Periodic Solutions of Second Order Degenerate Differential Equations with Delay in Banach Spaces

Bu, Shangquan ; Cai, Gang

Canadian mathematical bulletin, Tome 61 (2018) no. 4, pp. 717-737

Résumé

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 $X$ .

Détail

DOI : 10.4153/CMB-2017-057-6

Mots-clés : 34G10, 34K30, 43A15, 47D06, second order degenerate differential equation, Fourier multiplier theorem, wellposedness, Lebesgue-Bochner space, Besov space