德国学者HILGER^[1]于1988年首次提出了时间尺度上的分析理论, 并发表了将离散分析和连续分析相统一的数学理论^[2]，引起了学术界的广泛关注.之后, 学者们开始对时间尺度上动力方程的振动性进行了研究, 给出了一些非常好的结果^[1-18], 对中立型动力方程的研究也取得了一些结果^[9-18].

文献[12]研究了二阶中立型时滞动力方程

$ {\left[ {r\left( t \right){y^\Delta }\left( t \right)} \right]^\Delta } + f\left( {t,x\left( {t - \delta } \right)} \right) = 0,\;\;\;\;\;t \in T $

的振动性, 其中y(t)=x(t)+p(t)x(t-τ), 得到了一些新的结果.文献[13]研究了一类二阶非线性中立型时滞动力方程

$ {\left[ {r\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]^\Delta } + f\left( {t,x\left( {\delta \left( t \right)} \right)} \right) = 0,\;\;\;\;\;t \in T $

的振动性, 其中y(t)=x(t)+p(t)x(t-τ), 给出了几个方程振动的充分条件.文献[14]借助Riccati变换技术研究了二阶非线性中立型时滞动力方程

$ \begin{array}{l} {\left[ {A\left( t \right){y^\Delta }\left( t \right)} \right]^\Delta } + {P_1}\left( t \right){f_1}\left( {x\left( {\delta \left( t \right)} \right)} \right) + \\ \;\;\;\;\;\;\;{P_2}\left( t \right){f_2}\left( {x\left( {\delta \left( t \right)} \right)} \right) = 0,\;\;\;\;\;t \in T \end{array} $

的振动性, 其中y(t)=x(t)+B(t)g(x(τ(t))), 得到了方程振动的几个新的充分条件, 推广并改进了已有文献的相关结果.文献[15]研究了一类具有阻尼项的二阶非线性中立型变时滞动力方程

$ {\left[ {A\left( t \right){y^\Delta }\left( t \right)} \right]^\Delta } + b\left( t \right){y^\Delta }\left( t \right) + F\left( {t,x\left( {\delta \left( t \right)} \right)} \right) = 0 $

的振动性, 其中y(t)=x(t)+B(t)g(x(τ(t))), 得到了该方程的振动定理.

本文基于时间尺度上的微积分理论、Riccati变换、H函数法和不等式技巧，对一类新的具有阻尼项的二阶非线性时滞中立型动力方程

$ {\left( {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right)^\Delta } + b\left( t \right){\left( {{y^\Delta }\left( t \right)} \right)^\gamma } + F\left( {t,x\left( {\delta \left( t \right)} \right)} \right) = 0 $

(1)

的振动性进行研究, 其中0 < γ≤1, y(t)=x(t)+B(t)g(x(τ(t))).

由于我们感兴趣的是方程解的振动性, 所以假设时间尺度T是无界的, 即它是时间尺度上的无穷区间[t₀, ∞)_T=[t₀, ∞)∩T.以下假设贯穿全文：

(H₁) τ, δ：T→T都是滞量函数, τ(t)≤t, 且$\mathop {{\rm{lim}}}\limits_{t \to \infty } \tau (t) = \infty $ ; δ(t)≤t, $\mathop {{\rm{lim}}}\limits_{t \to \infty } \delta (t) = \infty $ 并且τ(δ(t))=δ(τ(t)).

(H₂)滞量τ是严格递增的, $\widetilde T = \tau \left(T \right) \subseteq T$ 是一时间尺度, τ°σ=σ°τ且τ^Δ(t)=τ₀>0, δ(t)≥τ(t).

(H₃) b∈C_rd(T, [0, ∞)), 即b(t)是定义在T上的实值rd-连续函数, B∈C_rd(T, R)，且0≤B(t)≤b₀ < ∞(这里b₀是常数).

(H₄) A∈C_rd(T, (0, ∞)), -b/A∈R ⁺, 并且$\int_{{t_0}}^\infty {{{\left({\frac{{{e_{-b/A}}(s, {t_0})}}{{A\left(s \right)}}} \right)}^{1/\gamma }}\Delta s{\rm{ = }}\infty } $ .

(H₅) g∈C(R, R)且当u≠0时ug(u)>0, g(u)/u≤η(这里0 < η≤1为常数).

(H₆) F∈(T×R, R), uF(t, u)>0(u≠0), 且$\exists $ p∈C_rd(T, (0, ∞))使得|F(t, u)|≥p(t)u^γ.

为了更好地证明本文的主要结果, 先引入2个引理.

引理1 ^[15]如果g∈R ⁺, 即g(t)∈C_rd(T, R)，且对任意的t∈[t₀, +∞)_T, 满足1+μ(t)g(t)>0.则初值问题y^Δ(t)=g(t)y(t), y(t₀)=y₀∈R在[t₀, +∞)_T上有唯一的正解e_g(t, t₀)，并满足半群性质e_g(a, b)e_g(b, c)=e_g(a, c).

