New Notion of Mild Solutions for Higher-order Riemann-Liouville Fractional Systems Involving Non-instantaneous Impulses

This paper is dedicated towards the inspection of impulsive nonlinear fractional evolution equations (NFEEs) for existence and uniqueness results in Banach spaces. The concerned NFEEs involves Riemann–Liouville higher-order time derivative operators with fixed lower limits along with non-instantaneous impulses. 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 Riemann–Liouville fractional resolvent operator Rγ(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathcal {R}_\gamma (t)$$\end{document}. Subsequently, existence and uniqueness results are established under sufficient assumptions utilising fixed point approach. An example integrating specific input parameters is presented at the end to demonstrate and validate the methodology proposed.