×

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
PDF BibTeX XML Cite
Full Text: EuDML