Invariant sets for nonlinear evolution equations, Cauchy problems and periodic problems

Abstract

In the case of $K\ne \overline{D\left(A\right)}$, we study Cauchy problems and periodic problems for nonlinear evolution equation $u\left(t\right)\in K$, $u^{\prime }\left(t\right)+Au\left(t\right)\ni f\left(t,u\left(t\right)\right)$, $0\le t\le T$, where $A$ is a maximal monotone operator on a Hilbert space $H$, $K$ is a closed, convex subset of $H$, $V$ is a subspace of $H$, and $f:\left[0,T\right]\times \left(K\cap V\right)\to H$ is of Carathéodory type.