On the $\varepsilon$-regular mild solution for fractional abstract integro-differential equations

TL;DR

This paper establishes the existence, regularity, and continuous dependence of epsilon-regular mild solutions for fractional abstract integro-differential equations in Banach spaces, providing foundational estimates and analytical results.

Contribution

It introduces new estimates and analytical techniques to prove the existence and regularity of epsilon-regular mild solutions for fractional integro-differential equations.

Findings

Proved existence of epsilon-regular mild solutions

Established regularity properties of solutions

Demonstrated continuous dependence on initial data

Abstract

In this present paper, we first obtained some estimates involving parts of $\varepsilon$-regular mild solutions of the fractional integro-differential equation. In this sense, through these preliminary results, we investigate the main results of this paper, i.e., the existence, regularity and continuous dependence of $\varepsilon$-regular mild solutions for fractional abstract integro-differential equations in Banach space.