The present manuscript is devoted to investigating some existence and uniqueness results on fixed points by employing generalized contractions in the context of metric space endued with a weak class of transitive relation. Our results improve, modify, enrich and unify several existing fixed point theorems, The results proved in this study are utilized to find a unique solution of certain fractional boundary value problems.

After the appearance of relation-theoretic contraction principle due to Alam and Imdad, the domain of fixed point theory applied to relational metric spaces has attracted much attention. Existence and uniqueness of fixed/coincidence points satisfying the different types of contractivity conditions in the framework of relational metric space have been studied in recent times. Such results have the great advantage to solve certain types of matrix equations and boundary value problems for ordinary differential equations, integral equations and fractional differential equations. This article is devoted to proving the coincidence and common fixed point theorems for a pair of mappings (T,S) employing relation-theoretic (ϕ,ψ)-contractions in a metric space equipped with a locally finitely T-transitive relation. Our results improve, modify, enrich and unify several existing coincidence points as well as fixed point results. Several examples are provided to substantiate the utility of our results.

This article discusses a variety of important notions, including ideal convergence and ideal Cauchyness of topological sequences produced by fuzzy normed spaces. Furthermore, the connections between the concepts of the ideal limit and ideal cluster points of a sequence in a fuzzy normed linear space are investigated. In a fuzzy normed space, we investigated additional effects, such as describing compactness in terms of ideal cluster points and other relevant but previously unresearched ideal convergence and adjoint ideal convergence aspects of sequences and nets. The countable compactness of a fuzzy normed space and its link to it were also defined. The terms ideal and its adjoint divergent sequences are then introduced, and specific aspects of them are explored in a fuzzy normed space. Our study supports the importance of condition (AP) in examining summability via ideals. It is suggested to use a fuzzy point symmetry-based genetic clustering method to automatically count the number of clusters in a data set and determine how well the data are fuzzy partitioned. As long as the clusters have the attribute of symmetry, they can be any size, form, or convexity. One of the crucial ways that symmetry is used in fuzzy systems is in the solution of the linear Fuzzy Fredholm Integral Equation (FFIE), which has symmetric triangular (Fuzzy Interval) output and any fuzzy function input.

In this article, some results on fixed points under quasicontractions in the framework of metric space endowed with binary relation are proved. Our newly proved results improve and extend several noted fixed point theorems of the existing literature besides their relation-theoretic analogues. We conclude this article by constructing an example to affirm the efficacy of our results.

The primary goal of this paper is to present and study an inertial projection algorithm for solving the split best proximity and mixed equilibrium problems. We find a solution of the best proximity problem in such a way that its image under a bounded linear operator is the solution of the mixed equilibrium problem under the setting of real Hilbert spaces. We construct an iterative algorithm for the proposed problem and prove a weak convergence theorem. Moreover, we deduce some consequences from the main convergence result. Finally, a numerical experiment is presented to demonstrate the convergence analysis of our algorithm. The methodology and results presented in this work improve and unify some previously published findings in this field.