Some convergence results on proximal contractions with application to nonlinear fractional differential equation

In this manuscript, we investigated the coincidence points, best proximity points, and fixed-points results endowed with $F$-contraction within the realm of suprametric spaces. The proximal point results obtained in this work show that our investigation is not purely theoretical; fundamental findings were supplemented with concrete examples that demonstrate their practical ramifications. Furthermore, this paper focuses on boundary value problems (BVPs) related to nonlinear fractional differential equations of order $2<\varpi \le 3$. By cleverly translating the BVP into an integral equation, we obtained conditions that confirm the existence and uniqueness of fixed points under $({\mathcal{F}}_{\tau }{)}_{F_{\mathcal{P}}}$-contraction. A relevant part of this work is the approximation of the Green's function, which is critical in proving the existence and uniqueness of solutions. Our work not only adds to the current body of knowledge but also provides strong approaches for dealing with hard mathematical problems in the field of fractional differential equations.