引理2 ^[15]若τ(t)是严格递增的, $\widetilde T \subseteq T$ 是一时间尺度, τ(σ(t))=σ(τ(t)).设x：$\widetilde T \to R$, 如果τ^Δ(t)和x^Δ(τ(t))存在(t∈T^k), 则(x(τ(t)))^Δ存在, 且

$ {\left( {x\left( {\tau \left( t \right)} \right)} \right)^\Delta } = {x^\Delta }\left( {\tau \left( t \right)} \right){\tau ^\Delta }\left( t \right). $

(2)

1 主要结果

定理1  若有函数φ(t)∈C_rd¹([t₀, ∞)_T, (0, ∞)), 使得

$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to \infty } \int_{{t_0}}^t {\left\{ {\xi \left( s \right)\varphi \left( s \right) - \frac{{A\left( {\tau \left( s \right)} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{{\left( {{\tau _0}\varphi \left( s \right)} \right)}^\gamma }}} \times } \right.} \\ \;\;\;\;\;\;\;\;\left[ {{{\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( s \right)}}{{{A^\sigma }\left( s \right)}}} \right)}^{\gamma + 1}} + } \right.\\ \;\;\;\;\;\;\;\;\left. {\left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}{{\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}{A^\sigma }\left( {\tau \left( s \right)} \right)}}} \right)}^{\gamma + 1}}} \right]} \right\}\Delta s = \infty , \end{array} $

(3)

其中, ξ(t)=min{p(t), p(τ(t))}, 则方程(1)在[t₀, ∞)_T上是振动的.

证明 假设方程(1)在[t₀, ∞)_T上有一非振动解x(t), 不失一般性, 不妨设最终解为正, 即存在t₁∈[t₀, ∞)_T使得x(t)>0, x(τ(t))>0, x(δ(t))>0, t∈[t₁, ∞)_T，则y(t)>0.由方程(1), 得

$ {\left( {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right)^\Delta } + b\left( t \right){\left( {{y^\Delta }\left( t \right)} \right)^\gamma } < 0. $

(4)

利用引理1, 可得

$ \begin{array}{l} {\left[ {\frac{{A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }}}{{{e_{ - b/A}}\left( {t,{t_0}} \right)}}} \right]^\Delta } = \frac{{{{\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]}^\Delta }{e_{ - b/A}}\left( {t,{t_0}} \right)}}{{{e_{ - b/A}}\left( {t,{t_0}} \right){e_{ - b/A}}\left( {\sigma \left( t \right),{t_0}} \right)}} - \\ \frac{{A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }{{\left[ {{e_{ - b/A}}\left( {t,{t_0}} \right)} \right]}^\Delta }}}{{{e_{ - b/A}}\left( {t,{t_0}} \right){e_{ - b/A}}\left( {\sigma \left( t \right),{t_0}} \right)}} = \frac{{{{\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]}^\Delta }}}{{{e_{ - b/A}}\left( {\sigma \left( t \right),{t_0}} \right)}} + \\ \frac{{b\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }}}{{{e_{ - b/A}}\left( {\sigma \left( t \right),{t_0}} \right)}} \leqslant - \frac{{p\left( t \right){x^\gamma }\left( {\delta \left( t \right)} \right)}}{{{e_{ - b/A}}\left( {\sigma \left( t \right),{t_0}} \right)}} < 0, \end{array} $

(5)

所以$\frac{{A(t){{({y^\Delta }(t))}^\gamma }}}{{{e_{-b/A}}(t, {t_0})}}$ 严格递减且定号, 可断言y^Δ(t)>0.否则存在t₂∈[t₁, ∞)_T, 使得当t∈[t₂, ∞)_T时, y^Δ(t) < 0.则有

$ \begin{array}{*{20}{c}} {\frac{{A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }}}{{{e_{ - b/A}}\left( {t,{t_0}} \right)}} \leqslant \frac{{A\left( {{t_2}} \right){{\left( {{y^\Delta }\left( {{t_2}} \right)} \right)}^\gamma }}}{{{e_{ - b/A}}\left( {{t_2},{t_0}} \right)}} = }\\ { - M < 0,\;\;\;t \in {{\left[ {{t_2},\infty } \right)}_T},} \end{array} $

(6)

其中，$M = \frac{{-A({t_2}){{({y^\Delta }({t_2}))}^\gamma }}}{{{e_{-b/A}}({t_2}, {t_0})}} > 0$ .

因此，${y^\Delta }(t) \leqslant {\left({-M\frac{{{e_{-b/A}}(t, {t_0})}}{{A(t)}}} \right)^{1/\gamma }}$ ,

进一步有

$ \begin{array}{l} y\left( t \right) \leqslant y\left( {{t_2}} \right) - {M^{1/\gamma }}\int_{{t_2}}^t {{{\left( {\frac{{{e_{ - b/A}}\left( {s,{t_0}} \right)}}{{A\left( s \right)}}} \right)}^{1/\gamma }}} \Delta s \to \\ \;\;\;\;\;\;\;\;\; - \infty ,\;\;\;t \to \infty . \end{array} $

(7)

这与y(t)>0矛盾, 故y^Δ(t)>0, t∈[t₁, ∞)_T.

利用式(4), 可得

$ \begin{array}{l} {\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]^\Delta }/{\left( {{\tau ^\Delta }\left( t \right)} \right)^\gamma } + \\ b\left( t \right){\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)^\gamma } + p\left( {\tau \left( t \right)} \right){x^\gamma }\left( {\delta \left( {\tau \left( t \right)} \right)} \right) \leqslant 0, \end{array} $

(8)

综合式(8)及式(4), 当t∈[t₁, ∞)_T时, 得

$ \begin{array}{l} {\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]^\Delta } + b\left( t \right){\left( {{y^\Delta }\left( t \right)} \right)^\gamma } + p\left( t \right){x^\gamma }\left( {\delta \left( t \right)} \right) + \\ \;\;\;\;\;\;\;{b_0}\eta \left\{ {\frac{{{{\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]}^\Delta }}}{{{{\left( {{\tau ^\Delta }\left( t \right)} \right)}^\gamma }}} + b\left( t \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma } + } \right.\\ \;\;\;\;\;\;\;\left. {p\left( {\tau \left( t \right)} \right){x^\gamma }\left( {\delta \left( {\tau \left( t \right)} \right)} \right)} \right\} \leqslant 0. \end{array} $

(9)

