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Oscillation criteria for second-order superlinear neutral differential equations. (English) Zbl 1217.34112
Summary: Some oscillation criteria are established for the second-order superlinear neutral differential equation $$(r(t)|z'(t)|^{\alpha-1} z'(t))'+f(t,x(\sigma(t)))=0,\quad t\ge t_0,$$ where $z(t)=x(t)+p(t)x(\tau(t))$, $\tau(t)\ge t$, $\sigma(t)\ge t$, $p\in C([t_0,\infty), [0,p_0])$, and $\alpha\ge 1$. Our results are based on the cases $\int^\infty_{t_0} 1/r^{1/\alpha}(t)\, dt=\infty$ or $\int^\infty_{t_0} 1/r^{1/\alpha}(t)\, dt< \infty$. Two examples are also provided to illustrate these results.

MSC:
34K11Oscillation theory of functional-differential equations
34K40Neutral functional-differential equations
WorldCat.org
Full Text: DOI EuDML
References:
[1] J. Hale, Theory of Functional Differential Equations, Springer, New York, NY, USA, 2nd edition, 1977. · Zbl 0461.05016 · doi:10.1016/0021-8693(77)90310-6
[2] J. Ba\vstinec, J. Diblík, and Z. \vSmarda, “Oscillation of solutions of a linear second-order discrete-delayed equation,” Advances in Difference Equations, vol. 2010, Article ID 693867, 12 pages, 2010. · Zbl 1200.39002 · doi:10.1155/2010/693867
[3] J. Ba\vstinec, L. Berezansky, J. Diblík, and Z. \vSmarda, “On the critical case in oscillation for differential equations with a single delay and with several delays,” Abstract and Applied Analysis, vol. 2010, Article ID 417869, pp. 1-20, 2010. · Zbl 1209.34080 · doi:10.1155/2010/417869
[4] J. Diblík, Z. Svoboda, and Z. \vSmarda, “Explicit criteria for the existence of positive solutions for a scalar differential equation with variable delay in the critical case,” Computers & Mathematics with Applications. An International Journal, vol. 56, no. 2, pp. 556-564, 2008. · Zbl 1155.34337 · doi:10.1016/j.camwa.2008.01.015
[5] R. P. Agarwal, S.-L. Shieh, and C.-C. Yeh, “Oscillation criteria for second-order retarded differential equations,” Mathematical and Computer Modelling, vol. 26, no. 4, pp. 1-11, 1997. · Zbl 0902.34061 · doi:10.1016/S0895-7177(97)00141-6
[6] J.-L. Chern, W.-C. Lian, and C.-C. Yeh, “Oscillation criteria for second order half-linear differential equations with functional arguments,” Publicationes Mathematicae Debrecen, vol. 48, no. 3-4, pp. 209-216, 1996. · Zbl 1274.34193
[7] J. D\vzurina and I. P. Stavroulakis, “Oscillation criteria for second-order delay differential equations,” Applied Mathematics and Computation, vol. 140, no. 2-3, pp. 445-453, 2003. · Zbl 1043.34071 · doi:10.1016/S0096-3003(02)00243-6
[8] T. Kusano and N. Yoshida, “Nonoscillation theorems for a class of quasilinear differential equations of second order,” Journal of Mathematical Analysis and Applications, vol. 189, no. 1, pp. 115-127, 1995. · Zbl 0823.34039 · doi:10.1006/jmaa.1995.1007
[9] T. Kusano and Y. Naito, “Oscillation and nonoscillation criteria for second order quasilinear differential equations,” Acta Mathematica Hungarica, vol. 76, no. 1-2, pp. 81-99, 1997. · Zbl 0906.34024 · doi:10.1007/BF02907054
[10] D. D. Mirzov, “The oscillation of the solutions of a certain system of differential equations,” Matematicheskie Zametki, vol. 23, no. 3, pp. 401-404, 1978. · Zbl 0423.34047
[11] Y. G. Sun and F. W. Meng, “Note on the paper of D\vzurina and Stavroulakis,” Applied Mathematics and Computation, vol. 174, no. 2, pp. 1634-1641, 2006. · Zbl 1096.34048 · doi:10.1016/j.amc.2005.07.008
[12] B. Baculíková, “Oscillation criteria for second order nonlinear differential equations,” Archivum Mathematicum, vol. 42, no. 2, pp. 141-149, 2006. · Zbl 1164.34499 · emis:journals/AM/06-2/index.html · eudml:130146
[13] Ch. G. Philos, “Oscillation theorems for linear differential equations of second order,” Archiv der Mathematik, vol. 53, no. 5, pp. 482-492, 1989. · Zbl 0661.34030 · doi:10.1007/BF01324723
[14] J. Diblík, Z. Svoboda, and Z. \vSmarda, “Retract principle for neutral functional differential equations,” Nonlinear Analysis. Theory, Methods & Applications, vol. 71, no. 12, pp. e1393-e1400, 2009. · doi:10.1016/j.na.2009.01.164
[15] L. Erbe, T. S. Hassan, and A. Peterson, “Oscillation of second order neutral delay differential equations,” Advances in Dynamical Systems and Applications, vol. 3, no. 1, pp. 53-71, 2008.
