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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.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 34C25 Periodic solutions to ordinary differential equations 47N20 Applications of operator theory to differential and integral equations
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##### References:
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