﻿ 一类具多滞量的广义Emden-Fowler中立型阻尼微分方程的振动性
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 浙江大学学报(理学版)  2017, Vol. 44 Issue (3): 270-273  DOI:10.3785/j.issn.1008-9497.2017.03.004 0

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LIN Wenxian. Oscillation for generalized Emden-Fowler neutral functional differential equations with damping terms and multiple delays[J]. Journal of Zhejiang University(Science Edition), 2017, 44(3): 270-273. DOI: 10.3785/j.issn.1008-9497.2017.03.004.
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### 文章历史

Oscillation for generalized Emden-Fowler neutral functional differential equations with damping terms and multiple delays
LIN Wenxian
College of Mathematics and Statistics, Hanshan Normal University, Chaozhou 521041, Guangdong Province, China
Abstract: Using Riccati transformation method and Young's inequality, some new interval oscillatory criterion for generalized Emden-Fowler neutral functional differential equations with damping terms and multiple de1ays are obtained. The results generalize and improve some known results.
Key words: generalized Emden-Fowler functional differential equations    oscillation criteria    damping terms
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)

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

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

 $\begin{array}{*{20}{c}} {\left( {{{\rm{H}}_3}} \right)r\left( t \right) \in {C^1}\left( {I,\left( {0,\infty } \right)} \right),}\\ {m\left( t \right) \in C\left( {I,\left[ {0,\infty } \right)} \right),r'\left( t \right) \ge 0,}\\ {\mathop {\lim }\limits_{t \to \infty } \int_{{t_0}}^t {{{\left[ {\frac{1}{{r\left( v \right)}}\exp \left( { - \int_{{t_0}}^v {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right)} \right]}^{\frac{1}{\alpha }}}{\rm{d}}v} = \infty ;} \end{array}$
 $\begin{array}{l} \left( {{{\rm{H}}_4}} \right)\;\;\;\tau \left( t \right),{\sigma _i}\left( t \right) \in C\left( {\left[ {{t_0},\infty } \right),R} \right),\tau \left( t \right) \le t,\\ {\sigma _i}\left( t \right) \le t,\;\;且\mathop {\lim }\limits_{t \to \infty } \tau \left( t \right) = \mathop {\lim }\limits_{t \to \infty } {\sigma _i}\left( t \right) = \infty ,i = 1,2, \cdots ,n. \end{array}$

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

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

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

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

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

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

t→∞, 由(H2), 有$\mathop {\lim }\limits_{t \to \infty } y\left( t \right) =-\infty$, 这与y(t)>0, tt1矛盾, 所以有y′(t)>0, tt1.

 $\int_{{t_0}}^\infty {\exp \left( {\int_{{t_0}}^t {\frac{{m\left( s \right)}}{{r\left( s \right)}}{\rm{d}}s} } \right){q_{{i_0}}}\left( t \right)\sigma _{{i_0}}^{{\beta _{{i_0}}}}} \left( t \right){\rm{d}}t = \infty ,$ (2)

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

 $\begin{array}{l} {\left[ {r\left( t \right){{\left( {y'\left( t \right)} \right)}^\alpha }} \right]^\prime } + m\left( t \right){\left( {y'\left( t \right)} \right)^\alpha } + \\ \;\;\;\;\;\;\;\sum\limits_{i = 1}^n {{q_i}\left( t \right){{\left( {1 - p} \right)}^{{\beta _i}}}x{{\left( {{\sigma _i}\left( t \right)} \right)}^{{\beta _i}}}} \le 0, \end{array}$ (3)

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

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

φ(t)单调增加且最终定号.现断言φ(t)>0.否则若φ(t)≤0，有

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

 $\frac{{y\left( {{\sigma _i}\left( t \right)} \right)}}{{{\sigma _i}\left( t \right)}} \ge {K_i},t \ge T,i = 1,2, \cdots ,n.$ (4)

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

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

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

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

t→∞, 与式(2) 矛盾.因此φ(t)>0成立.证毕.

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

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

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

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

 $\begin{array}{l} \mathop {\lim }\limits_{t \to \infty } \sup \int_{{t_0}}^t {\left( {\rho \left( s \right)Q\left( s \right) - {{\left( {\alpha + 1} \right)}^{ - \alpha - 1}}\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - } \right.} \right.} \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {{{\left. {\frac{{m\left( s \right)}}{{r\left( s \right)}}} \right]}^{\alpha + 1}}\rho \left( s \right)r\left( s \right)} \right){\rm{d}}s = \infty , \end{array}$ (5)

 $\begin{array}{l} Q\left( t \right) = \prod\limits_{i = 1}^n {{{\left[ {{q_i}\left( t \right)} \right]}^{{k_i}}}{{\left[ {\frac{{\left( {1 - p} \right)\sigma \left( t \right)}}{t}} \right]}^\alpha }} ,\\ \;\;\;\;\;\;\;\;\;\;\;\;\sigma \left( t \right) \le \min \left\{ {{\sigma _1}\left( t \right),{\sigma _2}\left( t \right), \cdots ,{\sigma _n}\left( t \right)} \right\}. \end{array}$ (6)

 $\begin{array}{l} {k_1} = \frac{{{\beta _n} - {\beta _1}}}{{{\beta _2} - \alpha }},{k_i} = \frac{{{\beta _n} - {\beta _1}}}{{{\beta _{i + 1}} - {\beta _{i - 1}}}},i = 2,3, \cdots ,n - 1,\\ \;\;\;\;\;\;{k_n} = \frac{{{\beta _n} - {\beta _1}}}{{\alpha - {\beta _{n - 1}}}}, \end{array}$ (7)

