Recent Advances in Proximity Point Theory Applied to Fractional Differential Equations

Abstract

This article introduces the concept of generalized (ffF,b,ϕ˘) contraction in the context of b-metric spaces by utilizing the idea of F contraction introduced by Dariusz Wardowski. The main findings of the research focus on the existence of best proximity points for multi-valued (ffF,b,ϕ˘) contractions in partially ordered b-metric spaces. The article provides examples to illustrate the main results and demonstrates the existence of solutions to a second-order differential equation and a fractional differential equation using the established theorems. Additionally, several corollaries are presented to show that the results generalize many existing fixed-point and best proximity point theorems.

Keywords

• best proximity point (<tt>BPP</tt>)
• fixed point (<tt>FP</tt>)
• banach contraction principle (BCP)
• partially ordered complete metric space (<tt>POCMS</tt>)
• partially ordered complete b-metric space (<tt>POCbMS</tt>)