Periodic solutions of damped differential systems with repulsive singular forces

We consider the periodic boundary value problem for the singular differential system: $u''+(\nabla F(u))'+\nabla G(u) = h(t),$ where $F\in C^{2}(\mathbb R ^{N}, \mathbb R )$, $G\in C^{1}(\mathbb R ^{N} \backslash \{0\}, \mathbb R )$, and $h\in L^{1}([0,T], \mathbb R ^{N})$. The singular potential $G(u)$ is of repulsive type in the sense that $G(u) \to +\infty$ as $u\to 0$. Under Habets-Sanchez's strong force condition on $G(u)$ at the origin, the existence results, obtained by coincidence degree in this paper, have no restriction on the damping forces $(\nabla F(u))'$. Meanwhile, some quadratic growth of the restoring potentials $G(u)$ at infinity is allowed.