# zbMATH — the first resource for mathematics

On the oscillation of higher-order half-linear delay differential equations. (English) Zbl 1223.34095
Summary: We study the oscillatory behavior of the following higher-order half-linear delay differential equation
$(r(t)(x^{(n-1)}(t))^\alpha)'+q(t)x^\beta(\tau(t))=0,\quad t\geq t_0,$
where we assume $$\int^\infty_{t_0}\frac{1}{r^{1/\alpha}(t)}\,dt<\infty$$. An example is given to illustrate the main results.

##### MSC:
 34K11 Oscillation theory of functional-differential equations
##### Keywords:
oscillation; delay differential equation; higher-order
Full Text:
##### References:
 [1] Hale, J.K., Theory of functional differential equations, (1977), Spring-Verlag New York · Zbl 0425.34048 [2] Agarwal, R.P.; Grace, S.R.; O’Regan, D., Oscillation theory for difference and functional differential equations, (2000), Marcel Dekker, Kluwer Academic Dordrecht · Zbl 0969.34062 [3] Ladde, G.S.; Lakshmikantham, V.; Zhang, B.G., Oscillation theory of differential equations with deviating arguments, (1987), Marcel Dekker New York · Zbl 0832.34071 [4] Erbe, L.; Kong, Q.; Zhang, B.G., Oscillation theory for functional differential eqautions, (1995), Marcel Dekker New York [5] Agarwal, R.P.; Grace, S.R.; O’Regan, D., Oscillation criteria for certain $$n$$th order differential equations with deviating arguments, J. math. appl. anal., 262, 601-622, (2001) · Zbl 0997.34060 [6] Agarwal, R.P.; Grace, S.R.; O’Regan, D., The oscillation of certain higher-order functional differential equations, Math. comput. modelling, 37, 705-728, (2003) · Zbl 1070.34083 [7] Dahiya, R.S., Oscillation criteria of even-order nonlinear delay differential equations, J. math. appl. anal., 54, 653-665, (1976) · Zbl 0322.34052 [8] Zhang, B.G., Oscillation of even order delay differential equations, J. math. appl. anal., 127, 140-150, (1987) · Zbl 0635.34046 [9] Grace, S.R., Oscillation theorems for $$n$$th-order differential equations with deviating arguments, J. math. appl. anal., 101, 268-296, (1984) · Zbl 0592.34046 [10] Grace, S.R.; Lalli, B.S., Oscillation of even order differential equations with deviating arguments, J. math. appl. anal., 147, 569-579, (1990) · Zbl 0711.34085 [11] Kartsatos, A.G., On oscillation of solutions of even order nonlinear differential equations, J. difference equ., 6, 232-237, (1969) · Zbl 0193.05705 [12] Mahfoud, W.E., Oscillation and asymptotic behavior of solutions of $$n$$th order nonlinear delay differential equations, J. difference equ., 24, 75-98, (1977) · Zbl 0341.34065 [13] Xu, Zhiting; Xia, Yong, Integral averaging technique and oscillation of certain even order delay differential equations, J. math. appl. anal., 292, 238-246, (2004) · Zbl 1062.34072 [14] Xu, Zhiting; Weng, Peixuan, Oscillation theorems for certain even order delay differential equations involving general means, Georgian math. J., 13, 383-394, (2006) · Zbl 1127.34042 [15] Philos, Ch.G., A new criterion for the oscillatory and asymptotic behavior of delay differential equations, Bull. Polish acad. sci. math., 39, 61-64, (1981) [16] Philos, Ch.G., On the existence of nonoscillatory solutions tending to zero at $$\infty$$ for differential equations with positive delays, Arch. math., 36, 168-178, (1981) · Zbl 0463.34050 [17] Shreve, W.E., Oscillation in first order nonlinear retarded argument differential equations, Proc. amer. math. soc., 41, 565-568, (1973) · Zbl 0254.34075 [18] Baculíková, B.; Džurina, J., Oscillation of third-order nonlinear differential equations, Appl. math. lett., 24, 466-470, (2011) · Zbl 1209.34042 [19] Baculíková, B.; Džurina, J., Oscillation of third-order functional differential equations, Electron. J. qual. theory differ. equ., 43, 1-10, (2010) · Zbl 1211.34077 [20] Grace, S.R.; Agarwal, R.P.; Pavani, R.; Thandapani, E., On the oscillation of certain third order nonlinear functional differential equations, Appl. math. comput., 202, 102-112, (2008) · Zbl 1154.34368
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.