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

Saadaoui, Bilel; Lazaiz, Samih; Aamri, Mohamed

doi:10.1155/2020/7481060

Cited by 4 publications

(3 citation statements)

References 9 publications

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“…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.…”

Section: Introductionmentioning

confidence: 87%

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

Saadaoui,

Harmouchi,

Mouline

et al. 2023

Moroccan Journal of Pure and Applied Analysis

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This paper explores the concept of normal structure by introducing a generalized form known as P-proximal normal structure. Focusing on the framework of Hausdorff locally convex spaces, the study establishes optimal proximity outcomes for both cyclic and noncyclic relatively P-nonexpansive mappings. Furthermore, the paper presents in the realm of probabilistic normed spaces, considered as instances of Hausdorff locally convex spaces, some theorems that address the existence of best proximity points within this context.

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Section: Introductionmentioning

confidence: 87%

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

Saadaoui,

Harmouchi,

Mouline

et al. 2023

Moroccan Journal of Pure and Applied Analysis

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“…We note also that [35] and [38] contain some results describing the behavior of nonexpansive mappings and best proximity pairs in the language of directed graphs.…”

Section: Introductionmentioning

confidence: 91%

Bipartite graphs and best proximity pairs

Chaira¹,

Dovgoshey²,

Lazaiz³

2021

Preprint

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In the present paper we introduce a proximinal graph as a bipartite graph G A,B with parts A, B for which (A, B) is a disjoint proximinal pair in a semimetric space (M, d) and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph iff G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph G A,B such that (A, B) is a proximinal pairs in an ultrametric space.

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“…A special kind of bipartite graphs, the trees, gives a natural language for the description of ultrametric spaces [1, 2, 6, 9-11, 13-20, 23, 24, 26-28, 32, 35, 36], but the authors are aware of only papers [3] and [36], in which complete bipartite and, more generally, complete multipartite graphs are systematically used to study ultrametric spaces. We note also that [37] and [40] contain some results describing the behavior of nonexpansive mappings and best proximity pairs in terms of directed graphs.…”

Section: Introductionmentioning

confidence: 91%

Bipartite graphs and best proximity pairs

2022

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We say that a bipartite graph G(A, B) with the fixed parts A and B is proximinal if there is a semimetric space (X, d) such that A and B are disjoint proximinal subsets of X and all edges {a, b} satisfy the equality d(a, b) = dist(A, B). It is proved that a bipartite graph G is not isomorphic to any proximinal graph if and only if G is finite and empty. It is also shown that the subgraph induced by all non-isolated vertices of a nonempty bipartite graph G is a disjoint union of complete bipartite graphs if and only if G is isomorphic to a nonempty proximinal graph for an ultrametric space.

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

Cited by 4 publications

References 9 publications

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

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

Bipartite graphs and best proximity pairs

Bipartite graphs and best proximity pairs

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