二阶非线性时标动态方程的振动准则  

Oscillation Criteria for a Second-order Nonlinear Neutral Dynamic Equations on Time Scales

作　　者：张少艳[1] 

机构地区：[1]广东金融学院应用数学系,广州510520

出　　处：《重庆师范大学学报（自然科学版）》2013年第5期80-83,共4页Journal of Chongqing Normal University:Natural Science

基　　金：广东省计算科学重点实验室开放基金(No.201206009)

摘　　要：时标理论在同时处理连续系统和离散系统方面具有非常广泛的应用。近年来有非常多的关于二阶中立型时标动态方程的振动性的结论,但已有结论均要求特殊的时标集,或r(t)函数递增。本文运用时标上积分及不等式的性质,得出x(t)/x(δ(t))≤α(t,T)的结论。利用该结论、Riccati变换技巧及配方法,得到了方程解的振动准则,即若方程能使得lim sup x→∞∫tT[Q(s)q(s)-r(s)(zΔ(s))2/4C(s)z(s)]Δs=∞或lim sup t→∞∫tt3[q(s)Q1(s)-(zΔ(s))2(r(s))1/γ(RT(s)r1/γ(s))1-γ]Δs=∞成立,则方程的解释振动所得到的结果去掉了时标集是特殊的及函数是递增的条件,其应用范围更为广泛。The theory of time scales has been widely used in the simultaneous processing of continuous system and discrete system. Therefore, many conclusions on the oscillation for second-order nonlinear neutral delay dynamic equations on time scales have been put forward in recent years. However, all these conclusions establish on special time scales or in the condition of increasing r （t）. Now, by using the characters of the integral and inequalities we get a conclusion, x （t） /x （δ （t）） ≤α （t, T）. With generalized Riccati technique and completing the square, we find the oscillation criterias for the equation. That is, if the equation can make limsupx→ ∞ ∫tTQ（s）q（s）-r（s）（zΔ（s））24C（s）z（s）Δs= ∞or lim supt→ ∞∫tt3q（s）Q1（s）-（zΔ（s））2（r（s））1γ（RT（s）r1γ（s））1-γΔs= ∞ to hold,the equation is oscillation. It is satisfactory that these results have a wider range of application because it needs no special time sales or increasing functions required in the former studies.

关 键 词：二阶时标动态方程 中立型 振动性 非线性 广义Riccati技巧 

分 类 号：O175[理学—数学]