Cauchy problems for Hilfer fractional evolution equations on an infinite interval |
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Authors: | Yong Zhou Jia Wei He |
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Affiliation: | 1. Faculty of Mathematics and Computational Science, Xiangtan University, Hunan, 411105 China;2. College of Mathematics and Information Science, Guangxi University, Nanning, 530004 China |
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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. |
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Keywords: | Ascoli-Arzelà theorem existence fractional evolution equations Hilfer derivative infinite interval |
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