## A periodic boundary value problem in Hilbert space.(English)Zbl 0815.34059

A periodic boundary value problem for the differential equation $$x'' - \alpha^ 2 x = f(t,x,x')$$, $$x \in H$$, is considered. Here $$H$$ is real Hilbert space, $$\alpha > 0$$ and $$f$$ satisfies inequalities of the type $$(f(t,x,y),x) \geq - a | x |^ 2 - b | x |\;| y | - c | x |$$. The author proves existence and uniqueness results for this BVP. Some additional properties (e.g. compactness, convexity) of the set of periodic solutions are studied.

### MSC:

 34G20 Nonlinear differential equations in abstract spaces 34B15 Nonlinear boundary value problems for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
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