\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2011 (2011), No. 96, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2011 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2011/96\hfil Positive periodic solutions]
{Positive periodic solutions of nonlinear first-order functional
difference equations with a parameter}

\author[Y. Lu\hfil EJDE-2011/96\hfilneg]
{Yanqiong Lu}

\address{Yanqiong Lu \newline
Department of Mathematics, 
Northwest Normal University
Lanzhou, 730070,  China}
\email{linmu8610@163.com}

\thanks{Submitted June 29, 2011. Published July 28, 2011.}
\thanks{Supported by grant 11061030 from the NSFC,  the
Fundamental Research Funds for the \hfill\break\indent
 Gansu Universities.}
\subjclass[2000]{34G20}
\keywords{Positive periodic solutions; existence; nonexistence;
\hfill\break\indent
difference equations; fixed point}

\begin{abstract}
 We obtain the existence and multiplicity of positive
 $T$-periodic solutions for the difference equations
 $$
 \Delta x(n)=a(n,x(n))-\lambda b(n)f(x(n-\tau(n)))
 $$
 and
 $$
 \Delta x(n)+a(n,x(n))=\lambda b(n)f(x(n-\tau(n))),
 $$
 where $f(\cdot)$ may be singular at $x=0$.
 Using a fixed point theorem in cones, we extend recent results
 in the literature.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}


\section{Introduction}

In recent years, there has been considerable interest in the
existence of  periodic solutions of the  equation
\begin{equation}
x'(t)=\tilde a(t, x(t))-\lambda \tilde b(t) \tilde
f(x(t-\tau(t))), \label{e1.1}
\end{equation}
where $\lambda>0$ is a positive parameter, $\tilde a$ is
continuous in $x$ and $T$-periodic in $t$,
$\tilde b\in C(\mathbb{R}, [0,\infty))$ and
$\tau\in C(\mathbb{R}, \mathbb{R})$ are $T$-periodic functions,
$\int^T_0\tilde  b(t)dt>0$, $f\in C([0,\infty),[0,\infty))$.
\eqref{e1.1} has been proposed as a model for a variety
of physiological processes and conditions including production
of blood cells, respiration, and cardiac arrhythmias.
See, for example, \cite{c1,c2,f1,g2,j1,w1,m3,w2,w3,y1}
and the references therein.

In this article, we study the existence of positive $T$-periodic
solutions of a discrete analogues to \eqref{e1.1} of the form
\begin{equation}
\Delta x(n)=a(n,x(n))-\lambda b(n)f(x(n-\tau(n))),\quad
n\in \mathbb{Z} \label{e1.2}
\end{equation}
and
\begin{equation}
\Delta x(n)+a(n,x(n))=\lambda b(n)f(x(n-\tau(n))),\quad
n\in \mathbb{Z}, \label{e1.3}
\end{equation}
where $\mathbb{Z}$ is the set of integer numbers,
$T\in \mathbb{N}$ is a fixed integer,
$a:\mathbb{Z}\times[0, \infty)\to[0, \infty)$ is continuous in
$x$ and $T$-periodic in $n$,
$b:\mathbb{Z}\to[0, +\infty)$, $\tau:\mathbb{Z}\to\mathbb{Z}$ are
$T$-periodic and $ \sum_{n=0}^{T-1}b(n)>0$,
$f\in C((0,+\infty),(0,+\infty))$ and may have a repulsive
singularity near $x=0$, $\lambda>0$ is a parameter.

So far, relatively little is known about the existence of
positive periodic solutions of \eqref{e1.2} and \eqref{e1.3}.
To our best knowledge,
Ma \cite{m2} dealt with the special equations of \eqref{e1.2}
and \eqref{e1.3} of the form
\begin{equation}
\Delta x(n)=a(n)g(x(n))x(n)-\lambda b(n)f(x(n-\tau(n)))
 \label{e1.4}
\end{equation}
and
\begin{equation}
\Delta x(n)+a(n)g(x(n))x(n)=\lambda b(n)f(x(n-\tau(n))),
 \label{e1.5}
\end{equation}
with certain values of  $\lambda$, for which there exist
positive $T$-periodic solutions of \eqref{e1.4} and
\eqref{e1.5}, respectively. If $g(x(n))\equiv1$, this special
case see \cite{l1,m1,r1}.
All these authors \cite{l1,m1,m2,r1}
focus their attention on the fact that the number of positive
$T$-periodic solutions can be determined by the behaviors
of the quotient of $f(x)/x$ at $\{0,+\infty\}$.
However, our main results show the number of positive
$T$-periodic solutions can be
determined by the behaviors of the quotient of
$f(x)/x$ at $[0,\infty]$.

