2. 梧州学院 复杂系统仿真与智能计算实验室, 广西 梧州 543002
2. Laboratory of Complex Systems Simulation and Intelligent Computing, Wuzhou University, Wuzhou 543002, Guangxi Zhuang Autonomous Region, China
在微分方程定性理论的研究中,振动性理论作为重要的研究方向之一,具有极为广泛的应用背景,近年来引起了国内外众多学者的极大兴趣和高度关注,该研究领域取得了大量成果[1-22].笔者考虑如下一类形式非常广泛的具有非线性中立项的二阶Emden-Fowler型微分方程:
$ \begin{array}{*{20}{c}} {{{\left\{ {a\left( t \right){{\left| {{{\left[ {x\left( t \right) + p\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right)} \right]}^\prime }} \right|}^{\beta - 1}}{{\left[ {x\left( t \right) + p\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right)} \right]}^\prime }} \right\}}^\prime } + }\\ {q\left( t \right){{\left| {x\left( {\delta \left( t \right)} \right)} \right|}^{\gamma - 1}}x\left( {\delta \left( t \right)} \right) = 0,\;\;\;\;\;t \ge {t_0}} \end{array} $ | (1) |
的振动性.为了叙述方便,假设:
(H1)常数0 < α≤1和γ>0为2个正奇数的商,而β>0;函数a∈C1([t0, +∞), (0, +∞))且
(H2)时滞函数τ, δ:[t0, +∞)→(0, +∞)满足:τ(t)≤t且
关于方程(1) 的解及其振动性的定义可参见文献[1]或[9].对具有非线性中立项的微分方程振动性的研究是一项很困难的工作,因此学者们或是回避这类方程,或是通过附加一些条件将其转化为线性中立项进行讨论[1-4, 6-17].仅有文献[5]直接研究具有一个拟线性中立项的一阶微分方程:
$ \begin{array}{l} {\left[ {x\left( t \right) - p{x^\alpha }\left( {t - \tau } \right)} \right]^\prime } + \\ \;\;\;\;\;\;\;q\left( t \right)\prod\limits_{j = 1}^m {{{\left| {x\left( {t - {\sigma _j}} \right)} \right|}^{{\beta _j}}}{\mathop{\rm sgn}} \left| {x\left( {t - {\sigma _j}} \right)} \right|} = 0, \end{array} $ |
得到了其解振动的一些判别准则.
最近,文献[15]研究了方程(1) 的特殊情形(即当α=1时,相当于中立项是线性的情形)的振动性,得到了如下结果:
定理A 设β≥γ, a′(t)≥0且
$ \int_{{t_0}}^{ + \infty } {\left\{ {\varphi \left( s \right)Q\left( s \right) - \frac{{a\left( {\delta \left( s \right)} \right){{\left( {\varphi '\left( s \right)} \right)}^{\gamma + 1}}}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{{\left( {k\varphi \left( s \right)\delta '\left( s \right)} \right)}^\gamma }}}} \right\}{\rm{d}}s} = + \infty , $ |
则方程
$ \begin{array}{l} {\left\{ {a\left( t \right){{\left| {z'\left( t \right)} \right|}^{\beta - 1}}z'\left( t \right)} \right\}^\prime } + \\ \;\;\;\;\;\;\;q\left( t \right){\left| {x\left( {\delta \left( t \right)} \right)} \right|^{\gamma - 1}}x\left( {\delta \left( t \right)} \right) = 0\left( {t \ge {t_0}} \right) \end{array} $ | (2) |
是振动的,其中函数Q(t)=q(t)[1-p(δ(t))]γ,z(t)=x(t)+p(t)x(τ(t)),k>0为常数.
定理A是文献[15]中的定理2.2,也是其主要结果.值得注意的是,当β < γ时,文献[15]没有得到方程(2) 的振动准则,且其条件“a′(t)≥0”似乎较为苛刻.受以上研究的启发,笔者利用Riccati变换技术和多种不等式(如Bernoulli不等式、Yang不等式和Hölder不等式等)技巧来研究具有非线性中立项的微分方程(1) 的振动性,得到了该方程振动的Kamenev型振动准则,而作为方程(1) 的特殊情形,即当α=β=γ=1或者α=1且β=γ时的情形,本文的这些振动准则改进了现有文献中的一系列结果.
1 方程振动的判别定理首先给出4个引理,其中引理1由函数f(x)=xλ(0 < λ≤1) 的凹凸性便可证得,引理2~4为公知的不等式,略去其证明.
引理1 设X,Y为非负实数,则当0 < λ≤1时,Xλ+Yλ≤21-λ(X+Y)λ.
引理2(Bernoulli不等式) 对任意实数x>-1,当0≤r≤1时,(1+x)r≤1+rx,当r≤0或r≥1时,(1+x)r≥1+rx.
