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

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.

In this paper we give sufficient conditions so that for every nonoscillatory

Cite this paper

A. Tiryaki, "Some Criteria for the Asymptotic Behavior of a Certain Second Order Nonlinear Perturbed Differential Equation,"*Advances in Pure Mathematics*, Vol. 2 No. 5, 2012, pp. 341-343. doi: 10.4236/apm.2012.25048.

A. Tiryaki, "Some Criteria for the Asymptotic Behavior of a Certain Second Order Nonlinear Perturbed Differential Equation,"

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.

[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.