New notion of mild solutions for nonlinear differential systems involving Riemann-Liouville derivatives of higher order with non-instantaneous impulses

TL;DR

This paper introduces a new framework for analyzing nonlinear fractional differential systems with Riemann-Liouville derivatives and non-instantaneous impulses, establishing conditions for the existence and uniqueness of mild solutions in Banach spaces.

Contribution

It develops a novel approach to define mild solutions for complex fractional systems with impulses, using fractional resolvent and fixed point methods, expanding the theoretical understanding in this area.

Findings

Established existence and uniqueness of mild solutions under new conditions.

Constructed solutions using fractional resolvent and integral initial conditions.

Validated the methodology with a practical example.

Abstract

The artefact is dedicated towards the inspection of nonlinear fractional differential systems involving Riemann-Liouville derivative with higher order and fixed lower limit, including non-instantaneous impulses for existence and uniqueness results in Banach spaces. The motive of the paper is to set sufficient conditions to guarantee the existence of mild solution in Banach spaces. Firstly, appropriate integral type initial conditions depending on the impulsive functions are chosen at suitable points. A mild solution of the concerned system is constructed using fractional resolvent. Subsequently, existence and uniqueness results are established under sufficient assumptions utilizing fixed point approach. An example is presented at the end to validate the methodology proposed.