0 引言

Emden-Fowler方程因其具有广泛的实际应用价值, 引发了众多学者的研究兴趣^[1-8].本文将讨论一类具阻尼项和多滞量的广义Emden-Fowler中立型泛函微分方程：

$ \begin{array}{l} {\left[ {r\left( t \right)\varphi \left( {y'\left( t \right)} \right)} \right]^\prime } + m\left( t \right)\varphi \left( {y'\left( t \right)} \right) + \\ \;\;\;\;\;\;\;{q_0}\left( t \right){\left| {x\left( {{\sigma _0}\left( t \right)} \right)} \right|^{\alpha - 1}}\left( {x\left( {{\sigma _0}\left( t \right)} \right)} \right) + \\ \;\;\;\;\;\;\;\sum\limits_{i = 1}^n {{q_i}\left( t \right)} {\left| {x\left( {{\sigma _i}\left( t \right)} \right)} \right|^{{\beta _i} - 1}}x\left( {{\sigma _i}\left( t \right)} \right) = 0, \end{array} $

(1)

其中，y(t)=x(t)+p(t)x(τ(t))，φ(s)=|s|^α-1s, α是常数，n(≥2) 是一个偶数，且有下列条件成立：

$ \begin{array}{l} \left( {{{\rm{H}}_1}} \right)p\left( t \right),{q_1}\left( t \right),{q_2}\left( t \right) \in C\left( {I,\left[ {0,\infty } \right)} \right),\\ I = \left[ {{t_0},\infty } \right),0 \le p\left( t \right) \le p < 1; \end{array} $

(H₂) β_n>β_n-2>…>β₂>α>β_n-1>β_n-3>…>β₁>0是常数；

$ \begin{array}{*{20}{c}} {\left( {{{\rm{H}}_3}} \right)r\left( t \right) \in {C^1}\left( {I,\left( {0,\infty } \right)} \right),}\\ {m\left( t \right) \in C\left( {I,\left[ {0,\infty } \right)} \right),r'\left( t \right) \ge 0,}\\ {\mathop {\lim }\limits_{t \to \infty } \int_{{t_0}}^t {{{\left[ {\frac{1}{{r\left( v \right)}}\exp \left( { - \int_{{t_0}}^v {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right)} \right]}^{\frac{1}{\alpha }}}{\rm{d}}v} = \infty ;} \end{array} $

$ \begin{array}{l} \left( {{{\rm{H}}_4}} \right)\;\;\;\tau \left( t \right),{\sigma _i}\left( t \right) \in C\left( {\left[ {{t_0},\infty } \right),R} \right),\tau \left( t \right) \le t,\\ {\sigma _i}\left( t \right) \le t,\;\;且\mathop {\lim }\limits_{t \to \infty } \tau \left( t \right) = \mathop {\lim }\limits_{t \to \infty } {\sigma _i}\left( t \right) = \infty ,i = 1,2, \cdots ,n. \end{array} $

当m(t)=q₀(t)=0, n=2时，式(1) 就是文献[6]所讨论的方程.本文的研究目的是要获得方程(1) 的一些振动性定理，使得文献[6]的结果成为本文结论的特例，并推广文献[7-8]的相应结论.

1 主要结果及证明

引理1 设x(t)是方程(1) 的最终正解，则存在t₁≥t₀，令y(t)>0, y′(t)>0, y″(t)≤0, t≥t₁.

证明 设x(t)是方程(1) 的最终正解，由条件(H₁)和(H₄)，有

$ \begin{array}{l} {\left[ {r\left( t \right){{\left| {y'\left( t \right)} \right|}^{\alpha - 1}}y'\left( t \right)} \right]^\prime } + m\left( t \right){\left| {y'\left( t \right)} \right|^{\alpha - 1}}y'\left( t \right) \le 0,\\ \;\;\;\;\;\;t \ge {t_0}. \end{array} $

于是

$ {\left[ {\exp \left( {\int_{{t_0}}^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right)r\left( t \right){{\left| {y'\left( t \right)} \right|}^{\alpha - 1}}y'\left( t \right)} \right]^\prime } \le 0. $

下证y′(t)>0.事实上，若存在t₁≥t₀，使y′(t₁) < 0，注意到$\exp \left( {\int_{{t_0}}^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\text{d}}s} } \right)r\left( t \right){\left| {y'\left( t \right)} \right|^{a-1}}y'\left( t \right) $是t的减函数，则有

