Oscillation of second-order variable delay differential equations with nonlinear neutral term

doi:10.3969/j.issn.1000-5641.2016.04.004

Abstract

Abstract:

We study the oscillatory behavior of a certain class of second-order variable delay nonlinear functional differential equations with a nonlinear neutral term in this article. By using the generalized Riccati transformation and some necessary analytic techniques, we establish two new oscillation criteria for the oscillation of the equations. In particular, the results obtained improve those presented in the literature.

Key words: oscillation, variable delay, functional differential equation, nonlinear neutral term, Riccati transformation

YANG Jia-shan. Oscillation of second-order variable delay differential equations with nonlinear neutral term[J]. Journal of East China Normal University(Natural Sc, doi: 10.3969/j.issn.1000-5641.2016.04.004.

References

[ 1 ] AGARWAL R P, BOHNER M, LI W T. Nonoscillation and Oscillation: Theory for Functional Differential
Equations [M]. New York: Marcel Dekker, 2004.
[ 2 ] BACUL´IKOV´A B, D ? ZURINA J. Oscillation theorems for second order neutral differential equations [J]. Comput
Math Appl, 2011, 61: 94-99.
[ 3 ] HASANBULLI M, ROGOVCHENKO YU V. Oscillation criteria for second order nonlinear neutral differential
equations [J], Appl Math Comput, 2010, 215: 4392-4399.
[ 4 ] LI T, AGARWAL R P, BOHNER M. Some oscillation results for second-order neutral differential equations [J].
J Indian Math Soc, 2012, 79: 97-106.
[ 5 ] LI T, AGARWAL R P, BOHNER M. Some oscillation results for second-order neutral dynamic equations [J].
Hacet J Math Stat, 2012, 41: 715-721.
[ 6 ] LI T X, HAN Z L, ZHANG C H, et al. Oscillation criteria for second-order superlinear neutral differential
equations [J]. Abstr Appl Anal, 2011 (1): 1-17.
[ 7 ] LI T X, HAN Z L, ZHANG C H, et al. On the oscillation of second-order Emden-Fowler neutral differential
equations [J]. J Appl Math Computing, 2011, 37: 601-610.
[ 8 ] LI T X, ROGOVCHENKO Y V, ZHANG C H. Oscillation of second-order neutral differential equations [J].
Funkc Ekvac, 2013, 56: 111-120.
[ 9 ] LIN X, TANG X. Oscillation of solutions of neutral differential equations with a superlinear neutral term [J].
Appl Math Lett, 2007, 20: 1016-1022.
[10] HAN Z L, LI T X, SUN S R, et al. Remarks on the paper [Appl. Math. Comput. 207 (2009) 388-396] [J]. Appl
Math Comput, 2010, 215(11): 3998-4007.
[11] LI T X, ROGOVCHENKO Y V. Oscillation theorems for second-order nonlinear neutral delay differential equa-
tions [J]. Abstract and Applied Analysis, 2014, 2014: 1-6.
[12] SUN S, LI T X, HAN Z L, et al. Oscillation theorems for second-order quasilinear neutral functional differential
equations [J]. Abstract and Applied Analysis, 2012, 2012: 1-17.
[13] ZHANG C H, AGARWAL R P, BOHNER M, et al. New oscillation results for second-order neutral delay dynamic
equations [J]. Advances in Difference Equations, 2012, 2012: 227.
[14] AGARWAL R P, BOHNER M, LI T X, et al. A new approach in the study of oscillatory behavior of even-order
neutral delay differential equations [J]. Appl Math Comput, 2013, 225: 787-794.
[15] YANG J S, QIN X W. Oscillation criteria for certain second-order Emden-Fowler delay functional dynamic
equations with damping on time scales [J]. Advances in Difference Equations, 2015, 2015: 97.
[16] AGARWAL R P, BOHNER M, LI T X, et al. Oscillation of second-order Emden-Fowler neutral delay differential
equations [J]. Annali di Matematica Pura ed Applicata, 2014, 193(6): 1861-1875.
[17] AGARWAL R P, BOHNER M, LI T X, et al. Oscillation of second-order differential equations with a sublinear
neutral term [J]. Carpathian Journal of Mathematics, 2014, 30(1): 1-6.
[18] 杨甲山, 黄劲. 时间模上一类二阶非线性动态方程振荡性的新准则~[J]. 华东师范大学学报 (自然科学版), 2015, 2015(3): 9-15.
[19] 杨甲山, 孙文兵. 一类多时滞二阶中立型微分方程的振动性~[J]. 中北大学学报 (自然科学版), 2012, 33(4): 363-368.
[20] 杨甲山, 方彬. 一类二阶中立型微分方程的振动和非振动准则~[J]. 四川师范大学学报 (自然科学版), 2012, 35(6): 776-780.
[21] 杨甲山, 方彬. 一类二阶中立型微分方程的振动性~[J]. 数学的实践与认识, 2013, 43(23): 193-197.
[22] 杨甲山, 覃学文. 具阻尼项的高阶\,Emden-Fowler\,型泛函微分方程的振荡性~[J]. 中山大学学报 (自然科学版), 2015, 54(4): 63-68.
[23] 杨甲山. 具正负系数和阻尼项的高阶泛函微分方程的振动性~[J]. 华东师范大学学报 (自然科学版), 2014(6): 25-34.

