Duality for nonlinear abstract evolution differential equations (Q1190760)

The authors consider the following \(T\)-periodic abstract problem in a real separable Hilbert space \(X\): \((d/dt)\partial\psi(t,x'(t))+\partial\varphi(t,x(t))\ni 0\), \(x(t+T)=x(t)\) for almost all \(t\in\mathbb{R}\), where \(T\) is a given positive number, \(\partial\psi\) and \(\partial\phi\) are subdifferentials of convex lower semicontinuous functions \(\psi(t,.)\), \(\varphi(t,.)\): \(X\to\mathbb{R}\). As a function of \(t\) the mappings \(\psi\) and \(\varphi\) are \(T\)-periodic. Finally, they are assumed to be measurable with respect to the \(\sigma\)- algebra generated in \(\mathbb{R}\times X\) by products of Lebesgue sets in \(\mathbb{R}\) and Borel sets in \(X\). The main results of the paper are an existence statement of periodic solutions and duality results under growth and representation conditions on \(\psi\) and \(\varphi\).