Non-instantaneous Impulsive Riemann-Liouville Fractional Differential Systems: Existence and Controllability Analysis

Abstract

The article is dedicated towards the study of fractional order non-linear differential systems with non-instantaneous impulses involving Riemann-Liouville derivatives with fixed lower limit and appropriate integral type initial conditions in Banach spaces. First, mild solution of the system is constructed and subsequently its existence is proven using Banach's fixed point theorem. Then, results of approximate controllability are established using concept of fractional semigroup and an iterative technique. Suitable examples are given in the end supporting the methodology along with pointing out correction in examples presented in previous articles.

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

“…Riemann-Liouville and Caputo type derivatives have been the focus of attention for many analysts in the field of fractional calculus (see [2,3,5,7,10,12,14,[18][19][20][23][24][25][26][27][28][29][30][31] and references therein). In the sense, that it allows the function to endure discontinuity at origin, the Riemann-Liouville derivative triumphs over Caputo.…”