Some Criteria for the Asymptotic Behavior of a Certain Second Order Nonlinear Perturbed Differential Equation

Aydin Tiryaki

Department of Mathematics and Computer Science, Izmir University, ?zmir, Turkey.

DOI: 10.4236/apm.2012.25048                

Abstract

In this paper we give sufficient conditions so that for every nonoscillatory u(t) solution of (r(t)ψ(u)u')'+Q(t,u,u'), we have lim inf|u(t)|=0. Our results contain the some known results in the literature as particular cases.

Keywords

Perturbed Differential Equations; Nonoscillatory Solution; Asymptotic Behavior

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	R. P. Agarwal, S. R. Grace and D. O. Regan, “Oscillation Theory for Second Order Linear, Half Linear, Superlinear, Sublinear Dynamic Equations,” Kluwer, Dordecht, 2002. Hdoi:10.1016/0022-247X(91)90185-3
[2]	S. R. Grace, “Oscillatory and Asymptotic Behavior of Certain Functional Differential Equations,” Journal of Mathematical Analysis and Applications, Vol. 62, No. 1, 1991, pp. 177-188
[3]	M. Remili, “Oscillation Criteria for Second Order Nonlinear Perturbed Differential Equations,” Electronic Journal of Qualitative Theory of Differential Equations, Vol. 25, 2010, pp. 1-11
[4]	P. Temtek and A. Tiryaki, “Oscillation Criteria for a Certain Second Order Nonlinear Perturbed Differential Equations,” submitted for publication.
[5]	Q. Zhang and L. Wang, “Oscillatory Behavior of Solutions for a Class of Second Order Nonlinear Differential Equation with Perturbation,” Acta Applicandae Mathematicae, Vol. 110, No. 2, 2010, pp. 885-893.doi:10.1007/s10440-009-9483-8
[6]	M. E. Hammet, “Nonoscillation Properties of a Nonlinear Differential Equation,” Proceedings of the American Mathematical Society, Vol. 30, No. 1, 1971, pp. 92-96. Hdoi:10.1090/S0002-9939-1971-0279384-5
[7]	R. P. Agarwal, S. R. Grace and D. O. Regan, “Oscillation Theory for Second Order Dynamic Equations,” Taylor and Francis, London, 2003. Hdoi:10.4324/9780203222898
[8]	Ch. G. Philos, “On the Existence of Nonoscillatory Solutions Tending to Zero at ∞ for Differential Equations with Positive Delays,” Archiv der Mathematik, Vol. 36, No. 1, 1981, pp. 168-170. Hdoi:10.1007/BF01223686
[9]	P. J. Y. Wong and R. P. Agarwal, “Oscillation Theorems and Existence Criteria of a Asymptotically Monotone Solution for Second Order Differential Equations,” Dynamic Systems & Applications, Vol. 4, 1995, pp. 477-496.
[10]	S. R. Grace and B. S. Lalli, “Oscillations in Second Order Differential Equations with Alternating Coefficients,” Periodica Mathematica Hungarica, Vol. 1, 1998, pp. 69-78.