Cauchy problems for Hilfer fractional evolution equations on an infinite interval

It is well known that the classical Ascoli-Arzela theorem is powerful technique to give a necessary and sufficient condition for investigating the relative compactness of a family of abstract continuous functions, while it is limited to finite compact interval. In this paper, we shall generalize the Ascoli-Arzela theorem on an infinite interval. As its application, we investigate an initial value problem for fractional evolution equations on infinite interval in the sense of Hilfer type, which is a generalization of both Riemann-Liuoville and Caputo fractional derivatives. Our methods are based on the Hausdorff theorem, classical/generalized Ascoli-Arzela theorem, Schauder fixed point theorem, Wright function, and Kuratowski measure of noncompactness. We obtain the existence of mild solutions on an infinite interval when the semigroup is compact as well as noncompact.

Automated summary

This paper generalizes the Ascoli-Arzela theorem and applies it to studying the initial value problem of fractional evolution equations on an infinite interval. By using the Hausdorff theorem, classical/generalized Ascoli-Arzela theorem, Schauder fixed point theorem, Wright function, and Kuratowski measure of noncompactness, we prove the existence of mild solutions on an infinite interval when the semigroup is both compact and noncompact.