## 时间尺度上带超线性中立项的二阶时滞动力方程的振动性

1 太原工业学院理学系 太原 030008

2 内蒙古科技大学信息工程学院 内蒙古包头 014010

## Oscillation of Second Order Delay Dynamic Equations with Superlinear Neutral Terms on Time Scales

Zhang Zhiyu,1, Zhao Cheng2, Li Yuyu1

1 Department of Sciences, Taiyuan Institute of Technology, Taiyuan 030008

2 Inner Mongolia University of Science and Technology, Inner Mongolia Baotou 014010

 基金资助: 国家自然科学基金.  11701528国家自然科学基金.  11647034国家自然科学基金.  11747098山西省自然科学基金.  2011011002-3

 Fund supported: the NSFC.  11701528the NSFC.  11647034the NSFC.  11747098the NSF of Shanxi Province.  2011011002-3

Abstract

In this paper, the oscillation of second order delay dynamic equations with super-linear neutral terms on time scales is studied. By using Riccati transformation and Bernoulli inequality techniques, several new oscillation theorems for the equation are obtained. The corresponding results in the existing literature are generalized and improved, some of which are new even for differential equations. Finally, some examples are given to verify the validity of the theorems.

Keywords： Oscillation ; Super-linear neutral term ; Second order ; Delay dynamic equation ; Time scale

Zhang Zhiyu, Zhao Cheng, Li Yuyu. Oscillation of Second Order Delay Dynamic Equations with Superlinear Neutral Terms on Time Scales. Acta Mathematica Scientia[J], 2021, 41(6): 1838-1852 doi:

## 1 引言

$$$\left( r( t ) [ x( t ) +p( t ) x^{1/\alpha}( \tau ( t ) ) ] ^{\Delta} \right) ^{\Delta}+ f(t, x( \delta ( t ) ) ) = 0, \; \; t\in I$$$

$( C_2 ) $$r\in C_{rd}^{1}( I, ( 0, \infty ) ) , \ p\in C_{rd}( I, [ p_0, \infty ) ) , \ p_0>1. ( C_3 )$$ f\in C( I\times {\Bbb R}, {\Bbb R} )$满足$xf( t, x ) >0 , \ f( t, x ) /( | x |^{\beta -1}x ) \geq q(t)>0, \; t\in I, \ x\ne 0 , $$q\in C_{rd}( I, ( 0, \infty ) ) 且最终不恒为0, \beta \in ( 0, \infty ). ( C_4 )$$ \tau , \delta \in C_{rd}^{1}( I, I ) , \ \tau ( t ) \leq t, \ \delta ( t ) \leq t, \ \tau ^{\Delta}( t ) >0 $$\lim\limits_{t\rightarrow \infty}\tau ( t ) = \lim\limits_{t\rightarrow \infty}\delta ( t ) = \infty . 本文仅记 \tau^{-1}(t) 为函数 \tau(t) 的反函数, 其他函数则不然, 例如 p^{-1}(t) = \frac{1}{p(t)}. 如果函数 x( t ) 满足方程(1.1)且在区间 [ T_x, \infty ) _{{\Bbb T}}, \ T_x\geq t_0 上满足 x\in C_{rd}^{1}( [ T_x, \infty ) _{{\Bbb T}}, {\Bbb R} ) , \$$ r( t ) \left( x( t ) +p( t ) x^{1/\alpha}\left( \tau ( t ) \right) \right) \in C_{rd}^{1}( [ T_x, \infty ) _{{\Bbb T}}, {\Bbb R} ),$则称函数$x( t )$是方程(1.1)的解, 而且本文只考虑方程(1.1)在$I = [ t_0, \infty ) _{{\Bbb T}}$上满足$\sup\left\{ \left| x( t ) \right|:\ t\geq T \right\} >0, \ T\geq T_x$的非平凡解.如果$x\left( t \right)$既不最终为正, 也不最终为负, 则称方程(1.1)的解$x\left( t \right)$是振动的, 否则, 称为非振动的.如果方程(1.1)的所有解都是振动的, 则称方程(1.1)是振动的.

