偶阶非线性中立型时滞微分方程的振动性
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摘要
本文研究一类新的偶阶非线性中立型时滞微分方程(r(t)|z~((n-1))(t)|~(a-1)z~((n-1))(t))'+F(t,x(g(t)))=0,t≥t_0(其中z(t)=x(t)+p(t)x(T(t)),α> 0为常数,n为偶数)的振动性.利用广义Riccati不等式和积分平均技巧得到方程一切解均为振动的若干新的振动准则,推广和改进了一些文献中的结果.
In this work, we investigate the oscillation of the new even order nonlinear neutral differential equation with distributed delay(r(t)|z~((n-1))(t)|~(a-1)z~((n-1))(t))'+F(t,x(g(t)))=0,t≥t_0,where z(t) = x(t) +p(t)x(T(t)), α > 0 is constant and n is even. By using the generalized Riccati transformation and integral averaging technique, we establish some new oscillation criteria. These results extend and improve some existing results in the literature.
In this work, we investigate the oscillation of the new even order nonlinear neutral differential equation with distributed delay(r(t)|z~((n-1))(t)|~(a-1)z~((n-1))(t))'+F(t,x(g(t)))=0,t≥t_0,where z(t) = x(t) +p(t)x(T(t)), α > 0 is constant and n is even. By using the generalized Riccati transformation and integral averaging technique, we establish some new oscillation criteria. These results extend and improve some existing results in the literature.
引文
[1]Agarwal R P, Grace S R, O'Regan D. Oscillation criteria for certain nth order differential equations with deviating arguments. J. Math. Anal. Appl., 2001, 262:601-622
[2]Dzurina J, Stavroulakis I P. Oscillation criteria for second order delay differential equations. Appl.Math. Comput., 2003, 140:445-453
[3]Sun Y G, Meng F W. Note on the paper of Dzurina and Stavroulakis. Appl. Math. Comput., 2006,174:1634-1641
[4]Erbe L, Hassan T S, Peterson A. Oscillation of second order neutral delay differential equations. Adv.Dyn. Syst. Appl., 2008, 3:53-71
[5]Li T, Han Z,Zhang C, Sun S. On the oscillation of second order Emden-Fowler neutral differential equations. J. Appl. Math. Comput., 2011, 37:601-610
[6]Liu H D, Meng F W, Liu P C. Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation. Appl. Math. Comput., 2012, 219:2739-2748
[7]曾云辉,罗李平,俞元洪.中立型Emden-Fowler时滞微分方程的振动性.数学物理学报,2015, 35A(4):803-814(Zeng Y H, Lou L P, Yu Y H. Oscillation for Emden-Fowler delay differential equations of neutral type. Acta Mathematica Scientia, 2015, 35A(4):803-814)
[8]Philos Ch G. A new criterion for the oscillatory and asymptotic behavior of delay differential equations.Bull. Polish. Acad. Sci. Math., 1981, 39:61-64
[9]Agarwal R P, Grace S R,O'Regan D. Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht, The Netherlands, 2000
[10]Meng F W, Xu R. Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments. Appl. Math. Comput., 2007, 190:458-464
[11]Tiryaki A. Oscillation criteria for a certain second order nonlinear differential equations with deviating arguments. Electron. J. Qual. Theory Differ. Equ., 2009, 61:1-11
[12]Xun R, Meng F W. Oscillation criteria for second order quasi-linear neutral delay differential equations.Appl. Math. Comput., 2007, 192:216-222
[13]Dong J G. Oscillation behavior of second order nonlinear neutral differential equations with with deviating arguments. Comput. Math. Appl., 2010, 59:3710-3717
[14]张全信,高丽,俞元洪.偶阶半线性中立型分布时滞微分方程的振动性.应用数学学报,2011,34:895-905(Zhang Q X, Gao L, Yu Y H. Oscillation of even order half-linear neutral differential equations with distributed delay. Acta Mathematica Applicatae Sinica, 2011, 34:895-950)
[2]Dzurina J, Stavroulakis I P. Oscillation criteria for second order delay differential equations. Appl.Math. Comput., 2003, 140:445-453
[3]Sun Y G, Meng F W. Note on the paper of Dzurina and Stavroulakis. Appl. Math. Comput., 2006,174:1634-1641
[4]Erbe L, Hassan T S, Peterson A. Oscillation of second order neutral delay differential equations. Adv.Dyn. Syst. Appl., 2008, 3:53-71
[5]Li T, Han Z,Zhang C, Sun S. On the oscillation of second order Emden-Fowler neutral differential equations. J. Appl. Math. Comput., 2011, 37:601-610
[6]Liu H D, Meng F W, Liu P C. Oscillation and asymptotic analysis on a new generalized Emden-Fowler equation. Appl. Math. Comput., 2012, 219:2739-2748
[7]曾云辉,罗李平,俞元洪.中立型Emden-Fowler时滞微分方程的振动性.数学物理学报,2015, 35A(4):803-814(Zeng Y H, Lou L P, Yu Y H. Oscillation for Emden-Fowler delay differential equations of neutral type. Acta Mathematica Scientia, 2015, 35A(4):803-814)
[8]Philos Ch G. A new criterion for the oscillatory and asymptotic behavior of delay differential equations.Bull. Polish. Acad. Sci. Math., 1981, 39:61-64
[9]Agarwal R P, Grace S R,O'Regan D. Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht, The Netherlands, 2000
[10]Meng F W, Xu R. Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments. Appl. Math. Comput., 2007, 190:458-464
[11]Tiryaki A. Oscillation criteria for a certain second order nonlinear differential equations with deviating arguments. Electron. J. Qual. Theory Differ. Equ., 2009, 61:1-11
[12]Xun R, Meng F W. Oscillation criteria for second order quasi-linear neutral delay differential equations.Appl. Math. Comput., 2007, 192:216-222
[13]Dong J G. Oscillation behavior of second order nonlinear neutral differential equations with with deviating arguments. Comput. Math. Appl., 2010, 59:3710-3717
[14]张全信,高丽,俞元洪.偶阶半线性中立型分布时滞微分方程的振动性.应用数学学报,2011,34:895-905(Zhang Q X, Gao L, Yu Y H. Oscillation of even order half-linear neutral differential equations with distributed delay. Acta Mathematica Applicatae Sinica, 2011, 34:895-950)