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Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces

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Abstract

We study the well-posedness of the fractional differential equations with infinite delay (P 2): \({D^\alpha u(t)=Au(t)+\int^{t}_{-\infty}a(t-s)Au(s)ds + f(t), (0\leq t \leq2\pi)}\), where A is a closed operator in a Banach space \({X, \alpha > 0, a\in {L}^1(\mathbb{R}_+)}\) and f is an X-valued function. Under suitable assumptions on the parameter α and the Laplace transform of a, we completely characterize the well-posedness of (P 2) on Lebesgue-Bochner spaces \({L^p(\mathbb{T}, X)}\) and periodic Besov spaces \({{B} _{p,q}^s(\mathbb{T}, X)}\) .

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Authors and Affiliations

  1. Department of Mathematical Science, University of Tsinghua, Beijing, 100084, China

    Shangquan Bu

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  1. Shangquan Bu
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Correspondence to Shangquan Bu.

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This work was supported by the NSF of China and the Specialized Research Fund for the Doctoral Program of Higher Education.

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Bu, S. Well-Posedness of Fractional Differential Equations on Vector-Valued Function Spaces. Integr. Equ. Oper. Theory 71, 259–274 (2011). https://doi.org/10.1007/s00020-011-1895-y

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  • DOI: https://doi.org/10.1007/s00020-011-1895-y

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