Quasi-periodicity of bounded solutions to some periodic evolution equations (Q2638498)

Let H be a real Hilbert space and for \(t\in {\mathbb{R}}\) let A(t) be a maximal monotone operator, and let \(C_ t\) denote the closure of its domain. The evolution equation \(0\in u'(t)+A(t)u(t)\) is investigated when A(t) is \(\tau\)-periodic, that is, \(A(t+\tau)=A(t)\) for almost all \(t\in {\mathbb{R}}.\) In the special case where \(C_ t=C_ 0\) and u(\({\mathbb{R}})\) is precompact in H, it has been proved in an earlier paper that u: \({\mathbb{R}}\to H\) is almost periodic. Later is was shown how to treat the general linear (non- monotone) case when \(H={\mathbb{R}}^ N\). The present paper establishes a similar result for \(H={\mathbb{R}}^ N\) when A(t) is a general, \(\tau\)- periodic, maximal monotone operator, and when A(t) is the subdifferential of a convex function. The proofs are made via solutions of some difference equations. A counterexample shows that the results do not extend to A(t) only almost periodic even for linear operators and with \(H={\mathbb{R}}^ 2\).