Solutions to fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions

Abstract

We are concerned with fractional Sobolev-type integro-differential equations in Banach spaces with operator pairs and impulsive conditions, where the operator pairs generate propagation families. With the help of the theory of propagation family and Laplace transforms, along with an estimate for a special sequence improved in this article, we introduce a definition of mild solutions to the impulsive problem for these abstract fractional Sobolev-type integro-differential equations and we establish general existence theorems and a continuous dependence theorem, which essentially extend some previous conclusions. In our results, the operator $B$ could be unbounded, and the existence of an operator $B^{-1}$ is not necessarily needed. Moreover, we give some examples to illustrate our main results.