Cauchy problems for Hilfer fractional evolution equations on an infinite interval

Abstract

It is well known that the classical Ascoli-Arzelà 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-Arzelà 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-Arzelà 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.

An analysis concerning to the existence of mild solution for Hilfer fractional neutral evolution system on infinite interval

“…The measure of noncompactness, fixed point methods, multivalued functions, impulsive systems, nonlocal conditions, and semigroup theory are applied to verify the existence and approximate controllability of Hilfer fractional differential equations in Gu and Trujillo [23]. Currently, the researchers have developed the existence and approximate controllability results for Hilfer fractional differential systems with or without delay by using a multivalued map, Sobolev type, Clarke subdifferential type, and various fixed point theorems in previous studies [24][25][26][27]. Due to their prevalence in real-world applications of mathematics, neutral differential equations have received a lot of attention lately (see previous studies [8,28,29]).…”

“…Additionally, Zhou and He [24] developed the Cauchy problem for fractional evolution equations with Hilfer fractional derivative on a semi-infinite interval by referring to the probability density function and generalized Ascoli-Arzela theorem. Very recently, in Zhou and He [25], the researchers discussed the Hilfer fractional differential systems on infinite interval by utilizing the almost sectorial operators, Kuratowski's measure of noncompactness, and Schauder's fixed point theorem. However, it should be emphasized that to the best of our knowledge, the existence of mild solutions for Hilfer neutral fractional differential equations on infinite interval in Banach spaces have not been investigated yet, and it is also the motivation of this paper.…”