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121.28.7.*

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\begin{document}
\vspace {0.8cm}
\title{{ Existence of Bounded Nonoscillatory Solution of Higher
Order Neutral Differential Equations with Positive and Negative
Coefficients
}\thanks{Research are supported by
the Natural Science Foundation of Hebei Province and Main
Foundation of Hebei Normal University.}}
\author{ Zhenguo Zhang, Haiyan Liang, Qiaoluan Li \\
}
\date{}
\maketitle
\begin{minipage}{120mm}
{\bf Abstract:} {\small In this paper, by applying the fixed point
theorem and with the condition of $f_i\,\,(i=1,2)$ satisfying the
Lipschitz condition,  we obtain the existence of bounded
nonoscillatory solution of the following higher order neutral
differential equations:
$$(x(t)-p(t)x(\tau(t)))^{(n)}+f_{1}(t,x(\sigma_{1}(t)))-f_{2}(t,x(\sigma_{2}(t)))=0, \eqno{(1.1)}$$
$$(x(t)-p(t)x(\tau(t)))^{(n)}+f_{1}(t,x(\sigma_{1}(t)))-f_{2}(t,x(\sigma_{2}(t)))=g(t),
\eqno{(1.2)}$$
$$(x(t)-p(t)x(\tau(t)))^{(n)}+\sum\limits_{j=1}^{m}q_{j}(t)x(\tau_{j}(t))=0,
\eqno{(1.3)}$$}
\vskip 0.0in
\noindent {\bf Keywords:} {\small  Neutral differential equation,
Nonoscillation.}
\\
   \noindent {\bf  2000MSC:}
34C10
\end{minipage} \vskip 0.2in
\begin{center}
{\bf 1. INTRODUCTION}
\end{center} \vskip 0.in
本文考虑具有正负项的高阶非线性微分方程
$$(x(t)-p(t)x(\tau(t)))^{(n)}+f_{1}(t,x(\sigma_{1}(t)))-f_{2}(t,x(\sigma_{2}(t)))=0, \eqno{(1.1)}$$
$$(x(t)-p(t)x(\tau(t)))^{(n)}+f_{1}(t,x(\sigma_{1}(t)))-f_{2}(t,x(\sigma_{2}(t)))=g(t),

1楼2009-06-17 17:37回复

121.28.7.*

\eqno{(1.2)}$$ 及具有振动系数的高阶中立型微分方程
$$(x(t)-p(t)x(\tau(t)))^{(n)}+\sum\limits_{j=1}^{m}q_{j}(t)x(\tau_{j}(t))=0,
\eqno{(1.3)}$$ 其中$\tau, \sigma_{i}, \tau_{j}\in
C([t_{0},\infty), R^{+}),\,i=1,2,\,j=1,2,\cdots,m.$
$\lim\limits_{t\rightarrow \infty}\tau(t)=\infty,$
$\lim\limits_{t\rightarrow\infty}\sigma_{i}(t)=\infty,$
$\lim\limits_{t\rightarrow\infty}\tau_{j} (t)=\infty$.
$p,f_{i},g,q_{j}\in C([t_{0},\infty),R),\,\,uf_{i}(t,u)>0 (u\neq
0).$ 在$[t_{0},\infty)\times [0,a]$ or $[t_{0},\infty)\times
[-a,0]$上,$f_{i}$满足 Lipschitz 条件，即
$$|f_{i}(t,u)-f_{i}(t,v)|\leqslant p_{i}(t)|u-v|, \eqno{(1.4)}$$
$$\int_{t}^{\infty}(s-t)^{n-1}p_{i}(s)ds < \infty,  \eqno{(1.5)}$$
$$\int_{t}^{\infty}(s-t)^{n-1}|g(s)|ds < \infty,  \eqno{(1.6)}$$
$$\int_{t}^{\infty}(s-t)^{n-1}|q_{j}(s)|ds < \infty,  \eqno{(1.7)}$$
It is well known that the oscillatory behavior of solutions of the
neutral differential equations has been extensively studied in the
literature since 1980, see [1], [2]. Recently, there have been a lot
of activities concerning the nonoscillations of neutral functional
differential equations. There have been some results about the
nonoscillatory behavior of solutions of first order neutral
differential equations, see [3]. In [4] and [5], the authors
investigated the following higher order neutral delay differential
equations respectively:
$$(x(t)+cx(t-\tau))^{(n)}+(-1)^{n+1}[P(t)x(t-\sigma)-Q(t)x(t-\delta)]=0,
\eqno{(1.8)}$$
$$(x(t)-x(t-\tau))^{(n)}+P(t)x(t-\sigma)=0, \eqno{(1.9)}$$
where $P,Q \in C([t_{0},\infty),R^+).$ In Eq.(1.9), $n$ is an  odd
number.
In this paper, we consider the more extensive equations
(1.1),(1.2) and (1.3), where $n\in N$. Our main goal in this paper will be to
obtain sufficient conditions for the existence of the bounded
nonoscillatory solutions of Eq.(1.1),(1.2),(1.3).
As is customary, a solution of Eq.(1.1),(1.2) and (1.3) is said to
be oscillatory, if it has  arbitrarily large zeros.Otherwise the solution is called
nonoscillatory.\\
对于$p(t)$,我们考虑下列四种情形:\\

