New Relation-Theoretic Fixed Point Theorems in Fuzzy Metric Spaces with an Application to Fractional Differential Equations

Abstract

In this paper, we introduce the notion of fuzzy R−ψ−contractive mappings and prove some relevant results on the existence and uniqueness of fixed points for this type of mappings in the setting of non-Archimedean fuzzy metric spaces. Several illustrative examples are also given to support our newly proven results. Furthermore, we apply our main results to prove the existence and uniqueness of a solution for Caputo fractional differential equations.

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

“…Mapping which involves such results verifies a weaker contraction condition relative to the usual contraction, as it must be held for comparative elements only. This restrictive nature enables such types of results to solve many complicated real world problems occurring in fractal spaces and fractional differential equations which employ specific auxiliary conditions, e.g., [44,45]. Very recently, Almarri et al [46] established the relation-theoretic analogue of Geraghty's fixed point theorem [29], which also remains an improvement of the results of Harandi and Emami [47].…”

Matkowski-Type Functional Contractions under Locally Transitive Binary Relations and Applications to Singular Fractional Differential Equations

“…In the recent past, fractional differential equations (FDEs for short) were discussed with regard to the impassable development and applicability of the area of fractional calculus. In recent years, some typical boundary value problems (BVPs for short) of FDEs have been solved using the fixed-point theorems proven in ordered metric space by Liang and Zhang [24] and Cabrera et al [25], relational metric space by Saleh et al [26] and Alamer et al [27], and orthogonal metric space by Abdou [28].…”

New Relation-Theoretic Fixed Point Theorems in Revised Fuzzy Metric Spaces With an Application to Fractional Differential Equations

“…The authors gave various interesting applications to boundary value problems and matrix equations in [10], which strongly supported their fixed point conclusions. Following that, a slew of fixed point theorems were developed, each with its own set of binary relation definitions (e.g., [2,8,16,19] and a slew of others).…”