Periodic solutions of second order differential equations in Banach spaces
The authors consider the maximal regularity for the second order periodic boundary value problem \[ \begin{gathered} u''(t)-aAu(t)-\alpha Au'(t)=f(t), \;0\leq t\leq 2\pi,\\ u(0)=u(2\pi),\;u'(0)=u'(2\pi),\end{gathered} \] on a Banach space \(X\). Necessary and sufficient conditions for the existence and uniqueness of periodic solutions in the spaces \(L_{2\pi}^{p}({\mathbb R},X)\) (\(1<p<\infty\)) and \(C_{2\pi}^{s}({\mathbb R},X)\) \((0<s<1)\) are given. Moreover, results on mild solutions are also presented. Two types of mild periodic solutions are considered. When the operator \(A\) is the generator of a strongly continuous cosine function, characterizations are given in terms of Fourier multipliers and spectral properties of the cosine function. Throughout the paper, examples involving elliptic partial differential operators with Dirichlet boundary conditions are discussed.
scientific article; zbMATH DE number 5071441
scientific article