引理3(Yang不等式) 设a>0及b>0均为常数,则
引理4(Hölder不等式)
方便起见,引入下列记号:
$ \begin{array}{*{20}{c}} {z\left( t \right) = x\left( t \right) + p\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right),}\\ {\varphi_+ \left( t \right) = \max \left\{ {\varphi \left( t \right),0} \right\},\;\;\;\;\mathit{\Theta }\left( t \right) = \int_{{t_0}}^t {\frac{1}{{{a^{1/\beta }}\left( s \right)}}{\rm{d}}s} .} \end{array} $ |
定理1 设(H1)和(H2)成立,若存在函数φ∈C1([t0, +∞), (0, +∞))及常数ω≥0,使得
$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to + \infty } \frac{1}{{{t^\omega }}}\int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\left\{ {\varphi \left( s \right)Q\left( s \right) - } \right.} \\ \left. {\frac{{{\beta ^\beta }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{{\left[ {\gamma \theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}}{{\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|}^{\beta + 1}}} \right\}{\rm{d}}s = + \infty , \end{array} $ | (3) |
其中函数
$ \begin{gathered} Q\left( t \right) = q\left( t \right){\left[ {1 - \left( {\alpha {2^{1 - \alpha }} + \frac{{\left( {{2^{1 - \alpha }} - 1} \right)}}{k}} \right)p\left( {\delta \left( t \right)} \right)} \right]^\gamma },\;\;\; \hfill \\ \;\;\;\;\theta \left( t \right) = \left\{ \begin{gathered} {k^{\left( {\gamma - \beta } \right)/\beta }},\beta < \gamma , \hfill \\ 1,\beta = \gamma , \hfill \\ {\left[ {m\mathit{\Theta }\left( {\delta \left( t \right)} \right)} \right]^{\left( {\gamma - \beta } \right)/\beta }},\beta > \gamma , \hfill \\ \end{gathered} \right. \hfill \\ \end{gathered} $ | (4) |
而t2≥t0,k>0和m>0均为常数,则方程(1) 是振动的.
证明 反证法:设方程(1) 存在一个非振动解x(t),不失一般性,设x(t)最终为正(当x(t)最终为负时类似可证),则
$ {\left[ {a\left( t \right){{\left| {z'\left( t \right)} \right|}^{\beta - 1}}z'\left( t \right)} \right]^\prime } = - q\left( t \right){x^\gamma }\left( {\delta \left( t \right)} \right) < 0, $ | (5) |
利用条件(H1),由式(5) 易得z′(t)>0(t≥t1).分别利用引理1及引理2,可得
$ \begin{array}{l} x\left( t \right) = z\left( t \right) - p\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right) = \\ \;\;\;\;\;\;\;\;\;\;z\left( t \right) - p\left( t \right)\left[ {1 + {x^\alpha }\left( {\tau \left( t \right)} \right)} \right] + p\left( t \right) \ge \\ \;\;\;\;\;\;\;\;\;\;z\left( t \right) - {2^{1 - \alpha }}p\left( t \right){\left[ {1 + x\left( {\tau \left( t \right)} \right)} \right]^\alpha } + p\left( t \right) \ge \\ \;\;\;\;\;\;\;\;\;\;z\left( t \right) - {2^{1 - \alpha }}p\left( t \right)\left[ {1 + \alpha x\left( {\tau \left( t \right)} \right)} \right] + p\left( t \right) = \\ \;\;\;\;\;\;\;\;\;\;z\left( t \right) - \alpha {2^{1 - \alpha }}p\left( t \right)x\left( {\tau \left( t \right)} \right) + \left( {1 - {2^{1 - \alpha }}} \right)p\left( t \right) \ge \\ \;\;\;\;\;\;\;\;\;\;z\left( t \right) - \alpha {2^{1 - \alpha }}p\left( t \right)z\left( {\tau \left( t \right)} \right) + \left( {1 - {2^{1 - \alpha }}} \right)p\left( t \right) \ge \\ \;\;\;\;\;\;\;\;\;\;\left[ {1 - \alpha {2^{1 - \alpha }}p\left( t \right)} \right]z\left( t \right) - \left( {{2^{1 - \alpha }} - 1} \right)p\left( t \right). \end{array} $ | (6) |
定义函数w(t)如下:
