A generalization to multifunctions of Fan’s best approximation theorem

Abstract

ABSTRACT. We prove a theorem for set valued mappings in an approximatively compact, convex subset of a locally convex space, and then derive results due to Ky Fan and S. Reich as corollaries.Let E be a locally convex Hausdorff topological vector sapee, S a nonempty subset of E and p a continuous seminorm on E. It is a well-known result (see the proof in Sehgal [8] or Ky Fan [1]) that if S is compact and convex and f:S->E is a continuous map, then there exists an a; G S satisfying (1) p(fx-x) =dp(fx,S) =min{p(fx,-y)\ yES}.Since then a number of authors have provided either an extension of the above theorem to set valued mappings or have weakened the compactness condition therein. Some of these results are (d) SEHGAL AND SINGH (1985). Let S Ç E with int(S) ^ 0 and cl(S) convex and let /: S -► 2E be a continuous condensing multifunction with convex, compact values and with a bounded range. Then for each w E int(S), there exists a continuous seminorm p = p(w) satisfying (1) [6].Our aim in this presentation is to prove (a) for multifonctions and derive some results as easy corollaries.For definitions and terminologies we refer to Reich [5] (see also [3]). DEFINITION. A subset S of E is approximatively p-compact iff for each y E E and a net {xa} in S satisfying p(xa -y)-+ dp(y, S) there is a subnet {xp} and an x E S such that Xß -► x.Clearly a compact set in E is approximatively compact. The converse, however, may fail. For example, the closed unit ball of an infinite dimensional uniformly convex Banach space is approximatively norm compact but not compact.

Some results on best proximity pair theorems

“…This result has been further extended by Sehgal and Singh [11], to the one for continuous multifunctions. Further, they have also proved a generalization of a best approximation theorem due to Prolla [6].…”

Optimal approximate solution theorems for Geraghty's proximal contractions in partially ordered sets via w-distances

“…Afterward, several authors, including Prolla [14], Reich [15], Sehgal and Singh [16,17], have derived extensions of Fan's theorem in many directions. In 2012, Sadiq Basha [1] extended Banach's fixed point theorem for self-mappings to the setting of non-self-mappings which resolve global optimization problems via best proximity theorems in the setting of partially ordered sets equipped with a metric.…”

On common best proximity points for generalized α−ψ -proximal contractions

“…In 2011, Sadiq Basha [17] stated some best proximity point theorems for proximal contractions. For some other results on best proximity points, see for example [1,3,5,6,7,8,9,10,11,12,13,14,15,16,18,19,21,22,23,24,25]. We recall some notations and definitions, which will be used in the sequel.…”