$ \begin{array}{l} \exp \left( {\int_{{t_0}}^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right)r\left( t \right){\left| {y'\left( t \right)} \right|^{\alpha - 1}}y'\left( t \right) \le \\ \;\;\;\;\;\;\;\exp \left( {\int_{{t_0}}^{{t_1}} {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right)r\left( {{t_1}} \right){\left| {y'\left( {{t_1}} \right)} \right|^{\alpha - 1}}y'\left( {{t_1}} \right) = :\\ \;\;\;\;\;\;\;M < 0,t \ge {t_1}. \end{array} $

因而

$ y'\left( t \right) \le {M^{\frac{1}{\alpha }}}{\left[ {\frac{1}{{r\left( t \right)}}\exp \left( { - \int_{{t_0}}^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right)} \right]^{\frac{1}{\alpha }}} < 0,t \ge {t_1}. $

然后从t₁到t积分得

$ y'\left( t \right) \le y\left( {{t_1}} \right) + {M^{\frac{1}{\alpha }}}\int_{{t_1}}^t {{{\left[ {\frac{1}{{r\left( t \right)}}\exp \left( { - \int_{{t_0}}^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right)} \right]}^{\frac{1}{\alpha }}}{\rm{d}}t} . $

令t→∞, 由(H₂), 有$ \mathop {\lim }\limits_{t \to \infty } y\left( t \right) =-\infty $, 这与y(t)>0, t≥t₁矛盾, 所以有y′(t)>0, t≥t₁.

注意到r′(t)>0，可得y″(t)≤0, t≥t₁.

引理2 设x(t)是方程(1) 的最终正解, 且存在某个i₀∈{1, 2, …, n}, 使得

$ \int_{{t_0}}^\infty {\exp \left( {\int_{{t_0}}^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right){q_{{i_0}}}\left( t \right)\sigma _{{i_0}}^{{\beta _{{i_0}}}}} \left( t \right){\rm{d}}t = \infty , $

(2)

则y(t)>ty′(t)且$ {\left( {\frac{{y\left( t \right)}}{t}} \right)^\prime } < 0$.

证明 设x(t)是方程(1) 的最终正解, 由引理1知y′(t)>0, 因此有x(t)>(1-p)y(t).则由方程(1) 得

$ \begin{array}{l} {\left[ {r\left( t \right){{\left( {y'\left( t \right)} \right)}^\alpha }} \right]^\prime } + m\left( t \right){\left( {y'\left( t \right)} \right)^\alpha } + \\ \;\;\;\;\;\;\;\sum\limits_{i = 1}^n {{q_i}\left( t \right){{\left( {1 - p} \right)}^{{\beta _i}}}x{{\left( {{\sigma _i}\left( t \right)} \right)}^{{\beta _i}}}} \le 0, \end{array} $

(3)

定义函数

$ \varphi \left( t \right) = y\left( t \right) - ty'\left( t \right), $

则

$ \varphi '\left( t \right) = - ty''\left( t \right) > 0, $

故φ(t)单调增加且最终定号.现断言φ(t)>0.否则若φ(t)≤0，有

$ {\left( {\frac{{y\left( t \right)}}{t}} \right)^\prime } = \frac{{ty'\left( t \right) - y\left( t \right)}}{{{t^2}}} \ge 0, $

因此，$\frac{{y\left( t \right)}}{t} $最终非减.故存在常数K_i>0, i=1, 2, …, n, T充分大, 使得

$ \frac{{y\left( {{\sigma _i}\left( t \right)} \right)}}{{{\sigma _i}\left( t \right)}} \ge {K_i},t \ge T,i = 1,2, \cdots ,n. $

(4)

联合式(3) 和(4), 得到

$ \begin{array}{l} {\left[ {r\left( t \right){{\left( {y'\left( t \right)} \right)}^\alpha }} \right]^\prime } + m\left( t \right){\left( {y'\left( t \right)} \right)^\alpha } + \\ \;\;\;\;\;\;\sum\limits_{i = 1}^n {{q_i}\left( t \right){{\left[ {{K_i}\left( {1 - p} \right)} \right]}^{{\beta _i}}}{{\left( {{\sigma _i}\left( t \right)} \right)}^{{\beta _i}}}} \le 0, \end{array} $

