A Unified Best Proximity Point Theory for Non-Self Mappings: G-C-Proximal Contractions in G-Metric Spaces and Application to Hammerstein Equations

Abstract:

The classical Banach Contraction Principle is inapplicable to non-self mappings where a fixed point does not exist, leading to the development of best proximity point theory. Simultaneously, Mustafa and Sims’ G-metric spaces and Ansari’s C-class functions have independently provided powerful generalizations of metric structures and contractive conditions, respectively. This paper aims to unify these distinct strands of nonlinear analysis by introducing a novel class of non-self mappings and establishing a comprehensive framework for their best proximity point (bpp) results. We define novel G-C-proximal contractive mappings, which synthetically integrate the multi-variable structure of G-metric spaces with the unifying flexibility of C-class functions to govern proximal contractions. We establish new theorems that guarantee both the existence and uniqueness of bpp for such mappings under a set of appropriate conditions. The derived results are shown to generalize and unify a significant body of existing contraction-type principles. We provide concrete, illustrative examples that validate the theoretical findings. Furthermore, we demonstrate the practical utility of our main theorem by applying it to solve a nonlinear Hammerstein-type integral equation, rigorously proving the existence and uniqueness of its optimal approximate solution. The proposed G-C-proximal contractive mappings offer a powerful new analytic tool that substantially extends the scope of bpp theory, opening new avenues for solving nonlinear operator equations in generalized metric frameworks.