It is the purpose of this paper to study more general equations
\eqref{e1.2} and \eqref{e1.3} and generalize the main results
of Ma \cite{m2}.  We also establish  some existence and multiplicity
for \eqref{e1.2} and \eqref{e1.3}, respectively.
The main tool we will use is the fixed point index theory
\cite{d1,g1}.
Throughout this paper, we denote the product of $x(n)$ from
$n=a$ to $n=b$ by $\prod_{n=a}^bx(n)$
with the understanding that $\prod_{n=a}^bx(n)=1$ for all $a>b$.

The rest of the paper is arranged as follows:
In Section 2, we give some preliminary results.
In Section 3 we state and prove some existence
results of positive periodic solutions for \eqref{e1.2}
and \eqref{e1.3}. Finally, Section 4 is devoted to improving
some results of Ma \cite{m2}. For related results on the
associated differential equations, see Weng and Sun \cite{w2}.

\section{Preliminaries}

In this article, we make the following assumptions:
\begin{itemize}
\item[(H1)] There exist functions $a_1, a_2:\mathbb{Z}\to[0,+\infty)$
 are $T-$periodic functions such that $\sum_{n=0}^{T-1}a_1(n)>0$,
$\sum_{n=0}^{T-1}a_2(n)>0$ and
$a_1(n)x(n)\leq a(n,x(n))\leq a_2(n)x(n)$ for $n\in\mathbb{Z}$ and
$x>0$.
In addition, $\lim_{x\to0}\frac{a(n,x)}{x}$ exists for
$n\in \mathbb{Z}$.


\item[(H2)] $a(n, x)$ is continuous in $x$ and $T$-periodic in $n$,
 $b:\mathbb{Z}\to[0, +\infty),\ \tau:\mathbb{Z}\to\mathbb{Z}$
are $T$-periodic and $B:=\sum_{n=0}^{T-1}b(n)>0$;
$f\in C((0,+\infty),(0,+\infty))$ and may have a repulsive
singularity near $x=0$.

\end{itemize}
Denote
$$
\sigma_i=\prod_{s=0}^{T-1}(1+a_i(s))^{-1},\quad i=1,2,\quad
 m=\frac{\sigma_2}{1-\sigma_2},\quad M=\frac{1}{1-\sigma_1}.
$$
From (H1), it is clear that $0<\frac{m}{M}<1$.
Let
$$
E:=\{x:\mathbb{Z}\to\mathbb{R}:x(n+T)=x(n)\}
$$
be the Banach space with the norm $\|x\|=\max_{n\in\mathbb{Z}}|x(n)|$.
Define the cone
$$
P:=\{x\in E : x(n)\geq0,\ x(n)\geq\frac{m}{M}\|x\|\},
$$
and the operator $A_{\lambda}:P\to E$ by
$$
(A_\lambda x)(n)=\lambda\sum_{s=n}^{n+T-1} G_x(n,s)b(s)f(x(s-\tau(s))),
\quad n\in\mathbb{Z}, \label{e2.1}
$$
where
$$
G_x(n,s)=\frac{\prod_{k=n}^{s}(1+\frac{a(k,x(k))}{x(k)})^{-1}}
{1-\prod_{k=1}^{T}(1+\frac{a(k,x(k))}{x(k)})^{-1}},\quad
n\leq s\leq n+T.
$$
It follows from (H1) that
$$
m\leq G_x(n,s)\leq M.
$$
If (H1) and (H2) hold and $x\in P$, then
\begin{equation}
\lambda m\sum_{s=n}^{n+T-1} b(s)f(x(s-\tau(s)))
\leq \|A_{\lambda}x\|\leq\lambda M\sum_{s=n}^{n+T-1}b(s)f(x(s-\tau(s))).
\label{e2.2}
\end{equation}

The construction of  $G_x(n,s)$ is due to Ma \cite{m2}.
Following the approach in \cite{m2}, we can easily prove the following two
Lemmas. Similar arguments have been also employed in \cite{r1}.
We remark that  the process of proofs are similar and are omitted.

