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

Abstract

• Approximate controllability associated to hybrid fractional differential inclusions with impulses conditions has not yet been addressed by any researcher until now. • The Wright function is used to find the solution of the studied problem and it is defined only when 0 ≤ σ ≤ 1 . • As we cannot apply the Wright function, observing that our problem deals with the special case 0 ≤ σ ≤ 2 . We make use of the general fractional resolvent operator and the Laplace transform to find the solution. • We make use of some facts about fractional calculus and set-valued maps, properties of resolvent operators, and a hybrid fixed point the- orem for three operators of Schaefer type. • First we prove the existence of a mild solution. • Then the approximate controllability of a problem governed by hybrid fractional differential inclusions under Hilfer derivative of order 1 ≤ σ ≤ 2 and type 0 ≤ ζ ≤ 1 , supplemented with non-instantaneous impulses in a weighted space is proved. • An example, illustrating the abstract theory, is given. 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.