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

Zhang, Meirong

Proc. Am. Math. Soc. 127, No. 2, 401-407 (1999).

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

Keywords:

singular force; strong force condition; damped system; coincidence degree