则有

$ \begin{array}{l} {\left[ {r\left( t \right){{\left( {y'\left( t \right)} \right)}^\alpha }} \right]^\prime } + m\left( t \right){\left( {y'\left( t \right)} \right)^\alpha } \le \\ \;\;\;\;\;\; - {q_{{i_0}}}\left( t \right){\left[ {{K_{{i_0}}}\left( {1 - p} \right)} \right]^{{\beta _{{i_0}}}}}\sigma _i^{{\beta _{{i_0}}}}\left( t \right),t \ge T, \end{array} $

即

$ \begin{array}{l} {\left[ {r\left( t \right)\exp \left( {\int_T^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right){{\left( {y'\left( t \right)} \right)}^\alpha }} \right]^\prime } \le \\ \;\;\;\;\;\; - \exp \left( {\int_T^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right){q_{{i_0}}}\left( t \right){\left[ {{K_{{i_0}}}\left( {1 - p} \right)} \right]^{{\beta _{{i_0}}}}}\sigma _i^{{\beta _{{i_0}}}}\left( t \right),t \ge T, \end{array} $

积分可得

$ \begin{array}{l} 0 < r\left( t \right)\exp \left( {\int_T^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right){\left( {y'\left( t \right)} \right)^\alpha } \le r\left( T \right){\left( {y'\left( T \right)} \right)^\alpha } - \\ \;\;\;\;\;\;\;{\left[ {{K_{{i_0}}}\left( {1 - p} \right)} \right]^{{\beta _{{i_0}}}}}\int_T^t {\exp \left( {\int_T^v {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right){q_{{i_0}}}\left( v \right)\sigma _i^{{\beta _{{i_0}}}}\left( v \right){\rm{d}}v} , \end{array} $

令t→∞, 与式(2) 矛盾.因此φ(t)>0成立.证毕.

引理3 设a_i, k_i是正数, 并且$ \sum\limits_{i = 1}^n {\frac{1}{{{k_i}}}} = 1$, i=1, 2, …, n, 那么

$ \prod\limits_{i = 1}^n {{a_i}} \le \sum\limits_{i = 1}^n {\frac{1}{{{k_i}}}a_i^{{k_i}}} . $

证明 设f(x)=ln x, 因f″(x) < 0，故当x>0时f(x)为严格凹函数, 所以有

$ \begin{array}{l} f\left( {\sum\limits_{i = 1}^n {\frac{1}{{{k_i}}}a_i^{{k_i}}} } \right) \ge \sum\limits_{i = 1}^n {\frac{1}{{{k_i}}}f\left( {a_i^{{k_i}}} \right)} = \sum\limits_{i = 1}^n {\frac{1}{{{k_i}}}\ln a_i^{{k_i}}} = \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\sum\limits_{i = 1}^n {\ln {a_i}} = \ln \left( {\prod\limits_{i = 1}^n {{a_i}} } \right). \end{array} $

即

$ \prod\limits_{i = 1}^n {{a_i}} \le \sum\limits_{i = 1}^n {\frac{1}{{{k_i}}}a_i^{{k_i}}} . $

引理3证毕.

引理4^[9] 设A>0, B>0, X≥0, 则

$ AX - B{X^{\frac{{\alpha + 1}}{\alpha }}} \le \frac{{{\alpha ^\alpha }}}{{{{\left( {\alpha + 1} \right)}^{\alpha + 1}}}}\frac{{{A^{\alpha + 1}}}}{{{B^\alpha }}}. $

定理1 设存在某个i₀∈{1, 2, …, n}，使得式(2) 成立, 且存在函数ρ(t)∈C¹(I, (0, ∞)), 使得

$ \begin{array}{l} \mathop {\lim }\limits_{t \to \infty } \sup \int_{{t_0}}^t {\left( {\rho \left( s \right)Q\left( s \right) - {{\left( {\alpha + 1} \right)}^{ - \alpha - 1}}\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - } \right.} \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {{{\left. {\frac{{m\left( s \right)}}{{r\left( s \right)}}} \right]}^{\alpha + 1}}\rho \left( s \right)r\left( s \right)} \right){\rm{d}}s = \infty , \end{array} $

(5)

其中,

$ \begin{array}{l} Q\left( t \right) = \prod\limits_{i = 1}^n {{{\left[ {{q_i}\left( t \right)} \right]}^{{k_i}}}{{\left[ {\frac{{\left( {1 - p} \right)\sigma \left( t \right)}}{t}} \right]}^\alpha }} ,\\ \;\;\;\;\;\;\;\;\;\;\;\;\sigma \left( t \right) \le \min \left\{ {{\sigma _1}\left( t \right),{\sigma _2}\left( t \right), \cdots ,{\sigma _n}\left( t \right)} \right\}. \end{array} $

