Solving a system of nonlinear Fractional Differential Equations via novel best proximity pair results in regular semimetric space

Abstract

This paper is devoted to examining the existence of optimal solutions for a coupled system of differential equations characterized by right sided-Hilfer fractional derivatives under initial conditions as form: \begin{align*} \begin{cases} \prescript{\mathsf{H}\!}{}{\mathbb{D}}_{b^+}^{p,q; \psi} \mu_1(\upkappa)= \lambda_1(\upkappa, \mu_2(\upkappa)),\\ \prescript{ \mathsf{H}\!}{}{\mathbb{D}}_{b^+}^{p,q; \psi} \mu_2( \upkappa)= \lambda_2( \upkappa, \mu_1 ( \upkappa)), \end{cases} \end{align*} for $b< \upkappa \leq v$. To this end, we develop a series of best proximity pair theorems for a new category of proximal contractions, referred to as the $\alpha$-generalized Geraghty proximal interpolative contraction pair, formulated within the framework of a regular semimetric space $(\mathfrak{Q},\rho,\Phi)$.