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 浙江大学学报(理学版)  2016, Vol. 43 Issue (3): 257-263  DOI:10.3785/j.issn.1008-9497.2016.03.002 0

### 引用本文 [复制中英文]

[复制中文]
YANG Jiashan. 2016. Oscillation of certain second-order nonlinear neutral functional differential equations with variable delay[J]. Journal of Zhejiang University(Science Edition), 43(3): 257-263. DOI: 10.3785/j.issn.1008-9497.2016.03.002.
[复制英文]

### 文章历史

Oscillation of certain second-order nonlinear neutral functional differential equations with variable delay
YANG Jiashan
School of Information and Electronic Engineering, Wuzhou University, Wuzhou 543002, Guangxi Zhuang Autonomous Region, China
Abstract: We study the oscillatory behavior of a class of second-order nonlinear neutral functional differential equations with variable delay. By using the generalized Riccati transformation and the inequality technique, we establish two new oscillation criteria for the oscillation of the equations. The examples are provided to illustrate that our result gives a sharper estimate for the oscillation of the equations.
Key words: oscillation    variable delay    functional differential equation    Riccati transformation

 ${\{ a(t)[(x(t) + p(t)x(t)))']^\gamma }\} ' + q(t)f(x(\delta (t))) = 0,\;t \geqslant {t_0}$ (1)

(H1) aC1([t0,+∞),(0,+∞))，q(t)>0,p(t)≥0．

(H2) τ,δ:[t0,+∞)→(0,+∞)并满足：τ(t)≤t，$\mathop {\lim }\limits_{t \to + \infty } \tau (t) = + \infty$；δ(t)≤t，$\mathop {\lim }\limits_{t \to + \infty } \delta (t) = + \infty$；τ°δ=δ°ττ′(t)≥τ0>0(这里τ0为常数)．

(H3) 当u≠0时，f(u)/uL(这里常数L>0)．

 $\int_{{t_0}}^{ + \infty } {{a^{ - 1/\gamma }}(t){\text{d}}t} = + \infty$ (2)

 $\int_{{t_0}}^{ + \infty } {{a^{ - 1/\gamma }}(t){\text{d}}t} < + \infty$ (3)

(i)当x>0时，$Ax - B{x^{\frac{{\lambda + 1}}{\lambda }}} \leqslant \frac{{{\lambda ^\lambda }{A^{\lambda + 1}}}}{{{{(\lambda + 1)}^{\lambda + 1}}{B^\lambda }}}$；

(ii)当x<0时，$Ax + B{x^{\frac{{\lambda + 1}}{\lambda }}} \geqslant \frac{{{\lambda ^\lambda }{A^{\lambda + 1}}}}{{{{(\lambda + 1)}^{\lambda + 1}}{B^\lambda }}}$．

1 主要结果及其证明

 $\begin{gathered} z(t) = x(t) + p(t)x(\tau (t)),\hfill \\ Q(t) = \min \{ q(t),\;q(\tau (t))\} ,\hfill \\ {\varphi _ + }(t) = \max \{ \varphi (t),\;0\} . \hfill \\ \end{gathered}$

 $\mathop {\lim \sup }\limits_{t \to + \infty } \int_T^t {\left[{\frac{{L\varphi (s)Q(s)\psi (s,\;{t_1})}}{{{{[b\theta (s)]}^{\gamma - 1}}}} - \frac{{\varphi (s)}}{{{{(\gamma + 1)}^{\gamma + 1}}}}{{\left( {\frac{{{\varphi _ + }'(s)}}{{\varphi (s)}}} \right)}^{\gamma + 1}} \times \left( {a(s) + \frac{{{p_0}a(\tau (s))}}{{\tau _0^{\gamma + 1}}}} \right)} \right]{\text{d}}s = + \infty } ,$ (5)

 $\begin{gathered} \psi (t,\;{t_0}) = \left( {\int_{{t_1}}^{\delta (t)} {{a^{ - 1/\gamma }}(s){\text{d}}s} } \right){\left( {\int_{{t_1}}^t {{a^{ - 1/\gamma }}(s){\text{d}}s} } \right)^{ - 1}},\hfill \\ \theta (t) = \int_{{t_0}}^t {{a^{ - 1/\gamma }}(s){\text{d}}s} ,\hfill \\ \end{gathered}$