(6)

并且

$ \begin{array}{l} {k_1} = \frac{{{\beta _n} - {\beta _1}}}{{{\beta _2} - \alpha }},{k_i} = \frac{{{\beta _n} - {\beta _1}}}{{{\beta _{i + 1}} - {\beta _{i - 1}}}},i = 2,3, \cdots ,n - 1,\\ \;\;\;\;\;\;{k_n} = \frac{{{\beta _n} - {\beta _1}}}{{\alpha - {\beta _{n - 1}}}}, \end{array} $

(7)

则方程(1) 是振动的.

证明 设x(t)是方程(1) 的非振动解，不失一般性，可设x(t)>0, t≥t₀.由引理1, 有y(t)>0, y′(t)>0, t≥t₁.令

$ W\left( t \right) = \rho \left( t \right)r\left( t \right){\left( {\frac{{y'\left( t \right)}}{{y\left( t \right)}}} \right)^\alpha },\;\;\;t \ge {t_1}, $

(8)

则

$ \begin{array}{l} W'\left( t \right) = \frac{{\rho '\left( t \right)}}{{\rho \left( t \right)}}W\left( t \right) + \frac{{\rho \left( t \right)}}{{{y^\alpha }\left( t \right)}}{\left( {r\left( t \right){{\left( {y'\left( t \right)} \right)}^\alpha }} \right)^\prime } - \\ \;\;\;\;\;\;\;\;\;\;\;\alpha \rho \left( t \right)r\left( t \right){\left( {\frac{{y'\left( t \right)}}{{y\left( t \right)}}} \right)^{\alpha + 1}}. \end{array} $

利用式(3) 和(8), 有

$ \begin{array}{l} W'\left( t \right) \le - \frac{{\rho \left( t \right)}}{{{y^\alpha }\left( t \right)}}\sum\limits_{i = 1}^n {{q_i}\left( t \right)x{{\left( {{\sigma _i}\left( t \right)} \right)}^{{\beta _i}}}} + \left[ {\frac{{\rho '\left( t \right)}}{{\rho \left( t \right)}} - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{m\left( t \right)}}{{r\left( t \right)}}} \right]W\left( t \right) - \frac{\alpha }{{{{\left[ {\rho \left( t \right)r\left( t \right)} \right]}^{\frac{1}{\alpha }}}}}{W^{\frac{{\alpha + 1}}{\alpha }}}\left( t \right), \end{array} $

(9)

由引理3, 可得

$ \begin{array}{l} \sum\limits_{i = 1}^n {{q_i}\left( t \right)x{{\left( {{\sigma _i}\left( t \right)} \right)}^{{\beta _i}}}} \ge {\left[ {{q_1}\left( t \right){{\left( {\left( {1 - p} \right)y\left( {{\sigma _1}\left( t \right)} \right)} \right)}^{{\beta _1}}}} \right]^{\frac{{{\beta _2} - \alpha }}{{{\beta _n} - {\beta _1}}}}} \times \\ \;\;\;\;\;\;\;\;\;{\left[ {{q_2}\left( t \right){{\left( {\left( {1 - p} \right)y\left( {{\sigma _2}\left( t \right)} \right)} \right)}^{{\beta _2}}}} \right]^{\frac{{{\beta _3} - {\beta _1}}}{{{\beta _n} - {\beta _1}}}}} \times \cdots \times \\ \;\;\;\;\;\;\;\;\;{\left[ {{q_{n - 1}}\left( t \right){{\left( {\left( {1 - p} \right)y\left( {{\sigma _{n - 1}}\left( t \right)} \right)} \right)}^{{\beta _{n - 1}}}}} \right]^{\frac{{{\beta _n} - {\beta _{n - 2}}}}{{{\beta _n} - {\beta _1}}}}} \times \\ \;\;\;\;\;\;\;\;\;{\left[ {{q_n}\left( t \right){{\left( {\left( {1 - p} \right)y\left( {{\sigma _n}\left( t \right)} \right)} \right)}^{{\beta _n}}}} \right]^{\frac{{\alpha - {\beta _{n - 1}}}}{{{\beta _n} - {\beta _1}}}}} \ge \\ \;\;\;\;\;\;\;\;\;\prod\limits_{i = 1}^n {{{\left[ {{q_i}\left( t \right)} \right]}^{{k_i}}}} \times {\left[ {\left( {1 - p} \right)y\left( {\sigma \left( t \right)} \right)} \right]^\alpha }, \end{array} $

其中，σ(t)≤min{σ₁(t), σ₂(t), …, σ_n(t)}, 且k_i(i=1, 2, …, n)由式(7) 所定义.

