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third-order; neutral functional differential equations; oscillation and asymptotic behavior
The aim of this paper is to study asymptotic properties of the third-order quasi-linear neutral functional differential equation \begin{equation*} \big [a(t)\big ([x(t)+p(t)x(\delta (t))]^{\prime \prime }\big )^\alpha \big ]^{\prime }+q(t)x^\alpha (\tau (t))=0\,, E \end{equation*} where $\alpha >0$, $0\le p(t)\le p_0<\infty $ and $\delta (t)\le t$. By using Riccati transformation, we establish some sufficient conditions which ensure that every solution of () is either oscillatory or converges to zero. These results improve some known results in the literature. Two examples are given to illustrate the main results.
[1] Baculíková, B., Džurina, J.: Oscillation of third-order functional differential equations. Electron. J. Qual. Theory Differ. Equ. 43 (2010), 1–10. MR 2678385 | Zbl 1211.34077
[2] Baculíková, B., Džurina, J.: Oscillation of third-order neutral differential equations. Math. Comput. Modelling 52 (2010), 215–226. DOI 10.1016/j.mcm.2010.02.011 | MR 2645933 | Zbl 1201.34097
[3] Baculíková, B., Džurina, J.: Oscillation of third-order nonlinear differential equations. Appl. Math. Lett. 24 (2011), 466–470. DOI 10.1016/j.aml.2010.10.043 | MR 2749728 | Zbl 1209.34042
[4] Baculíková, B., Elabbasy, E. M., Saker, S. H., Džurina, J.: Oscillation criteria for third-order nonlinear differential equations. Math. Slovaca 58 (2008), 1–20. DOI 10.2478/s12175-008-0068-1 | MR 2391214 | Zbl 1174.34052
[5] Erbe, L.: Existence of oscillatory solutions and asymptotic behavior for a class of third order linear differential equations. Pacific J. Math. 64 (1976), 369–385. DOI 10.2140/pjm.1976.64.369 | MR 0435508 | Zbl 0339.34030
[6] Grace, S. R., Agarwal, R. P., Pavani, R., Thandapani, E.: On the oscillation of certain thir d order nonlinear functional differential equations. Appl. Math. Comput. 202 (2008), 102–112. DOI 10.1016/j.amc.2008.01.025 | MR 2437140
[7] Han, Z., Li, T., Zhang, C., Sun, S.: An oscillation criterion for third order neutral delay differential equations. J. Appl. Anal., to appear. MR 2740504
[8] Hanan, M.: Oscillation criteria for third order differential equations. Pacific J. Math. 11 (1961), 919–944. DOI 10.2140/pjm.1961.11.919 | MR 0145160
[9] Hartman, P., Winter, A.: Linear differential and difference equations with monotone solutions. Amer. J. Math. 75 (1953), 731–743. DOI 10.2307/2372548 | MR 0057404
[10] Karpuz, B., Öcalan, Ö., Öztürk, S.: Comparison theorems on the oscillation and asymptotic behavior of higher-order neutral differential equations. Glasgow Math. J. 52 (2010), 107–114. DOI 10.1017/S0017089509990188 | MR 2587820
[11] Philos, Ch. G.: Oscillation theorems for linear differential equations of second order. Arch. Math. 53 (1989), 482–492. DOI 10.1007/BF01324723 | MR 1019162 | Zbl 0661.34030
[12] Saker, S. H., Džurina, J.: On the oscillation of certain class of third-order nonlinear delay differential equations. Math. Bohem. 135 (2010), 225–237. MR 2683636 | Zbl 1224.34217
[13] Zhong, J., Ouyang, Z., Zou, S.: Oscillation criteria for a class of third-order nonlinear neutral differential equations. J. Appl. Anal. (2010), to appear.
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