Existence and controllability of fractional evolution inclusions with impulse and sectorial operator

Abstract

Many evolutionary operations fromdiverse fields of engineering and physical sciences go through abrupt modifications of state at specific moments of time among periods of non-stop evolution. These operations are more conveniently modeled via impulsive differential equations and inclusions. In this work, firstly we address the existence of mild solutions for nonlocal fractional impulsive semilinear differential inclusions related to Caputo derivative in Banach spaces when the linear part is sectorial. Secondly, we determine the enough, conditions for the controllability of the studied control problem. We apply effectively fixed point theorems, contraction mapping, multivalued analysis and fractional calculus. Moreover, we enhance our results by introducing an illustrative examples.

Existence of mild solutions of fractional measure evolution inclusions

“…It is clear that g is non-decreasing, but discontinuous. Thus our main results also extend some existence results in [1,2]. Moreover, there is no need to consider whether the number of jumps are finite in this case.…”

“…Our results also extend and unify the differential inclusions with impulses [1] and the difference inclusions [2]. There is an interesting case that the solutions of the impulsive differential inclusion should be discontinuous (see [1]), however we find that any mild solution to system (1.1) is an impulsive mild solution to system (3.2) since we can put F(t, u) = G(t, u) for t = t i and F(t, u) = I i (u(t)) whenever t = t i . F, G are essentially the same in continuity.…”

“…which was studied by Alsarori and Ghadle [1]. When (H1), (H2), and (H6) are satisfied, system (3.2) has mild solutions on [0, 1].…”