因此, 式(9) 成为

$ \begin{array}{l} W'\left( t \right) \le - \rho \left( t \right){\left( {{q_1}\left( t \right)} \right)^{\frac{{{\beta _2} - \alpha }}{{{\beta _2} - {\beta _1}}}}}{\left( {{q_2}\left( t \right)} \right)^{\frac{{\alpha - {\beta _1}}}{{{\beta _2} - {\beta _1}}}}}{\left[ {\frac{{\left( {1 - p} \right)y\left( {\sigma \left( t \right)} \right)}}{{y\left( t \right)}}} \right]^\alpha } + \\ \;\;\;\;\;\;\;\;\;\;\;\;\left[ {\frac{{\rho '\left( t \right)}}{{\rho \left( t \right)}} - \frac{{m\left( t \right)}}{{r\left( t \right)}}} \right]W\left( t \right) - \frac{\alpha }{{{{\left[ {\rho \left( t \right)r\left( t \right)} \right]}^{\frac{1}{\alpha }}}}}{W^{\frac{{\alpha + 1}}{\alpha }}}\left( t \right), \end{array} $

(10)

由引理2知，函数$\frac{{y\left( t \right)}}{t} $单调减小, 故有

$ \frac{{y\left( {\sigma \left( t \right)} \right)}}{{y\left( t \right)}} \ge \frac{{\sigma \left( t \right)}}{t}. $

(11)

联合式(7)、(10) 和(11), 得到

$ \begin{array}{l} W'\left( t \right) \le - \rho \left( t \right)Q\left( t \right) + \left[ {\frac{{\rho '\left( t \right)}}{{\rho \left( t \right)}} - \frac{{m\left( t \right)}}{{r\left( t \right)}}} \right]W\left( t \right) - \\ \;\;\;\;\;\;\;\;\;\;\frac{\alpha }{{{{\left[ {\rho \left( t \right)r\left( t \right)} \right]}^{\frac{1}{\alpha }}}}}{W^{\frac{{\alpha + 1}}{\alpha }}}\left( t \right), \end{array} $

(12)

由引理4, 有

$ \begin{array}{l} W'\left( t \right) \le - \rho \left( t \right)Q\left( t \right) + {\left( {\alpha + 1} \right)^{ - \alpha - 1}}{\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - \frac{{m\left( s \right)}}{{r\left( s \right)}}} \right]^{\alpha + 1}} \times \\ \;\;\;\;\;\;\;\;\;\;\rho \left( t \right)r\left( t \right),\;\;\;\;\;t \ge {t_1}, \end{array} $

(13)

对式(13) 积分可得

$ \begin{array}{l} W\left( t \right) \le W\left( {{t_1}} \right) - \int_{{t_1}}^t {\left( {\rho \left( s \right)Q\left( s \right) - } \right.} \\ \;\;\;\;\;\;\;\;\;\;\left. {{{\left( {\alpha + 1} \right)}^{ - \alpha - 1}}{{\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - \frac{{m\left( s \right)}}{{r\left( s \right)}}} \right]}^{\alpha + 1}}\rho \left( s \right)r\left( s \right)} \right){\rm{d}}s, \end{array} $

令t→∞, 注意到式(5), 有W(t)→-∞，这与W(t)>0矛盾.因此, 方程(1) 没有最终正解.故方程(1) 是振动的.定理1证毕.

推论1 设存在某个i₀∈{1, 2, …, n}，使得式(2) 成立, 且

$ \mathop {\lim }\limits_{t \to \infty } \sup \int_{{t_0}}^t {\left( {Q\left( s \right) - {{\left[ { - \frac{{m\left( s \right)}}{{\alpha + 1}}} \right]}^{\alpha + 1}}\frac{1}{{{r^\alpha }\left( s \right)}}} \right){\rm{d}}s} = \infty , $

(14)

则方程(1) 是振动的.