 $[a(t){(z'(t))^\gamma }]' = - q(t)f(x(\delta (t))) \leqslant - Lq(t)x(\delta (t)) < 0,$ (6)

 $\frac{{[a(\tau (t)){{(z'(\tau (t)))}^\gamma }]'}}{{\tau '(t)}} + Lq(\tau (t))x(\delta (\tau (t))) \leqslant 0,$ (7)

 $\begin{gathered} [a(t){(z'(t))^\gamma }]' + Lq(t)x(\delta (t)) + \hfill \\ {p_0}Lq(\tau (t))x(\delta (\tau (t))) + \hfill \\ {p_0}\frac{{[a(\tau (t)){{(z'(\tau (t)))}^\gamma }]'}}{{\tau '(t)}} \leqslant 0,\hfill \\ \end{gathered}$

 $\begin{gathered} [a(t){(z'(t))^\gamma }]' + \frac{{{p_0}}}{{{\tau _0}}}[a(\tau (t)){(z'(\tau (t)))^\gamma }]' \leqslant \hfill \\ - LQ(t)[x(\delta (t)) + {p_0}x(\delta (\tau (t)))] \leqslant \hfill \\ - LQ(t)z(\delta (t)) \leqslant 0,\hfill \\ \end{gathered}$ (8)

 $w(t) = \varphi (t)\frac{{a(t){{(z'(t))}^\gamma }}}{{{z^{\gamma (t)}}}},\;t \geqslant {t_1}$ (9)

w(t)>0(tt1)，利用式(9)及引理1(i)，可得

 $\begin{gathered} w'(t) = \varphi '(t)\frac{{a(t){{(z'(t))}^\gamma }}}{{{z^\gamma }(t)}} + \varphi (t) \times \hfill \\ \frac{{[a(t){{(z'(t))}^\gamma }]'{z^\gamma }(t) - a(t){{(z'(t))}^\gamma }\gamma {z^{\gamma - 1}}(t)z'(t)}}{{{z^{2\gamma }}(t)}} \hfill \\ = \frac{{\varphi '(t)}}{{\varphi (t)}}w(t) + \frac{{\varphi (t)}}{{{z^\gamma }(t)}}[a(t){(z'(t))^\gamma }]' - \hfill \\ \gamma \frac{{\varphi (t)a(t){{(z'(t))}^{\gamma + 1}}}}{{{z^{\gamma + 1}}(t)}} \leqslant \frac{{\varphi (t)}}{{{z^\gamma }(t)}}[a(t){(z'(t))^\gamma }]' + \hfill \\ \frac{{\varphi {'_ + }(t)}}{{\varphi (t)}}w(t) - \frac{{\gamma {w^{(\gamma + 1)/\gamma }}(t)}}{{{{[\varphi (t)a(t)]}^{1/\gamma }}}} \leqslant \hfill \\ \frac{{\varphi (t)}}{{{z^\gamma }(t)}}[a(t){(z'(t))^\gamma }]' + \frac{{\varphi (t)a(t)}}{{{{(\gamma + 1)}^{\gamma + 1}}}}{\left( {\frac{{\varphi {'_ + }(t)}}{{\varphi (t)}}} \right)^{\gamma + 1}}. \hfill \\ \end{gathered}$ (10)

 $v(t) = \varphi (t)\frac{{a(\tau (t)){{(z'(\tau (t)))}^\gamma }}}{{{z^\gamma }(\tau (t))}},\;t \geqslant {t_1},$ (11)

