## Oscillation theorems related to averaging technique for second order emden-fowler type neutral differential equations.(English)Zbl 1167.34028

Summary: Some oscillation theorems are established by the averaging techniques for the second order Emden-Fowler type neutral delay differential equation
$(r(t)x'(t))'+q_1(t)|y(t-\sigma_1)|^{\alpha-1} y(t-\sigma_1)+ q_2(t)|y(t-\sigma_2)|^{\beta-1} y(t-\sigma_2)=0,\quad t\geq t_0,$
where $$x(t)=y(t)+p(t)y(t-\tau)$$, $$\tau$$, $$\sigma_1$$ and $$\sigma_2$$ are nonnegative constants, $$0<\alpha<1$$, $$\beta>1$$, and $$r$$, $$p$$, $$q_1,q_2\in C(t_0,\infty),\mathbb R)$$. These theorems obtained here extend and improve some known results. In particular, two interesting examples that point out the applications of our results are also included.

### MSC:

 34K11 Oscillation theory of functional-differential equations 34K40 Neutral functional-differential equations
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### References:

 [1] S.H. Abdallah, Oscillatory and non-oscillatory of second order neutral delay differential equations , Appl. Math. Coumput. 135 (2003), 333-344. · Zbl 1079.34062 · doi:10.1016/S0096-3003(01)00335-6 [2] R.P. Agarwal, S.R. Grace and D. O’Regan, Oscillation theory for second order dynamic equations , Taylor & Francis, London, 2003. · Zbl 1043.34032 [3] ——–, Nonoscillatory solutions of delay and neutral singular differential equations , Appl. Anal. 81 (2002), 1221-1244. · Zbl 1049.34079 · doi:10.1080/0003681021000029891 [4] F.V. Atkinson, On second order nonlinear oscillation , Pacific J. Math. 5 (1955), 643-647. · Zbl 0065.32001 · doi:10.2140/pjm.1955.5.643 [5] D.D. Bainov and D.P. Mishev, Oscillation theorey for neutral equations with delay , Adam Hilger IOP Publishing Ltd., New York, 1991. · Zbl 0747.34037 [6] S. Belohorec, Oscillatory solutions of certain nonlinear differential equations of the second order , Math. Fyz. Casopis Sloven Akad. Vied. 11 (1961), 250-255. · Zbl 0108.09103 [7] J. Dzurina and B. Mihalikova, Oscillation criteria for second order neutral differential equations , Math. Boh. 125 (2000), 145-153. · Zbl 0973.34057 [8] L.H. Erbe, Q. Kong and B.G. Zhang, Oscillation theorey for functional differential equations , Marcel Dekker, New York, 1995. · Zbl 0821.34067 [9] M.K. Grammatikopoulos, G. Ladas and A. Meimaidou, Oscillation of second order neutral delay differential equations , Radovi Mat. 1 (1985), 267-274. · Zbl 0581.34051 [10] I. Györi and G. Ladas, Oscillation theory of delay differential equations with applications , Clarendon Press, Oxford, 1991. · Zbl 0780.34048 [11] J.K. Hale, Theory of functional differential equations , Springer-Verlag, New York, 1977. · Zbl 0352.34001 [12] G.H. Hardy, J.E. Littlewood and G. Polyu, Inequalities , 2nd editor, Cambridge University Press, Cambridge, 1988. [13] I.V. Kamenev, An integral criterion for oscillation of linear differential equations , Mat. Z. 23 (1978), 249-251 (in Russian). · Zbl 0386.34032 [14] H.J. Li, Oscillation theorems for second order neutral delay differential equations , Nonlinear Anal. TMA 26 (1996), 1397-1409. · Zbl 0852.34065 · doi:10.1016/0362-546X(94)00346-J [15] H.J. Li and W.L. Liu, Oscillation criteria for second order neutral differential equations , Canad. J. Math. 48 (1996), 871-886. · Zbl 0859.34055 · doi:10.4153/CJM-1996-044-6 [16] Ch.G. Philos, Oscillation theorems for linear differential equation of second order , Arch. Math (Basel) 53 (1989), 482-492. · Zbl 0661.34030 · doi:10.1007/BF01324723 [17] S.G. Ruan, Oscillation of second order neutral differential equations , Canad. Math. Bull. 36 (1993), 485-496. · Zbl 0798.34079 · doi:10.4153/CMB-1993-064-4 [18] Y. Sahiner, On oscillation of second order neutral type delay differential equations , Appl. Math. Coumput. 150 (2004), 697-706. · Zbl 1045.34038 · doi:10.1016/S0096-3003(03)00300-X [19] S.H. Saker, Oscillation for second order neutral delay differential equations of Emden-Fowler type , Acta Math. Hungar. 100 (2003), 37-62. · Zbl 1051.34052 · doi:10.1023/A:1024699900047 [20] S.H. Saker and J.V. Manojlovć, Oscillation criteria for second order superlinear neutral delay differential equations , Elect. J. Qualitative Theory Differential Equations 10 (2004), 1-22. · Zbl 1082.34057 [21] J.S.W. Wong, Necessary and sufficient conditions for oscillation for second order neutral differential equations , J. Math. Anal. Appl. 252 (2000), 342-352. · Zbl 0976.34057 · doi:10.1006/jmaa.2000.7063 [22] ——–, On Kamenev-tye oscillation theorems for second order differential equations , J. Math. Anal. Appl. 258 (2001), 244-257. · Zbl 0987.34024 · doi:10.1006/jmaa.2000.7376 [23] J. Yan, Oscillation theorems for second order linear differential equations with damping , Proc. Amer. Math. Soc. 98 (1980), 276-282. JSTOR: · Zbl 0622.34027 · doi:10.2307/2045698 [24] Q. Yang, L. Yang and S. Zhu, Interval criteria for oscillation of second order nonlinear neutral differential equations , Comput. Math. Appl. 46 (2003), 903-918. · Zbl 1057.34088 · doi:10.1016/S0898-1221(03)90152-0 [25] Y.H. Yu, Leighton type oscillation criterion and Sturm type comparison theorem , Math. Nachr. 153 (1991), 137-143. · Zbl 0795.34025 · doi:10.1002/mana.19911530114
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