Relation-Theoretic Fixed Point Theorems Involving Certain Auxiliary Functions with Applications

Abstract

This article includes some fixed point results for (φ,ψ,θ)-contractions in the context of metric space endowed with a locally H-transitive relation. We constructed an example for attesting to the credibility of our results. We also discussed the existence and uniqueness of the solution of a Fredholm integral equation using our results.

Relation-Preserving Functional Contractions Involving a Triplet of Auxiliary Functions with an Application to Integral Equations

“…To demonstrate our findings, we deliver an example and a possible application to Fredholm integral equations. Our findings also generalize the corresponding results of Altaweel and Khan [18], Sk et al [19], and Jleli et al [21].…”

“…Through the implementation of our results, we discussed an existence and uniqueness theorem for certain FIEs prescribed with some additional conditions. The contractivity condition utilized in our results and the class of BR (i.e., locally finitely S-transitive BR) remain more general than those used in Altaweel and Khan [18]. If we take θ = 0, then our results achieve the corresponding results of Sk et al [19].…”

“…In 2015, Alam and Imdad [10] initiated a relation-theoretic formulation of the BCP which has attracted the attention of numerous researchers, e.g., Refs. [11][12][13][14][15][16][17][18][19]. In order to ascertain the existence of fixed points for a nonlinear contraction, the transitivity of the underlying relation is also required.…”

Geraghty Type Contractions in Relational Metric Space with Applications to Fractional Differential Equations

“…Following Geraghty [29], a self-map S on a complete metric space (M, σ) verifying, for some β ∈ B and for all r, s ∈ M, In 2015, Alam and Imdad [30] obtained an intense and flexible version of the BCP in a metric space endued with a binary relation. Following this novel work, various researchers have established a multiple of fixed point results in the framework of relational metric space under different types of contractivity conditions, viz., Boyd-Wong type contractions [31,32], Matkowski contractions [33,34], Meir-Keeler contractions [35], weak contractions [36,37], F-contractions [38], θ-contractions [39], almost contractions [40], rational type contractions [41], (ψ, φ)-contractions [42], (ψ, φ, θ)-contractions [43], etc. Mapping which involves such results verifies a weaker contraction condition relative to the usual contraction, as it must be held for comparative elements only.…”