Existence and multiplicity of periodic solutions for a Liénard differential system (Q1355051)

The author proves existence and multiplicity results for the periodic boundary value problem (Liénard type system) \[ x''(t) +{{d}\over{dt}}\text{ grad} F(x(t)) + g(t,x(t)) = s,\;x(0)= x(2 \pi).\tag{*} \] Here \(s\) is a real bifurcation parameter, \(F:\mathbb{R}^n \to \mathbb{R}\) is of \(C^2\) and \(g: \mathbb{R} \times \mathbb{R}^n \to \mathbb{R}^n\) is continuous and \(2\pi\)-periodic in \(t\). Using the techniques of a priori estimates and the coincidence degree theory, the author establishes that, under some additional conditions imposed on \(F, g\), there exist \(s_0 \leq s_1 \in \mathbb{R}^n\) such that BVP \((*)\) 1) has no solution for \(s < s_0\); 2) has at least one solution for \(s = s_1\); 3) has at least two solutions for \(s > s_1\).