[1]	Wenxian LIN. Forced oscillation of fractional damped partial differential equation solutions with impulsive delays [J]. Journal of East China Normal University(Natural Science), 2024, 2024(2): 33-41.
[2]	Zhongyue HAN, Yuanhong YU. Asymptotic properties of a class of delay differential equations with a sub-linear neutral term [J]. Journal of East China Normal University(Natural Science), 2021, 2021(1): 1-7.
[3]	LI Ji-meng. Oscillation of second-order generalized Emden-Fowler-type differential equations [J]. Journal of East China Normal University(Natural Sc, 2019, 2019(4): 11-18.
[4]	LU Wei. Oscillation and asymptotics for damped fractional difference equations [J]. Journal of East China Normal University(Natural Sc, 2017, (4): 34-40,51.
[5]	YANG Jia-shan, ZHANG Xiao-jian. New results of oscillation for certain second-order nonlinear dynamic equations on time scales [J]. Journal of East China Normal University(Natural Sc, 2017, (3): 54-63.
[6]	LIN Wen-xian. Oscillation of certain third-order half linear neutral functional differential equations with damping [J]. Journal of East China Normal University(Natural Sc, 2017, (3): 48-53.
[7]	LI Mo-han. Oscillation of certain third-order variable delay damped dynamic equations on time scales [J]. Journal of East China Normal University(Natural Sc, 2016, 2016(4): 11-24.
[8]	YANG Jia-shan,HUANG Jin. New criteria for oscillation of certain second-order nonlinear dynamic equations on time scales [J]. Journal of East China Normal University(Natural Sc, 2015, 2015(3): 9-15.
[9]	YANG Jia-Shan. Oscillation of higher order functional differential equations with positive and negative coefficients and damping term [J]. Journal of East China Normal University(Natural Sc, 2014, 2014(6): 25-34.
[10]	ZHANG Xiao-jian, YANG Jia-shan. Oscillation and asymptotic behaviors for third-order delay dynamic equations on time scales [J]. Journal of East China Normal University(Natural Sc, 2014, 2014(3): 51-59.
[11]	YANG Jia-shan. Oscillation criteria of a class of second-order dynamic equations on time scales [J]. Journal of East China Normal University(Natural Sc, 2012, 2012(3): 17-23.
[12]	HAN Zhong-yue. Oscillations of a class of third order nonlinear neutral functional differential [J]. Journal of East China Normal University(Natural Sc, 2012, 2012(1): 113-120.
[13]	CAI Jiang-tao. H-oscillation of a class of second-order vector neutral partial differential equations with damped terms [J]. Journal of East China Normal University(Natural Sc, 2011, 2011(5): 73-78.
[14]	YANG Jia-shan. Oscillation theorems of second order neutral differential equations with positive and negative coefficients [J]. Journal of East China Normal University(Natural Sc, 2011, 2011(2): 10-16，3.
[15]	XU Yan-cong;MENG Fan-wei. Oscillation Theorems for Certain Second-Order Nonlinear Matrix Differential Equations(English) [J]. Journal of East China Normal University(Natural Sc, 2007, 2007(5): 34-38.