由于y(t)≤x(t)+b₀ηx(τ(t)), 并利用ξ(t)的定义及τ^Δ(t)=τ₀>0, δ(t)≥τ(t), 得

$ \begin{array}{l} {\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]^\Delta } + {\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]^\Delta }\frac{{{b_0}\eta }}{{\tau _0^\gamma }} + \\ \;\;\;\;\;\;\;b\left( t \right){\left( {{y^\Delta }\left( t \right)} \right)^\gamma } + {b_0}\eta b\left( t \right){\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)^\gamma } \leqslant \\ \;\;\;\;\;\;\; - \xi \left( t \right){y^\gamma }\left( {\delta \left( t \right)} \right) \leqslant - \xi \left( t \right){y^\gamma }\left( {\tau \left( t \right)} \right). \end{array} $

(10)

由式(4)和y^Δ(t)>0可知，[A(t)(y^Δ(t))^γ]^Δ≤0, 故

$ A\left( t \right){\left( {{y^\Delta }\left( t \right)} \right)^\gamma } \geqslant A\left( {\sigma \left( t \right)} \right){\left( {{y^\Delta }\left( {\sigma \left( t \right)} \right)} \right)^\gamma }, $

$ A\left( {\tau \left( t \right)} \right){\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)^\gamma } \geqslant A\left( {\tau \left( {\sigma \left( t \right)} \right)} \right){\left( {{y^\Delta }\left( {\tau \left( {\sigma \left( t \right)} \right)} \right)} \right)^\gamma }. $

因此, 式(10)可写成

$ \begin{array}{l} {\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]^\Delta } + {\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]^\Delta }\frac{{{b_0}\eta }}{{\tau _0^\gamma }} \leqslant \\ - \xi \left( t \right){y^\gamma }\left( {\tau \left( t \right)} \right) - \frac{{b\left( t \right)}}{{A\left( t \right)}}{\left( {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right)^\sigma } - \\ \frac{{{b_0}\eta }}{{A\left( {\tau \left( t \right)} \right)}} \times b\left( {\tau \left( t \right)} \right){\left( {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right)^\sigma }. \end{array} $

(11)

定义函数

$ w\left( t \right) = \varphi \left( t \right) \times \frac{{A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}},\;\;\;\;\;t \in {\left[ {{t_1},\infty } \right)_T}, $

(12)

则w(t)>0.由引理2和式(12)得

$ \begin{array}{l} {w^\Delta }\left( t \right) = \frac{{\varphi \left( t \right){{\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]}^\Delta }}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}} + {\left[ {\frac{{\varphi \left( t \right)}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}}} \right]^\Delta } \times \\ {\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]^\sigma } = \frac{{\varphi \left( t \right)}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}}{\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]^\Delta } + \\ \frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}{w^\sigma }\left( t \right) - \frac{{{{\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]}^\sigma }\varphi \left( t \right){{\left[ {{y^\gamma }\left( {\tau \left( t \right)} \right)} \right]}^\Delta }}}{{{y^\gamma }\left( {\tau \left( t \right)} \right){y^{\gamma \sigma }}\left( {\tau \left( t \right)} \right)}}. \end{array} $

由链式法则和0 < γ≤1知

$ {\left[ {{y^\gamma }\left( {\tau \left( t \right)} \right)} \right]^\Delta } \geqslant \gamma {\tau _0}{\left( {{y^\sigma }\left( {\tau \left( t \right)} \right)} \right)^{\gamma - 1}}{y^\Delta }\left( {\tau \left( t \right)} \right), $

即得

$ \begin{array}{l} {w^\Delta }\left( t \right) \leqslant \frac{{\varphi \left( t \right){{\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]}^\Delta }}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}} + \frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}} \times {w^\sigma }\left( t \right) - \\ \frac{{\gamma {\tau _0}\varphi \left( t \right){{\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]}^\sigma }{{\left( {{y^\sigma }\left( {\tau \left( t \right)} \right)} \right)}^{\gamma - 1}}{y^\Delta }\left( {\tau \left( t \right)} \right)}}{{{y^\gamma }\left( {\tau \left( t \right)} \right){y^{\gamma \sigma }}\left( {\tau \left( t \right)} \right)}} \leqslant \\ \frac{{\varphi \left( t \right)}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}}{\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]^\Delta } + \frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}{w^\sigma }\left( t \right) - \\ \frac{{\gamma {\tau _0}\varphi \left( t \right){{\left( {{w^\sigma }\left( t \right)} \right)}^{1 + 1/\gamma }}}}{{{\varphi ^{1 + 1/\gamma }}\left( {\sigma \left( t \right)} \right){A^{1/\gamma }}\left( {\tau \left( t \right)} \right)}}. \end{array} $

(13)

另一方面, 定义函数

$ v\left( t \right) = \varphi \left( t \right) \times \frac{{A\left( t \right)\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}},\;\;\;\;\;t \in {\left[ {{t_1},\infty } \right)_T}. $

(14)

则v(t)>0.和上面的情形类似, 得

$ \begin{array}{l} {v^\Delta }\left( t \right) = {\left[ {\frac{{\varphi \left( t \right)}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}}} \right]^\Delta }{\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]^\sigma } + \\ \;\;\;\;\;\;\;\frac{{\varphi \left( t \right){{\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]}^\Delta }}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}} \leqslant \frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}{v^\sigma }\left( t \right) + \\ \;\;\;\;\;\;\;\frac{{\varphi \left( t \right){{\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]}^\Delta }}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}} - \gamma {\tau _0}\varphi \left( t \right) \times \\ \;\;\;\;\;\;\;{\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]^\sigma }\frac{{{{\left( {{y^\sigma }\left( {\tau \left( t \right)} \right)} \right)}^{\gamma - 1}}{y^\Delta }\left( {\tau \left( t \right)} \right)}}{{{y^\gamma }\left( {\tau \left( t \right)} \right){y^{\gamma \sigma }}\left( {\tau \left( t \right)} \right)}} \leqslant \\ \;\;\;\;\;\;\;\frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}{v^\sigma }\left( t \right) + \frac{{\varphi \left( t \right)}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}}{\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]^\Delta } - \\ \;\;\;\;\;\;\;\left( {\gamma {\tau _0}\varphi \left( t \right){{\left( {{v^\sigma }\left( t \right)} \right)}^{1 + 1/\gamma }}} \right)/\left( {{\varphi ^{1 + 1/\gamma }}\left( {\sigma \left( t \right)} \right){A^{1/\gamma }}\left( {\tau \left( t \right)} \right)} \right). \end{array} $

