1 results
Periodic Solutions of Second Order Degenerate Differential Equations with Delay in Banach Spaces
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- Journal:
- Canadian Mathematical Bulletin / Volume 61 / Issue 4 / 01 December 2018
- Published online by Cambridge University Press:
- 20 November 2018, pp. 717-737
- Print publication:
- 01 December 2018
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- Article
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- You have access
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We give necessary and sufficient conditions of the
${{L}^{p}}$-well-posedness (resp. 
$B_{p,\,q}^{s}$-wellposedness) for the second order degenerate differential equation with finite delays
$${{\left( Mu \right)}^{\prime \prime }}\left( t \right)+B{u}'\left( t \right)+Au\left( t \right)=G{{{u}'}_{t}}+F{{u}_{t}}+f\left( t \right),\left( t\in \left[ 0,2\pi \right] \right)$$with periodic boundary conditions
$\left( Mu \right)\,\left( 0 \right)\,=\,\left( Mu \right)\,\left( 2\pi \right),\,{{\left( Mu \right)}^{\prime }}\left( 0 \right)\,=\,{{\left( Mu \right)}^{\prime }}\left( 2\pi \right)$, where 
$A,\,B,\,\text{and}\,M$ are closed linear operators on a complex Banach space 
$X$ satisfying 
$D\left( A \right)\,\cap \,D\left( B \right)\,\subset \,D\left( M \right)$, 
$F\,\text{and}\,G$ are bounded linear operators from 
${{L}^{p}}\left( \left[ -2\pi ,\,0 \right];\,X \right)\,\left( \text{resp}\text{.}\,\text{B}_{p,q}^{s}\left( \left[ -2\pi ,\,0 \right];\,X \right) \right)$ into $X$.