v(t)>0(tt1)．由于τ′(t)≥τ0>0，z′(t)>0，由引理1(i),类似地可得

 $\begin{gathered} v'(t) = \varphi '(t)\frac{{a(\tau (t)){{(z'(\tau (t)))}^\gamma }}}{{{z^\gamma }(\tau (t))}} + \hfill \\ \varphi (t)\frac{{[a(\tau (t)){{(z'(\tau (t)))}^\gamma }]'{z^\gamma }(\tau (t))}}{{{z^{2\gamma }}(\tau (t))}} - \hfill \\ \frac{{\varphi (t)a(\tau (t)){{(z'(\tau (t)))}^\gamma }\gamma {z^{\gamma - 1}}(\tau (t))z'(\tau (t))\tau '(t)}}{{{z^{2\gamma }}(\tau (t))}} \hfill \\ \leqslant \frac{{\varphi (t)}}{{{z^\gamma }(\tau (t))}}[a(\tau (t)){(z'(\tau (t)))^\gamma }]' + \frac{{\varphi '(t)}}{{\varphi (t)}}v(t) - \hfill \\ \gamma {\tau _0}\frac{{\varphi (t)a(\tau (t)){{(z'(\tau (t)))}^{\gamma + 1}}}}{{{z^{\gamma + 1}}(\tau (t))}} \leqslant \hfill \\ \frac{{\varphi (t)}}{{{z^\gamma }(t)}}[a(\tau (t)){(z'(\tau (t)))^\gamma }]' + \frac{{\varphi {'_ + }(t)}}{{\varphi (t)}}v(t) - \hfill \\ \frac{{\gamma {\tau _0}{{(v(t))}^{(\gamma + 1)/\gamma }}}}{{{{(\varphi (t)a(\tau (t)))}^{1/\gamma }}}} \hfill \\ \frac{{\varphi (t)}}{{{z^\gamma }(t)}}[a(\tau (t)){(z'(\tau (t)))^\gamma }]' + \hfill \\ \frac{{\varphi (t)a(\tau (t))}}{{{{(\gamma + 1)}^{\gamma + 1}}\tau _0^\gamma }}{\left( {\frac{{\varphi {'_ + }(t)}}{{\varphi (t)}}} \right)^{\gamma + 1}}. \hfill \\ \end{gathered}$ (12)

 $\begin{gathered} w'(t) + \frac{{{p_0}}}{{{\tau _0}}}v'(t) \leqslant \frac{{\varphi (t)}}{{{z^\gamma }(t)}}[a(t){(z'(t))^\gamma }]' + \hfill \\ \frac{{\varphi (t)a(t)}}{{{{(\gamma + 1)}^{\gamma + 1}}}}{\left( {\frac{{\varphi {'_ + }(t)}}{{\varphi (t)}}} \right)^{\gamma + 1}} + \frac{{{p_0}}}{{{\tau _0}}}\left[{\frac{{\varphi (t)}}{{{z^\gamma }(t)}}[a(\tau (t)){{(z'(\tau (t)))}^\gamma }]' + \frac{{\varphi (t)a(\tau (t))}}{{{{(\gamma + 1)}^{\gamma + 1}}\tau _0^\gamma }}{{\left( {\frac{{\varphi {'_ + }(t)}}{{\varphi (t)}}} \right)}^{\gamma + 1}}} \right] \hfill \\ = \frac{{\varphi (t)}}{{{z^\gamma }(t)}}\left\{ {[a(t){{(z'(t))}^\gamma }]' + \frac{{{p_0}}}{{{\tau _0}}}[a(\tau (t)){{(z'(\tau (t)))}^\gamma }]'} \right\} + \hfill \\ \frac{{\varphi (t)}}{{{{(\gamma + 1)}^{\gamma + 1}}}}{\left( {\frac{{\varphi {'_ + }(t)}}{{\varphi (t)}}} \right)^{\gamma + 1}}\left( {a(t) + \frac{{{p_0}a(\tau (t))}}{{\tau _0^{\gamma + 1}}}} \right) \leqslant \hfill \\ - \frac{{\varphi (t)}}{{{z^\gamma }(t)}}LQ(t)z(\delta (t)) + \frac{{\varphi (t)}}{{{{(\gamma + 1)}^{\gamma + 1}}}}{\left( {\frac{{\varphi {'_ + }(t)}}{{\varphi (t)}}} \right)^{\gamma + 1}}^{\gamma + 1}\left( {a(t) + \frac{{{p_0}a(\tau (t))}}{{\tau _0^{\gamma + 1}}}} \right). \hfill \\ \end{gathered}$ (13)