$ \begin{array}{l} w\left( t \right) = \varphi \left( t \right)\frac{{a\left( t \right){{\left| {z'\left( t \right)} \right|}^{\beta - 1}}z'\left( t \right)}}{{{{\left| {z\left( {\delta \left( t \right)} \right)} \right|}^{\gamma - 1}}z\left( {\delta \left( t \right)} \right)}} = \\ \;\;\;\;\;\;\;\;\;\;\varphi \left( t \right)\frac{{a\left( t \right){{\left( {z'\left( t \right)} \right)}^\beta }}}{{{z^\gamma }\left( {\delta \left( t \right)} \right)}},t \ge {t_1}, \end{array} $ | (7) |
则有w(t)>0(t≥t1).利用式(5),(6) 及a(t)(z′(t))β≤a(δ(t))(z′(δ(t)))β,由式(7) 可推得
$ \begin{array}{l} w'\left( t \right) = \varphi '\left( t \right)\frac{{a\left( t \right){{\left( {z'\left( t \right)} \right)}^\beta }}}{{{z^\gamma }\left( {\delta \left( t \right)} \right)}} + \varphi \left( t \right) \times \\ \frac{{{{\left[ {a\left( t \right){{\left( {z'\left( t \right)} \right)}^\beta }} \right]}^\prime }{z^\gamma }\left( {\delta \left( t \right)} \right) - a\left( t \right){{\left( {z'\left( t \right)} \right)}^\beta }\gamma {z^{\gamma - 1}}\left( {\delta \left( t \right)} \right)z'\left( {\delta \left( t \right)} \right)\delta '\left( t \right)}}{{{z^{2\gamma }}\left( {\delta \left( t \right)} \right)}} \le \\ \frac{{\varphi '\left( t \right)}}{{\varphi \left( t \right)}}w\left( t \right) - \varphi \left( t \right)\frac{{q\left( t \right){x^\gamma }\left( {\delta \left( t \right)} \right)}}{{{z^\gamma }\left( {\delta \left( t \right)} \right)}} - \gamma \frac{{\varphi \left( t \right)a\left( t \right){{\left( {z'\left( t \right)} \right)}^\beta }\delta '\left( t \right)}}{{{z^{\gamma + 1}}\left( {\delta \left( t \right)} \right)}} \times \\ \frac{{{a^{1/\beta }}\left( t \right)z'\left( t \right)}}{{{a^{1/\beta }}\left( {\delta \left( t \right)} \right)}} \le \frac{{\varphi '\left( t \right)}}{{\varphi \left( t \right)}}w\left( t \right) - \varphi \left( t \right)q\left( t \right) \times \\ {\left[ {\frac{{\left( {1 - \alpha {2^{1 - \alpha }}p\left( {\delta \left( t \right)} \right)} \right)z\left( {\delta \left( t \right)} \right) - \left( {{2^{1 - \alpha }} - 1} \right)p\left( {\delta \left( t \right)} \right)}}{{z\left( {\delta \left( t \right)} \right)}}} \right]^\gamma } - \\ \gamma \frac{{\varphi \left( t \right){a^{\left( {\beta + 1} \right)/\beta }}\left( t \right){{\left( {z'\left( t \right)} \right)}^{\beta + 1}}\delta '\left( t \right)}}{{{z^{\gamma + 1}}\left( {\delta \left( t \right)} \right){a^{1/\beta }}\left( {\delta \left( t \right)} \right)}}. \end{array} $ | (8) |
注意到z(t)>0, z′(t)>0(t≥t1),因此
$ z\left( {\delta \left( t \right)} \right) \ge z\left( {\delta \left( {{t_1}} \right)} \right) = k\left( {t \ge {t_1}} \right). $ | (9) |
其中常数k>0.于是,利用式(7)、(9) 和函数Q(t)的定义,由式(8),可得
$ \begin{array}{l} w'\left( t \right) \le \frac{{\varphi '\left( t \right)}}{{\varphi \left( t \right)}}w\left( t \right) - \varphi \left( t \right)Q\left( t \right) - \\ \;\;\;\;\;\;\;\;\;\;\;\;\gamma \frac{{{z^{\left( {\gamma - \beta } \right)/\beta }}\left( {\delta \left( t \right)} \right)\delta '\left( t \right)}}{{{\varphi ^{1/\beta }}\left( t \right){a^{1/\beta }}\left( {\delta \left( t \right)} \right)}}{w^{\left( {\beta + 1} \right)/\beta }}. \end{array} $ | (10) |
(ⅰ)若β=γ,则z(γ-β)/β(δ(t))=1.
(ⅱ)若β < γ,则由式(9) 知,z(γ-β)/β(δ(t))≥k(γ-β)/β.
