## 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.

### MSC:

 34C25 Periodic solutions to ordinary differential equations
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### References:

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