(15)

由式(11)、(13)、(15)和y(τ(t))≤y(σ(τ(t)))得

$ \begin{array}{l} {w^\Delta }\left( t \right) + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}{v^\Delta }\left( t \right) \leqslant \frac{{\varphi \left( t \right)}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}} \times \left\{ {{{\left[ {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right]}^\Delta } + } \right.\\ \left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}{{\left[ {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right]}^\Delta }} \right\} + \frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}{w^\sigma }\left( t \right) - \\ \frac{{\gamma {\tau _0}\varphi \left( t \right){{\left( {{w^\sigma }\left( t \right)} \right)}^{1 + 1/\gamma }}}}{{{\varphi ^{1 + 1/\gamma }}\left( {\sigma \left( t \right)} \right){A^{1/\gamma }}\left( {\tau \left( t \right)} \right)}} + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}\frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}{v^\sigma }\left( t \right) - \\ \frac{{{b_0}\eta }}{{\tau _0^\gamma }} \times \frac{{\gamma {\tau _0}\varphi \left( t \right){{\left( {{v^\sigma }\left( t \right)} \right)}^{1 + 1/\gamma }}}}{{{\varphi ^{1 + 1/\gamma }}\left( {\sigma \left( t \right)} \right){A^{1/\gamma }}\left( {\tau \left( t \right)} \right)}} \leqslant - \xi \left( t \right)\varphi \left( t \right) - \\ \frac{{\varphi \left( t \right)b\left( t \right)}}{{A\left( t \right)}} \times \frac{{{{\left( {A\left( t \right){{\left( {{y^\Delta }\left( t \right)} \right)}^\gamma }} \right)}^\sigma }}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}} - \frac{{{b_0}\eta \varphi \left( t \right)b\left( {\tau \left( t \right)} \right)}}{{A\left( {\tau \left( t \right)} \right)}} \times \\ \frac{{{{\left( {A\left( {\tau \left( t \right)} \right){{\left( {{y^\Delta }\left( {\tau \left( t \right)} \right)} \right)}^\gamma }} \right)}^\sigma }}}{{{y^\gamma }\left( {\tau \left( t \right)} \right)}} + \frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}{w^\sigma }\left( t \right) - \\ \frac{{\gamma {\tau _0}\varphi \left( t \right){{\left( {{w^\sigma }\left( t \right)} \right)}^{1 + 1/\gamma }}}}{{{\varphi ^{1 + 1/\gamma }}\left( {\sigma \left( t \right)} \right){A^{1/\gamma }}\left( {\tau \left( t \right)} \right)}} + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}\frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}{v^\sigma }\left( t \right) - \\ \frac{{{b_0}\eta }}{{\tau _0^\gamma }}\frac{{\gamma {\tau _0}\varphi \left( t \right){{\left( {{\sigma ^\sigma }\left( t \right)} \right)}^{1 + 1/\gamma }}}}{{{\varphi ^{1 + 1/\gamma }}\left( {\sigma \left( t \right)} \right){A^{1/\gamma }}\left( {\tau \left( t \right)} \right)}} \leqslant - \xi \left( t \right)\varphi \left( t \right) - \\ \frac{{\varphi \left( t \right)b\left( t \right)}}{{A\left( t \right){\varphi ^\sigma }\left( t \right)}}{w^\sigma }\left( t \right) - \frac{{{b_0}\eta \varphi \left( t \right)b\left( {\tau \left( t \right)} \right)}}{{A\left( {\tau \left( t \right)} \right){\varphi ^\sigma }\left( t \right)}}{v^\sigma }\left( t \right) + \\ \frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}{w^\sigma }\left( t \right) - \frac{{\gamma {\tau _0}\varphi \left( t \right){{\left( {{w^\sigma }\left( t \right)} \right)}^{1 + 1/\gamma }}}}{{{\varphi ^{1 + 1/\gamma }}\left( {\sigma \left( t \right)} \right){A^{1/\gamma }}\left( {\tau \left( t \right)} \right)}} + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}\frac{{{\varphi ^\Delta }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}} \times \\ {v^\sigma }\left( t \right) - \frac{{{b_0}\eta }}{{\tau _0^\gamma }} \times \frac{{\gamma {\tau _0}\varphi \left( t \right){{\left( {{v^\sigma }\left( t \right)} \right)}^{1 + 1/\gamma }}}}{{{\varphi ^{1 + 1/\gamma }}\left( {\sigma \left( t \right)} \right){A^{1/\gamma }}\left( {\tau \left( t \right)} \right)}}, \end{array} $

