Relation-theoretic metrical fixed point theorems under nonlinear contractions

TL;DR

This paper develops new fixed point theorems for nonlinear contractions in metric spaces with arbitrary binary relations, extending and unifying several existing results and providing sharper conditions and illustrative examples.

Contribution

It introduces generalized fixed point theorems under nonlinear contractions on metric spaces with arbitrary binary relations, broadening previous frameworks and improving known results.

Findings

Extended fixed point theorems to non-complete metric spaces with binary relations

Unified various existing fixed point results into a broader framework

Provided examples demonstrating the improvements over prior theorems

Abstract

We establish fixed point theorems for nonlinear contractions on a metric space (not essentially complete) endowed with an arbitrary binary relation. Our results extend, generalize, modify and unify several known results especially those contained in Samet and Turinici [Commun. Math. Anal. 13, 82-97 (2012)] and Alam and Imdad [J. Fixed Point Theory Appl. 17(4), 693-702 (2015)]. Interestingly a corollary to one of our main results proved under symmetric closure of any binary relation remains a sharpened version of a theorem due to Samet and Turinici. Finally, we use examples to highlight the realized improvements in the results proved in this paper.