Periodic solutions for scalar Lieńard equations (Q1187239)

The paper deals with periodic solutions of the scalar Liénard equation \(u''+cu'+g(u)=e\), where \(g: \mathbb{R}\to\mathbb{R}\) and \(e: [0,T]\to\mathbb{R}\) arc continuous and \(c\in\mathbb{R}\). The authors prove the existence of solutions provided \(g\) is either decreasing or increasing. The proofs are based on the upper and lower solutions method and an abstract existence theorem for problems at resonance of \textit{L. Cesari} and \textit{R. Kannan} [Proc. Am. Math. Soc. 63, 221-225 (1977; Zbl 0361.47021)]. Further, using results of \textit{J. Bebernes} and \textit{M. Martelli} [Nonlinear Anal., Theory Methods Appl. 4, 821-830 (1980; Zbl 0453.34019)] and \textit{J. J. Nieto} [J. Differ. Equations 60, 90-102 (1985; Zbl 0537.35049)], the authors study the structure of the solution set. They obtain for example the following result: Let \(\omega=(1/T)\int^ T_ 0 e(t)dt\), \(g\) be decreasing on \(\mathbb{R}\). Then the set of solutions is nonempty, compact, connected, and acyclic provided \(\omega\in\text{Int}(\text{Range }g)\).