整理后得

$ \begin{array}{l} {w^\Delta }\left( t \right) + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}{v^\Delta }\left( t \right) \leqslant - \xi \left( t \right)\varphi \left( t \right) + \frac{{{w^\sigma }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}} \times \\ \;\;\;\;\;\;\;\;\left( {{\varphi ^\Delta }\left( t \right) - \frac{{\varphi \left( t \right)b\left( t \right)}}{{A\left( t \right)}}} \right) - \frac{{\gamma {\tau _0}\varphi \left( t \right)}}{{{A^{1/\gamma }}\left( {\tau \left( t \right)} \right)}}{\left( {\frac{{{w^\sigma }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}} \right)^{1 + 1/\gamma }} - \\ \;\;\;\;\;\;\;\;\frac{{{b_0}\eta \gamma {\tau _0}\varphi \left( t \right)}}{{\tau _0^\gamma {A^{1/\gamma }}\left( {\tau \left( t \right)} \right)}}{\left( {\frac{{{v^\sigma }\left( t \right)}}{{{\varphi ^\sigma }\left( t \right)}}} \right)^{1 + 1/\gamma }} + \frac{{{b_0}\eta {v^\sigma }\left( t \right)}}{{\tau _0^\gamma {\varphi ^\sigma }\left( t \right)}} \times \\ \;\;\;\;\;\;\;\;\left( {{\varphi ^\Delta }\left( t \right) - \frac{{\varphi \left( t \right)b\left( {\tau \left( t \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( t \right)} \right)}}} \right). \end{array} $

(16)

不妨令$\lambda = \frac{{\gamma + 1}}{\gamma }$ , 再利用不等式

$ \lambda A{B^{\lambda - 1}} - {A^\lambda } \leqslant \left( {\lambda - 1} \right){B^\lambda }, $

则式(16)变为

$ \begin{array}{l} {w^\Delta }\left( t \right) + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}{v^\Delta }\left( t \right) \leqslant - \xi \left( t \right)\varphi \left( t \right) + \\ \;\;\;\;\;\;\;\;\frac{{A\left( {\tau \left( t \right)} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{{\left( {{\tau _0}\varphi \left( t \right)} \right)}^\gamma }}}{\left( {{\varphi ^\Delta }\left( t \right) - \frac{{\varphi \left( t \right)b\left( t \right)}}{{A\left( t \right)}}} \right)^{\gamma + 1}} + \\ \;\;\;\;\;\;\;\;\frac{{{b_0}\eta }}{{\tau _0^\gamma }}{\left( {{\varphi ^\Delta }\left( t \right) - \frac{{\varphi \left( t \right)b\left( {\tau \left( t \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( t \right)} \right)}}} \right)^{\gamma + 1}}. \end{array} $

(17)

对式(17)两边从t₁到t(t≥t₁)积分, 有

$ \begin{array}{l} \int_{{t_1}}^t {\left\{ {\xi \left( s \right)\varphi \left( s \right) - \frac{{A\left( {\tau \left( s \right)} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{{\left( {{\tau _0}\varphi \left( s \right)} \right)}^\gamma }}} \times } \right.} \\ \left[ {\frac{{{b_0}\eta }}{{\tau _0^\gamma }} \times {{\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( s \right)} \right)}}} \right)}^{\gamma + 1}} + } \right.\\ \left. {\left. {{{\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( s \right)}}{{A\left( s \right)}}} \right)}^{\gamma + 1}}} \right]} \right\}\Delta s \leqslant \\ - \int_{{t_1}}^t {\left[ {{w^\Delta }\left( s \right) + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}{v^\Delta }\left( s \right)} \right]} = w\left( {{t_1}} \right) + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}v\left( {{t_1}} \right) - \\ w\left( t \right) - \frac{{{b_0}\eta }}{{\tau _0^\gamma }}{v^\Delta }\left( t \right) \leqslant w\left( {{t_1}} \right) + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}v\left( {{t_1}} \right), \end{array} $

(18)

这与已知矛盾, 故方程(1)是振动的.

注1 选取的φ(t)不同, 则会得到不同的振动准则, 例如, 在定理1中分别取φ(t)=1和φ(t)=t, 就会得到以下推论.

推论1 若

$ \begin{array}{*{20}{c}} {\mathop {\lim \sup }\limits_{t \to \infty } \int_{{t_0}}^t {\left\{ {\xi \left( s \right) - \frac{{A\left( {\tau \left( s \right)} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}\tau _0^\gamma }}\left[ {{{\left( {\frac{{b\left( s \right)}}{{A\left( s \right)}}} \right)}^{\gamma + 1}} + } \right.} \right.} }\\ {\left. {\left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}{{\left( {\frac{{b\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( s \right)} \right)}}} \right)}^{\gamma + 1}}} \right]} \right\}\Delta s = \infty ,} \end{array} $

其中ξ(t)的定义同定理1, 则方程(1)在[t₀, ∞)_T上是振动的.

推论2 若

$ \begin{array}{*{20}{c}} {\mathop {\lim \sup }\limits_{t \to \infty } \int_{{t_0}}^t {\left\{ {s\xi \left( s \right) - \frac{{A\left( {\tau \left( s \right)} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{{\left( {{\tau _0}s} \right)}^\gamma }}}\left[ {{{\left( {1 - \frac{{sb\left( s \right)}}{{A\left( s \right)}}} \right)}^{\gamma + 1}} + } \right.} \right.} }\\ {\left. {\left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}{{\left( {1 - \frac{{sb\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( s \right)} \right)}}} \right)}^{\gamma + 1}}} \right]} \right\}\Delta s = \infty ,} \end{array} $

其中ξ(t)的定义如定理1, 则方程(1)在[t₀, ∞)_T上是振动的.

注2 当γ=1时, 由定理1则可得文献[15]中的定理1.下面给出philos型振动准则.

