By using the topological degree theory and some analytic methods, we consider the periodic boundary value problem for the singular dissipative dynamical systems with p-Laplacian: ${\left({\varphi }_p\left(x^{\text{'}}\right)\right)}^{\text{'}}+d/dtgradF\left(x\right)+gradG\left(x\right)=e\left(t\right)$, x(0) = x(T), x’(0) = x’(T). Sufficient conditions to guarantee the existence of solutions are obtained under no restriction on the damping forces d/dt gradF(x).

A translation along trajectories approach together with averaging procedure and topological degree are used to derive effective criteria for existence of periodic solutions for nonautonomous evolution equations with periodic perturbations. It is shown that a topologically nontrivial zero of the averaged right hand side is a source of periodic solutions for the equations with increased frequencies. Our setting involves equations on closed convex cones, therefore it enables us to study positive solutions...