Existence and multiplicity of periodic solutions of the second order Liénard equation with Lipschtzian condition (Q1612542)

The equation (1) \(x''+ f(x) x'+ g(x)= p(t)\) is considered. In (1), \(f,g,p:\mathbb{R}\to \mathbb{R}\) are continuous functions, \(p(t)\) is \(2\pi\)-periodic. Assuming the conditions \((g_1)\) \(\lim_{|x|\to+\infty}\text{sgn}(x) g(x)= +\infty\), \((g_2)\) \(g(x)\) satisfies the globally Lipschitzian condition and employing also some other conditions (already used in previous investigations), four theorems concerning existence and multiplicity of periodic solutions to equation (1) are proved.