$$$\int_{t_0}^{\infty}{\frac{\Delta t}{r\left( t \right)} = \infty}$$$

$$$\int_{t_0}^{\infty}{\frac{\Delta t}{r\left( t \right)}<\infty}.$$$

$$$\int_{t_0}^{\infty}{\frac{{\rm d}t}{a\left( t \right)} = \infty}$$$

$$$\int_{t_0}^{\infty}{\frac{{\rm d}t}{a\left( t \right)}<\infty}$$$

## 2 重要引理

$$$x^{1/\alpha}( t ) \geq Q_1( t ) u( \tau ^{-1}( t ) ) ,$$$

假设$x( t )$是方程(1.1)的最终正解, 则存在一个$t_1\in I , $$t\geq t_1 时有 x( t) >0, \ x( \tau ( t ) ) >0 .$$ u( t )$的定义和条件$( C_2 ),$可得到$u( t ) \geq x( t ), \; \; t\geq t_1\geq t_0 , $$\tau ( t ) 的逆 \tau ^{-1}( t) 存在, 所以 再由 因为 0< \alpha\leq 1 , 由引理2.3中的(2.3)式, 可得 $$x^{1/\alpha}( t ) \geq \frac{u( \tau ^{-1}( t ) ) - p^{-\alpha}( \tau ^{-1}( \tau ^{-1}( t ) ) )\left( \alpha u( \tau ^{-1}( \tau ^{-1}( t ) ) ) +1-\alpha \right)}{p( \tau ^{-1}( t ) )}.$$ 又因为 u^{\Delta}( t ) >0 ,$$ \varphi ( t )$非增且趋于0, 则存在$T\geq t_1, \ T\in I,$使得

$$$u( t ) \geq \varphi ( t ) , \ t\geq T.$$$

$$$u( t ) \geq u( t_1 ) +\int_{t_1}^t{\frac{r( t ) u^{\Delta}( t )}{r( s )}}\Delta s>R( t ) r( t ) u^{\Delta}( t ).$$$

$$$\left( \frac{u( t )}{R( t )} \right) ^{\Delta} = \frac{u^{\Delta}( t ) R( t ) -\frac{u( t )}{r( t )}}{R( t ) R( \sigma ( t ) )} <0.$$$

$$$A\left( t \right) u^{\Delta}\left( t \right) >-\frac{u\left( t \right)}{r\left( t \right)}, \ t\geq t_1.$$$

$$$\left( \frac{u( t )}{A( t )} \right) ^{\Delta} = \frac{u^{\Delta}( t ) A( t ) +\frac{u( t )}{r( t )}}{A\left( \sigma ( t ) \right) A( t )}>0,$$$

$$$\frac{u\left( t \right)}{A\left( t \right)}\geq \phi \left( t \right) , \ t\geq T.$$$

## 3 振动性定理

$$$\limsup\limits_{t\rightarrow \infty}\int_T^t{\left[ \eta ( s ) q( \sigma ( s ) ) Q_{1}^{\alpha\beta}( \delta ( \sigma ( s ) ) ) \frac{R^{\alpha\beta}( \tau ^{-1}( \delta ( \sigma ( s ) ) ) )}{R^{\alpha\beta}( \sigma ( s ) )} -\frac{\left( \eta ^{\Delta}( s ) \right) ^2r( s )}{4M^{\alpha\beta-1}\eta ( s )} \right] \Delta}s = \infty ,$$$

假设$x( t )$是方程(1.1)的最终正解$( x( t )$为最终负解时类似可证), 则存在$t_1\in I ,$使得当$t\geq t_1$时有$x( t ) >0, \ x\left( \tau ( t ) \right) >0, \ x\left( \delta ( t ) \right) >0 .$由于条件(1.2)成立, 从而$u( t )$满足(2.1)式.又由引理2.4中的(2.4)式和方程(1.1)知, 存在$T\geq t_1, \ T\in I ,$满足

$$$\left( r( t ) u^{\Delta}( t ) \right) ^{\Delta}+q( t ) Q_{1}^{\alpha\beta}\left( \delta ( t ) \right) u^{\alpha\beta}\left( \tau _{}^{-1}\left( \delta ( t ) \right) \right) \leq 0, \ t\geq T.$$$

$$$\limsup\limits_{t\rightarrow \infty}\int_T^t{\left[ \eta ( s ) q( \sigma ( s ) ) Q_{1}^{\alpha\beta}( \delta ( \sigma ( s ) ))\frac{R^{\alpha\beta}\left( \tau ^{-1}( \delta ( \sigma ( s ) ) ) \right)}{R\left( \sigma ( s ) \right)} -\frac{\left( \eta ^{\Delta}( s ) \right) ^2 r( s )}{4M^{1-\alpha\beta}\eta ( s )} \right] \Delta s} = \infty ,$$$