 $z(t) \geqslant z(t) - z({t_1}) = \int_{{t_1}}^t {\frac{{{a^{1/\gamma }}(s)z'(s)}}{{{a^{1/\gamma }}(s)}}} ds \geqslant {a^{1/\gamma }}(t)z'(t)\int_{{t_1}}^t {{a^{ - 1/\gamma }}(s)} ds,$

 $\frac{{\text{d}}}{{{\text{dt}}}}\frac{{z(t)}}{{\int_{{t_1}}^t {{a^{ - 1/\gamma }}(s)} {\text{d}}s}} = \frac{{z'(t)\int_{{t_1}}^t {{a^{ - 1/\gamma }}(s)} {\text{d}}s - z(t)\int_{{t_1}}^t {{a^{ - 1/\gamma }}(t)} }}{{{{\left( {\int_{{t_1}}^t {{a^{ - 1/\gamma }}(s)} {\text{d}}s} \right)}^2}}} \leqslant 0,$

 $\frac{{z(\delta (t))}}{{z(t)}} \geqslant \psi (t,\;{t_1}).$ (14)

 $\begin{gathered} z(t) \leqslant z({t_1}) + M_0^{1/\gamma }\int_{{t_1}}^t {{a^{ - 1/\gamma }}(s)} {\text{d}}s \leqslant \hfill \\ z({t_1}) + M_0^{1/\gamma }\int_{{t_0}}^t {{a^{ - 1/\gamma }}(s)} {\text{d}}s,\hfill \\ \end{gathered}$

 $z(t) \leqslant b\int_{{t_0}}^t {{a^{ - 1/\gamma }}(s)} {\text{d}}s = b\theta (t)$ (15)

 $\begin{gathered} w'(t) + \frac{{{p_0}}}{{{\tau _0}}}v'(t) \leqslant - \frac{{L\varphi (t)Q(t)\psi (t,\;{t_1})}}{{{{[b\theta (t)]}^{\gamma - 1}}}} + \hfill \\ \frac{{\varphi (t)}}{{{{(\gamma + 1)}^{\gamma + 1}}}}{\left( {\frac{{\varphi {'_ + }(t)}}{{\varphi (t)}}} \right)^{\gamma + 1}}\left( {a(t) + \frac{{{p_0}a(\tau (t))}}{{\tau _0^{\gamma + 1}}}} \right),\hfill \\ \end{gathered}$

 $\begin{gathered} \int_{{t_2}}^{{t_1}} {\left[{\frac{{L\varphi (s)Q(s)\psi (s,\;{t_1})}}{{{{[b\theta (s)]}^{\gamma - 1}}}} - \frac{{\varphi (s)}}{{{{(\gamma + 1)}^{\gamma + 1}}}}{{\left( {\frac{{\varphi {'_ + }s)}}{{\varphi (s)}}} \right)}^{\gamma + 1}} \times \left( {a(s) + \frac{{{p_0}a(\tau (s))}}{{\tau _0^{\gamma + 1}}}} \right)} \right]} ds \hfill \\ \leqslant - w(t) + w({t_2}) - \frac{{{p_0}}}{{{\tau _0}}}v(t) + \frac{{{p_0}}}{{{\tau _0}}}v({t_2}) \leqslant w({t_2}) + \frac{{{p_0}}}{{{\tau _0}}}v({t_2}),\hfill \\ \end{gathered}$

 $\mathop {\lim \sup }\limits_{t \to + \infty } \int_T^t {\left[{LQ(s)\eta (s){\zeta ^\gamma }(s) - \left( {1 + \frac{{{p_0}}}{{{\tau _0}}}} \right)\frac{{{\gamma ^{\gamma + 1}}{a^{ - 1/\gamma }}(s)}}{{{{(\gamma + 1)}^{\gamma + 1}}\zeta (s)}}} \right]} {\text{d}}s = + \infty$ (16)

 $\eta (t) = \left\{ {\begin{array}{*{20}{l}} {k,\;\gamma > 1,} \\ {1,\;\gamma = 1,} \\ {k{\zeta ^{1 - \gamma }}(t),\;\gamma < 1,} \end{array}} \right.\;\;k > 0为常数,$