(ⅲ)若β>γ,则由式(5) 知,当s≥t1时,a(s)(z′(s))β≤a(t1)(z′(t1))β=M(这里M>0为常数),即
$ \begin{array}{l} z\left( t \right) \le z\left( {{t_1}} \right) + {M^{1/\beta }}\int_{{t_1}}^t {\frac{1}{{{a^{1/\beta }}\left( s \right)}}{\rm{d}}s} \le \\ \;\;\;\;\;\;\;\;\;z\left( {{t_1}} \right) + {M^{1/\beta }}\int_{{t_0}}^t {\frac{1}{{{a^{1/\beta }}\left( s \right)}}{\rm{d}}s} , \end{array} $ |
因此存在常数m>0,使得当t2≥t1充分大时,有
$ z\left( t \right) \le m\int_{{t_0}}^t {\frac{1}{{{a^{1/\beta }}\left( s \right)}}{\rm{d}}s} = m\mathit{\Theta }\left( t \right)\left( {t \ge {t_2}} \right), $ |
于是
$ {z^{\left( {\gamma - \beta } \right)/\beta }}\left( {\delta \left( t \right)} \right) \ge {\left[ {m\mathit{\Theta }\left( {\delta \left( t \right)} \right)} \right]^{\left( {\gamma - \beta } \right)/\beta }}. $ |
综合上述3种情形,并注意到函数θ(t)的定义,由式(10),得
$ \begin{array}{l} w'\left( t \right) \le - \varphi \left( t \right)Q\left( t \right) + \frac{{\varphi '\left( t \right)}}{{\varphi \left( t \right)}}w\left( t \right) - \\ \;\;\;\;\;\;\;\;\;\;\;\frac{{\gamma \theta \left( t \right)\delta '\left( t \right)}}{{{\varphi ^{1/\beta }}\left( t \right){a^{1/\beta }}\left( {\delta \left( t \right)} \right)}}{w^{\left( {\beta + 1} \right)/\beta }}\left( t \right),\;\;\;\;t \ge {t_2}. \end{array} $ | (11) |
现将式(11) 中的t改成s,两边同乘以(t-s)ω,再从t2到t(t≥t2)积分,得
$ \begin{array}{l} \int_{{t_2}}^t {\varphi \left( s \right)Q\left( s \right){{\left( {t - s} \right)}^\omega }{\rm{d}}s} \le - \int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }w'\left( s \right){\rm{d}}s} + \\ \;\;\;\;\;\;\;\;\;\;\;\;\int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}}w\left( s \right){\rm{d}}s} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\frac{{\gamma \theta \left( s \right)\delta '\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}{w^{\left( {\beta + 1} \right)/\beta }}\left( s \right){\rm{d}}s} = \\ \;\;\;\;\;\;\;\;\;\;\;\;{\left( {t - {t_2}} \right)^\omega }w\left( {{t_2}} \right) + \int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\left\{ {\left[ {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right]} \right.w\left( s \right)} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{\gamma \theta \left( s \right)\delta '\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}{w^{\left( {\beta + 1} \right)/\beta }}\left( s \right)} \right\}{\rm{d}}s \le {\left( {t - {t_2}} \right)^\omega }w\left( {{t_2}} \right) + \\ \;\;\;\;\;\;\;\;\;\;\;\;\int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\left\{ {\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|} \right.w\left( s \right)} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{\gamma \theta \left( s \right)\delta '\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}{w^{\left( {\beta + 1} \right)/\beta }}\left( s \right)} \right\}{\rm{d}}s. \end{array} $ | (12) |
令
$ \begin{array}{*{20}{c}} {a = \frac{{{{\left[ {\left( {\left( {\beta + 1} \right)\gamma /\beta } \right)\theta \left( s \right)\delta '\left( s \right)} \right]}^{\beta /\left( {\beta + 1} \right)}}}}{{{{\left[ {\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)} \right]}^{1/\left( {\beta + 1} \right)}}}}w\left( s \right),}\\ {b = \frac{{{{\left[ {\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)} \right]}^{1/\left( {\beta + 1} \right)}}}}{{{{\left[ {\left( {\left( {\beta + 1} \right)\gamma /\beta } \right)\theta \left( s \right)\delta '\left( s \right)} \right]}^{\beta /\left( {\beta + 1} \right)}}}}\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|,} \end{array} $ |
代入引理3的不等式:
$ \begin{array}{l} \left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|w\left( s \right) - \frac{{\gamma \theta \left( s \right)\delta '\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}{w^{\left( {\beta + 1} \right)/\beta }}\left( s \right) \le \\ \;\;\;\;\;\;\;\;\;\;\;\frac{{{\beta ^\beta }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{{\left[ {\gamma \theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}}{\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|^{\beta + 1}}. \end{array} $ |
代入式(12),得
$ \begin{array}{l} \int_{{t_2}}^t {\varphi \left( s \right)Q\left( s \right){{\left( {t - s} \right)}^\omega }{\rm{d}}s} \le {\left( {t - {t_2}} \right)^\omega }w\left( {{t_2}} \right) + \\ \;\;\;\;\;\;\;\;\int_{{t_2}}^t {\frac{{{\beta ^\beta }{{\left( {t - s} \right)}^\omega }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{{\left[ {\gamma \theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}}} {\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|^{\beta + 1}}{\rm{d}}s, \end{array} $ |
所以
$ \begin{array}{l} \frac{1}{{{t^\omega }}}\int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\left\{ {\varphi \left( s \right)Q\left( s \right) - \frac{{{\beta ^\beta }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{{\left[ {\gamma \theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}}} \right.} \times \\ \;\;\;\;\;\;\;\;\left. {{{\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left(s \right)}} - \frac{\omega }{{t - s}}} \right|}^{\beta + 1}}} \right\}{\rm{d}}s \le {\left( {1 - \frac{{{t_2}}}{t}} \right)^\omega }w\left( {{t_2}} \right) \le w\left( {{t_2}} \right), \end{array} $ | (13) |
与式(4) 矛盾.定理证毕.
推论1 设(H1)和(H2)成立,若存在函数φ∈C1([t0, +∞), (0, +∞)),使得
$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to + \infty } \int_{{t_2}}^t {\left\{ {\varphi \left( s \right)Q\left( s \right) - \frac{{{\beta ^\beta }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{{\left[ {\gamma \theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}} \times } \right.} \\ \;\;\;\;\;\;\;\left. {{{\left( {\frac{{{{\varphi }_ + }'\left( s \right)}}{{\varphi \left( s \right)}}} \right)}^{\beta + 1}}} \right\}{\rm{d}}s = + \infty , \end{array} $ |
其中常数t2,k,m及函数Q(t)和θ(t)的定义都同定理1,则方程(1) 是振动的.