假设$x( t )$是方程(1.1)的最终正解$( x( t )$为最终负解时可类似证明), 则存在$t_1\in I ,$使得当$t\geq t_1$时有$x( t) >0, \ x( \tau ( t ) ) >0, \ x\left( \delta \left( t \right) \right) >0 .$由于条件(1.2)成立, 从而$u\left( t \right)$满足(2.1)式.于是类似于定理3.1的证明, 可得(3.8)式.因为$\left( \frac{u( t )}{R( t )} \right) ^{\Delta}<0 ,$所以有$\left( \frac{R( t )}{u( t )} \right) ^{\Delta}>0 , $$\frac{R( t )}{u( t )} 是递增的.由于 \alpha\beta \leq 1,$$ M_2 = \min \left\{ \frac{R\left( T \right)}{u\left( T \right)}, 1 \right\},$故有$\frac{R( t )}{u( t )}\geq M_2\in ( 0, 1 ] , \ t\geq T.$所以, 又有

$$$\frac{u^{\alpha\beta-1}( t )}{R^{\alpha\beta-1}( t )} = \frac{R^{1-\alpha\beta}(t)}{u^{1-\alpha\beta}(t)}\geq M_2^{1-\alpha\beta}, \ t\geq T,$$$

$T $$t\geq T 积分上式, 得 上式两端令 t\rightarrow \infty 取上极限, 得与(3.9)式矛盾.证毕. 注3.1 显然当 {\Bbb T} = {\Bbb R} 时动力方程 (1.1) 就退化为微分方程, 因此, 本文定理 3.1 和定理 3.2 已经完全包含并改进了文献[25]中的定理 2.1 和定理 2.2 , 因为本文只需存在某个(而非任意的)非增正函数 \varphi ( t ) \in C_{rd}( I, {\Bbb R}^+ ), \ \lim\limits_{t\rightarrow \infty}\varphi ( t ) = 0. 此外本文还可以取 {\Bbb T} = {\Bbb Z}^+ , 得到相应的差分方程的振动定理.进一步本文还可以令 {\Bbb T} = h{\Bbb N} , \ q^{{\Bbb N}}, \ {\Bbb N}^2 , \ {\Bbb T}_n 等诸多时间尺度, 分别得到对应的动力方程的振动定理, 这里从略了. 如果在定理3.1和定理3.2中取 \eta ( t ) = C 为常数, 则可以得到如下简捷的推论. 推论3.1 设条件 (1.2) 成立且 \alpha\beta \geq 1 , \ \delta ( t ) \leq \tau ( t ) , \ t\geq t_0. 如果存在某非增正函数 \varphi ( t ) \in C_{rd}( I, {\Bbb R}^+ ), \ \lim\limits_{t\rightarrow \infty}\varphi ( t ) = 0, 使得对于充分大的 T\geq t_0, \ T\in I , 其中 Q_1( t ) 如引理 2.4 的定义, 则方程 (1.1) 振动. 推论3.2 设条件 (1.2) 成立且 \alpha \beta \leq 1, \ \delta ( t ) \leq \tau ( t ) , \ t\geq t_0 . 如果存在非增正函数 \varphi ( t ) \in C_{rd}( I, {\Bbb R}^+ t), \ \lim\limits_{t\rightarrow \infty}\varphi ( t ) = 0 , 使得对于充分大的 T\geq t_0, \ T\in I, 其中 Q_1(t) 如引理 2.4 的定义, 则方程 (1.1) 振动. 注3.2 不难看出, 当 \alpha = 1 时方程 (1.1) 退化成为带线性中立项的动力方程, 显然定理 3.1 、定理 3.2 及其推论 3.1 和推论 3.2 均不需要"某个非增正连续函数 \varphi(t)\rightarrow 0\; (t\rightarrow \infty) 存在"的条件, 因此也不需要" \lim\limits_{t\to \infty}p(t) = \infty "的假设, 从而减弱了文献[25]的这一条件.这是带超线性中立项方程和带线性中立项或带次线性中立项方程的本质区别. 定理3.3 设条件 (1.3) 成立且 \alpha\beta \geq 1, \ \delta ( t ) \leq \tau ( t ) , \ t\in I . 如果存在某可导正函数 \eta ( t ) \in C_{rd}^{1}( I, \ {\Bbb R}^+ ) 和某非增正函数 \varphi ( t ) , \ \phi ( t ) \in C_{rd}( I, \ {\Bbb R}^+ ), \ \lim\limits_{t\rightarrow \infty}\varphi(t) = \lim\limits_{t\rightarrow \infty}\phi ( t ) = 0, 使得对充分大的 T\in I 和任意常数 M, \ L\in ( 0, 1 ] ( 特别地, 当 \alpha \beta = 1 时, M = L = 1 ),$$ (3.1)$式和

