Fractional differential equations of Sobolev type with sectorial operators

Abstract

This paper treats the asymptotic behavior of resolvent operators of Sobolev type and its applications to the existence and uniqueness of mild solutions to fractional functional evolution equations of Sobolev type in Banach spaces. We first study the asymptotic decay of some resolvent operators (also called solution operators) and next, by using fixed point results, we obtain the existence and uniqueness of solutions to a class of Sobolev type fractional differential equation. We notice that, the existence or compactness of an operator $$E^{-1}$$ is not necessarily needed in our results.