Theory and Computation of Fixed Points for Generalized Enriched Multi-Valued Contractions

This paper introduces the concepts of inward-β-enriched Reich multi-valued contractions and outward-β-enriched Reich multi-valued contractions in the context of normed linear spaces. We establish fixed point existence theorems for these generalized enriched contraction mappings under appropriate conditions and develop algorithms to computationally approximate fixed points of these mappings. The presented theorems guarantee the convergence of these algorithms. Another important aspect of the article is that our proofs employ a technique that handles the non-standard nature of enriched contractions without transforming them into standard contractions via corresponding averaging mappings. The results extend and unify several existing fixed point theorems in the literature, including those for enriched contraction mappings, enriched Kannan mappings, and enriched multi-valued contractions. We provide illustrative examples to demonstrate the applicability of our main results and highlight their significance through comparative remarks. To emphasize the importance of the algorithms, several simulations are incorporated that provide the approximate fixed points of non-trivial examples. Moreover, we also approximate the solution of Erdélyi–Kober-type fractional integral inclusion.