Periodic solutions of damped differential systems with repulsive singular forces.(English)Zbl 0908.34024

Summary: The author considers the periodic boundary value problem for the singular differential system $$u''+(\nabla F(u))'+\nabla G(u) = h(t)$$, with $$F\in C^{2}(\mathbb{R} ^{N}, \mathbb{R})$$, $$G\in C^{1}(\mathbb{R} ^{N} \backslash \{0\}, \mathbb{R})$$, 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.

MSC:

 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations
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