On periodic solutions of multivalued nonlinear evolution equations (Q1917982)

Two existence theorems for periodic solutions of finite dimensional differential inclusions are given. It is proved that the differential inclusion \(x'(t) \in F(x(t))\), \(x(0) = x(r)\), defined on a closed convex bounded subset \(K\) of a finite dimensional Banach space admits a periodic solution. It is assumed that \(F\) is an upper semicontinuous set-valued map with compact convex values and \(F(x) \cap T_K (x) \neq \emptyset\) for all \(x \in K\). Using Galerkin approximations, this result is generalized to the evolution inclusion \(x'(t) + A(x(t)) \in F(x(t))\), \(x(0) = x(r)\), where \(A\) is a monotone operator.