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Periodic solutions for evolution equations. (Solutions périodiques des équations d’évolution.) (French) Zbl 1047.34065
Theorem 1: Let $$H$$ be a Hilbert space, $$A$$ a linear unbounded maximal monotone symmetric operator and $$f\in C^1(\mathbb{R},H)$$ a $$T$$-periodic function, then there is a periodic function $$x$$ such that $$x'+ Ax=f$$ if and only if $${1\over T}\int^T_0 f\,dt\in \text{im}(A)$$.
Theorem 2: If $$g: \mathbb{R}\to\mathbb{R}$$ is a Lipschitz increasing function and $$f:\mathbb{R}\to\mathbb{R}$$ is a $$T$$-periodic continuous function, then there is a periodic function $$x$$ such that $$x'+ g\circ x= f$$ if and only if $${1\over T} \int^T_0 fdt\in g(\mathbb{R})$$; if $$g$$ is strictly increasing, then the solution is unique.
Moreover, the author adapts the results to boundary problems.

##### MSC:
 34G10 Linear differential equations in abstract spaces 34C25 Periodic solutions to ordinary differential equations
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