Journal of Integral Equations and Applications

Well-posedness of fractional degenerate differential equations with infinite delay in vector-valued functional spaces

Shangquan Bu and Gang Cai

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We study the well-posedness of degenerate fractional differential equations with infinite delay $(P_\alpha ): D^\alpha (Mu)(t) =Au(t)+\int _{-\infty }^t a(t-s)Au(s)\,ds+f(t)$, $0\leq t\leq 2\pi $, in Lebesgue-Bochner spaces $L^p(\mathbb {T}; X)$ and Besov spaces $B_{p,q}^s(\mathbb {T}; X)$, where $A$ and $M$ are closed linear operators on a Banach space~$X$ satisfying $D(A)\subset D(M)$, $\alpha >0$ and $a\in L^1(\mathbb {R}_+)$ are fixed. Using well known operator-valued Fourier multiplier theorems, we completely characterize the well-posedness of $(P_\alpha )$ in the above vector-valued function spaces on $\mathbb {T}$.

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J. Integral Equations Applications, Volume 29, Number 2 (2017), 297-323.

First available in Project Euclid: 17 June 2017

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Primary: 26A33: Fractional derivatives and integrals 34C25: Periodic solutions 34K37: Functional-differential equations with fractional derivatives 43A15: $L^p$-spaces and other function spaces on groups, semigroups, etc. 45N05: Abstract integral equations, integral equations in abstract spaces

Well-posedness Fourier multiplier degenerate fractional differential equation vector-valued function spaces


Bu, Shangquan; Cai, Gang. Well-posedness of fractional degenerate differential equations with infinite delay in vector-valued functional spaces. J. Integral Equations Applications 29 (2017), no. 2, 297--323. doi:10.1216/JIE-2017-29-2-297.

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