Abstract Cauchy problems for the generalized fractional calculus

TL;DR

This paper investigates abstract Cauchy problems involving generalized fractional derivatives, establishing local existence, uniqueness, and a generalized Grönwall inequality, along with spectral properties of these derivatives.

Contribution

It introduces a generalized Grönwall inequality and analyzes eigenvalues and eigenfunctions of generalized fractional derivatives, advancing the theoretical framework for such problems.

Findings

Proved local existence and uniqueness for the problems

Established a generalized Grönwall inequality

Analyzed spectral properties of generalized fractional derivatives

Abstract

We focus on eventually non-linear abstract Cauchy problems with a generalized fractional derivative in time. First we prove a local existence and uniqueness result, then we focus on a generalized Gr\"onwall inequality. Before addressing the inequality, we study some properties of eigenvalues and eigenfunctions of the generalized fractional derivatives. Finally, we prove some consequences of the generalized Gr\"onwall inequality.