Existence results for a generalized fractional boundary value problem in $b$-metric space

TL;DR

This paper investigates the existence and uniqueness of solutions for a nonlinear fractional boundary value problem in a b-metric space, using fixed point theorems and properties of Green's function.

Contribution

It introduces new existence and uniqueness results for a fractional boundary value problem within a b-metric space framework, employing advanced contraction techniques.

Findings

Existence and uniqueness of solutions established.

Green's function properties analyzed.

Theoretical results supported by examples.

Abstract

This paper is concerned with a class of nonlinear boundary value problem involving fractional derivative in the $\varphi$-Riemann-Liouville sense. Some Properties of the Green's function for this problem are mentioned. By means of the Banach contraction principle in $b$-metric space and the technique of the $\gamma$-$\psi$- Geraghty contractive maps, existence and uniqueness results are obtained. Two examples are given to support the theoretical results.