 $w(t) = \frac{{a(t){{(z'(t))}^\gamma }}}{{{z^\gamma }(t)}},\;t \geqslant {t_1},$ (17)

w(t)<0(tt1)，并且

 $\begin{gathered} w'(t) = \frac{{[a(t){{(z'(t))}^\gamma }]'}}{{{z^\gamma }(t)}} - \frac{{a(t){{(z'(t))}^\gamma }\gamma {z^{\gamma - 1}}(t)z'(t)}}{{{z^{2\gamma }}(t)}} \hfill \\ = \frac{{[a(t){{(z'(t))}^\gamma }]'}}{{{z^\gamma }(t)}} - \frac{{\gamma a(t){{(z'(t))}^{\gamma + 1}}}}{{{z^{\gamma + 1}}(t)}} \hfill \\ = \frac{{[a(t){{(z'(t))}^\gamma }]'}}{{{z^\gamma }(t)}} - \frac{{\gamma {{(w(t))}^{(\gamma + 1)/\gamma }}}}{{{a^{1/\gamma }}(t)}}. \hfill \\ \end{gathered}$ (18)

 $z'(s) \leqslant {a^{1/\gamma }}(t)z'(t){a^{ - 1/\gamma }}(s),$

 $z(u) - z(t) \leqslant {a^{1/\gamma }}(t)z'(t)\int_t^u {{a^{ - 1/\gamma }}(s)} {\text{d}}s,$

 $z(t) + {a^{1/\gamma }}(t)z'(t)\int_t^{ + \infty } {{a^{ - 1/\gamma }}(s)} {\text{d}}s \geqslant 0,\;t \geqslant {t_1}.$

 $- 1 \leqslant {w^{1/\gamma }}(t)\zeta (t) \leqslant 0,\;t \geqslant {t_1}.$ (19)

 $v(t) = \frac{{a(\tau (t)){{(z'(\tau (t)))}^\gamma }}}{{{z^\gamma }(t)}},\;t \geqslant {t_1},$ (20)

v(t)<0(tt1)．再利用a(t)[z′(t)]γ的单调递减性，有

 $a(\tau (t)){[z'(\tau (t))]^\gamma } \geqslant a(t){[z'(t)]^\gamma },$

 $z'(t) \leqslant {\left( {\frac{{a(\tau (t))}}{{a(t)}}} \right)^{1/\gamma }}z'(\tau (t)).$

 $\begin{gathered} v'(t) = \frac{{[a(\tau (t)){{(z'(\tau (t)))}^\gamma }]'}}{{{z^\gamma }(t)}} - \hfill \\ \frac{{a(\tau (t)){{(z'(\tau (t)))}^\gamma }\gamma {z^{\gamma - 1}}(t)z'(t)}}{{{z^{2\gamma }}(t)}} \leqslant \hfill \\ \frac{{[a(\tau (t)){{(z'(\tau (t)))}^\gamma }]'}}{{{z^\gamma }(t)}} - \hfill \\ \frac{{\gamma a(\tau (t)){{(z'(\tau (t)))}^\gamma }}}{{{z^{\gamma + 1}}(t)}}{\left( {\frac{{a(\tau (t))}}{{a(t)}}} \right)^{1/\gamma }}z'(\tau (t)) = \hfill \\ \frac{{[a(\tau (t)){{(z'(\tau (t)))}^\gamma }]'}}{{{z^\gamma }(t)}} - \frac{{\gamma {{(v(t))}^{(\gamma + 1)/\gamma }}}}{{{a^{1/\gamma }}(t)}}. \hfill \\ \end{gathered}$ (21)

a(τ(t))[z′(τ(t))]γa(t)[z′(t)]γ，得v(t)≥w(t)，利用式(19)，同样有

 $- 1 \leqslant {v^{1/\gamma }}(t)\zeta (t) \leqslant 0,\;t \geqslant {t_1}.$ (22)

