Approximate controllability of hybrid Hilfer fractional differential inclusions with non-instantaneous impulses

Abstract

This paper deals with the approximate controllability of a class of non-instantaneous impulsive hybrid systems for fractional differential inclusions under Hilfer derivative of order 1<σ<2 and type 0≤ζ≤1, on weighted spaces. As an alternative to the Wright function which is defined only when 0<σ<1, we make use of a family of general fractional resolvent operators to give a proper form of the mild solution. This latter is consequently formulated by Laplace transform, improving and extending important results on this topic. Based on known facts about fractional calculus and set-valued maps, properties of the resolvent operator, and a hybrid fixed point theorem for three operators of Schaefer type, the existence result and the approximate controllability of our system is achieved. An example is given to demonstrate the effectiveness of our result