A general class of noninstantaneous impulsive fractional differential inclusions in Banach spaces

Abstract

In this paper we introduce the concept of a PC-mild solution to a general new class of noninstantaneous impulsive fractional differential inclusions involving the generalized Caputo derivative with the lower bound at zero in infinite dimensional Banach spaces. Using the formula of a PC-mild solution, we give two classes of sufficient conditions to guarantee the existence of PC-mild solutions via fixed point theorems for multivalued functions. Also we characterize the compactness of the solution set. We introduce the concept of generalized Ulam-Hyers stability and present a generalized Ulam-Hyers stability result using multivalued weakly Picard operator theory. Examples are given to illustrate the theoretical results.