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On periodic solutions of nonlinear evolution equations in Banach spaces. (English) Zbl 1029.34045
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.

34G20 Nonlinear differential equations in abstract spaces
34C25 Periodic solutions to ordinary differential equations
47N20 Applications of operator theory to differential and integral equations
Full Text: DOI
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