定理2 考虑集合D≡(t, s)∈T²：t≥s≥t₀, 如果存在函数φ(t)∈C_rd([t₀, ∞)_T, R)和H, h∈C_rd(D, R), 使得

$ H\left( {t,t} \right) = 0,t \geqslant {t_0},\;\;\;H\left( {t,s} \right) > 0,t > s \geqslant {t_0}, $

(19)

且H关于第二变元s有一非正的Δ偏导数H^Δs(t, s), 并满足

$ {H^{\Delta s}}\left( {t,s} \right) + \frac{{H\left( {t,s} \right){\varphi ^\Delta }\left( s \right)}}{{\varphi \left( s \right)}} = - h\left( {t,s} \right){H^{\gamma /\left( {1 + \gamma } \right)}}\left( {t,s} \right), $

(20)

若对充分大的t₁, 有

$ \begin{array}{l} \mathop {\lim }\limits_{t \to \infty } \sup \frac{1}{{H\left( {t,{t_1}} \right)}}\int_{{t_1}}^t {\left\{ {\xi \left( s \right)\varphi \left( s \right)H\left( {t,s} \right) - } \right.} \\ \;\;\;\;\;\;\;\;\;\left. {\psi \left( s \right)\left[ {{{\left( {{\varphi _1}\left( s \right)} \right)}^{\gamma + 1}} + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}{{\left( {{\varphi _2}\left( s \right)} \right)}^{\gamma + 1}}} \right]} \right\}\Delta s = \infty , \end{array} $

(21)

其中,

$ {\varphi _1} = {h_ - }\left( {t,s} \right){\varphi ^\sigma }\left( s \right){H^{\gamma /\left( {\gamma + 1} \right)}}\left( {t,s} \right) - \frac{{H\left( {t,s} \right)\varphi \left( s \right)b\left( s \right)}}{{A\left( s \right)}}, $

$ {\varphi _2} = {h_ - }\left( {t,s} \right){\varphi ^\sigma }\left( s \right){H^{\gamma /\left( {\gamma + 1} \right)}}\left( {t,s} \right) - \frac{{H\left( {t,s} \right)\varphi \left( s \right)b\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( s \right)} \right)}}, $

$ \psi = \frac{{A\left( {\tau \left( s \right)} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}\tau _0^\gamma {\varphi ^\gamma }\left( s \right){{\left( {H\left( {t,s} \right)} \right)}^\gamma }}}, $

则方程(1)在[t₀, ∞)_T上是振动的.

证明 假设方程(1)在[t₀, ∞)_T上有一非振动解x(t), 不妨设最终为正, 即存在t₁∈[t₀, ∞)_T, 使得x(t)>0, x(τ(t))>0, x(δ(t))>0, t∈[t₁, ∞)_T, 则y(t)>0.由定理1的证明得式(16)成立.即

$ \begin{array}{l} \xi \left( s \right)\varphi \left( s \right) \leqslant - {w^\Delta }\left( s \right) - \frac{{{b_0}\eta }}{{\tau _0^\gamma }}{v^\Delta }\left( s \right) - \frac{{\gamma {\tau _0}\varphi \left( s \right)}}{{{A^{1/\gamma }}\left( {\tau \left( s \right)} \right)}} \times \\ \;\;\;\;\;\;\;\;{\left( {\frac{{{w^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}} \right)^{1 + 1/\gamma }} + \frac{{{w^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( s \right)}}{{A\left( s \right)}}} \right) + \\ \;\;\;\;\;\;\;\;\frac{{{b_0}\eta }}{{\tau _0^\gamma }}\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( s \right)} \right)}}} \right)\frac{{{v^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}} - \\ \;\;\;\;\;\;\;\;\frac{{{b_0}\eta }}{{\tau _0^\gamma }}\frac{{\gamma {\tau _0}\varphi \left( s \right)}}{{{A^{1/\gamma }}\left( {\tau \left( s \right)} \right)}}{\left( {\frac{{{v^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}} \right)^{1 + 1/\gamma }}, \end{array} $

