Controllability of nonlocal impulsive functional differential equations with measure of noncompactness in Banach spaces

Abstract

This paper is concerned with the controllability of impulsive differential equations with nonlocal conditions. First, we establish a property of measure of noncompactness in the space of piecewise continuous functions. Then, by using this property and Darbo-Sadovskii's fixed point theorem, we get the controllability of nonlocal impulsive differential equations under compactness conditions, Lipschitz conditions, and mixed-type conditions, respectively.

Analysis on Controllability Results for Wellposedness of Impulsive Functional Abstract Second-Order Differential Equation with State-Dependent Delay

“…and it has a bounded inverse 4) is the abstraction formulation of ( 33)- (37). Furthermore, the bounded linear operators…”

“…By using the previous result and Theorem 1, we get the next result. In this consequence, y ∈ C([−γ, a]; X) is called a mild solution of (33)- (37) defined in [−γ, a] if y(•) is the mild solution of ( 1)-( 4) in the interval [−γ, a]. Thus, the system (33)-( 37) is controllable on [−γ, a].…”

“…Our prior work investigates the existence of mild solution for controllability of nonlocal impulsive differential equations via measure of noncompactness, see [37]. The measure of noncompactness property and Darbo-Sadovskii's fixed point theorem are used to determine the controllability of nonlocal impulsive differential equations under Lipschitz conditions.…”