$$$\limsup\limits_{t\rightarrow \infty}\int_T^t \left[ q( s ) Q_{2}^{\alpha\beta}\left( \delta ( s ) \right) A( \sigma ( s ) ) A^{\alpha\beta-1}( s ) -\frac{1}{4L^{\alpha\beta-1}A( \sigma ( s ) ) r( s )} \right] \Delta s = \infty$$$

$$$w^{\Delta}\left( t \right) = \frac{u\left( t \right) \left( r\left( t \right) u^{\Delta}\left( t \right) \right) ^{\Delta}-u^{\Delta}\left( t \right) r\left( t \right) u^{\Delta}\left( t \right)}{u\left( t \right) u\left( \sigma \left( t \right) \right)}, \ t\geq T.$$$

$\begin{eqnarray} w^{\Delta}( t )& \leq &-\frac{q( t ) Q_{2}^{\alpha\beta}\left( \delta ( t ) \right) u^{\alpha\beta}\left( \tau ^{-1}\left( \delta ( t ) \right) \right)}{u\left( \sigma \left( t \right) \right)}-\left( w\left( t \right) \right) ^2\frac{u\left( t \right)}{r\left( t \right) u\left( \sigma \left( t \right) \right)}, {} \\ & \leq &-q( t ) Q_{2}^{\alpha\beta}\left( \delta \left( t \right) \right) u^{\alpha\beta-1} \left( \tau ^{-1}\left( \delta \left( t \right) \right) \right) -\frac{w^2\left( t \right)} {r\left( t \right)}{} \\ & \leq &-q( t ) Q_{2}^{\alpha\beta}\left( \delta \left( t \right) \right) u^{\alpha\beta-1}\left( t \right) -\frac{w^2\left( t \right)}{r\left( t \right)}. \end{eqnarray}$

$0>w( t ) A( t ) >-1, \ t\geq T,$所以有

假设$x( t )$是方程(1.1)的最终正解($x( t )$为最终负解时可类似证明), 则存在$t_1\in I ,$使得当$t\geq t_1$时有$x( t ) >0, \ x( \tau ( t ) ) >0, \ x( \delta ( t ) ) >0.$由于条件(1.3)成立, 从而$u( t )$满足(2.1)或(2.2)式, 下面将分两种情形证明之.

$$$u^{\alpha\beta-1}(t) = \frac{1}{u^{1-\alpha\beta}(t)}\geq L_2^{1-\alpha\beta}, {\quad} t\geq T.$$$

$$$\limsup\limits_{t\to \infty} \int_T^{t}{q(s)Q_1^{\alpha\beta}(\delta(s)) \Delta s = \infty}$$$

$$$\limsup\limits_{t\to \infty} \int_T^{t}\frac{1}{r( v )}\left( \int_T^v q( s ) Q_{2}^{\alpha\beta}\left( \delta ( s ) \right) A^{\alpha\beta}\left( \tau ^{-1}\left( \delta ( s ) \right) \right) \Delta s \right) \Delta v = \infty,$$$

$$$-r( t ) u^{\Delta}( t ) \geq \int_T^t{q( s ) Q_{2}^{\alpha\beta}\left( \delta ( s ) \right) u^{\alpha\beta}\left( \tau ^{-1}\left( \delta ( s ) \right) \right) \Delta s}, \ t\geq T.$$$

## 参考文献 原文顺序 文献年度倒序 文中引用次数倒序 被引期刊影响因子

Hilger S .

