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# Periodic solutions of second order degenerate differential equations with delay in Banach spaces

Published:2018-01-17

• Shangquan Bu,
Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China
• Gang Cai,
School of Mathematical Sciences, Chongqing Normal University, Chongqing, 401331, China
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## Abstract

We give necessary and sufficient conditions of the $L^p$-well-posedness (resp. $B_{p,q}^s$-well-posedness) for the second order degenerate differential equation with finite delays: $(Mu)''(t)+Bu'(t)+Au(t)=Gu'_t+Fu_t+f(t),(t\in [0,2\pi])$ with periodic boundary conditions $(Mu)(0)=(Mu)(2\pi)$, $(Mu)'(0)=(Mu)'(2\pi)$, where $A, B, M$ are closed linear operators on a complex Banach space $X$ satisfying $D(A)\cap D(B)\subset D(M)$, $F$ and $G$ are bounded linear operators from $L^p([-2\pi,0];X)$ (resp. $B_{p,q}^s([-2\pi,0];X)$) into $X$.
 Keywords: second order degenerate differential equation, Fourier multiplier theorem, well-posedness, Lebesgue-Bochner space, Besov space
 MSC Classifications: 34G10 - Linear equations [See also 47D06, 47D09] 34K30 - Equations in abstract spaces [See also 34Gxx, 35R09, 35R10, 47Jxx] 43A15 - $L^p$-spaces and other function spaces on groups, semigroups, etc. 47D06 - One-parameter semigroups and linear evolution equations [See also 34G10, 34K30]

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