 $\begin{gathered} w'(t) + \frac{{{p_0}}}{{{\tau _0}}}v'(t) \leqslant \frac{1}{{{z^\gamma }(t)}}\left\{ {[a(t){{(z'(t))}^\gamma }]' + \frac{{{p_0}}}{{{\tau _0}}}[a(\tau (t)){{(z'(\tau (t)))}^\gamma }]'} \right\} - \hfill \\ \frac{\gamma }{{{a^{1/\gamma }}(t)}}\left[{(w{{(t)}^{(\gamma + 1)/\gamma }} + \frac{{{p_0}}}{{{\tau _0}}}{{(v(t))}^{(\gamma + 1)/\gamma }}} \right] \leqslant - \frac{{LQ(t)z(\delta (t))}}{{{z^\gamma }(t)}} - \hfill \\ \frac{\gamma }{{{a^{1/\gamma }}(t)}}\left[{(w{{(t)}^{(\gamma + 1)/\gamma }} + \frac{{{p_0}}}{{{\tau _0}}}{{(v(t))}^{(\gamma + 1)/\gamma }}} \right] \leqslant - LQ(t){z^{1 - \gamma }}(t) - \hfill \\ \frac{\gamma }{{{a^{1/\gamma }}(t)}}\left[{(w{{(t)}^{(\gamma + 1)/\gamma }} + \frac{{{p_0}}}{{{\tau _0}}}{{(v(t))}^{(\gamma + 1)/\gamma }}} \right]. \hfill \\ \end{gathered}$ (23)

γ>1时，因为z(t)>0，z′(t)<0(tt1)，所以z(t)≤z(t1)，即z1-γ(t)≥z1-γ(t1)=k

γ=1时，显然有z1-γ(t)=1．

γ<1时，由于a(t)[z′(t)]γ是单调减小的，所以当stt1时，有

 $a(s){[z'(s)]^\gamma } \leqslant a(t){[z'(t)]^\gamma },$

 $z'(s) \leqslant {\{ a(t){[z'(t)]^\gamma }\} ^{1/\gamma }}{a^{ - 1/\gamma }}(s),$

 $z(u) - z(t) \leqslant {\{ a(t){[z'(t)]^\gamma }\} ^{1/\gamma }}\int_t^u {{a^{ - 1/\gamma }}(s)} {\text{d}}s,$

 $\begin{gathered} z(t) \geqslant - {\{ a(t){[z'(t)]^\gamma }\} ^{1/\gamma }}\int_t^u {{a^{ - 1/\gamma }}(s)} {\text{d}}s \geqslant \hfill \\ - {\{ a({t_1}){[z'({t_1})]^\gamma }\} ^{1/\gamma }}\int_t^u {{a^{ - 1/\gamma }}(s)} {\text{d}}s = \hfill \\ M\int_t^u {{a^{ - 1/\gamma }}(s)} {\text{d}}s,\hfill \\ \end{gathered}$

 $z(t) \geqslant M\int_t^{ + \infty } {{a^{ - 1/\gamma }}(s)} {\text{d}}s = M\zeta (t),$

 $\begin{gathered} LQ(t)\eta (t) \leqslant - w'(t) - \frac{{{p_0}}}{{{\tau _0}}}v'(t) - \hfill \\ \frac{\gamma }{{{a^{1/\gamma }}(t)}}\left[{(w{{(t)}^{(\gamma + 1)/\gamma }} + \frac{{{p_0}}}{{{\tau _0}}}{{(v(t))}^{(\gamma + 1)/\gamma }}} \right],\hfill \\ \end{gathered}$

