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Cauchy problem for fractional non-autonomous evolution equations

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Abstract

This paper deals with the Cauchy problem to a class of nonlinear time fractional non-autonomous integro-differential evolution equation of mixed type via measure of noncompactness in infinite-dimensional Banach spaces. Combining the theory of fractional calculus and evolution families, the fixed point theorem with respect to convex-power condensing operator and a new estimation technique of the measure of noncompactness, we obtained the existence of mild solutions under the situation that the nonlinear function satisfy some appropriate local growth condition and a noncompactness measure condition. Our results generalize and improve some previous results on this topic, since the condition of uniformly continuity of the nonlinearity is not required, and also the strong restriction on the constants in the condition of noncompactness measure is completely deleted. As samples of applications, we consider the initial value problem to a class of time fractional non-autonomous partial differential equation with homogeneous Dirichlet boundary condition at the end of this paper.

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Acknowledgements

We are extremely grateful to the critical comments and invaluable suggestions made by anonymous honorable reviewers. This work is supported by National Natural Science Foundations of China (no. 11501455, no. 11661071) and Doctoral Research Fund of Northwest Normal University (no. 6014/0002020209).

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Authors and Affiliations

  1. Department of Mathematics, Northwest Normal University, Lanzhou, 730070, People’s Republic of China

    Pengyu Chen, Xuping Zhang & Yongxiang Li

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  1. Pengyu Chen
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  2. Xuping Zhang
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  3. Yongxiang Li
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Correspondence to Pengyu Chen.

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Communicated by Joseph Ball.

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Chen, P., Zhang, X. & Li, Y. Cauchy problem for fractional non-autonomous evolution equations. Banach J. Math. Anal. 14, 559–584 (2020). https://doi.org/10.1007/s43037-019-00008-2

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