A global optimality result using Geraghty type contraction

Abstract

In this paper we prove two proximity point results for finding the distance between two sets. Unlike the best approximation theorems they provide with globally optimal values. Here our approach is to reduce the problem to that of finding optimal approximate solutions of some fixed point equations. We use Geraghty type contractive inequalities in our theorem. Two illustrative examples are given.

Best proximity points of roximal contractive mappings in partially ordered complete metric spaces

“…, by combining this equation with (10) and by the fact that T is an α−β−ψ−proximal contractive mapping, we have…”

“…In this regard, the best proximity point evolves as a generalization of the best approximation. The authors Basha [5], Choudhury, Maity and Konar, [9,10] and Kutbi, Chandok and Sintunavarat [16] tried to reduce the problem of finding approximate solutions to that of finding optimal approximate solutions.…”

Coupled Optimal Results with an Application to Nonlinear Integral Equations

“…V. Sankar Raj [27] obtained an interesting result on best proximity for weakly contractive non-self mappings. Many discussions related with the existence of fixed point through the consideration of order relation with the underneath metric and of best approximation are investigated in [2,20,[27][28][29][30][31][32][33][34][35][36][37][38]. Contraction mapping procedures have been also continuously employing in differential equations and integral equations as cornerstone instruments to prove the existence of related solutions (see [39][40][41]).…”

Multivalued weak cyclic δ-contraction mappings

“…Cyclic contractions are contractions useful to obtain fixed point and optimality results for non-self-mappings. Some coupling over the study of fixed points can be obtained through cyclic contractions; for details see [13]. The other utility of cyclic contractions is related to optimality problems; for details see [14].…”