\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 146, pp. 1--7.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/146\hfil Existence of non-oscillatory solutions] {Existence of non-oscillatory solutions for a higher-order nonlinear neutral difference equation} \author[Z. Guo, M. Liu,\hfil EJDE-2010/146\hfilneg] {Zhenyu Guo, Min Liu} % in alphabetical order \address{Zhenyu Guo \newline School of Sciences, Liaoning Shihua University\\ Fushun, Liaoning 113001, China} \email{guozy@163.com} \address{Min Liu \newline School of Sciences, Liaoning Shihua University, Fushun, Liaoning 113001, China} \email{min\_liu@yeah.net} \thanks{Submitted July 30, 2010. Published October 14, 2010.} \subjclass[2000]{34K15, 34C10} \keywords{Nonoscillatory solution; neutral difference equation; \hfill\break\indent Krasnoselskii fixed point theorem} \begin{abstract} This article concerns the solvability of the higher-order nonlinear neutral delay difference equation $$\Delta\Big(a_{kn}\dots\Delta\big(a_{2n} \Delta(a_{1n}\Delta(x_n+b_nx_{n-d}))\big)\Big) +\sum_{j=1}^s p_{jn}f_j(x_{n-r_{jn}})=q_n,$$ where $n\geq n_0\ge0$, $d,k,j,s$ are positive integers, $f_j:\mathbb{R}\to \mathbb{R}$ and $xf_j(x)\geq 0$ for $x\ne 0$. Sufficient conditions for the existence of non-oscillatory solutions are established by using Krasnoselskii fixed point theorem. Five theorems are stated according to the range of the sequence $\{b_n\}$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction and preliminaries} Interest in the solvability of difference equations has increased lately, as inferred from the number of related publications; see for example the references in this article and their references. Authors have examined various types difference equations, as follows: \begin{gather} \Delta(a_n\Delta x_n)+p_nx_{g(n)}=0,\quad n\ge0, \quad \text{in \cite{z2}}, \label{e1.1}\\ \Delta(a_n\Delta x_n)=q_nx_{n+1},\quad\Delta(a_n\Delta x_n)=q_nf(x_{n+1}),\quad n\ge0, \quad \text{in \cite{t1}}, \label{e1.2}\\ \Delta^2(x_n+px_{n-m})+p_nx_{n-k}-q_nx_{n-l}=0,\quad n\geq n_0, \quad \text{in \cite{c2}}, \label{e1.3}\\ \Delta^2(x_n+px_{n-k})+f(n,x_n)=0,\quad n\ge1, \quad \text{in \cite{m2}},\label{e1.4}\\ \Delta^2(x_n-px_{n-\tau})=\sum_{i=1}^{m}q_if_i(x_{n-\sigma_i}),\quad n\geq n_0, \quad \text{in \cite{m1}}, \label{e1.5} \\ \Delta(a_n\Delta(x_n+bx_{n-\tau}))+f(n,x_{n-d_{1n}},x_{n-d_{2n}}, \dots,x_{n-d_{kn}})=c_n,\notag \\ n\geq n_0, \quad \text{in \cite{l1}}, \label{e1.6} \\ \Delta^m(x_n+cx_{n-k})+p_nx_{n-r}=0, n\geq n_0, \quad \text{in \cite{z3}}, \label{e1.7} \\ \Delta^m(x_n+c_nx_{n-k})+p_nf(x_{n-r})=0,\quad n\geq n_0, \quad \text{in \cite{a3,a4,y1,z1}}, \label{e1.8} \\ \Delta^m(x_n+cx_{n-k})+\sum_{s=1}^up_n^{s}f_s(x_{n-r_s})=q_n,\quad n\geq n_0, \quad \text{in \cite{z4}}, \label{e1.9} \\ \Delta^m(x_n+cx_{n-k})+p_nx_{n-r}-q_nx_{n-l}=0,\quad n\geq n_0, \quad \text{in \cite{z5}}. \label{e1.10} \end{gather} Motivated by the above