\begin{lemma} \label{lem2.1}
 Assume that {\rm (H1), (H2)} hold. Then
$A_\lambda(P)\subset P$ and $A_\lambda:P\to P$ is compact
and continuous.
\end{lemma}

\begin{lemma} \label{lem2.2}
 Assume that {\rm (H1), (H2)} hold. Then $x\in P$ is a solution
of  \eqref{e1.2} if and only
if $x$ is a fixed point of $A_\lambda$ in $P$.
\end{lemma}

The following well-known result of the fixed point index is crucial
in our arguments.


\begin{lemma}[\cite{d1,g1}] \label{lem2.3}
 Let $E$ be a Banach space and $K$ be
a cone in $E$. For $r>0$, define $K_r=\{u\in K : \|u\|<r\}$. Assume
that $T:\overline{K}_r\to K$ is completely continuous such
that $Tu\neq u$ for $u\in\partial K_r=\{u\in K : \|u\|=r\}.$
\begin{itemize}
\item[(i)] If $\|Tu\|\geq\|u\|$ for $u\in\partial K_r$, then
$i(T, K_r, K)=0$.

\item[(ii)] If $\|Tu\|\leq\|u\|$ for $u\in\partial K_r$, then
$i(T, K_r, K)=1$.
\end{itemize}
\end{lemma}


\section{Existence of positive periodic solutions
for \eqref{e1.2} and \eqref{e1.3}}

In this section, we shall provide two explicit intervals
of $\lambda$ such that \eqref{e1.2} and \eqref{e1.3} have at
least one positive $T$-periodic solution.

\begin{theorem} \label{thm3.1}
 Assume that {\rm (H1), (H2)} hold and there exist $R$, $r$
such that $R>r>0$ and
\begin{equation}
m^2\min_{x\in[\frac{m}{M}r,r]}\frac{f(x)}{x}>M^2
\max_{x\in[R,\frac{M}{m}R]}\frac{f(x)}{x}.\label{e3.1}
\end{equation}
Then, for each $\lambda$ satisfying
\begin{equation}
\frac{M}{m^2B\min_{x\in[\frac{m}{M}r,r]}
\frac{f(x)}{x}}<\lambda
\leq\frac{1}{MB\max_{x\in[R,\frac{M}{m}R]}
\frac{f(x)}{x}},\label{e3.2}
\end{equation}
equation \eqref{e1.2} has a positive $T$-periodic solution $x$
satisfying $r<x\leq\frac{M}{m}R$.
\end{theorem}

\begin{proof}
 According to \eqref{e3.1}, the set $\{\lambda:\lambda
\text{ satisfies \eqref{e3.2}}\}$ is nonempty.
It follows from \eqref{e3.2} that
$$
\frac{f(x)}{x}>\frac{M}{\lambda m^2 B},\quad\forall
 x\in[\frac{m}{M}r,r] \quad \text{and}\quad
\frac{f(x)}{x}\leq\frac{1}{\lambda MB} ,\quad
 x\in[R,\frac{M}{m}R].
$$
Define the open sets
$$
\Omega_1:=\{x\in E: \|x\|<r\},\quad
\Omega_2:=\{x\in E: \|x\|<\frac{M}{m}R\}.
$$
If $x\in \partial\Omega_1\cap P$, then $\|x\|=r$ and
$\frac{m}{M}r\leq x\leq r$. According to \eqref{e2.2},
it follows that
\begin{align*}
A_\lambda x(n)
&\geq\lambda m\sum_{s=n}^{n+T-1}b(s)f(x(s-\tau(s)))\\
&>\lambda m\sum_{s=n}^{n+T-1}b(s)\frac{M}{\lambda m^2 B}
 x(s-\tau(s))) \\
&\geq\frac{M}{mB}\sum_{s=0}^{T-1}b(s)\frac{m}{M}r=r=\|x\|.
\end{align*}
Hence $\|A_\lambda x\|>\|x\|$, $x\in \partial\Omega_1\cap P$.
From Lemma \ref{lem2.3}, we have that
$$
i(A_\lambda,\Omega_1\cap P,P)=0.
$$
If $x\in \partial\Omega_2\cap P$,
then $\|x\|=\frac{M}{m}R$ and $R\leq x\leq\frac{M}{m}R$.
According to \eqref{e2.2}, it follows that
\begin{align*}
\|A_\lambda x\|
&\leq\lambda M\sum_{s=n}^{n+T-1}b(s)f(x(s-\tau(s)))\\
&\leq\lambda M\sum_{s=n}^{n+T-1}b(s)\frac{1}{\lambda MB}x(s-\tau(s))) \\
&\leq\frac{1}{B}\sum_{s=0}^{T-1}b(s)\frac{M}{m}R=\frac{M}{m}R=\|x\|.
\end{align*}
Hence $\|A_\lambda x\|\leq\|x\|$, $x\in \partial\Omega_2\cap P$.
From Lemma \ref{lem2.3}, we have that
$$
i(A_\lambda,\Omega_2\cap P,P)=1.
$$
Thus $i(A_\lambda,\Omega_{2}\backslash\bar{\Omega}_{1},P)=1$
and $A_\lambda$ has a fixed point in $\Omega_{2}\backslash
\bar{\Omega}_{1}$, which is a positive $T$-periodic solution
of \eqref{e1.2} and
$$
r<x(n)\leq \frac{M}{m}R, \quad  n\in\mathbb{Z}.
$$
\end{proof}


