Proximal Contractions for Multivalued Mappings with an Application to 2D Volterra Integral Equations

In this paper, we delve into the ideas of Geraghty-type proximal contractions and their relation to multivalued, single-valued, and self mappings. We begin by introducing the notions of (ψω)MCP-proximal Geraghty contraction and rational (ψω)RMCP-proximal Geraghty contraction for multivalued mappings, aimed at establishing coincidence point results. To enhance our understanding and illustrate the concepts, practical examples are provided with each definition. This study extends these contractions to single-valued mappings with the introduction of (ψω)SCP-proximal Geraghty contraction and rational (ψω)RSCP-proximal Geraghty contraction, supported by relevant examples to reinforce the main results. Then, we explore (ψω)SFP Geraghty contraction and rational (ψω)RSFP contraction for self-mappings, obtaining fixed point theorems and clearly illustrating them through examples. Finally, we apply the theoretical framework developed to investigate the existence and uniqueness of solutions to certain two-dimensional Volterra integral equations. Specifically, we consider the transformation of first-kind Volterra integral equations, which play crucial roles in modeling memory in diverse scientific fields like biology, physics, and engineering. This approach provides a powerful tool for solving difficult integral equations and furthering applied mathematics research.