publications, we investigate the higher-order nonlinear neutral difference equation $$\Delta\Big(a_{kn}\dots\Delta\big(a_{2n}\Delta(a_{1n}\Delta(x_n+b_nx_{n-d}))\big)\Big) +\sum_{j=1}^sp_{jn}f_j(x_{n-r_{jn}})=q_n, \label{e1.11}$$ where $n\geq n_0\geq 0$, $d,k,j,s$ are positive integers, $\{a_{in}\}_{n\geq n_0}$ ($i=1,2,\dots,k$), $\{b_n\}_{n\geq n_0}$, $\{p_{jn}\}_{n\geq n_0}$ ($1\leq j\leq s$) and $\{q_n\}_{n\geq n_0}$ are sequences of real numbers, $r_{jn}\in \mathbb{Z}$ ($1\leq j\leq s,n_0\leq n$), $f_j:\mathbb{R}\to \mathbb{R}$ and $xf_j(x)\geq 0$ for $x\ne0$ ($j=1,2,\dots,s$). Clearly, difference equations \eqref{e1.1}--\eqref{e1.10} are special cases of \eqref{e1.11}, for which we use Krasnoselskii fixed point theorem to obtain non-oscillatory solutions. \begin{lemma}[Krasnoselskii Fixed Point Theorem] \label{lem1.1} Let $\Omega$ be a bounded closed convex subset of a Banach space $X$ and $T_1,T_2:S\to X$ satisfy $T_1x+T_2y\in \Omega$ for each $x,y\in \Omega$. If $T_1$ is a contraction mapping and $T_2$ is a completely continuous mapping, then the equation $T_1x+T_2x=x$ has at least one solution in $\Omega$. \end{lemma} As usual, the forward difference $\Delta$ is defined as $\Delta x_n=x_{n+1}-x_n$, and for a positive integer $m$ the higher-order difference is defined as $$\Delta^mx_n=\Delta(\Delta^{m-1}x_n),\quad \Delta^0x_n=x_n.$$ In this article, $\mathbb{R}=(-\infty,+\infty)$, $\mathbb{N}$ is the set of positive integers, $\mathbb{Z}$ is the sets of all integers, $\alpha=\inf\{n-r_{jn}:1\leq j\leq s,n_0\leq n\}$, $\beta=\min\{n_0-d,\alpha\}$, $\lim_{n\to\infty}(n-r_{jn})=+\infty$, $1\leq j\leq s$, $l_{\beta}^{\infty}$ denotes the set of real-valued bounded sequences $x=\{x_n\}_{n\ge\beta}$. It is well known that $l_{\beta}^{\infty}$ is a Banach space under the supremum norm $\|x\|=\sup_{n\geq\beta}|x_n|$. For $N>M>0$, let $$A(M,N)=\big\{x=\{x_n\}_{n\ge\beta}\in l_{\beta}^{\infty}: M\leq x_n\leq N,n\ge\beta\big\}.$$ Obviously, $A(M,N)$ is a bounded closed and convex subset of $l_{\beta}^{\infty}$. Put $$\overline{b}=\limsup_{n\to\infty} b_n\quad\text{and}\quad \underline{b}=\liminf_{n\to\infty} b_n.$$ \begin{definition}[\cite{c1}] \label{def1.1} \rm A set $\Omega$ of sequences in $l_{\beta}^{\infty}$ is uniformly Cauchy (or equi-Cauchy) if for every $\varepsilon>0$, there exists an integer $N_0$ such that $$|x_i-x_j|<\varepsilon,$$ whenever $i,j>N_0$ for any $x={x_k}$ in $\Omega$. \end{definition} \begin{lemma}[{Discrete Arzela-Ascoli's theorem \cite{c1}}] \label{lem1.2} A bounded, uniformly Cauchy subset $\Omega$ of $l_{\beta}^{\infty}$ is relatively compact. \end{lemma} By a solution of \eqref{e1.11}, we mean a sequence $\{x_n\}_{n\ge\beta}$ with a positive integer $N_0\geq n_0+d+|\alpha|$ such that \eqref{e1.11} is satisfied for all $n\geq N_0$. As is customary, a solution of \eqref{e1.11} is said to be oscillatory about zero, or simply oscillatory, if the terms $x_n$ of the sequence $\{x_n\}_{n\ge\beta}$ are neither eventually all positive nor eventually all negative. Otherwise, the solution is called non-oscillatory. \section{Existence of non-oscillatory solutions} In this section, we will give five sufficient conditions of the existence of non-oscillatory solutions of \eqref{e1.11}. \begin{theorem} \label{thm2.1} If there exist constants $M$ and $N$ with $N>M>0$ and such that \begin{gather} |b_n|\leq b<\frac{N-M}{2N},\quad \text{eventually},\label{e2.1} \\ \sum_{t=n_0}^{\infty}\max\big\{\frac{1}{|a_{it}|},|p_{jt}|,|q_t|: 1\leq i\leq k,1\leq j\leq s\big\}<+\infty, \label{e2.2} \end{gather} then \eqref{e1.11} has a non-oscillatory solution in $A(M,N)$. \end{theorem} \begin{proof} Choose $L\in(M+bN,N-bN)$. By \eqref{e2.1} and \eqref{e2.2}, an integer $N_0>n_0+d+|\alpha|$ can be chosen such that $$|b_n|\leq b<\frac{N-M}{2N},\ \forall n\geq N_0,\label{e2.3}$$ and $$\sum_{t_1=N_0}^{\infty}\sum_{t_2=t_1}^{\infty}\dots \sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{F\big|\sum_{j=1}^sp_{jt}\big|+|q_t|} {\big|\prod_{i=1}^{k}a_{it_i}\big|} \leq \min\{L-bN-M,N-bN-L\}, \label{e2.4}$$ where $F=\max_{M\leq x\leq N}\{f_j(x):1\leq j\leq s\}$. Define two mappings $T_{1},T_{2}:A(M,N)\to X$ by \begin{gather} (T_{1}x)_n=\begin{cases} L-b_nx_{n-d}, & n\geq N_0,\\ (T_{1}x)_{N_0}, & \beta\leq n0$, take$N_1\geq N_0$large enough, $$\sum_{t_1=N_1}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{F\big|\sum_{j=1}^sp_{jt}\big|+|q_t|} {\big|\prod_{i=1}^{k}a_{it_i}\big|}<\frac{\varepsilon}{2}.\label{e2.7}$$ Then, for any$x=\{x_n\}\in A(M,N)$and$n_1,n_2\geq N_1, \eqref{e2.7} ensures that \begin{align*} \big|T_{2}x_{n_1}-T_{2}x_{n_2}\big| &\leq \sum_{t_1=n_1}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{\big|\sum_{j=1}^sp_{jt}f_j(y_{t-r_{jt}})-q_t\big|} {\big|\prod_{i=1}^{k}a_{it_i}\big|}\\ &\quad +\sum_{t_1=n_2}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{\big|\sum_{j=1}^sp_{jt}f_j(y_{t-r_{jt}})-q_t\big|} {\big|\prod_{i=1}^{k}a_{it_i}\big|}\\ &\leq \sum_{t_1=N_1}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{F\big|\sum_{j=1}^sp_{jt}\big|+|q_t|} {\big|\prod_{i=1}^{k}a_{it_i}\big|}\\ &\quad +\sum_{t_1=N_1}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{F\big|\sum_{j=1}^sp_{jt}\big|+|q_t|} {\big|\prod_{i=1}^{k}a_{it_i}\big|}\\ &< \frac{\varepsilon}{2}+\frac{\varepsilon}{2}=\varepsilon, \end{align*} which impliesT_{2}A(M,N)$begin uniformly Cauchy. Therefore, by Lemma \ref{lem1.2}, the set$T_{2}A(M,N)$is relatively compact. By Lemma \ref{lem1.1}, there exists$x=\{x_n\}\in A(M,N)$such that$T_{1}x+T_{2}x=x$, which is a bounded non-oscillatory solution to \eqref{e1.11}. This completes the proof. \end{proof} \begin{theorem} \label{thm2.2} If \eqref{e2.2} holds, $$b_n\ge0\ \text{eventually, }\ 0\leq \underline{b}\leq \overline{b}<1, \label{e2.8}$$ and there exist constants$M$and$N$with$N>\frac{2-\underline{b}}{1-\overline{b}}M>0$then \eqref{e1.11} has a non-oscillatory solution in$A(M,N)$. \end{theorem} \begin{proof} Choose$L\in(M+\frac{1+\overline{b}}{2}N,N+\frac{\underline{b}}{2}M)$. By \eqref{e2.2} and \eqref{e2.8}, an integer$N_0>n_0+d+|\alpha|can be chosen such that $$\frac{\underline{b}}{2}\leq b_n\leq \frac{1+\overline{b}}{2},\ \forall n\geq N_0\label{e2.9}$$ and \begin{aligned} &\sum_{t_1=N_0}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{F\big|\sum_{j=1}^sp_{jt}\big|+|q_t|} {\big|\prod_{i=1}^{k}a_{it_i}\big|}\\ &\leq \min\Big\{L-M-\frac{1+\overline{b}}{2}N,N-L +\frac{\underline{b}}{2}M\Big\}, \end{aligned}\label{e2.10} whereF=\max_{M\leq x\leq N}\{f_j(x):1\leq j\leq s\}$. Then define$T_1,T_2:A(M,N)\to X$as \eqref{e2.5} and \eqref{e2.6}. The rest proof is similar to that of Theorem \ref{thm2.1}, and it is omitted. \end{proof} \begin{theorem} \label{thm2.3} If \eqref{e2.2} holds, $$b_n\leq 0 \text{ eventually},\quad -1< \underline{b}\leq \overline{b}\leq 0, \label{e2.11}$$ and there exist constants$M$and$N$with$N>\frac{2+\overline{b}}{1+\underline{b}}M>0$, then \eqref{e1.11} has a non-oscillatory solution in$A(M,N)$. \end{theorem} \begin{proof} Choose$L\in(\frac{2+\overline{b}}{2}M,\frac{1+\underline{b}}{2}N)$. By \eqref{e2.2} and {\eqref{e2.11}}, an integer$N_0>n_0+d+|\alpha|can be chosen such that $$\frac{\underline{b}-1}{2}\leq b_n\leq \frac{\overline{b}}{2},\ \forall n\geq N_0,\label{e2.12}$$ and \begin{aligned} &\sum_{t_1=N_0}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{F\big|\sum_{j=1}^sp_{jt}\big|+|q_t|} {\big|\prod_{i=1}^{k}a_{it_i}\big|}\\ &\leq \min\Big\{L-\frac{2+\overline{b}}{2}M, \frac{1+\underline{b}}{2}N-L\Big\}, \end{aligned}\label{e2.13} whereF=\max_{M\leq x\leq N}\{f_j(x):1\leq j\leq s\}$. Then define$T_1,T_2:A(M,N)\to X$by \eqref{e2.5} and \eqref{e2.6}. The rest proof is similar to that of Theorem \ref{thm2.1}, and is omitted. \end{proof} \begin{theorem} \label{thm2.4} If \eqref{e2.2} holds, $$b_n>1 \text{ eventually},\quad 1<\underline{b},\text{ and } \overline{b}<\underline{b}^2<+\infty, \label{e2.14}$$ and there exist constants$M$and$N$with$N>\frac{\underline{b}(\overline{b}^2-\underline{b})} {\overline{b}(\underline{b}^2-\overline{b})}M>0$, then \eqref{e1.11} has a non-oscillatory solution in$A(M,N)$. \end{theorem} \begin{proof} Take$\varepsilon\in(0,\underline{b}-1)$sufficiently small satisfying $$1<\underline{b}-\varepsilon<\overline{b}+\varepsilon< (\underline{b}-\varepsilon)^2 \label{e2.15}$$ and $$\big((\overline{b}+\varepsilon)(\underline{b}-\varepsilon)^2 -(\overline{b}+\varepsilon)^2\big)N >\big((\overline{b}+\varepsilon)^2(\underline{b}-\varepsilon) -(\underline{b}-\varepsilon)^2\big)M. \label{e2.16}$$ Choose$L\in\big((\overline{b}+\varepsilon)M+\frac{\overline{b}+\varepsilon} {\underline{b}-\varepsilon}N, (\underline{b}-\varepsilon)N+\frac{\underline{b}-\varepsilon} {\overline{b}+\varepsilon}M\big)$. By \eqref{e2.2} and {\eqref{e2.15}}, an