\begin{theorem} \label{thm3.2}
 Assume that {\rm (H1), (H2)} hold and there exist $R$, $r$
such that $R>r>0$ and
\begin{equation}
m^2\min_{x\in[R,\frac{M}{m}R]}f(x)/x>M^2\max_{x\in[\frac{m}{M}r,r]}
f(x)/x.\label{e3.3}
\end{equation}
Then, for each $\lambda$ satisfying
\begin{equation}
\frac{M}{m^2B\min_{x\in[R,MR/m]} f(x)/x}
\leq\lambda<\frac{1}{MB\max_{x\in[m/rM,r]}f(x)/x},\label{e3.4}
\end{equation}
equation \eqref{e1.2} has a positive $T$-periodic solution
$x$ satisfying $r<x\leq\frac{M}{m}R$.
\end{theorem}

\begin{proof} By \eqref{e3.3}, the set
$\{\lambda:\lambda  \text{ satisfies \eqref{e3.4}}\}$ is nonempty.
It follows from \eqref{e3.4} that
$$
\frac{f(x)}{x}<\frac{1}{\lambda M B},\quad \forall
 x\in[\frac{m}{M}r,r] \quad\text{and}\quad
 \frac{f(x)}{x}\geq\frac{M}{\lambda m^2B} ,\quad
 x\in[R,\frac{M}{m}R].
$$
The rest of the proof is similar to the proof of Theorem \ref{thm3.1}
and is omitted.
\end{proof}

Next we turn our attention to \eqref{e1.3}; i.e.,
\begin{equation}
x(n+1)=[1-\frac{a(n,x(n))}{x(n)}]x(n)+\lambda b(n)f(x(n-\tau(n))),
\quad n\in\mathbb{Z},\label{e3.5}
\end{equation}
where $\lambda, a(n),  b(n),  f(x(n-\tau(n)))$ satisfy the same
assumptions stated for \eqref{e1.2} except that
$$
0<\prod_{k=0}^{T-1}(1-a_2(k))\leq \prod_{k=0}^{T-1}(1-a_1(k))<1,
$$
for all $n\in\mathbb{Z}$. In view of \eqref{e1.3} we have
\begin{equation}
x(n)=\lambda\sum_{s=n}^{n+T-1}K_x(n, s)b(s)f(x(s-\tau(s))),\label{e3.6}
\end{equation}
where
\begin{equation}
K_x(n, s)=\frac{\prod_{k=s+1}^{n+T-1}(1-\frac{a(k,x(k))}{x(k)})}
{1-\prod_{k=0}^{T-1}(1-\frac{a(k,x(k))}{x(k)})},\quad
 s\in[n, n+T-1].\label{e3.7}
\end{equation}
Note that since $0\leq a_1(n)\leq a(n,x(n))\leq a_2(n)<1$ for all
$n\in\mathbb{Z}$, we have
$$
\bar{m}:=\frac{\rho_2}{1-\rho_2}\leq K_x(n, s)\leq\frac{1}{1-\rho_1}
:=\bar{M}, \quad   n\leq s\leq n+T-1,
$$
here
$$
\rho_i=\prod_{k=0}^{T-1}(1-a_i(k)),\quad  i=1, 2 \quad
 \text{and}\quad  0<\frac{\rho_2(1-\rho_1)}{1-\rho_2}<1.
$$
Similarly,  we can get the following theorems.