上式两边同乘H(t, s), 并从t₁到t积分, 利用分部积分法得

$ \begin{array}{l} \int_{{t_1}}^t {\xi \left( s \right)\varphi \left( s \right)H\left( {t,s} \right)\Delta s} \leqslant \int_{{t_1}}^t { - {w^\Delta }\left( s \right)H\left( {t,s} \right)\Delta s} - \\ \frac{{{b_0}\eta }}{{\tau _0^\gamma }}\int_{{t_1}}^t {H\left( {t,s} \right){v^\Delta }\left( s \right)\Delta s} + \int_{{t_1}}^t {\left[ {\frac{{{w^\sigma }\left( s \right)H\left( {t,s} \right)}}{{{\varphi ^\sigma }\left( s \right)}} \times } \right.} \\ \left. {\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( s \right)}}{{A\left( s \right)}}} \right)} \right]\Delta s + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}\int_{{t_1}}^t {\left[ {\frac{{{v^\sigma }\left( s \right)H\left( {t,s} \right)}}{{{\varphi ^\sigma }\left( s \right)}} \times } \right.} \\ \left. {\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( s \right)} \right)}}} \right)} \right]\Delta s - \\ \int_{{t_1}}^t {\left\{ {H\left( {t,s} \right)\left[ {\frac{{\gamma {\tau _0}\varphi \left( s \right)}}{{{A^{1/\gamma }}\left( {\tau \left( s \right)} \right)}}{{\left( {\frac{{{w^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}} \right)}^{1 + 1/\gamma }} + } \right.} \right.} \\ \left. {\left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}\frac{{\gamma {\tau _0}\varphi \left( s \right)}}{{{A^{1/\gamma }}\left( {\tau \left( s \right)} \right)}}{{\left( {\frac{{{v^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}} \right)}^{1 + 1/\gamma }}} \right]} \right\}\Delta s = H\left( {t,{t_1}} \right)w\left( {{t_1}} \right) + \\ \frac{{{b_0}\eta }}{{\tau _0^\gamma }}H\left( {t,{t_1}} \right)v\left( {{t_1}} \right) + \int_{{t_1}}^t {{w^\sigma }\left( s \right){H^{\Delta s}}\left( {t,s} \right)\Delta s} + \\ \frac{{{b_0}\eta }}{{\tau _0^\gamma }}\int_{{t_1}}^t {{H^{\Delta s}}\left( {t,s} \right){v^\sigma }\left( s \right)\Delta s} + \\ \frac{{{b_0}\eta }}{{\tau _0^\gamma }}\int_{{t_1}}^t {\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( s \right)} \right)}}} \right)\frac{{{v^\sigma }\left( s \right)H\left( {t,s} \right)}}{{{\varphi ^\sigma }\left( s \right)}}\Delta s} + \\ \int_{{t_1}}^t {\frac{{{w^\sigma }\left( s \right)H\left( {t,s} \right)}}{{{\varphi ^\sigma }\left( s \right)}}\left( {{\varphi ^\Delta }\left( s \right) - \frac{{\varphi \left( s \right)b\left( s \right)}}{{A\left( s \right)}}} \right)\Delta s} - \\ \int_{{t_1}}^t {\left\{ {H\left( {t,s} \right) \times \frac{{\gamma {\tau _0}\varphi \left( s \right)}}{{{A^{1/\gamma }}\left( {\tau \left( s \right)} \right)}}\left[ {{{\left( {\frac{{{w^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}} \right)}^{1 + 1/\gamma }} + } \right.} \right.} \\ \left. {\left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}{{\left( {\frac{{{v^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}} \right)}^{1 + 1/\gamma }}} \right]} \right\}\Delta s \leqslant \\ H\left( {t,{t_1}} \right)w\left( {{t_1}} \right) + \int_{{t_1}}^t {\left\{ {\left( {{h_ - }\left( {t,s} \right){\varphi ^\sigma }\left( s \right){{\left( {H\left( {t,s} \right)} \right)}^{\gamma /\left( {\gamma + 1} \right)}}} \right) + } \right.} \\ \left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}\left( {H\left( {t,{t_1}} \right)v\left( {{t_1}} \right) - \frac{{H\left( {t,s} \right)\varphi \left( s \right)b\left( s \right)}}{{A\left( s \right)}}} \right)\frac{{{w^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}} \right\}\Delta s\\ \frac{{{b_0}\eta }}{{\tau _0^\gamma }}\int_{{t_1}}^t {\left\{ {\frac{{{v^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}\left( {{h_ - }\left( {t,s} \right){\varphi ^\sigma }\left( s \right){{\left( {H\left( {t,s} \right)} \right)}^{\gamma /\left( {\gamma + 1} \right)}} - } \right.} \right.} \\ \left. {\left. {\frac{{H\left( {t,s} \right)\varphi \left( s \right)b\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}A\left( {\tau \left( s \right)} \right)}}} \right)} \right\}\Delta s - \\ \int_{{t_1}}^t {\left\{ {H\left( {t,s} \right)\left[ {\frac{{\gamma {\tau _0}\varphi \left( s \right)}}{{{A^{1/\gamma }}\left( {\tau \left( s \right)} \right)}}{{\left( {\frac{{{w^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}} \right)}^{1 + 1/\gamma }} + } \right.} \right.} \\ \left. {\left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}\frac{{\gamma {\tau _0}\varphi \left( s \right)}}{{{A^{1/\gamma }}\left( {\tau \left( s \right)} \right)}}{{\left( {\frac{{{v^\sigma }\left( s \right)}}{{{\varphi ^\sigma }\left( s \right)}}} \right)}^{1 + 1/\gamma }}} \right]} \right\}\Delta s, \end{array} $

再利用不等式λAB^λ-1-A^λ≤(λ-1)B^λ, 得

$ \begin{array}{l} \int_{{t_1}}^t {\xi \left( s \right)\varphi \left( s \right)H\left( {t,s} \right)\Delta s} \leqslant \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;H\left( {t,{t_1}} \right)w\left( {{t_1}} \right) + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}H\left( {t,{t_1}} \right)v\left( {{t_1}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\int_{{t_1}}^t {\left\{ {\psi \left( s \right)\left[ {{{\left( {{\varphi _1}\left( s \right)} \right)}^{\gamma + 1}} + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}{{\left( {{\varphi _2}\left( s \right)} \right)}^{\gamma + 1}}} \right]} \right\}\Delta s} , \end{array} $

即

$ \begin{array}{l} \frac{1}{{H\left( {t,{t_1}} \right)}}\int_{{t_1}}^t {\left\{ {\xi \left( s \right)\varphi \left( s \right)H\left( {t,s} \right) - \psi \left( s \right)\left[ {{{\left( {{\varphi _1}\left( s \right)} \right)}^{\gamma + 1}} + } \right.} \right.} \\ \;\;\;\;\;\left. {\left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}{{\left( {{\varphi _2}\left( s \right)} \right)}^{\gamma + 1}}} \right]} \right\}\Delta s \leqslant w\left( {{t_1}} \right) + \frac{{{b_0}\eta }}{{\tau _0^\gamma }}v\left( {{t_1}} \right). \end{array} $

与式(21)矛盾, 故方程(1)的解是振动的.

注3 选择不同的H(t, s)及φ(t), 就会得到不同的振动准则; 同时当γ=1时得到了文献[15]中的定理2.

显然, 本文给出了一类新的具有阻尼项的二阶非线性时滞中立型动力方程的振动性, 改进并推广了文献[15]的相关结果.