证明 只需在定理1中取ρ(t)=1.

定理2 设除式(5) 外定理1的全部假设都成立, 当n>1时, 有

$ \begin{array}{l} \mathop {\lim }\limits_{t \to \infty } \sup \frac{1}{{{t^n}}}\int_{{t_0}}^t {{{\left( {t - s} \right)}^n}\left( {\rho \left( s \right)Q\left( s \right) - } \right.} \\ \left. {\;\;\;\;\;{{\left( {\alpha + 1} \right)}^{ - \alpha - 1}}{{\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - \frac{{m\left( s \right)}}{{r\left( s \right)}}} \right]}^{\alpha + 1}}\rho \left( s \right)r\left( s \right)} \right){\rm{d}}s = \infty , \end{array} $

(15)

其中，Q(t)由式(7) 定义, 则方程(1) 是振动的.

证明 如同定理1的证明, 设x(t)是方程(1) 的非振动解，不失一般性，设x(t)>0, x(τ(t))>0，x(σ₁(t))>0, x(σ₂(t))>0, t≥t₁, 故有y(t)>0.令W(t)的定义同式(8), 则W(t)>0, t≥t₁, 且式(13) 成立.由式(13) 得

$ \begin{array}{l} \int_{{t_1}}^t {{{\left( {t - s} \right)}^n}\left( {\rho \left( s \right)Q\left( s \right) - {{\left( {\alpha + 1} \right)}^{ - \alpha - 1}} \times } \right.} \\ \;\;\;\;\;\;\;\left. {{{\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - \frac{{m\left( s \right)}}{{r\left( s \right)}}} \right]}^{\alpha + 1}}\rho \left( s \right)r\left( s \right)} \right){\rm{d}}s \le \\ \;\;\;\;\;\;\; - \int_{{t_1}}^t {{{\left( {t - s} \right)}^n}W'\left( s \right){\rm{d}}s} ,\;\;\;\;n > 1. \end{array} $

从而有

$ \begin{array}{l} \frac{1}{{{t^n}}}\int_{{t_1}}^t {{{\left( {t - s} \right)}^n}R\left( s \right){\rm{d}}s} = W\left( {{t_1}} \right){\left( {\frac{{t - {t_1}}}{t}} \right)^n} - \\ \frac{n}{{{t^n}}}\int_{{t_1}}^t {{{\left( {t - s} \right)}^{n - 1}}W\left( s \right){\rm{d}}s} , \end{array} $

其中，

$ R\left( s \right) = \rho \left( s \right)Q\left( s \right) - {\left( {\alpha + 1} \right)^{ - \alpha - 1}}{\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - \frac{{m\left( s \right)}}{{r\left( s \right)}}} \right]^{\alpha + 1}}\rho \left( s \right)r\left( s \right), $

因此

$ \frac{1}{{{t^n}}}\int_{{t_1}}^t {{{\left( {t - s} \right)}^n}R\left( s \right){\rm{d}}s} \le W\left( {{t_1}} \right){\left( {\frac{{t - {t_1}}}{t}} \right)^n}, $

则

$ \mathop {\lim }\limits_{t \to \infty } \sup \frac{1}{{{t^n}}}\int_{{t_1}}^t {{{\left( {t - s} \right)}^n}R\left( s \right){\rm{d}}s} \le W\left( {{t_1}} \right) < \infty , $

上式与条件(15) 矛盾.定理2证毕.

令D={(t, s)|t≥s≥t₀}, D₀={(t, s)|t>s≥t₀}.函数H(t, s)∈C(D, R)属于$\wp $类, 记作H∈ $\wp $, 如果

(ⅰ)H(t, t)=0, t≥t₀; H(t, s)>0, (t, s)∈D₀;

(ⅱ)$\frac{{\partial H}}{{\partial s}} \leqslant 0 $，(t, s)∈D₀.且存在函数h(t, s)∈C(D₀, R)和ρ(t)∈C¹(I, (0, ∞)), 使得

$ \begin{array}{l} \frac{{\partial H\left( {t,s} \right)}}{{\partial s}} + H\left( {t,s} \right)\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - \frac{{m\left( s \right)}}{{r\left( s \right)}}} \right] = \\ \;\;\;\;\;\; - h\left( {t,s} \right){H^{\frac{{\alpha + 1}}{\alpha }}}\left( {t,s} \right),\;\;\left( {t,s} \right) \in {D_0}. \end{array} $