\begin{theorem} \label{thm3.3}
Assume that {\rm (H1), (H2)} hold and $0\leq a_1(n)\leq a_2(n)<1$
for $n\in\mathbb{Z}$. Moreover,
there exist $R$, $r$ such that $R>r>0$ and
$$
\bar{m}^2\min_{x\in[\frac{\bar{m}}{\bar{M}}r,r]}
\frac{f(x)}{x}>\bar{M}^2\max_{x\in[R,\frac{\bar{M}}{\bar{m}}R]}
\frac{f(x)}{x}.
$$
Then, for each $\lambda$ satisfying
$$
\frac{\bar{M}}{\bar{m}^2B\min_{x\in[\frac{\bar{m}}{\bar{M}}r,r]}
\frac{f(x)}{x}}<\lambda\leq\frac{1}{\bar{M}B\max_{x\in[R,
\frac{\bar{M}}{\bar{m}}R]}\frac{f(x)}{x}},
$$
equation \eqref{e1.3} has a positive $T$-periodic solution $x$
satisfying $r<x\leq\frac{\bar{M}}{\bar{m}}R$.
\end{theorem}

\begin{theorem} \label{thm3.4}
 Assume that {\rm (H1)-(H2)} hold and $0\leq a_1(n)\leq a_2(n)<1$
for $n\in\mathbb{Z}$. In addition, there exist $R$, $r$ such that
$R>r>0$ and
$$
\bar{m}^2\min_{x\in[R,\frac{\bar{M}}{\bar{m}}R]}
\frac{f(x)}{x}>\bar{M}^2\max_{x\in[\frac{\bar{m}}{\bar{M}}r,r]}
\frac{f(x)}{x}.
$$
Then, for each $\lambda$ satisfying
$$
\frac{\bar{M}}{\bar{m}^2B\min_{x\in[R,\frac{\bar{M}}{\bar{m}}R]}
\frac{f(x)}{x}}\leq\lambda<\frac{1}{\bar{M}B\max_{x\in[\frac{\bar{m}}
{\bar{M}}r,r]}\frac{f(x)}{x}},
$$
equation \eqref{e1.3} has a positive $T$-periodic solution $x$
satisfying $r<x\leq\frac{\bar{M}}{\bar{m}}R$.
\end{theorem}


\section{Multiplicity  of positive periodic solutions for \eqref{e1.2}
and \eqref{e1.3}}

To illustrate applications of Theorems \ref{thm3.1}-\ref{thm3.4},
we will provide four corollaries in this section.
For convenience, we introduce the notation
\begin{gather*}
i_0=\text{number of zeros in the set }\{f_0, f_\infty\}, \\
i_\infty=\text{number of infinities in the set }\{f_0, f_\infty\}.
\end{gather*}
It is clear that $i_0, i_\infty=0, 1$ or $2$. Then we shall
show that \eqref{e1.2} has $i_0$ or $i_\infty$
positive $T$-periodic solution(s) for sufficiently large or
small $\lambda$, respectively.



\begin{corollary} \label{coro4.1}
Assume that {\rm (H1), (H2)} hold and $c\in(0, \infty)$ is a
fixed constant, then
\begin{itemize}
\item[(i)] If $i_0=1$ or $2$, then \eqref{e1.2} has $i_0$ positive
$T$-periodic solution(s) for
$\lambda>\frac{M}{m^2B\min_{x\in[mc/M,c]}f(x)/x}$.