 $\begin{gathered} \int_{{t_1}}^t {LQ(s)\eta (s){\zeta ^\gamma }(s)} {\text{d}}s \leqslant \hfill \\ - \int_{{t_1}}^t {{\zeta ^\gamma }(s)w'(s)} {\text{d}}s - \frac{{{p_0}}}{{{\tau _0}}}\int_{{t_1}}^t {{\zeta ^\gamma }(s)v'(s)} {\text{d}}s - \hfill \\ \int_{{t_1}}^t {\frac{{\gamma {\zeta ^\gamma }(s)}}{{{a^{1/\gamma }}(s)}}\left[{{{(w(s))}^{(\gamma + 1)/\gamma }} + \frac{{{p_0}}}{{{\tau _0}}}{{(v(s))}^{(\gamma + 1)/\gamma }}} \right]} {\text{d}}s = \hfill \\ - [{\zeta ^\gamma }(s)w(s)]_{{t_1}}^t + \int_{{t_1}}^t {\gamma {\zeta ^{\gamma - 1}}(s)\zeta '(s)} w(s){\text{d}}s - \hfill \\ \frac{{{p_0}}}{{{\tau _0}}}\left\{ {[{\zeta ^\gamma }(s)v(s)]_{{t_1}}^t - \int_{{t_1}}^t {\gamma {\zeta ^{\gamma - 1}}(s)\zeta '(s)} v(s){\text{d}}s} \right\} - \hfill \\ \int_{{t_1}}^t {\frac{{\gamma {\zeta ^\gamma }(s)}}{{{a^{1/\gamma }}(s)}}\left[{{{(w(s))}^{(\gamma + 1)/\gamma }} + \frac{{{p_0}}}{{{\tau _0}}}{{(v(s))}^{(\gamma + 1)/\gamma }}} \right]} {\text{d}}s = \hfill \\ - {\zeta ^\gamma }(t)w(t) + {\zeta ^\gamma }({t_1})w({t_1}) - \hfill \\ \frac{{{p_0}}}{{{\tau _0}}}[{\zeta ^\gamma }(t)v(t) + {\zeta ^\gamma }({t_1})v({t_1})] - \hfill \\ \int_{{t_1}}^t {\left[{\gamma {\zeta ^{\gamma - 1}}(s){a^{ - 1/\gamma }}(s)w(s) + \frac{{\gamma {\zeta ^\gamma }(s)}}{{{a^{1/\gamma }}(s)}}{{(w(s))}^{(\gamma + 1)/\gamma }}} \right]} {\text{d}}s - \hfill \\ \frac{{{p_0}}}{{{\tau _0}}}\int_{{t_1}}^t {\left[{\gamma {\zeta ^{\gamma - 1}}(s){a^{ - 1/\gamma }}(s)v(s) + \frac{{\gamma {\zeta ^\gamma }(s)}}{{{a^{1/\gamma }}(s)}}{{(v(s))}^{(\gamma + 1)/\gamma }}} \right]} {\text{d}}s \leqslant \hfill \\ 1 + {\zeta ^\gamma }({t_1})w({t_1}) + \frac{{{p_0}}}{{{\tau _0}}}[1 + {\zeta ^\gamma }({t_1})v({t_1})] + \hfill \\ \int_{{t_1}}^t {\frac{{{\gamma ^{\gamma + 1}}{a^{ - 1/\gamma }}(s)}}{{{{(\gamma + 1)}^{\gamma + 1}}\zeta (s)}}} {\text{ds + }}\frac{{{p_0}}}{{{\tau _0}}}\int_{{t_1}}^t {\frac{{{\gamma ^{\gamma + 1}}{a^{ - 1/\gamma }}(s)}}{{{{(\gamma + 1)}^{\gamma + 1}}\zeta (s)}}} {\text{ds}},\hfill \\ \hfill \\ \hfill \\ \end{gathered}$

 $\begin{gathered} \int_{{t_1}}^t {\left[{LQ(s)\eta (s){\zeta ^\gamma }(s) - \left( {1 + \frac{{{p_0}}}{{{\tau _0}}}} \right)\frac{{{\gamma ^{\gamma + 1}}{a^{ - 1/\gamma }}(s)}}{{{{(\gamma + 1)}^{\gamma + 1}}\zeta (s)}}} \right]} {\text{d}}s \hfill \\ \leqslant 1 + {\zeta ^\gamma }({t_1})w({t_1}) + \frac{{{p_0}}}{{{\tau _0}}}[1 + {\zeta ^\gamma }({t_1})v({t_1})],\hfill \\ \end{gathered}$

2 实例分析

 $\left( {x(t) + \frac{2}{3}x(t - 1)} \right)'' + \frac{\alpha }{{{t^2}}}x(t) = 0,\;t \geqslant 1,$ (24)

