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.