\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
\begin{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}
\end{equation}
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
\begin{equation}
|b_n|\leq b<\frac{N-M}{2N},\ \forall n\geq N_0,\label{e2.3}
\end{equation}
and
\begin{equation}
\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}
\end{equation}
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 n<N_0,
\end{cases} \label{e2.5}
\\
 (T_{2}x)_n=\begin{cases} (-1)^k
\sum_{t_1=n}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\\[2pt]
\sum_{t_k=t_{k-1}}^{\infty}
\sum_{t=t_k}^{\infty}\frac{\sum_{j=1}^sp_{jt}f_j(x_{t-r_{jt}})-q_t}
{\prod_{i=1}^{k}a_{it_i}} ,& n\geq N_0,\\[4pt]
(T_{2}x)_{N_0}, &\beta\leq n<N_0,
\end{cases}
\label{e2.6}
\end{gather}
for all $x\in A(M,N)$.

(i) Note that $T_{1}x+T_{2}y\in A(M,N)$ for all $x,y\in A(M,N)$.
In fact, for every $x,y\in A(M,N)$ and $n\geq N_0$, by \eqref{e2.4},
we have
\begin{align*}
 (T_{1}x+T_{2}y)_n
&\geq L-bN- \sum_{t_1=n}^{\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|}\\
&\geq L-bN-\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|}
\geq  M
\end{align*}
and
\begin{align*}
 (T_{1}x+T_{2}y)_n
&\leq L+bN+\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  N.
\end{align*}
That is, $(T_{1}x+T_{2}y)(A(M,N))\subseteq A(M,N)$.

(ii) W show that $T_{1}$ is a contraction mapping on $A(M,N)$.
For any $x,y\in A(M,N)$ and $n\geq N_0$, it is easy to
derive that
$$
 \big|(T_{1}x)_n-(T_{1}y)_n\big| \leq
|b_n\|x_{n-d}-y_{n-d}|\leq b\|x-y\|,
$$
which implies
$$
\|T_{1}x-T_{1}y\|\leq b\|x-y\|.
$$
Then $b<\frac{N-M}{2N}<1$ ensures that $T_{1}$ is a contraction
mapping on $A(M,N)$.

(iii) We show that $T_{2}$ is completely continuous.
First, we show $T_{2}$ that is continuous.
Let $x^{(u)}=\{x_n^{(u)}\}\in A(M,N)$ be a sequence such
that $x_n^{(u)}\to x_n$ as $u\to\infty$.
Since $A(M,N)$ is closed, $x=\{x_n\}\in A(M,N)$. Then, for
$n\geq N_0$,
$$
 \big|T_{2}x_n^{(u)}-T_{2}x_n\big|\leq
\sum_{t_1=N_0}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\sum_{t_k=t_{k-1}}^{\infty}
\sum_{t=t_k}^{\infty}
\frac{\big|\sum_{j=1}^sp_{jt}\big\|f_j(x^{(u)}_{t-r_{jt}})-f_j(x_{t-r_{jt}})|}
{\big|\prod_{i=1}^{k}a_{it_i}\big|}.
$$
Since
\begin{align*}
\frac{\big|\sum_{j=1}^sp_{jt}\big\|f_j(x^{(u)}_{t-r_{jt}})-f_j(x_{t-r_{jt}})|}
{\big|\prod_{i=1}^{k}a_{it_i}\big|}
&\leq \frac{\big|\sum_{j=1}^sp_{jt}\big|\big(|f_j(x^{(u)}_{t-r_{jt}})|
+|f_j(x_{t-r_{jt}})|\big)} {\big|\prod_{i=1}^{k}a_{it_i}\big|}\\
&\leq \frac{2F\big|\sum_{j=1}^sp_{jt}\big|}
{\big|\prod_{i=1}^{k}a_{it_i}\big|}
\end{align*}
and $|f_j(x^{(u)}_{t-r_{jt}})-f_j(x_{t-r_{jt}})|\to0$ as
$u\to \infty$ for $j=1,2,\dots,s$, it follows from \eqref{e2.2}
and the Lebesgue dominated convergence theorem that
$\lim_{u\to\infty}\|T_{2}x^{(u)}-T_{2}x\|=0$, which means that
$T_{2}$ is continuous.