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121.28.7.*

方程(1.1)存在有界非振动解.}\\
{\bf 证明:} 选取常数 $M_{1},\,M_{2},\,\alpha$,满足
$0<p_{2}M_{1}<\alpha <(p_{1}-1)M_{2}$.  设
$$c=\min\{\frac{(p_{1}-1)M_{2}-\alpha}{M_{2}},\,\,\,\,\,
\frac{p_1
\alpha-p_1p_{2}M_{1}}{p_{2}M_{2}},\,\,\,\,\,\frac{p_{1}-1}{2}\}$$
由式(1.5)知,存在$T\geqslant t_{0}$, 当 $t\geqslant T$时,有
$$\frac{1}{(n-1)!}\int_{\tau^{-1}(t)}^{\infty}(s-\tau^{-1}(t))^{n-1}(p_{1}(s)+p_{2}(s))ds\leqslant
c$$
Consider the Banach space BC of all continuously bounded
functions defined on $[t_{0},\infty)$ with the sup norm
$||x||=sup\{|x(t)|:t\geqslant t_{0}\}.$ Set
$$\Omega =\{x\in BC:M_{1}\leqslant x(t)\leqslant M_{2},\,
for\,\,\,\,\, t\geqslant t_{0}\}$$ Then $\Omega$ is a nonempty,
bounded, closed and
convex subset of BC.\\
Define a mapping $\Gamma$ on $\Omega$ as follows:
$$ (\Gamma x)(t)=\left\{\begin{array}{lll}\frac{\alpha+x(\tau^{-1}(t))}{p(\tau^{-1}(t))}
+ \frac{(-1)^{n-1}}{(n-1)!p(\tau^{-1}(t))}\times &
\\
\int_{\tau^{-1}(t)}^{\infty}(s-\tau^{-1}(t))^{n-1}
[f_{1}(s,x(\sigma_{1}(s)))-f_{2}(s,x(\sigma_{2}(s)))]ds &t\geqslant
T\\ (\Gamma x)(T) & t_{0}\leqslant t <T
\end{array}\right.$$
对任意$x\in \Omega$, $$(\Gamma
x)(t)\leqslant\frac{\alpha}{p_{1}}
+\frac{M_{2}}{p_{1}}+\frac{cM_{2}}{p_1}\leqslant M_{2}$$ and
$$(\Gamma x)(t)\geqslant \frac{\alpha}{p_{2}}-\frac{M_{2}c}{p_1} \geqslant
M_{1}$$ 因此 $\Gamma \Omega\subset\Omega$. 上的
下面我们证明$\Gamma$是$\Omega$上的压缩映射.\\
对任意 $x,y\in \Omega$,
$$|(\Gamma x)(t)-(\Gamma y)(t)|\leqslant \frac{1}{p_{1}}||x-y||+\frac{c}{p_1}||x-y||$$
则对任意$t\in [t_{0},\infty)$,有 $$||\Gamma x-\Gamma y||\leqslant
\frac{1+c}{p_{1}}||x-y||$$  因此$\Gamma$是$\Omega$上的压缩映射.
因此，存在 $x\in\Omega$, 使得$\Gamma x=x$. 容易验证
$x(t)$就是方程(1.1)的有界正解.\\
{\bf 定理 2.3}  {\it 假设条件(iii)和式(1.4),(1.5)成立,则方程
(1.1)存在有界非振动解。}\\
{\bf 证明:} 选取常数$M_{1},\,M_{2},\,\alpha$, 满足
$0<M_{1}<M_{2},\,\,M_{1}-pM_{2}<\alpha <M_{2}$.  设
$$c=\min\{\frac{M_{2}-\alpha}{M_{2}},\,\,\,\,\,\frac{\alpha
+pM_{2}-M_{1}}{M_{2}},\,\,\,\,\,\frac{1+p}{2}\}$$ 由式(1.5)知,
存在$T\geqslant t_{0}$,当$t\geqslant T$时,有
$$\frac{1}{(n-1)!}\int_{t}^{\infty}(s-t)^{n-1}(p_{1}(s)+p_{2}(s))ds\leqslant
c$$
Consider the Banach space BC of all continuously bounded
functions defined on $[t_{0},\infty)$ with the sup norm