(16)

定理3 设存在某个i₀∈{1, 2, …, n}，使得式(2) 成立, 且存在函数ρ(t)∈C¹(I, (0, ∞))和H∈$\wp $，使得

$ \begin{array}{l} \mathop {\lim \sup }\limits_{t \to \infty } \frac{1}{{H\left( {t,{t_0}} \right)}}\int_{{t_0}}^t {\left[ {H\left( {t,s} \right)\rho \left( s \right)Q\left( s \right) - } \right.} \\ \;\;\;\;\;\;\;\;\left. {{{\left( {\frac{{\left| {h\left( {s,a} \right)} \right|}}{{\alpha + 1}}} \right)}^{\alpha + 1}}\rho \left( s \right)r\left( s \right)} \right]{\rm{d}}s = \infty , \end{array} $

(17)

其中，h(t, s)由式(16) 定义，则方程(1) 是振动的.

证明 设x(t)是方程(1) 的非振动解，不妨设x(t)>0, x(τ(t))>0，x(σ₁(t))>0, x(σ₂(t))>0, t≥t₁, 故有y(t)>0.令W(t)的定义同式(8), 则W(t)>0, t≥t₁, 且式(12) 成立.记

$ A\left( t \right) = \frac{\alpha }{{{{\left[ {\rho \left( s \right)r\left( s \right)} \right]}^{\frac{1}{\alpha }}}}},\;\;\;\;B\left( s \right) = \frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - \frac{{m\left( s \right)}}{{r\left( s \right)}}, $

则由式(12) 得

$ \begin{array}{l} \int_{{t_1}}^t {H\left( {t,s} \right)\rho \left( s \right)Q\left( s \right){\rm{d}}s} \le \int_{{t_1}}^t {H\left( {t,s} \right)\left[ { - W'\left( s \right) + } \right.} \\ \;\;\;\;\;\;\left. {B\left( s \right)W\left( s \right) - A\left( s \right){W^{\frac{{\alpha + 1}}{\alpha }}}\left( s \right)} \right]{\rm{d}}s = H\left( {t,{t_1}} \right)W\left( {{t_1}} \right) + \\ \;\;\;\;\;\;\int_{{t_1}}^t {\left[ { - h\left( {t,s} \right){H^{\frac{{\alpha + 1}}{\alpha }}}\left( {t,s} \right)W\left( s \right) - A\left( s \right)H\left( {t,s} \right){W^{\frac{{\alpha + 1}}{\alpha }}}\left( s \right)} \right]{\rm{d}}s} \le \\ \;\;\;\;\;\;H\left( {t,{t_1}} \right)W\left( {{t_1}} \right) + \int_{{t_1}}^t {\left[ {\left| {h\left( {t,s} \right)} \right|{H^{\frac{{\alpha + 1}}{\alpha }}}\left( {t,s} \right)W\left( s \right) - } \right.} \\ \;\;\;\;\;\;\left. {A\left( s \right)H\left( {t,s} \right){W^{\frac{{\alpha + 1}}{\alpha }}}\left( s \right)} \right]{\rm{d}}s. \end{array} $

由引理4及A(s)的定义, 可得

$ \begin{array}{l} \int_{{t_1}}^t {H\left( {t,s} \right)\rho \left( s \right)Q\left( s \right){\rm{d}}s} \le H\left( {t,{t_1}} \right)W\left( {{t_1}} \right) + \\ \;\;\;\;\;\int_{{t_1}}^t {\frac{{\rho \left( s \right)r\left( s \right)}}{{{{\left( {\alpha + 1} \right)}^{\alpha + 1}}}}} {\left| {h\left( {s,a} \right)} \right|^{\alpha + 1}}{\rm{d}}s. \end{array} $

因此,

$ \begin{array}{l} \frac{1}{{H\left( {t,{t_1}} \right)}}\int_{{t_1}}^t {\left( {H\left( {t,s} \right)\rho \left( s \right)Q\left( s \right) - \frac{{\rho \left( s \right)r\left( s \right)}}{{{{\left( {\alpha + 1} \right)}^{\alpha + 1}}}}{{\left| {h\left( {s,a} \right)} \right|}^{\alpha + 1}}} \right){\rm{d}}s} \le \\ \;\;\;\;\;\;W\left( {{t_1}} \right), \end{array} $

与条件(17) 矛盾.定理3证毕.