A multiplicity result for periodic solutions of forced nonlinear second order ordinary differential equations. (English) Zbl 0586.34038

Fabry, C.; Mawhin, J.; Nkashama, M. N.

Bull. Lond. Math. Soc. 18, 173-180 (1986).

The main result of this paper is a multiplicity result concerning \(2\pi\)- periodic solutions of the Liénard differential equation (1) \(x''+f(x)x'+g(t,x)=s\) where s is a real parameter, f and g are continuous functions, and g is \(2\pi\)-periodic in t. Assuming that g satisfies the condition \(\lim_{| x| \to +\infty}g(t,x)=+\infty\) (uniformly in t), it is shown that a number \(s_ 0\) exists with the following property: i) for \(s<s_ 0\), equation (1) has no periodic solution; ii) for \(s=s_ 0\), equation (1) has at least one \(2\pi\)-periodic solution; iii) \(s>s_ 0\), equation (1) has at least two \(2\pi\)-periodic solutions. That result is prepared by partial results along the same lines for more general second order equations. The method of proof is based on the use of upper and lower solutions and on the additivity property of the topological degree.

MSC:

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

Keywords:

upper solutions; Liénard differential equation; second order equations; lower solutions