证明 在定理1中,取ω=0,即可得推论1成立.
推论2 设(H1)和(H2)成立,如果
$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to + \infty } \int_{{t_0}}^t {\left\{ {{\mathit{\Theta }^\beta }\left( {\delta \left( s \right)} \right)q\left( s \right) - \frac{{{{\left[ {\beta /\left( {\beta + 1} \right)} \right]}^{\beta + 1}}\delta '\left( s \right)}}{{{a^{1/\beta }}\left( {\delta \left( s \right)} \right)\mathit{\Theta }\left( {\delta \left( s \right)} \right)}}} \right\}{\rm{d}}s} = \\ \;\;\;\;\;\;\;\;\;\; + \infty , \end{array} $ |
则方程
$ \begin{array}{l} {\left[ {a\left( t \right){{\left| {x'\left( t \right)} \right|}^{\beta - 1}}x'\left( t \right)} \right]^\prime } + \\ \;\;\;\;\;\;q\left( t \right){\left| {x\left( {\delta \left( t \right)} \right)} \right|^{\beta - 1}}x\left( {\delta \left( t \right)} \right) = 0\left( {t \ge {t_0}} \right) \end{array} $ | (14) |
是振动的.
证明 在方程(1) 中,令p(t)≡0且β=γ,并在推论1中取φ(t)=Θβ(δ(t)),即可得推论2.
注1 当α=1(即中立项是线性的情形)且β≥γ时,由推论1可得定理A(即文献[15]中的定理2.2),但这里去掉了文献[15]中的限制条件“a′(t)≥0”,且当β≤γ时也满足方程(1) 的振动准则;而推论2就是SUN等[16]得到的关于方程(14) 振动的主要判别定理.
若定理1中的条件(3) 不成立,则方程(1) 的振动准则如下:
定理2设(H1)和(H2)成立,若存在函数φ(t)∈C1([t0, +∞), (0, +∞))及ξ1(t), ξ2(t)∈L2([t0, +∞), R),使得对
$ \mathop {\lim \sup }\limits_{t \to + \infty } \frac{1}{{{t^\omega }}}\int_u^t {{{\left( {t - s} \right)}^\omega }\varphi \left( s \right)Q\left( s \right){\rm{d}}s} \ge {\xi _1}\left( u \right), $ | (15) |
$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to + \infty } \frac{1}{{{t^\omega }}}\int_u^t {\frac{{{{\left( {t - s} \right)}^\omega }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left[ {\theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}} \times } \\ \;\;\;\;\;\;\;\;\;\;\;{\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|^{\beta + 1}}{\rm{d}}s \le {\xi _2}\left( u \right), \end{array} $ | (16) |
且ξ1和ξ2满足
$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to + \infty } \frac{1}{{{t^\omega }}}\int_{{t_2}}^t {\frac{{{{\left( {t - s} \right)}^\omega }\theta \left( s \right)\delta '\left( s \right)\left[ {{\xi _1}\left( s \right) - \zeta {\xi _2}\left( s \right)} \right]_ + ^{\left( {\beta + 1} \right)/\beta }}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}{\rm{d}}s} = \\ \;\;\;\;\;\;\;\; + \infty , \end{array} $ | (17) |
其中,常数t2≥t0,
证明 同定理1的证明,可得式(12) 和(13).首先,根据式(13),当t≥u≥t2≥t0时,有
$ \begin{array}{l} \frac{1}{{{t^\omega }}}\int_u^t {{{\left( {t - s} \right)}^\omega }\left\{ {\varphi \left( s \right)Q\left( s \right) - \frac{{{\beta ^\beta }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{{\left[ {\gamma \theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}} \times } \right.} \\ \;\;\;\;\;\;\;\;\;\left. {{{\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|}^{\beta + 1}}} \right\}{\rm{d}}s \le w\left( u \right), \end{array} $ |
便可得
$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to + \infty } \frac{1}{{{t^\omega }}}\int_u^t {{{\left( {t - s} \right)}^\omega }\varphi \left( s \right)Q\left( s \right){\rm{d}}s} \le w\left( u \right) + \\ \mathop {\lim \sup }\limits_{t \to + \infty } \frac{1}{{{t^\omega }}}{\int_u^t {\frac{{{\beta ^\beta }{{\left( {t - s} \right)}^\omega }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{{\left[ {\gamma \theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}}\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|} ^{\beta + 1}}{\rm{d}}s, \end{array} $ |
利用式(15) 和(16),即可得
$ \begin{array}{l} {\xi _1}\left( u \right) \le w\left( u \right) + \frac{{{\beta ^\beta }}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{\gamma ^\beta }}}{\xi _2}\left( u \right),\\ {\xi _1}\left( u \right) - \zeta {\xi _2}\left( u \right) \le w\left( u \right),u \ge {t_2} \ge {t_0}. \end{array} $ | (18) |
其次,由式(12) 可得
