On the existence of solutions of fractional differential equations in Banach spaces

TL;DR

This paper investigates the conditions under which fractional differential equations in Banach spaces have solutions, using measures of non-compactness and Kamke functions, with applications to fractional PDE discretizations.

Contribution

It introduces new sufficient conditions for local solutions of fractional differential equations in Banach spaces, extending to systems from fractional PDE semi-discretization.

Findings

Established existence criteria for fractional differential equations in Banach spaces.

Applied the main theorem to systems from fractional PDE semi-discretization.

Provided a framework for analyzing solvability using measures of non-compactness.

Abstract

Utilising the notion of measures of non-compactness and Kamke function of order $\alpha$, we address the question of solvability of fractional differential equations in Banach spaces. In particular, we provide sufficient conditions ensuring the existence of a local solution. Our main existence theorem is then applied on countable systems of fractional differential equations arising from semi-discretisation of fractional PDEs with $p$-Laplacian.