Analysis on measure chains–a unified approach to continuous and discrete calculus

Results in Mathematics, 1990, 18 (1/2): 18- 56

Bohner M , Georgiev S G . Multivariable Dynamic Calculus on Time Scales. Switzerland: Springer, 2016

Bohner M , Peterson A . Advances in Dynamic Equations on Time Scales. Boston: Birkhauser, 2003

Agarwal R , O'Regan D , Saker S . Dynamic Inequalities on Time Scales. Switzerland: Springer, 2014

Saker S H .

Oscillation of second-order nonlinear neutral delay dynamic equations on time scales

Journal of Computational and Applied Mathematics, 2006, 187 (2): 123- 141

Wu H W , Zhuang R K , Mathsen R M .

Oscillation criteria for second-order nonlinear neutral variable delay dynamic equations

Applied Mathematics and Computation, 2006, 178 (2): 321- 331

Saker S H , Agarwal R , O'Regan D .

Oscillation results for second-order nonlinear neutral delay dynamic equations on time scales

Applicable Analysis, 2007, 86 (1): 1- 17

Han Z , Li T , Sun S , et al.

Oscillation for second-order nonlinear delay dynamic equations on time scales

Advances in Difference Equations, 2009, 2009, 1- 13

Zhang S Y , Wang Q R .

Oscillation of second-order nonlinear neutral dynamic equations on time scales

Applied Mathematics and Computation, 2010, 216 (10): 2837- 2848

Li T , Han Z , Zhang C , et al.

Oscillation criteria for second-order superlinear neutral differential equations

Abstract and Applied Analysis, 2011, 2011, 367541

Li T , Agarwal R , Bohner M .

Some oscillation results for second-order neutral differential equations

J Indian Math Soc (N S), 2012, 79 (1): 97- 106

Li T , Rogovchenko Yu V .

Oscillation criteria for second-order superlinear Emden-Fowler neutral differential equations

Monatsh Math, 2017, 184 (3): 489- 500

Bohner M , Hassan T S , Li T .

Fite-Hille-Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments

Indian Math(N S), 2018, 29 (2): 548- 560

Li T, Pintus N, Viglialoro G. Properties of solutions to porous medium problems with different sources and boundary conditions. Z Angew Math Phys, 2019, 70(3), Article number: 86

Li T , Rogovchenko Yu V .

On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations

Applied Mathematics Letters, 2020, 105, 106293

Agarwal R , Bohner M , Li T , et al.

Oscillation of second-order differential equations with a sublinear neutral term

Carpathian Journal of Mathematics, 2014, 2014, 1- 6

Tamilvanan S , Thandapani E , Dzurina J .

Oscillation of second order nonlinear differential equation with sublinear neutral term

Differ Equ Appl, 2017, 9 (1): 29- 35

Grace S R , Graef J R .

Oscillatory behavior of second order nonlinear differential equations with a sublinear neutral term

Mathematical Modelling and Analysis, 2018, 23 (2): 217- 226

Dzurina J , Grace S R , Jadlovska I , et al.

Oscillation criteria for second-order Emden-Fowler delay differential equations with a sublinear neutral term

Math Nachr, 2020, 293 (5): 910- 922

Tamilvanan S , Thandapani E , Grace S R .

Oscillation theorems for second-order non-linear differential equation with a non-linear neutral term

International Journal of Dynamical Systems and Differential Equations, 2017, 7 (4): 316- 327

Zhang Z Y , Yu Y H , Li S P , et al.

Oscillation of second order nonlinear Differential equations with neutral delay

Acta Math Sci, 2019, 39A (4): 797- 811

Zhang Z Y , Song F F , Yu Y H .

Oscillation and asymptotic behavior for second order nonlinear neutral Emden-Fowler dfferential equations with explicit damping

Mathematica Applicata, 2020, 33 (3): 770- 781

Zhang Z Y , Song F F , Li T , et al.

Oscillation criteria of second nonlinear neutral Emden-Fowler differential equation with damping

Acta Math Sci, 2020, 40A (4): 934- 946

Zhang Z Y .

Oscillation criteria of second-order generalized Emden-Fowler delay differential equations with a sub-linear term

Acta Math Sci, 2021, 41A (3): 811- 826

Bohner M , Sudha B , Tangavelu K , et al.

Oscillation criteria for second-order differential equations with superlinear neutral term

Nonlinear Studies, 2019, 26 (2): 425- 434

/

 〈 〉