Non-Local Conformable Differential Inclusions Generated by Semigroups of Linear Bounded Operators or by Sectorial Operators with Impulses in Banach Spaces

This paper aims to explore the sufficient conditions for assuring that, the set of mild solutions to two types of non-local semilinear fractional differential inclusions involving the conformable derivative, in the existence of non-instantaneous impulses, is not empty and compact. We will consider the case when the linear part in the studied problem is the infinitesimal generator of a C₀ - semigroup or a sectorial operator. We give the definition of mild solutions, and then, by using appropriate fixed point theorems for multi-valued functions and the properties of both the conformable derivative, and the measure of noncompactness, we achieve to our findings. Since the most of the known fractional derivatives do not satisfy many basic properties that usual derivatives have, the conformable derivative is introduced in a previous paper, and it is shows that it is the most natural definition. Therefore, many works have been done on differential equation with the conformable. But, works on semilinear differential inclusions are not reported until now. We will do not assume that the semigroup generated by the linear term is not compact,also, we will examine the case when the values of the multi-valued function are convex, also nonconvex. So, our work is novel, and interested. We give examples of the application of our theoretical resultsThis paper aims to explore the sufficient conditions for assuring that, the set of mild solutions to two types of non-local semilinear fractional differential inclusions involving the conformable derivative, in the existence of non-instantaneous impulses, is not empty and compact. We will consider the case when the linear part in the studied problem is the infinitesimal generator of a C₀ - semigroup or a sectorial operator. We give the definition of mild solutions, and then, by using appropriate fixed point theorems for multi-valued functions and the properties of both the conformable derivative, and the measure of noncompactness, we achieve to our findings. Since the most of the known fractional derivatives do not satisfy many basic properties that usual derivatives have, the conformable derivative is introduced in a previous paper, and it is shows that it is the most natural definition. Therefore, many works have been done on differential equation with the conformable. But, works on semilinear differential inclusions are not reported until now. We will do not assume that the semigroup generated by the linear term is not compact,also, we will examine the case when the values of the multi-valued function are convex, also nonconvex. So, our work is novel, and interested. We give examples of the application of our theoretical results.