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.


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