Best proximity pair theorems for multifunctions with open fibres
type: Item
instance of: scholarly article
title: Best proximity pair theorems for multifunctions with open fibres
zbMATH Open document ID: 0965.41020
author: S. Sadiq Basha
author: P. Veeramani
published in: Journal of Approximation Theory
publication date: 2000-08-28T00:00:00Z
review text: Let \(A\) and \(B\) be non-empty subsets of a normed linear space \(E\), and let \(T:A\to 2^B\) be a convex multi-valued function with open fibres \(T^{-1}(y)\) (i.e.) \(\{x\in X:y\in Tx\}\). For an element \(x_0\in A\) sufficient conditions are found so that \(\text{dist}(x_0, Tx_0)= \text{dist}(A,B)\). This is the case if, say, \(A\) is a non-empty approximately compact, and convex proximinal subset of \(E\), and \(B\) is a non-empty, closed and convex subset of \(E\), and \(A_0\) is compact, while \(T(A_0)\subset B_0\). Here \(A_0=\{a\in A: \text{dist}(a,b)= \text{dist}(A,B)\) for some \(b\in B\}\). Consequences include special cases of the Brouwer's fixed point theorem.
Mathematics Subject Classification ID: 41A65
zbMATH DE Number: 1423077
zbMATH Keywords: best proximity pairs
zbMATH Keywords: multifunctions with open fibres
zbMATH Keywords: best approximant
reviewed by: Thomas V. Tonev
MaRDI profile type: Publication
full work available at URL: jath.1999.3415
OpenAlex ID: W2064008630
cites work: Proximal maps, prox maps and coincidence points
cites work: The fixed point theory of multi-valued mappings in topological vector spaces
cites work: Extensions of two fixed point theorems of F. E. Browder
cites work: Contractibility and generalized convexity
cites work: Fixed points for Kakutani factorizable multifunctions
cites work: Fixed-point theorems for set-valued mappings and existence of best approximants
cites work: Q4220449
cites work: Q3954260
cites work: A Generalization to Multifunctions of Fan's Best Approximation Theorem
cites work: A theorem on best approximations
cites work: Q5595759
cites work: Some extensions of fan's best approximation theorem
cites work: A result on best proximity pair of two sets
DOI: 10.1006/JATH.1999.3415
description: scientific article; zbMATH DE number 1423077