Research Article

Well-posedness of fractional differential equations with finite delay in complex Banach spaces

Published in: Quaestiones Mathematicae
Volume 47 , issue 2 , 2024 , pages: 413–435

DOI: 10.2989/16073606.2023.2229553

Author(s): Shangquan Bu Tsinghua University, China , Gang Cai Chongqing Normal University, China

Keywords: 34K13 , 34K30 , 47D06 , 47A10 , Fractional differential equation , Fourier multiplier , well-posedness , Lebesgue-Bochner spaces , Besov spaces

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Abstract

We study the well-posedness of the fractional diﬀerential equations with finite delay: D^αu(t) + BD^βu(t) = Au(t) + Fu_t + f(t), (0 ≤ t ≤ 2π) on Lebesgue-Bochner spaces and periodic Besov spaces , where A and B are closed linear operators in a complex Banach space X satisfying D(A) ⊂ D(B), 0 ≤ β < α, α ≥ 1 and F is a bounded linear operator from L^p ([−2π, 0]; X) (resp. into X. We give necessary and sufficient conditions for the L^p -well-posedness and the -well-posedness of above equations by using known operator-valued Fourier multiplier theorems.

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