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Well-Posedness of Second-Order Degenerate Differential Equations with Finite Delay

Part of: Differential equations in abstract spaces Functional-differential and differential-difference equations Groups and semigroups of linear operators, their generalizations and applications

Published online by Cambridge University Press:  19 September 2016

Shangquan Bu  and
Gang Cai
Shangquan Bu
Affiliation:
Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, People's Republic of China (sbu@math.tsinghua.edu.cn)
Gang Cai*
Affiliation:
School of Mathematical Sciences, Chongqing Normal University, Chongqing 401331, People's Republic of China (caigang-aaaa@163.com)
*
*Corresponding author.
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Abstract

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We give necessary and sufficient conditions for the Lp-well-posedness of the second-order degenerate differential equations with finite delay

with periodic boundary conditions (Mu)(0) = (Mu)(2π), (Mu)′(0) = (Mu)′(2π). Here A and M are closed operators on a complex Banach space X satisfying D(A) ⊂ D(M), α ∈ ℂ is fixed, F is a bounded linear operator from Lp([−2π, 0],X) into X, and ut is given by ut(s) = u(t + s) when s ∈ [−2π, 0].

Keywords

degenerate differential equation with delayLp-well-posednessLebesgue–Bochner spacesFourier multipliers

MSC classification

Secondary: 34G10: Linear equations 47D06: One-parameter semigroups and linear evolution equations 34K13: Periodic solutions
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 60 , Issue 2 , May 2017 , pp. 349 - 360
Copyright
Copyright © Edinburgh Mathematical Society 2017