Periodic solutions for evolution equations (Q2721638)

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 \({1\over T}\int^T_0 f\,dt\in \text{im}(A)\).NEWLINENEWLINETheorem 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.NEWLINENEWLINEMoreover, the author adapts the results to boundary problems.