2 例子

例1考虑下列非线性二阶动力方程

$ \begin{array}{l} {\left[ {{{\left( {{{\left( {x\left( t \right) + \left( {2 + \cos t} \right)g\left( {x\left( {\frac{t}{2}} \right)} \right)} \right)}^\Delta }} \right)}^\gamma }} \right]^\Delta } + \frac{1}{{{t^{5/2}}}} \times \\ {x^\gamma }\left( {\frac{t}{2}} \right) + \frac{1}{{{t^2}}}{\left( {{{\left( {x\left( t \right) + \left( {2 + \cos t} \right)g\left( {x\left( {\frac{t}{2}} \right)} \right)} \right)}^\Delta }} \right)^\gamma } = 0, \end{array} $

T=2^Z的振动性.

解  显然, 上述方程是具有非线性的二阶2-差分方程, 其中, t≥t₀=2, B(t)=2+cos t, A(t)=1, $\delta (t) = \tau (t) = \frac{t}{2}$ , $b(t) = \frac{1}{{{t^{5/2}}}}, $ , $p(t) = \frac{7}{{{t^2}}}, $ , $g(u) = \frac{u}{{\sqrt {1 + si{n^2}\left({u + 1} \right)} }}, $ , $\delta (t) = \tau (t) = \frac{t}{2} \leqslant t$ 并且limt→∞ δ(t)=$\mathop {{\rm{lim}}}\limits_{t \to \infty } \delta (t) = \mathop {{\rm{lim}}}\limits_{t \to \infty } \tau (t) = \infty $ , τ是严格递增的且

$ \begin{array}{*{20}{c}} {{\tau ^\Delta }\left( t \right) = \frac{1}{2} > 0,0 < B\left( t \right) \leqslant 3 = {b_0} < \infty ,}\\ {\frac{{g\left( u \right)}}{u} = \frac{1}{{\sqrt {1 + {{\sin }^2}\left( {u + 1} \right)} }} \leqslant 1 = \eta .} \end{array} $

因为$1-\mu (t)\frac{{b(t)}}{{A(t)}} = 1-t\cdot\frac{1}{{{t^{5/2}}}} > 0$ , 则$\frac{{-b}}{A}$ ∈R ⁺,

$ \begin{array}{*{20}{c}} {{e_{ - b/A}}\left( {s,{t_0}} \right) \geqslant 1 - \int_2^t {\frac{{b\left( s \right)}}{{A\left( s \right)}}\Delta s} = 1 - \int_2^t {{s^{ - 5/2}}\Delta s} = 1 - }\\ {\frac{{{t^{ - 3/2}} - {2^{ - 3/2}}}}{{{2^{ - 3/2}} - 1}} \geqslant {t^{ - 3/2}} - {2^{ - 1/2}} + 1 \geqslant \frac{1}{4},} \end{array} $

故$\int_{{t_0}}^{^{ + \infty }} {{{\left({\frac{{{e_{-b/A}}(s, {t_0})}}{{A\left(s \right)}}} \right)}^{1/\gamma }}\Delta s \geqslant \int_{{t_0}}^{^{ + \infty }} {{{\left({\frac{1}{4}} \right)}^{1/\gamma }}\Delta s = \infty } } $ .

满足(H₁)~(H₆)，再根据ξ(t)的定义, 得ξ(t)=$\frac{7}{{{t^2}}}$ , 且

$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to \infty } \int_{{t_0}}^t {\left\{ {s\xi \left( s \right) - \frac{{A\left( {\tau \left( s \right)} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{{\left( {{\tau _0}s} \right)}^\gamma }}}\left[ {{{\left( {1 - \frac{{sb\left( s \right)}}{{{A^\sigma }\left( s \right)}}} \right)}^{\gamma + 1}} + } \right.} \right.} \\ \left. {\left. {\frac{{{b_0}\eta }}{{\tau _0^\gamma }}{{\left( {1 - \frac{{sb\left( {\tau \left( s \right)} \right)}}{{\tau _0^{ - 1}{A^\sigma }\left( {\tau \left( s \right)} \right)}}} \right)}^{\gamma + 1}}} \right]} \right\}\Delta s = \\ \mathop {\lim \sup }\limits_{t \to \infty } \int_{{t_0}}^t {\left\{ {\frac{7}{s} - \frac{{{2^\gamma }}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{s^\gamma }}}\left[ {{{\left( {1 - {s^{ - 3/2}}} \right)}^{\gamma + 1}} + } \right.} \right.} \\ \left. {\left. {3 \times {2^\gamma }{{\left( {1 - {2^{3/2}}{s^{ - 3/2}}} \right)}^{\gamma + 1}}} \right]} \right\}\Delta s \geqslant \\ \mathop {\lim \sup }\limits_{t \to \infty } \int_{{t_0}}^t {\left\{ {\frac{7}{s} - \frac{{{2^\gamma }\left( {1 + 3 \times {2^\gamma }} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{s^\gamma }}}{{\left( {1 - {2^{3/2}}{s^{ - 3/2}}} \right)}^{\gamma + 1}}} \right\}\Delta s} \geqslant \\ \mathop {\lim \sup }\limits_{t \to \infty } \int_{{t_0}}^t {\left\{ {\frac{7}{s} - \frac{{{2^\gamma }\left( {1 + 3 \times {2^\gamma }} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{s^\gamma }}}} \right\}\Delta s} = \infty . \end{array} $

由推论2可知该方程的解是振动的.

3 结论

基于时间尺度上的微积分理论、Riccati变换、H函数法和不等式技巧, 并利用假设和引理讨论了时间尺度上一类新的具有阻尼项的二阶非线性时滞中立型动力方程的振动性, 得到该方程的4个振动准则.特别地, 当γ=1时, 本文结果便是文献[15]的相关结果, 因此在理论上是对文献[15]的推广; 同时, 本文首次研究了方程(1)的振动性, 丰富了二阶时滞动力方程的振动结果.