The paper deals with the existence of p-periodic solutions \((p>0)\) for the Liénard differential system in \({\mathbb{R}}^ m\) (1) \(x''(t)+(d/dt)\phi (x(t))+Ag(x(t))=h(t)\) (t\(\in {\mathbb{R}})\), where \(\phi\) : \({\mathbb{R}}^ m\to {\mathbb{R}}^ m\) is a \(C^ 1\)-map, \(g: {\mathbb{R}}^ m\to {\mathbb{R}}^ m\) is continuous, A is an \(m\times m\) constant matrix (possibly singular), \(h: {\mathbb{R}}\to {\mathbb{R}}^ m\) is continuous and p- periodic. Assuming conditions on the growth of the damping term \(\phi\) (x), for \(| x|\) large, the classical theorem of Mizohata and Yamaguti, for the scalar equation, is extended, in various directions, to the vector equation (1). Even a functional counterpart of equation (1), \[ (2)\quad x''(t)+(d/dt)\phi (x(t))+\int^{s}_{-r}d\eta (\vartheta)g(x(t+\vartheta))=h(t), \] with \(\eta\) (\(\vartheta)\), -r\(\leq \vartheta \leq s\), an \(m\times m\) function matrix whose entries have bounded variation, is considered. Topological degree techniques in function spaces are exploited for deriving the results.