On Proximal contractions via implicit relations and Best proximity points

Abstract

In this paper, we employ two types of implicit relations to define some new kind of proximal contractions and study about their best proximity points. More precisely, we use two class of functions A and A ′ to explore proximal A, A ′ -contractions of first and second kind and strong proximal A, A ′ -contractions. We investigate the existence of best proximity points results of the same. It is worth mentioning that the well-known results of Sadiq Basha [J. Approx. Theory, 2011] on proximal contractions are the special cases of our obtained results. We authenticate our results by suitable examples. Finally, we point out some areas where our obtained results can be applied.

On Common fixed point results via implicit relations

“…For examples of particular mappings 'g' belonging to the above two collections, the readers are referred to see [1,7,10,11]. Now, we introduce the notion of A-contraction in case of a couple and a family of mappings.…”