integer$N_0>n_0+d+|\alpha|can be chosen such that $$\underline{b}-\varepsilon< b_n< \overline{b}+\varepsilon,\quad \forall b\geq N_0\label{e2.17}$$ and \begin{aligned} &\sum_{t_1=N_0}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{F\big|\sum_{j=1}^sp_{jt}\big|+|q_t|} {\big|\prod_{i=1}^{k}a_{it_i}\big|}\\ &\leq \min\Big\{\frac{\underline{b}-\varepsilon}{\overline{b}+\varepsilon}L -(\underline{b}-\varepsilon)M-N, \frac{\underline{b}-\varepsilon}{\overline{b}+\varepsilon}M +(\underline{b}-\varepsilon)N-L\Big\}, \end{aligned}\label{e2.18} whereF=\max_{M\leq x\leq N}\{f_j(x):1\leq j\leq s\}$. Define two mappings$T_{1},T_{2}:A(M,N)\to X$by \begin{gather} (T_{1}x)_n=\begin{cases} \frac{L}{b_{n+d}}-\frac{x_{n+d}}{b_{n+d}},& n\geq N_0,\\ (T_{1}x)_{N_0}, &\beta\leq n\frac{1+\underline{b}}{1+\overline{b}}M>0$, then \eqref{e1.11} has a non-oscillatory solution in $A(M,N)$. \end{theorem} \begin{proof} Take $\epsilon\in\big(0,-(1+\overline{b})\big)$ sufficiently small satisfying $$\underline{b}-\epsilon<\overline{b}+\epsilon<-1 \label{e2.22}$$ and $$(1+\overline{b}+\epsilon)N<(1+\underline{b}-\epsilon)M. \label{e2.23}$$ Choose $L\in\big((1+\overline{b}+\epsilon)N, (1+\underline{b}-\epsilon)M\big)$. By \eqref{e2.2} and {\eqref{e2.22}}, an integer $N_0>n_0+d+|\alpha|$ can be chosen such that $$\underline{b}-\epsilon< b_n< \overline{b}+\epsilon,\quad \forall n\geq N_0,\label{e2.24}$$ and \begin{aligned} &\sum_{t_1=N_0}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty} \sum_{t=t_k}^{\infty}\frac{F\big|\sum_{j=1}^sp_{jt}\big|+|q_t|} {\big|\prod_{i=1}^{k}a_{it_i}\big|}\\ &\leq\min\Big\{\Big(\overline{b}+\epsilon+\frac{\overline{b}+\epsilon} {\underline{b}-\epsilon}\Big)M- \frac{\overline{b}+\epsilon}{\underline{b}-\epsilon}L, L-(1+\overline{b}+\epsilon)N\Big\}, \end{aligned}\label{e2.25} where $F=\max_{M\leq x\leq N}\{f_j(x):1\leq j\leq s\}$. Then define $T_1,T_2:A(M,N)\to X$ as \eqref{e2.19} and \eqref{e2.20}. The rest proof is similar to that in Theorem \ref{thm2.1}, and is omitted. \end{proof} \begin{remark} \label{rmk2.1} \rm Theorems \ref{thm2.1}--\ref{thm2.5} extend the results in Cheng \cite[Theorem 1]{c2}, Liu, Xu and Kang \cite[Theorems 2.3-2.7]{l1}, Zhou and Huang \cite[Theorems 1-5]{z4} and corresponding theorems in \cite{a3,a4,m1,m2,t1,y1,z1,z2,z3}. \end{remark} \subsection*{Acknowledgments} The authors are grateful to the anonymous referees for their careful reading, editing, and valuable comments and suggestions. \begin{thebibliography}{00} \bibitem{a1} R. P. Agarwal; Difference equations and inequalities, 2nd ed., \emph{Dekker, New York} (2000). \bibitem{a2} R. P. Agarwal, S. R. Grace, D. O'Regan; Oscillation theory for difference and functional differential equations, \emph{Kulwer Academic} (2000). \bibitem{a3} R. P. Agarwal, E. Thandapani, P. J. Y. Wong; Oscillations of higher-order neutral difference equations, \emph{Appl. Math. 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