On the Existence and Computation of Best Proximity Points for Proximal Enriched Contractions

This article introduces the concept of proximal enriched contractions for nonself mappings by integrating the enrichment technique of Berinde and Păcurar with the relation-theoretic approach of Alam and Imdad. The existence of best proximity points for this notion is ensured by two distinct results. The proofs are aligned with the proximal enriched contraction condition and are based on a constructive iteration scheme that generates a sequence converging to the best proximity point of T. Furthermore, the sequence generated by the constructive iteration satisfies the Cauchy property via a nonstandard inequality between consecutive elements. These theoretical results are utilized to derive an algorithm for computing an approximate best proximity point of T. Numerical examples in R3 are presented to demonstrate the effectiveness of both the main results and the proposed algorithm. Additionally, an application section is included wherein the fixed point result derived from our best proximity point theorems is employed to establish the existence of solutions for a nonlinear fractional integral equation.