$ \begin{array}{l} \frac{1}{{{t^\omega }}}\int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\left\{ {\frac{{\gamma \theta \left( s \right)\delta '\left( s \right){w^{\left( {\beta + 1} \right)/\beta }}\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}} - } \right.} \\ \;\;\;\;\;\;\;\left. {\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|w\left( s \right)} \right\}{\rm{d}}s \le \\ \;\;\;\;\;\;\;{\left( {1 - \frac{{{t_2}}}{t}} \right)^\omega }w\left( {{t_2}} \right) - \frac{1}{{{t^\omega }}}\int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\varphi \left( s \right)Q\left( s \right){\rm{d}}s} \le \\ \;\;\;\;\;\;\;w\left( {{t_2}} \right) - \frac{1}{{{t^\omega }}}\int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\varphi \left( s \right)Q\left( s \right){\rm{d}}s} . \end{array} $ |
注意到式(15),有
$ \begin{array}{l} \mathop {\lim \inf }\limits_{t \to + \infty } \frac{1}{{{t^\omega }}}\int_{{t_2}}^t {{{\left( {t - s} \right)}^\omega }\left\{ {\frac{{\gamma \theta \left( s \right)\delta '\left( s \right){w^{\left( {\beta + 1} \right)/\beta }}\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}} - } \right.} \\ \;\;\;\;\;\;\;\left. {\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{t - s}}} \right|w\left( s \right)} \right\}{\rm{d}}s \le \\ \;\;\;\;\;\;\;w\left( {{t_2}} \right) - {\xi _1}\left( {{t_2}} \right) \le {M_0}, \end{array} $ | (19) |
其中M0是常数.这样,根据式(19) 就可断言:
$ \mathop {\lim \inf }\limits_{t \to + \infty } \int_{{t_2}}^t {\frac{{{{\left( {t - s} \right)}^\omega }}}{{{t^\omega }}}\frac{{\theta \left( s \right)\delta '\left( s \right){w^{\left( {\beta + 1} \right)/\beta }}\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}{\rm{d}}s} < + \infty . $ | (20) |
事实上,若式(20) 不成立,则一定存在序列
$ \mathop {\lim }\limits_{u \to + \infty } \int_{{t_2}}^{{T_n}} {\frac{{\gamma {{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\theta \left( s \right)\delta '\left( s \right){w^{\left( {\beta + 1} \right)/\beta }}\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}} {\rm{d}}s = + \infty . $ | (21) |
于是,综合式(19) 和(21) 可推得
$ \mathop {\lim }\limits_{n \to + \infty } \int_{{t_2}}^{{T_n}} {{{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }} \left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{{T_n} - s}}} \right|w\left( s \right){\rm{d}}s = + \infty . $ | (22) |
则对充分大的正整数n,有
$ \begin{array}{l} \int_{{t_2}}^{{T_n}} {\frac{{\gamma {{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\theta \left( s \right)\delta '\left( s \right){w^{\left( {\beta + 1} \right)/\beta }}\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}} {\rm{d}}s - \\ \;\;\;\;\;\int_{{t_2}}^{{T_n}} {{{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }} \left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{{T_n} - s}}} \right|w\left( s \right){\rm{d}}s < {M_0} + 1, \end{array} $ |
进一步,对正数0 < ε < 1及充分大的正整数n,可推得
$ \frac{{\int_{{t_2}}^{{T_n}} {{{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }} \left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{{T_n} - s}}} \right|w\left( s \right){\rm{d}}s}}{{\int_{{t_2}}^{{T_n}} {\frac{{\gamma {{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\theta \left( s \right)\delta '\left( s \right){w^{\left( {\beta + 1} \right)/\beta }}\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}} {\rm{d}}s}} > 1 - \varepsilon > 0. $ | (23) |
另一方面,应用引理4中的Hölder不等式,得
$ \begin{array}{l} \int_{{t_2}}^{{T_n}} {{{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }} \left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{{T_n} - s}}} \right|w\left( s \right){\rm{d}}s = \\ \;\;\;\;\;\;{\int_{{t_2}}^{{T_n}} {\left[ {\frac{{\gamma {{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\theta \left( s \right)\delta '\left( s \right){w^{\frac{{\left( {\beta + 1} \right)}}{\beta }}}\left( s \right)}}{{{\varphi ^{\frac{1}{\beta }}}\left( s \right){a^{\frac{1}{\beta }}}\left( {\delta \left( s \right)} \right)}}} \right]} ^{\frac{\beta }{{\beta + 1}}}} \times \\ \;\;\;\;\;\;{\left( {1 - \frac{s}{{{T_n}}}} \right)^{\frac{\omega }{{\beta + 1}}}}{\left[ {\frac{{{\varphi ^{\frac{1}{\beta }}}\left( s \right){a^{\frac{1}{\beta }}}\left( {\delta \left( s \right)} \right)}}{{\gamma \theta \left( s \right)\delta '\left( s \right)}}} \right]^{\frac{\beta }{{\beta + 1}}}} \times \\ \;\;\;\;\;\;\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{{T_n} - s}}} \right|{\rm{d}}s \le \\ \;\;\;\;\;\;{\left[ {\int_{{t_2}}^{{T_n}} {\frac{{\gamma {{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\theta \left( s \right)\delta '\left( s \right){w^{\frac{{\beta + 1}}{\beta }}}\left( s \right)}}{{{\varphi ^{\frac{1}{\beta }}}\left( s \right){a^{\frac{1}{\beta }}}\left( {\delta \left( s \right)} \right)}}{\rm{d}}s} } \right]^{\frac{\beta }{{\beta + 1}}}} \times \\ \;\;\;\;\;\;\left[ {\int_{{t_2}}^{{T_n}} {\frac{{{{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\gamma \theta \left( s \right)\delta '\left( s \right)} \right)}^\beta }}} \times } } \right.\\ \;\;\;\;\;\;{\left. {{{\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{{T_n} - s}}} \right|}^{\beta + 1}}{\rm{d}}s} \right]^{\frac{1}{{\beta + 1}}}}, \end{array} $ | (24) |
由式(24),并利用式(23),可得
$ \begin{array}{l} 0 < {\left( {1 - \varepsilon } \right)^\gamma }\int_{{t_2}}^{{T_n}} {{{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{{T_n} - s}}} \right|w\left( s \right){\rm{d}}s < } \\ \;\;\;\;\;\;\;\;\frac{{{{\left[ {\int_{{t_2}}^{{T_n}} {{{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{{T_n} - s}}} \right|w\left( s \right){\rm{d}}s} } \right]}^{\beta + 1}}}}{{{{\left[ {\frac{{\gamma {{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\theta \left( s \right)\delta '\left( s \right){w^{\frac{{\beta + 1}}{\beta }}}\left( s \right)}}{{{\varphi ^{\frac{1}{\beta }}}\left( s \right){a^{\frac{1}{\beta }}}\left( {\delta \left( s \right)} \right)}}} \right]}^\beta }}} \le \\ \;\;\;\;\;\;\;\;\int_{{t_2}}^{{T_n}} {{{\left( {1 - \frac{s}{{{T_n}}}} \right)}^\omega }\frac{{\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\gamma \theta \left( s \right)\delta '\left( s \right)} \right)}^\beta }}} \times } \\ \;\;\;\;\;\;\;\;{\left| {\frac{{\varphi '\left( s \right)}}{{\varphi \left( s \right)}} - \frac{\omega }{{{T_n} - s}}} \right|^{\beta + 1}}{\rm{d}}s, \end{array} $ |
但由式(16) 知,上式右边是有界的,这与式(22) 矛盾!式(20) 得证.
于是,分别利用式(18) 和(20),得
$ \begin{array}{l} \mathop {\lim \inf }\limits_{t \to + \infty } \frac{1}{{{t^\omega }}}\int_{{t_2}}^t {\frac{{{{\left( {t - s} \right)}^\omega }\theta \left( s \right)\delta '\left( s \right)\left[ {{\xi _1}\left( s \right) - \zeta {\xi _2}\left( s \right)} \right]_ + ^{\left( {\beta + 1} \right)/\beta }}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}{\rm{d}}s \le } \\ \;\;\;\;\;\;\;\;\;\;\mathop {\lim \inf }\limits_{t \to + \infty } \int_{{t_2}}^t {\frac{{{{\left( {t - s} \right)}^\omega }}}{{{t^\omega }}}\frac{{\theta \left( s \right)\delta '\left( s \right){w^{\left( {\beta + 1} \right)/\beta }}\left( s \right)}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}{\rm{d}}s < } + \infty , \end{array} $ |
这与式(17) 矛盾!定理证毕.
推论3 设(H1)和(H2)成立,若存在函数φ(t)∈C1([t0, +∞), (0, +∞))及ξ1(t), ξ2(t)∈L2([t0, +∞), R),使得对
$ \mathop {\lim \sup }\limits_{t \to + \infty } \int_u^t {\varphi \left( s \right)Q\left( s \right){\rm{d}}s} \ge {\xi _1}\left( u \right), $ | (25) |
$ \mathop {\lim \sup }\limits_{t \to + \infty } \int_u^t {\frac{{\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left[ {\theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}}} {\left( {\frac{{{{\varphi }_ + }'\left( s \right)}}{{\varphi \left( s \right)}}} \right)^{\beta + 1}}{\rm{d}}s \le {\xi _2}\left( u \right), $ | (26) |
且ξ1和ξ2满足
$ \mathop {\lim \sup }\limits_{t \to + \infty } \int_{{t_2}}^t {\frac{{\theta \left( s \right)\delta '\left( s \right)\left[ {{\xi _1}\left( s \right) - \zeta {\xi _2}\left( s \right)} \right]_ + ^{\left( {\beta + 1} \right)/\beta }}}{{{\varphi ^{1/\beta }}\left( s \right){a^{1/\beta }}\left( {\delta \left( s \right)} \right)}}{\rm{d}}s = + \infty ,} $ | (27) |
其中常数t2,ζ及函数[ξ1(s)-ζξ2(s)]+,Q(t)和θ(t)的定义均同定理2,则方程(1) 是振动的.