Next, we show that  $T_{2}A(M,N)$ is relatively compact.
By {\eqref{e2.2}}, for any $\varepsilon>0$, take $N_1\geq N_0$
large enough,
\begin{equation}
\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}
\end{equation}
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 implies $T_{2}A(M,N)$ 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,
\begin{equation}
b_n\ge0\ \text{eventually, }\   0\leq \underline{b}\leq
\overline{b}<1, \label{e2.8}
\end{equation}
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
\begin{equation}
\frac{\underline{b}}{2}\leq b_n\leq \frac{1+\overline{b}}{2},\ \forall
n\geq N_0\label{e2.9}
\end{equation}
and
\begin{equation}
\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}
\end{equation}
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{e2.5}
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,
\begin{equation}
b_n\leq 0  \text{ eventually},\quad
-1< \underline{b}\leq \overline{b}\leq 0, \label{e2.11}
\end{equation}
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
\begin{equation}
\frac{\underline{b}-1}{2}\leq b_n\leq \frac{\overline{b}}{2},\ \forall
n\geq N_0,\label{e2.12}
\end{equation}
and
\begin{equation}
\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}
\end{equation}
where $F=\max_{M\leq x\leq N}\{f_j(x):1\leq j\leq s\}$.
Then define  $T_1,T_2:A(M,N)\to X$ 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,
\begin{equation}
b_n>1 \text{ eventually},\quad  1<\underline{b},\text{ and }
\overline{b}<\underline{b}^2<+\infty, \label{e2.14}
\end{equation}
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
\begin{equation}
1<\underline{b}-\varepsilon<\overline{b}+\varepsilon<
(\underline{b}-\varepsilon)^2 \label{e2.15}
\end{equation}
and
\begin{equation}
\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}
\end{equation}
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
\begin{equation}
\underline{b}-\varepsilon< b_n< \overline{b}+\varepsilon,\quad
\forall b\geq N_0\label{e2.17}
\end{equation}
and
\begin{equation}
\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}
\end{equation}
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}
\frac{L}{b_{n+d}}-\frac{x_{n+d}}{b_{n+d}},& n\geq N_0,\\
(T_{1}x)_{N_0}, &\beta\leq n<N_0,
\end{cases}
\label{e2.19}\\
 T_{2}x)_n=\begin{cases}
\frac{(-1)^k}{b_{n+d}}
\sum_{t_1=n}^{\infty}\sum_{t_2=t_1}^{\infty}\dots\\
\sum_{t_k=t_{k-1}}^{\infty}
\sum_{t=t_k}^{\infty}\frac{\sum_{j=1}^sp_{jt}f_j(x_{t-r_{jt}})-q_t}
{\prod_{i=1}^{k}a_{it_i}} ,& n\geq N_0,\\[4pt]
(T_{2}x)_{N_0}, &\beta\leq n<N_0,
\end{cases}
\label{e2.20}
\end{gather}
for all $x\in A(M,N)$. The rest proof is similar to that in Theorem
\ref{thm2.1}, and is omitted.
\end{proof}

\begin{theorem} \label{thm2.5}
 If  \eqref{e2.2} holds,
\begin{equation}
b_n<-1 \text{ eventually},\quad  -\infty<\underline{b}, \
\overline{b}<-1 \label{e2.21}
\end{equation}
and  there exist constants $M$ and $N$ with
$N>\frac{1+\underline{b}}{1+\overline{b}}M>0$, 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
\begin{equation}
\underline{b}-\epsilon<\overline{b}+\epsilon<-1 \label{e2.22}
\end{equation}
and
\begin{equation}
(1+\overline{b}+\epsilon)N<(1+\underline{b}-\epsilon)M. \label{e2.23}
\end{equation}
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
\begin{equation}
\underline{b}-\epsilon< b_n< \overline{b}+\epsilon,\quad \forall
n\geq N_0,\label{e2.24}
\end{equation}
and
\begin{equation}
\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}
\end{equation}
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.

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\end{document}