variety of frameworks.…”

Best Proximity Points for Monotone Relatively Nonexpansive Mappings in Ordered Banach Spaces

“…Furthermore, we find in [2] a similar result without invoking Zorn's lemma, i.e., without proximal normal structure. Recently, Chaira and Lazaiz [3] gave an extension of this last result in modular spaces. For a recent account of the theory we refer the reader to [4][5][6].…”

On Best Proximity Point Theorems in Locally Convex Spaces Endowed with a Graph

“…is situation motivates the researchers to develop the notion of best proximity point theory. It is worth to note that the best proximity point theorems can be viewed as a generalization of fixed point theorems, since most fixed point theorems can be derived as corollaries of best proximity point results (for more details, see [1][2][3][4][5][6]).…”