 $\begin{gathered} \mathop {\lim \sup }\limits_{t \to + \infty } \int_T^t {\left[{\frac{{L\varphi (s)Q(s)\psi (s,\;{t_1})}}{{{{[b\theta (s)]}^{\gamma - 1}}}} - \frac{{\varphi (s)}}{{{{(\gamma + 1)}^{\gamma + 1}}}}{{\left( {\frac{{\varphi {'_ + }(s)}}{{\varphi (s)}}} \right)}^{\gamma + 1}} \times \left( {a(s) + \frac{{{p_0}a(\tau (s))}}{{\tau _0^{\gamma + 1}}}} \right)} \right]} {\text{d}}s \hfill \\ = \mathop {\lim \sup }\limits_{t \to + \infty } \int_1^t {\left[{\frac{{s\alpha }}{{{s^2}}} - \frac{s}{{{2^2}}}{{\left( {\frac{1}{s}} \right)}^2}\left( {1 + \frac{2}{3}} \right)} \right]} {\text{d}}s = \left( {\alpha - \frac{5}{{12}}} \right)\mathop {\lim \sup }\limits_{t \to + \infty } \int_1^t {\frac{1}{s}{\text{d}}s} ,\hfill \\ \end{gathered}$

 $\begin{gathered} \mathop {\lim \sup }\limits_{t \to + \infty } \int_{{t_0}}^t {\left\{ {\varphi (s)q(s){{[1 - p(\delta (s))]}^\gamma } - \frac{{{{(\varphi {'_ + }(s))}^{\gamma + 1}}a(\delta (s))}}{{k{{(\gamma + 1)}^{\gamma + 1}}{{[\varphi (s)\delta '(s)]}^\gamma }}}} \right\}{\text{d}}s} \hfill \\ = \left( {\frac{\alpha }{3} - \frac{1}{4}} \right)\mathop {\lim \sup }\limits_{t \to + \infty } \int_1^t {\frac{1}{s}{\text{d}}s} ,\hfill \\ \end{gathered}$

 $\begin{gathered} \left\{ {\frac{1}{{{t^{1/3}}}}{{\left[{\left( {x(t) + (5 - \sin {t^2})x\left( {\frac{t}{2}} \right)} \right)'} \right]}^{1/3}}} \right\}' + \hfill \\ \frac{1}{{{t^{1/4}}}}f\left( {x\left( {\frac{t}{2}} \right)} \right) = 0,\;\;t \geqslant 1,\hfill \\ \end{gathered}$ (25)

f(u)=u[1+ln(1+u4)]，因为

 $\begin{gathered} 0 < p(t) = 5 - \sin {t^2} \leqslant 6 = {p_0},\hfill \\ \frac{{f(u)}}{u} = 1 + \ln (1 + {u^4}) \geqslant 1 = L,\hfill \\ \end{gathered}$

 $Q(t) = \min \{ q(t),\;q(\tau (t))\} = {t^{ - 1/4}},$

 $\begin{gathered} \mathop {\lim \sup }\limits_{t \to + \infty } \int_T^t {\left[{k\varphi (s)Q(s){\theta ^{ - \gamma }}(\tau (s)) - \left( {1 + \frac{{{p_0}}}{{{\tau _0}}}} \right)\frac{{\varphi (s)a(\tau (s))}}{{{{(\gamma + 1)}^{\gamma + 1}}\tau _0^\gamma }}{{\left( {\frac{{\varphi '(s)}}{{\varphi (s)}}} \right)}^{\gamma + 1}}} \right]} {\text{d}}s \hfill \\ = \mathop {\lim \sup }\limits_{t \to + \infty } \int_3^t {\left[{k\frac{1}{{{s^{1/4}}}}{{\left( {\frac{1}{8}({s^2} - 4)} \right)}^{ - 1/3}} - 0} \right]{\text{d}}s} \hfill \\ = \mathop {\lim \sup }\limits_{t \to + \infty } \int_3^t {\frac{{2k}}{{{s^{1/4}}{{({s^2} - 4)}^{1/3}}}}{\text{d}}s = + \infty } ,\hfill \\ \end{gathered}$

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