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Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces

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Abstract

In this paper, we study a class of backward problems for nonlinear fractional super-diffusion equations in Banach spaces. We consider the time fractional derivative in the sense of Caputo type. First, we establish some results for the existence of the mild solutions. Moreover, we obtain regularity results of the first order and fractional derivatives of mild solutions. These conclusions are mainly based on fixed point theorems and properties of \(\alpha \)-resolvent family as well as Mittag-Leffler functions. Finally, two applications are provided to illustrate the efficiency of our results.

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Acknowledgements

The work was supported by the Fundo para o Desenvolvimento das Ciências e da Tecnologia of Macau (No. 0092/2022/A) and Natural Science Foundation of Hunan Province (2023JJ40616).

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Authors and Affiliations

  1. Faculty of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, China

    Xuan X. Xi

  2. School of Computer Science and Engineering, Macau University of Science and Technology, Macau, 999078, China

    Yong Zhou

  3. School of Mathematical Sciences, Huaibei Normal University, Anhui, 235000, China

    Mimi Hou

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  1. Xuan X. Xi
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  2. Yong Zhou
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  3. Mimi Hou
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Correspondence to Yong Zhou.

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Xi, X.X., Zhou, Y. & Hou, M. Existence and regularity of mild solutions to backward problem for nonlinear fractional super-diffusion equations in Banach spaces. Fract Calc Appl Anal 27, 1073–1111 (2024). https://doi.org/10.1007/s13540-024-00286-0

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  • DOI: https://doi.org/10.1007/s13540-024-00286-0

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