<abstract><p>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.</p></abstract>

<abstract><p>In this paper, we prove some coincidence point theorems for weak C-contractions and K-contractions involving a new auxiliary function in a metric space endowed with a locally $ f $-transitive binary relation. In this context, we generalize some relevant fixed point results in the literature. Further, we give an example to substantiate the utility of our results.</p></abstract>

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 T -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.

In this paper, we extend the idea of α-ψ contraction mapping to the product spaces by introducing Prešić–Ćirić-type α-ψ contractions and utilize them to prove some coincidence and common fixed-point theorems in the context of ordered metric spaces using α-admissibility of the mapping. Our newly established results generalize a number of well-known fixed-point theorems from the literature. Moreover, we give some examples that attest to the credibility of our results. Further, we give an application to the nonlinear integral equations, which can be employed to study the existence and uniqueness of solutions to the integral equations.