Relation-theoretic metrical coincidence theorems

TL;DR

This paper extends classical coincidence point theorems by generalizing metrical notions within a relation-theoretic framework, leading to broader and unified results in fixed point theory.

Contribution

It introduces weaker, relation-based notions of completeness, closedness, and continuity, and proves generalized coincidence point theorems that encompass and extend many existing results.

Findings

Generalized classical coincidence point theorems.

Unified framework for various fixed point results.

Extended applicability to arbitrary binary relations.

Abstract

In this article, we generalize some frequently used metrical notions such as: completeness, closedness, continuity, g-continuity and compatibility to relation-theoretic setting and utilize these relatively weaker notions to prove results on the existence and uniqueness of coincidence points involving a pair of mappings defined on a metric space endowed with an arbitrary binary relation. Particularly, under universal relation our results deduce the classical coincidence point theorems of Goebel (Bull. Acad. Pol. Sci. Ser. Sci. Math. Astron. Phys. 16 (1968) 733-735) and Jungck (Int. J. Math. Math. Sci. 9 (4) (1986) 771-779). In process our results generalize, extend, modify and unify several well-known results especially those obtained in Alam and Imdad (J. Fixed Point Theory Appl. 17 (4) (2015) 693-702), Karapinar et al: (Fixed Point Theory Appl. 2014:92 (2014) 16 pp), Alam et al: (Fixed Point Theory Appl. 2014:216 (2014) 30 pp), Alam and Imdad (Fixed Point Theory, in press) and Berzig (J. Fixed Point Theory Appl. 12 (1-2) (2012) 221-238.