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Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments. (English) Zbl 1131.34319
The authors study the oscillatory behaviour of $n$-th order neutral differential equations of the form $$(r(t)\vert (x(t)+p(t)x(t-\tau))^{(n-1)}\vert^{\alpha-1}(x(t)+p(t)x(t-\tau))^{(n-1)})' +q(t)f(x(\sigma(t)))=0,\tag1$$ where $t\geq t_0$, $n$ is an even integer, $\alpha>0$ and $\tau\geq 0$ are constants. Throughout the paper they assume that: ($H_1$) $p,q\in C([t_0,\infty),\Bbb R))$, $0\leq p(t)\leq 1$, $q(t)\geq 0$; ($H_2$) $r\in C^1[t_0,\infty)$, $r(t)>0$, $r'(t)\geq 0$, $R(t)=\int_{t_0}^tr^{-1/\alpha}(s)ds\to\infty$ as $t\to\infty$; ($H_3$) $f\in C(\Bbb R,\Bbb R)$, $\frac{f(x)}{\vert x\vert^{\alpha-1}x}\geq \beta>0$ for $x\neq 0$ ($\beta$ is a constant); ($H_4$) $\sigma\in C^1[t_0,\infty)$, $\sigma(t)\leq t$, $\sigma'(t)>0$, $\lim_{t\to\infty}\sigma(t)=\infty$. Main result of the paper (Theorem 2.1) gives sufficient conditions for equation (1) to be oscillatory and it is applied to the second-order neutral differential equation (2) $(\vert(x(t)+p(t)x(t-\tau))'\vert^{\alpha-1}(x(t)+p(t)x(t-\tau))')'+q(t)\vert x(\sigma(t))\vert^{\alpha-1}x(\sigma(t))=0$. The purpose of the paper is to improve and extend several known results.

34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations
Full Text: DOI
[1] Agarwal, R. P.; Grace, S. R.; O’regan, D.: Oscillation criteria for certain nth order differential equations with deviating arguments. J. math. Anal. appl. 262, 601 (2001)
[2] Xu, Z.; Xia, Y.: Integral averaging technique and oscillation of certain even order delay differential equtions. J. math. Anal. appl. 292, 238-246 (2004) · Zbl 1062.34072
[3] Dzurina, J.; Stavroulakis, I. P.: Oscillation criteria for second-order delay differential equations. Appl. math. Comput. 140, 445-453 (2003) · Zbl 1043.34071
[4] Sun, Y. G.; Meng, F. W.: Note on the paper of dzurina and stavroulakis. Appl. math. Comput. 174, 1634-1641 (2006) · Zbl 1096.34048
[5] Mirzov, D. D.: On the oscillation of system of nonlinear differential equations. Diferencianye uravnenija 9, 581-583 (1973)
[6] Mirzov, D. D.: On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems. J. math. Anal. appl. 53, 418-425 (1976) · Zbl 0327.34027
[7] Mirzov, D. D.: On the oscillation of solutions of a system of differential equations. Mat. zametki, 401-404 (1978) · Zbl 0423.34047
[8] A. Elbert, A half-linear second order differential equation, Colloquia Math. Soc. Janos Bolyai, Qualitative Theory of differential Equations, 30 (1979) 153 -- 180.
[9] A. Elbert, Oscillation and nonoscillation theorems for some nonlinear differential equations, in: Ordinary and Partial Differential Equations, Lecture Notes in Mathematics, vol. 964, 1982, pp.187 -- 212. · Zbl 0528.34034
[10] Agarwal, R. P.; Shieh, S. H.; Yeh, C. C.: Oscillation criteria for second order retarded differential equations. Math. comput. Modell. 26, 1-11 (1997) · Zbl 0902.34061
[11] Chern, J. L.; Lian, W. Ch.; Yeh, C. C.: Oscillation criteria for second order half-linear differential equations with functional arguments. Publ. math. Ddbrecen 48, 209-216 (1996) · Zbl 1274.34193
[12] Xu, R.; Meng, F.: Some new oscillation criteria for second order quasi-linear neutral delay differential equations. Appl. math. Comput. 182, 797-803 (2006) · Zbl 1115.34341
[13] Philos, Ch.G.: A new criteria for the oscillatory and asymptotic behavior of delay differential equations. Bull. acad. Pol sci. Ser. sci. Mat. 39, 61-64 (1981)