Periodic solutions for nonlinear differential inclusions with multivalued perturbations (Q482741)

The authors study the periodic boundary value problem \[ \begin{aligned} &\dot{x}(t) \in -Dx(t) - \partial \varphi(x(t)) + F(t,x(t)), \\ & x(0) = x(\omega), \end{aligned} \] where \(D \in \mathbb{R}^{n\times n}\) is a positive definite matrix, \(\varphi : \mathbb{R}^n \to \mathbb{R}\) is a convex function, \(\partial \varphi\) denotes its subdifferential, and \(F: \mathbb{R} \times \mathbb{R}^n \multimap \mathbb{R}^n\) is a multimap satisfying \(F(0,x) = F(\omega,x)\) for each \(x \in \mathbb{R}^n\). For the case \(F(t,x) \equiv f(t),\) \(f \in L^2((0,\omega);\mathbb{R}^n)\), the existence and uniqueness of a solution to the problem is proved. The existence of a periodic solution is demonstrated for the cases when \(F\) satisfies lower and upper semicontinuity type assumptions. The authors establish also the existence of an extremal solution and prove a relaxation theorem.