×

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
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Boccardo, L.; Drabek, P.; Giachetti, D.; Kucera, M., Generalization of Fredholm alternative for nonlinear differential operators, Nonlinear analysis, 10, 1083-1103, (1986) · Zbl 0623.34031
[2] Drabek, P.; Invernizzi, S., On the periodic BVP for the forced Duffing equations with jumping nonlinearity, Nonlinear analysis, 10, 643-650, (1986) · Zbl 0616.34010
[3] {\scDEL} P{\scINO} M., E{\scLGUETA} M. & M{\scANASEVICH} R., Sturm’s comparison theorem and a Hartman’s type oscillation criterion for (|u′|p−2u′)′ + c(t)|u|p−2u = 0, preprint.
[4] del Pino, M.; Manasevich, R., A homotopic deformation along p of a Leray-Schauder degree result and existence for (|u′|p−2u′)′ + f(t, u) = 0, u(0) = u(T) = 0, p > 1, J. diff. eqns, 80, 1-13, (1989) · Zbl 0708.34019
[5] Ding, W.Y., A generalization of the Poincaré-Birkhoff theorem, Proc. am. math. soc., 88, 341-346, (1983) · Zbl 0522.55005
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.