例1 考虑二阶时滞微分方程:
$ {\left( {x\left( t \right) + \frac{1}{5}x\left( {\frac{t}{5}} \right)} \right)^{\prime \prime }} + \frac{{{q_0}}}{{{t^2}}}x\left( t \right) = 0,t \ge 1, $ | (28) |
其中常数q0>0.这相当于方程(1) 中a(t)≡1,p(t)=1/5,q(t)=q0/t2,τ(t)=t/5,δ(t)=t,α=1,β=1,γ=1.容易验证条件(H1)与(H2)均满足.现取φ(t)=t,则当q0>5/16=0.312 5时,
$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to + \infty } \int_{{t_2}}^t {\left\{ {\varphi \left( s \right)Q\left( s \right) - } \right.} \\ \;\;\;\;\;\;\;\left. {\frac{{{\beta ^\beta }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{{\left[ {\gamma \theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}}{{\left( {\frac{{{{\varphi }_ + }'\left( s \right)}}{{\varphi \left( s \right)}}} \right)}^{\beta + 1}}} \right\}{\rm{d}}s = \\ \;\;\;\;\;\;\;\frac{4}{5}\left( {{q_0} - \frac{5}{{16}}} \right)\mathop {\lim \sup }\limits_{t \to + \infty } \int_1^t {\frac{1}{s}{\rm{d}}s} = + \infty , \end{array} $ |
因此,由推论1知,当q0>0.312 5时方程(28) 是振动的.
注2 现用文献[7]中的定理3.4来判定方程(28) 的振动性:因为当q0>2.5时,
$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to + \infty } \int_{{t_0}}^t {\left\{ {\frac{{\varphi \left( s \right)q\left( s \right)}}{{{2^{\gamma - 1}}}} - \frac{{\left( {1 + p_0^\gamma /{\tau _0}} \right){{\left( {\varphi ' + \left( s \right)} \right)}^{\gamma + 1}}a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\gamma + 1} \right)}^{\gamma + 1}}{{\left[ {{\tau _0}\varphi \left( s \right)} \right]}^\gamma }}}} \right\}{\rm{d}}s} = \\ \;\;\;\;\;\;\;\left( {{q_0} - 2.5} \right)\mathop {\lim \sup }\limits_{t \to + \infty } \int_1^t {\frac{1}{s}{\rm{d}}s} = + \infty . \end{array} $ |
所以当q0>2.5时方程(28) 是振动的.这说明本文定理的特殊情形即当α=1(相当于方程(1) 的中立项是线性的)时的振动准则要比文献[7]的有关结论“精细”得多.
例2 考虑具非线性中立项的二阶微分方程
$ \begin{array}{l} {\left\{ {t{{\left[ {{{\left( {x\left( t \right) + \frac{1}{5}\sqrt[3]{{x\left( {t/2} \right)}}} \right)}^\prime }} \right]}^{5/3}}} \right\}^\prime } + \frac{{{q_0}}}{t}{x^{7/5}}\left( {t/3} \right) = 0,\\ \;\;\;\;\;\;t \ge 1, \end{array} $ | (29) |
其中常数q0>0.这相当于方程(1) 中α=1/3, β=5/3, γ=7/5, a(t)=t, p(t)=1/5, q(t)=q0/t, τ(t)=t/2, δ(t)=t/3.显然条件(H1)和(H2)均满足.现取φ(t)=1,由于β>γ,且
$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to + \infty } \int_{{t_2}}^t {\left\{ {\varphi \left( s \right)Q\left( s \right) - } \right.} \\ \;\;\;\;\left. {\frac{{{\beta ^\beta }\varphi \left( s \right)a\left( {\delta \left( s \right)} \right)}}{{{{\left( {\beta + 1} \right)}^{\beta + 1}}{{\left[ {\gamma \theta \left( s \right)\delta '\left( s \right)} \right]}^\beta }}}{{\left( {\frac{{{{\varphi }_ + }'\left( s \right)}}{{\varphi \left( s \right)}}} \right)}^{\beta + 1}}} \right\}{\rm{d}}s = \\ \;\;\;\;\mathop {\lim \sup }\limits_{t \to + \infty } \int_1^t {\left\{ {\frac{{{q_0}}}{s}{{\left[ {1 - \left( {\frac{1}{3}{2^{2/3}} + \frac{{\left( {{2^{2/3}} - 1} \right)}}{k}} \right)\frac{1}{5}} \right]}^{7/5}} - 0} \right\}{\rm{d}}s} = \\ \;\;\;\; + \infty , \end{array} $ |
所以由推论1知,方程(29) 是振动的.
注3由于方程(29) 是具有非线性中立项的微分方程,并且β≠γ,所以文献[1-4, 6-17, 19-22]中的定理均不能用于方程(29).
从以上例子可看出,即使中立项是线性的,即当α=1且β=γ时,本文的振动准则也是较“精准”的,几乎是方程(1) 振动的“sharp”条件,所以本文定理推广、改进并丰富了现有文献的结果.
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