On the existence of periodic solutions for nonlinear evolutions in Hilbert spaces (Q5940174)

The authors study the existence of periodic solutions to nonlinear evolution inclusions of the form NEWLINE\[NEWLINE{du\over dt}\in -Au(t)+F \bigl(t, u(t)\bigr),\;t\in\mathbb{R},NEWLINE\]NEWLINE in a real, separable Hilbert space \(H\). The man result is theorem 1.1, which proves the existence of \(T\)-periodic solutions under the conditions \(F:\mathbb{R}\times H\to 2^H\) is measurable and periodic with period \(T\) in \(t\), upper semicontinuous in \(x\), nonempty compact convex-valued and satisfies a growth condition, \(A:H\to H^2\) is a maximal monotone operator, plus a couple of conditions on the resolvent of \(A\). The proof is an application of the topological degree theory for set-valued \((S_+)\) type operators introduced by Chang and Chen (1990). (The single-valued \((S_+)\) operator was introduced by \textit{W. V. Petryshyn} and \textit{P. M. Fitzpatrick} [Trans. Am. Math. Soc. 160, 39-63 (1971; Zbl 0236.47057)]. An example is given in which the conditions of theorem 1.1 are satisfied even though the coercive condition is not satisfied.