On periodic solutions of nonlinear evolution equations in Banach spaces (Q1856968)

Here, the existence of \(T\)-periodic solutions to the nonlinear evolution equation \((*)\quad x'(t)+A(t,x(t))=f(t,x(t)), \quad t\in (0,T)=:I\), is proved under the following conditions: Let \(V\hookrightarrow{H}\hookrightarrow{V^*}\) be an evolution triple, where \(H\) is a real separable Hilbert space, \(V\) is a dense subspace of \(H\) and \(V^*\) is the topological dual space of \(V.\) Then \(A:I\times V\to V^*\) is such that for each \(t\in I\) the operator \(A(t,\cdot)\) is uniformly monotone and coercive. The nonlinearity \(f(t,x)\) is a Carathéodory function which is Hölder continuous with respect to \(x\in H\) and with exponent \(\alpha\in(0,1]\) uniformly in \(t.\) After transforming \((*)\) into an operator equation, the authors use the Leray-Schauder fixed-point theorem and prove the existence of periodic solutions. An application is given to a quasi-linear parabolic partial differential equation of even order.