In this article, we generalize some frequently used metrical notions such as: completeness, closedness, continuity, 1-continuity and compatibility to relation-theoretic setting and utilize these relatively weaker notions to prove our 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, Jungck and others. Furthermore, our results generalize, modify, unify and extend several well-known results of the existing literature.

After the appearance of relation-theoretic contraction principle proved in a metric space equipped with an amorphous binary relation (often termed as relational metric space), various core fixed point results have been proved in the setting of different relational distance spaces by varying underlying contraction conditions. In proving such results, the notions of completeness of ambient space, continuity of involved mapping and d-self-closedness of underlying binary relation are of paramount importance. The aim of this paper is to further refine the relation-theoretic contraction principle by relaxing the conditions of completeness and continuity by replacing their respective relation-theoretic analogues. Moreover, we observe that the notion of d-self-closedness utilized in relation-theoretic contraction principle is more general than the concepts of regularity and strong regularity utilized by earlier authors.

In this paper, we present some fixed point theorems for generalized nonlinear contractions involving a new pair of auxiliary functions in a metric space endowed with a locally finitely $ T $-transitive binary relation. Our newly proved results generalize some well-known fixed point theorems existing in the literature. We also provide an example which substantiates the utility of our results.

In this article, we discuss the relation-theoretic aspects of weakly contractive mappings to prove fixed point results in the setting of metric spaces endowed with a certain binary relation. We also provide an example and an application to validate of our results. The results proved herewith unify, generalize, improve, extend, sharpen, subsume and enrich some well-known fixed point theorems of the existing literature.

In this article, we carry out some observations on existing metrical coincidence theorems of Karapinar et al. (Fixed Point Theory Appl. 2014:92, 2014 ) and Erhan et al. (J. Inequal. Appl. 2015:52, 2015 ) proved for Lakshmikantham–Ćirić-type nonlinear contractions involving \((f,g)\)-closed transitive sets after proving some coincidence theorems satisfying Boyd–Wong-type nonlinear contractivity conditions employing the idea of \((f,g)\)-closed locally f-transitive binary relation.

In this article, we prove some coincidence and common fixed point theorems under the relation-theoretic Meir-Keeler contractions in a metric space endowed with a locally finitely -transitive binary relation. Our newly proved results generalize, extend, and sharpen some existing coincidence point as well as fixed point theorems existing in the literature. Moreover, we give some examples to affirm the efficacy of our results.