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121.28.7.*

\sum\limits_{j=1}^{m}q_{j}(s)ds &t\geqslant T\\
(\Gamma x)(T) & t_{0}\leqslant t <T
\end{array}\right.$$
The others are similar with Theorem 2.4.\\
Then we complete the proof.\\
\begin{center}
{\bf References}
\end {center}
1. L.H.Erbe, Q.K.Kong and B.G.Zhang, {\it Oscillation Theory for
Functional Differential Equations}, Marcel Kekker, (1995).\\
2. D.D.Bainov and D.P.Mishev, {\it Oscillation Theory for Neutral
Differential Equations with Delay}, Adam Hilger, (1991).\\
3. S.G.Ruan, {\it Oscillations for First-Order Neutral Differential
Equations with Variable Coefficients}, Bull.Austral.Math.Soc.43,
147-152, (1991).\\
4. Y.Zhou and B.G.Zhang, {\it Existence of Nonoscillatory Solutions
of Higher-Order Neutral Differential Equations with Positive and
Negative Coefficients}, Appl.Math.Letters 15, 867-874,
(2002).\\
5. Q.K.Kong, Y.J.Sun and B.G.Zhang, {\it Nonoscillation of a Class
of Neutral Differential Equations}, Computers
Math.Applic.44,643-654, (2002).\\
6. B.G.Zhang and J.S.Yu, {\it The Existence of Positive Solutions of
Neutral Differential Equations}, Sci.Sinica 8, 785-790,
(1992).\\
7. M.R.S.Kulenovi\'{c} and S.Had\v{z}iomerspahi\'{c}, {\it Existence
of Nonoscillatory Solution of Second Order Linear Neutral
Delay Equation}, J.Math.Anal.Appl.228, 436-448, (1998).\\
8. J.H.Shen and J.S.Yu, {\it Asymptotic Behavior of Solutions of
Neutral Differential Equations with Posituve and Negative
Coefficients}, J.Math.Anal.Appl.195, 517-526, (1995).\\
9. S.Tanaka, {\it Existence of Positive Solutions of Higher Order
Nonlinear Neutral Differential Equations}, Rocky Mountain J.Math.30,
1139-1149, (2000).\\
Zhenguo Zhang, Haiyan Liang, Qiaoluan Li\\
{\small College of Mathematics and Information
Science,\\} \mbox{}\hspace{0.6cm}{\small Hebei Normal University,\\}
\mbox{}\hspace{0.6cm}{\small Shijiazhuang,\,\,050016,\,\,
P.R.\,China\\} \mbox{}\hspace{0.6cm}{\small E-mail:
liang730110@eyou.com}
\end {document}

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