Existence and multiplicity of solutions with prescribed period for a second order quasilinear O.D.E. (English) Zbl 0761.34032

The authors present some results about the existence and multiplicity of \(T\)-periodic solutions for the nonlinear ordinary differential equation (1) \((\varphi_ p(u'))'+f(t,u)=0\), where \(\varphi_ p: \mathbb R\to \mathbb R\) is given by \(\varphi_ p(s)=| s|^{p-2}s\), \(p>1\) and \(f:\mathbb R\times \mathbb R\to \mathbb R\) is continuous and \(T\)-periodic in \(t\), \(T>0\). The main results of this article are proved in Section 4 and Section 5. In Section 4, Theorem 4.1, the authors prove that equation (1) possesses at least one \(T\)-periodic solution and in Section 5, Theorem 5.1 gives a result concerning the existence of nontrivial \(T\)-periodic solutions of (1). For the proofs they use the degree theory, the Poincaré-Birkhoff theorem and a Sturmian comparison result which is proved in Section 3.


34C25 Periodic solutions to ordinary differential equations
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