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Oscillation theorems for second-order nonlinear neutral differential equations. (English) Zbl 1236.34092

Summary: The purpose of this paper is to study the oscillation of the second-order neutral differential equations of the form

${\left(a\left(t\right){\left[{z}^{\text{'}}\left(t\right)\right]}^{\gamma }\right)}^{\text{'}}+q\left(t\right){x}^{\beta }\left(\sigma \left(t\right)\right)=0,$

where $z\left(t\right)=x\left(t\right)+p\left(t\right)x\left(\tau \left(t\right)\right)$. We explore properties of given equations by examining those of associated first-order delay equations. New comparison theorems essentially simplify the examination of the equations studied as they allow us to deduce the oscillation of the second-order delay differential equation by applying the oscillation criteria obtained to the first-order delay equations. The results obtained are easy to verify.

##### MSC:
 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations 34K12 Growth, boundedness, comparison of solutions of functional-differential equations
##### References:
 [1] Agarwal, R. P.; Grace, S. R.: Oscillation theorems for certain neutral functional differential equations, Comput. math. Appl. 38, 1-11 (1999) · Zbl 0981.34059 · doi:10.1016/S0898-1221(99)00280-1 [2] Baculíková, B.: Oscillation criteria for second order nonlinear differential equations, Arch. math. 42, 141-149 (2006) · Zbl 1164.34499 · doi:emis:journals/AM/06-2/index.html [3] Baculíková, B.; Lackova, D.: Oscillation criteria for second order retarded differential equations, Stud. univ. z̭ilina math. Ser. 20, 11-18 (2006) [4] Baculíková, B.; Džurina, J.: Oscillation theorems for second order neutral differential equations, Comput. math. Appl. 61, 94-99 (2011) · Zbl 1207.34081 · doi:10.1016/j.camwa.2010.10.035 [5] Bainov, D. D.; Mishev, D. P.: Oscillation theory for nonlinear differential equations with delay, (1991) · Zbl 0747.34037 [6] Džurina, J.; Stavroulakis, I. P.: Oscillation criteria for second order delay differential equations, Appl. math. Comput. 140, 445-453 (2003) · Zbl 1043.34071 · doi:10.1016/S0096-3003(02)00243-6 [7] Džurina, J.; Hudáková, D.: Oscillation of second order neutral delay differential equations, Math. bohem. 134, 31-38 (2009) · Zbl 1212.34190 · doi:emis:journals/MB/134.1/index.html [8] Erbe, L. H.; Kong, Q.; Zhang, B. G.: Oscillation theory for functional differential equations, (1995) [9] Grace, S. R.; Lalli, B. S.: Oscillation of nonlinear second order neutral delay differential equations, Rad. math. 3, 77-84 (1987) · Zbl 0642.34059 [10] Grammatikopoulos, M. K.; Ladas, G.; Meimaridou, A.: Oscillation of second order neutral delay differential equation, Rad. math. 1, 267-274 (1985) · Zbl 0581.34051 [11] Ladde, G. S.; Lakshmikantham, V.; Zhang, B. G.: Oscillation theory of differential equations with deviating arguments, (1987) [12] Kiguradze, I. T.; Chaturia, T. A.: Asymptotic properties of solutions of nonautonomous ordinary differential equations, (1993) · Zbl 0782.34002 [13] Şahiner, Y.: On oscillation of second order neutral type delay differential equations, Appl. math. Comput. 150, 697-706 (2007) · Zbl 1045.34038 · doi:10.1016/S0096-3003(03)00300-X [14] Liu, L. H.; Bai, Y. Z.: New oscillation criteria for second-order nonlinear neutral delay differential equations, J. comput. Appl. math. 231, 657-663 (2009) · Zbl 1175.34087 · doi:10.1016/j.cam.2009.04.009 [15] Hasanbulli, M.; Rogovchenko, Y.: Oscillation criteria for second order nonlinear neutral differential equations, Appl. math. Comput. 215, 4392-4399 (2010) · Zbl 1195.34098 · doi:10.1016/j.amc.2010.01.001 [16] Rogovchenko, Y.; Tuncay, F.: Oscillation criteria for second order nonlinear differential equations with damping, Nonlinear anal. TMA 69, 208-221 (2008) · Zbl 1147.34026 · doi:10.1016/j.na.2007.05.012 [17] Xu, R.; Xia, Y.: A note on the oscillation of second-order nonlinear neutral functional differential equations, Int. J. Contemp. math. Sci. 3, 1441-1450 (2008) · Zbl 1176.34078 · doi:http://www.m-hikari.com/ijcms-password2008/29-32-2008/index.html [18] 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 · doi:10.1016/j.amc.2006.04.042 [19] Xu, R.; Meng, F.: Oscillation criteria for second order quasi-linear neutral delay differential equations, Appl. math. Comput. 192, 216-222 (2007) · Zbl 1193.34137 · doi:10.1016/j.amc.2007.01.108 [20] Dong, J. G.: Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments, Comput. math. Appl. 59, 3710-3717 (2010) · Zbl 1198.34132 · doi:10.1016/j.camwa.2010.04.004 [21] Philos, Ch.G.: On the existence of nonoscillatory solutions tending to zero at $\infty$ for differential equations with positive delay, Arch. math. 36, 168-178 (1981) · Zbl 0463.34050 · doi:10.1007/BF01223686 [22] Li, T.; Han, Z.; Zhao, P.; Sun, S.: Oscillation of even-order neutral delay differential equations, Adv. difference equ. 2010, 1-9 (2010) · Zbl 1209.34082 · doi:10.1155/2010/184180 [23] Han, Z.; Li, T.; Sun, S.; Chen, W.: On the oscillation of second-order neutral delay differential equations, Adv. difference equ. 2010, 1-8 (2010) · Zbl 1192.34074 · doi:10.1155/2010/289340 [24] Han, Z.; Li, T.; Sun, S.; Chen, W.: Oscillation criteria for second-order nonlinear neutral delay differential equations, Adv. difference equ. 2010, 1-23 (2010)