[16] L. Liu and Y. Bai, “New oscillation criteria for second-order nonlinear neutral delay differential equations,” Journal of Computational and Applied Mathematics, vol. 231, no. 2, pp. 657-663, 2009. · Zbl 1175.34087 · doi:10.1016/j.cam.2009.04.009
[17] R. Xu and F. Meng, “Some new oscillation criteria for second order quasi-linear neutral delay differential equations,” Applied Mathematics and Computation, vol. 182, no. 1, pp. 797-803, 2006. · Zbl 1115.34341 · doi:10.1016/j.amc.2006.04.042
[18] R. Xu and F. Meng, “Oscillation criteria for second order quasi-linear neutral delay differential equations,” Applied Mathematics and Computation, vol. 192, no. 1, pp. 216-222, 2007. · Zbl 1193.34137 · doi:10.1016/j.amc.2007.01.108
[19] J.-G. Dong, “Oscillation behavior of second order nonlinear neutral differential equations with deviating arguments,” Computers & Mathematics with Applications, vol. 59, no. 12, pp. 3710-3717, 2010. · Zbl 1198.34132 · doi:10.1016/j.camwa.2010.04.004
[20] B. Baculíková and D. Lacková, “Oscillation criteria for second order retarded differential equations,” Studies of the University of \vZilina. Mathematical Series, vol. 20, no. 1, pp. 11-18, 2006. · Zbl 05375291
[21] J. Jiang and X. Li, “Oscillation of second order nonlinear neutral differential equations,” Applied Mathematics and Computation, vol. 135, no. 2-3, pp. 531-540, 2003. · Zbl 1026.34081 · doi:10.1016/S0096-3003(02)00066-8
[22] Q. Wang, “Oscillation theorems for first-order nonlinear neutral functional differential equations,” Computers & Mathematics with Applications, vol. 39, no. 5-6, pp. 19-28, 2000. · Zbl 0954.34058 · doi:10.1016/S0898-1221(00)00042-0
[23] M. Hasanbulli and Y. V. Rogovchenko, “Oscillation criteria for second order nonlinear neutral differential equations,” Applied Mathematics and Computation, vol. 215, no. 12, pp. 4392-4399, 2010. · Zbl 1195.34098 · doi:10.1016/j.amc.2010.01.001
[24] S. H. Saker and D. O’Regan, “New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitution,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 1, pp. 423-434, 2011. · Zbl 1221.34245 · doi:10.1016/j.cnsns.2009.11.032
[25] B. Baculíková and J. D\vzurina, “Oscillation of third-order neutral differential equations,” Mathematical and Computer Modelling, vol. 52, no. 1-2, pp. 215-226, 2010. · Zbl 1201.34097 · doi:10.1016/j.mcm.2010.02.011
[26] Z. Han, T. Li, S. Sun, and Y. Sun, “Remarks on the paper, (Applied Mathematics and Computation), 2009, vol. 207, 388-396,” Applied Mathematics and Computation, vol. 215, pp. 3998-4007, 2010.
[27] Q. Zhang, J. Yan, and L. Gao, “Oscillation behavior of even-order nonlinear neutral differential equations with variable coefficients,” Computers & Mathematics with Applications, vol. 59, no. 1, pp. 426-430, 2010. · Zbl 1189.34135 · doi:10.1016/j.camwa.2009.06.027
[28] Z. Xu, “Oscillation theorems related to averaging technique for second order Emden-Fowler type neutral differential equations,” The Rocky Mountain Journal of Mathematics, vol. 38, no. 2, pp. 649-667, 2008. · Zbl 1167.34028 · doi:10.1216/RMJ-2008-38-2-649
[29] Z. Han, T. Li, S. Sun, and W. Chen, “On the oscillation of second-order neutral delay differential equations,” Advances in Difference Equations, vol. 2010, Article ID 289340, pp. 1-8, 2010. · Zbl 1192.34074 · doi:10.1155/2010/289340 · eudml:227157
[30] T. Li, Z. Han, P. Zhao, and S. Sun, “Oscillation of even-order neutral delay differential equations,” Advances in Difference Equations, vol. 2010, Article ID 184180, pp. 1-9, 2010. · Zbl 1209.34082 · doi:10.1155/2010/184180 · eudml:232103
[31] Z. Han, T. Li, S. Sun, C. Zhang, and B. Han, “Oscillation criteria for a class of second-order neutral delay dynamic equations of Emden-Fowler type,” Abstract and Applied Analysis, vol. 2011, Article ID 653689, pp. 1-26, 2011. · Zbl 1210.34134 · doi:10.1155/2011/653689 · eudml:230229
[32] Z. Han, T. Li, S. Sun, and C. Zhang, “An oscillation criteria for third order neutral delay differential equations,” Journal of Analytical and Applied, vol. 16, pp. 295-303, 2010. · Zbl 1276.34025 · doi:10.1515/JAA.2010.020
[33] S. Sun, T. Li, Z. Han, and Y. Sun, “Oscillation of second-order neutral functional differential equations with mixed nonlinearities,” Abstract and Applied Analysis, vol. 2011, pp. 1-15, 2011. · Zbl 1210.34094 · doi:10.1155/2011/927690 · eudml:229318
[34] Z. Han, T. Li, S. Sun, and W. Chen, “Oscillation criteria for second-order nonlinear neutral delay differential equations,” Advances in Difference Equations, vol. 2010, Article ID 763278, pp. 1-23, 2010. · Zbl 1203.34104 · doi:10.1155/2010/763278 · eudml:226993
[35] Z. Han, T. Li, C. Zhang, and Y. Sun, “Oscillation criteria for a certain second-order nonlinear neutral differential equations of mixed type,” Abstract and Applied Analysis, vol. 2011, pp. 1-8, 2011. · Zbl 1217.34111