 $W\left( t \right) = \rho \left( t \right)r\left( t \right){\left( {\frac{{y'\left( t \right)}}{{y\left( t \right)}}} \right)^\alpha },\;\;\;t \ge {t_1},$ (8)

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

 $\begin{array}{l} W'\left( t \right) \le - \frac{{\rho \left( t \right)}}{{{y^\alpha }\left( t \right)}}\sum\limits_{i = 1}^n {{q_i}\left( t \right)x{{\left( {{\sigma _i}\left( t \right)} \right)}^{{\beta _i}}}} + \left[ {\frac{{\rho '\left( t \right)}}{{\rho \left( t \right)}} - } \right.\\ \;\;\;\;\;\;\;\;\;\;\;\left. {\frac{{m\left( t \right)}}{{r\left( t \right)}}} \right]W\left( t \right) - \frac{\alpha }{{{{\left[ {\rho \left( t \right)r\left( t \right)} \right]}^{\frac{1}{\alpha }}}}}{W^{\frac{{\alpha + 1}}{\alpha }}}\left( t \right), \end{array}$ (9)

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

 $\begin{array}{l} W'\left( t \right) \le - \rho \left( t \right){\left( {{q_1}\left( t \right)} \right)^{\frac{{{\beta _2} - \alpha }}{{{\beta _2} - {\beta _1}}}}}{\left( {{q_2}\left( t \right)} \right)^{\frac{{\alpha - {\beta _1}}}{{{\beta _2} - {\beta _1}}}}}{\left[ {\frac{{\left( {1 - p} \right)y\left( {\sigma \left( t \right)} \right)}}{{y\left( t \right)}}} \right]^\alpha } + \\ \;\;\;\;\;\;\;\;\;\;\;\;\left[ {\frac{{\rho '\left( t \right)}}{{\rho \left( t \right)}} - \frac{{m\left( t \right)}}{{r\left( t \right)}}} \right]W\left( t \right) - \frac{\alpha }{{{{\left[ {\rho \left( t \right)r\left( t \right)} \right]}^{\frac{1}{\alpha }}}}}{W^{\frac{{\alpha + 1}}{\alpha }}}\left( t \right), \end{array}$ (10)

 $\frac{{y\left( {\sigma \left( t \right)} \right)}}{{y\left( t \right)}} \ge \frac{{\sigma \left( t \right)}}{t}.$ (11)

 $\begin{array}{l} W'\left( t \right) \le - \rho \left( t \right)Q\left( t \right) + \left[ {\frac{{\rho '\left( t \right)}}{{\rho \left( t \right)}} - \frac{{m\left( t \right)}}{{r\left( t \right)}}} \right]W\left( t \right) - \\ \;\;\;\;\;\;\;\;\;\;\frac{\alpha }{{{{\left[ {\rho \left( t \right)r\left( t \right)} \right]}^{\frac{1}{\alpha }}}}}{W^{\frac{{\alpha + 1}}{\alpha }}}\left( t \right), \end{array}$ (12)

 $\begin{array}{l} W'\left( t \right) \le - \rho \left( t \right)Q\left( t \right) + {\left( {\alpha + 1} \right)^{ - \alpha - 1}}{\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - \frac{{m\left( s \right)}}{{r\left( s \right)}}} \right]^{\alpha + 1}} \times \\ \;\;\;\;\;\;\;\;\;\;\rho \left( t \right)r\left( t \right),\;\;\;\;\;t \ge {t_1}, \end{array}$ (13)

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

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.

 $\mathop {\lim }\limits_{t \to \infty } \sup \int_{{t_0}}^t {\left( {Q\left( s \right) - {{\left[ { - \frac{{m\left( s \right)}}{{\alpha + 1}}} \right]}^{\alpha + 1}}\frac{1}{{{r^\alpha }\left( s \right)}}} \right){\rm{d}}s} = \infty ,$ (14)

 $\begin{array}{l} \mathop {\lim }\limits_{t \to \infty } \sup \frac{1}{{{t^n}}}\int_{{t_0}}^t {{{\left( {t - s} \right)}^n}\left( {\rho \left( s \right)Q\left( s \right) - } \right.} \\ \left. {\;\;\;\;\;{{\left( {\alpha + 1} \right)}^{ - \alpha - 1}}{{\left[ {\frac{{\rho '\left( s \right)}}{{\rho \left( s \right)}} - \frac{{m\left( s \right)}}{{r\left( s \right)}}} \right]}^{\alpha + 1}}\rho \left( s \right)r\left( s \right)} \right){\rm{d}}s = \infty , \end{array}$ (15)

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

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

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

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

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

D={(t, s)|tst0}, D0={(t, s)|t>st0}.函数H(t, s)∈C(D, R)属于$\wp$类, 记作H$\wp$, 如果

(ⅰ)H(t, t)=0, tt0; 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)

 $\begin{array}{l} \mathop {\lim \sup }\limits_{t \to \infty } \frac{1}{{H\left( {t,{t_0}} \right)}}\int_{{t_0}}^t {\left[ {H\left( {t,s} \right)\rho \left( s \right)Q\left( s \right) - } \right.} \\ \;\;\;\;\;\;\;\;\left. {{{\left( {\frac{{\left| {h\left( {s,a} \right)} \right|}}{{\alpha + 1}}} \right)}^{\alpha + 1}}\rho \left( s \right)r\left( s \right)} \right]{\rm{d}}s = \infty , \end{array}$ (17)

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

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

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

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

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