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Existence of periodic solutions for nonlinear evolution equations in Hilbert spaces. (English) Zbl 0795.34051
Let $$H$$ be a Hilbert space and $$A\subset H\times H$$ be an $$m$$-accretive operator. In the paper there is considered the existence of $$T$$-periodic solutions for equations $du/dt+ Au\ni g(t,u), \qquad t\in \mathbb{R}^ 1, \quad \text{where}\tag{1}$ $$g: \mathbb{R}^ 1\times H\to H$$ is a Carathéodory function. The main result asserts: Let hold (a) For some $$\lambda>0$$, $$(I+\lambda^{-1} A)^{-1}$$ is a compact mapping on $$H$$; (b) $$g$$ satisfies that for some $$M_ 1, M_ 2>0$$ $$\| g(t,v)\|\leq M_ 1\| v\|+ M_ 2$$ for all $$t\in\mathbb{R}^ 1$$, $$v\in H$$. Assume further that $$g$$ is $$T$$-periodic with respect to the first variable and satisfies that there exist positive constants $$a$$ and $$b$$ such that $$\langle z- g(t,v), v\rangle\geq a\| v\|^ 2-b$$ for all $$v\in D(A)$$, $$z\in Av$$. Then (1) has at least one $$T$$-periodic mild solution.

##### MSC:
 34G20 Nonlinear differential equations in abstract spaces 34C25 Periodic solutions to ordinary differential equations 35K55 Nonlinear parabolic equations
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##### References:
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