\item[(ii)] If $i_\infty=1$ or $2$, then \eqref{e1.2} has
$i_\infty$ positive $T$-periodic solution(s) for
$0<\lambda<\frac{1}{MB\max_{x\in[c,Mc/m]}f(x)/x}$.
\end{itemize}
\end{corollary}


\begin{proof}
 (i) If $f_0=0$, then there exists small enough $r_1$ such
that $c>r_1>0$ and
$$
m^2\min_{x\in[mc/M,c]}\frac{f(x)}{x}
\geq M^2\max_{x\in[\frac{m^2}{M^2}r_1,\frac{m}{M}r_1]}
\frac{f(x)}{x}\to0
\quad(\text{as } r_1\to0).
$$
By applying Theorem \ref{thm3.2} with $R=\frac{m}{M}c$ and $r=\frac{m}{M}r_1$,
 Equation \eqref{e1.2} has a positive $T$-periodic solution $x$
satisfying
$$\frac{m}{M}r_1<x\leq c.$$

If $f_\infty=0$, then there exists large enough $R_1$ such that $R_1>c>0$ and
$$
m^2\min_{x\in[mc/M,c]}\frac{f(x)}{x}\geq M^2
\max_{x\in[\frac{M}{m}R_1,\frac{M^2}{m^2}R_1]}\frac{f(x)}{x}\to0 \quad
(\text{as }R_1\to\infty).
$$
Thus, by applying Theorem \ref{thm3.1} with $R=\frac{M}{m}R_1$ and $r=c$,
there exists a positive $T$-solution $x$ of Eq.\eqref{e1.2} satisfying
$$
c<x\leq \frac{M^2}{m^2}R.
$$

(ii) If $f_0=\infty$, then there exists small enough $r_2$
such that $c>r_2>0$ and
$$
M^2\max_{x\in[c,\frac{M}{m}c]}\frac{f(x)}{x}
\leq m^2\min_{x\in[\frac{m^2}{M^2}r_2,\frac{m}{M}r_2]}
\frac{f(x)}{x}\to\infty \quad(\text{as } r_2\to0).
$$
Thus, by applying Theorem \ref{thm3.1} with $R=c$ and $r=\frac{m}{M}r_2$,
Equation \eqref{e1.2} has a positive $T$-periodic solution $x$
satisfying
$$
\frac{m}{M}r_2<x\leq \frac{M}{m}c.
$$
If $f_\infty=\infty$, then there exists large enough  $R_2>c>0$
such that
$$
M^2\max_{x\in[c,\frac{M}{m}c]}\frac{f(x)}{x}
\leq m^2\min_{x\in[\frac{M}{m}R_2,\frac{M^2}{m^2}R_2]}
\frac{f(x)}{x}\to\infty \quad
(\text{as } R_2\to\infty ).
$$
Thus, by applying Theorem \ref{thm3.2} with $R=\frac{M}{m}R_2$ and
$r=\frac{M}{m}c$, there exists a positive $T$-solution $x$
of \eqref{e1.2} satisfying
$$
\frac{M}{m}c<x\leq \frac{M^2}{m^2}R_2.
$$
\end{proof}

\begin{corollary} \label{coro4.2}
 Assume that {\rm (H1), (H2)} hold and $i_0=i_\infty=0$, then
\begin{itemize}
\item[(1)] If $m^2f_0>M^2f_\infty$, Equation \eqref{e1.2} has
a positive $T$-periodic solution for
$$
\frac{M}{m^2Bf_0}<\lambda<\frac{1}{MBf_{\infty}}.
$$

\item[(2)] If $m^2f_\infty>M^2f_0$, Equation \eqref{e1.2} has
a positive $T$-periodic solution for
$$
\frac{M}{m^2Bf_\infty}<\lambda<\frac{1}{MBf_0}.
$$
\end{itemize}
\end{corollary}

 \begin{proof}
 (1) Since $m^2f_0>M^2f_\infty$, inequality \eqref{e3.1} is satisfied
by taking $r$ small enough and $R$ large enough.
According to Theorem \ref{thm3.1}, Equation \eqref{e1.2} has a positive
$T$-periodic solution for
$$
\frac{M}{m^2B(f_0+\epsilon)}<\lambda<\frac{1}{MB(f_{\infty}-\epsilon)},
$$
where $\epsilon>0$ is sufficiently small.

(2) Since $m^2f_\infty>M^2f_0$, inequality \eqref{e3.3}
is satisfied by taking $r$ small enough and
$R$ large enough. As a consequence of Theorem \ref{thm3.2},
Equation \eqref{e1.2} has a positive $T$-periodic solution for
$$
\frac{M}{m^2B(f_\infty+\epsilon)}<\lambda
<\frac{1}{MB(f_{0}-\epsilon)},
$$
where $\epsilon>0$ is sufficiently small.
\end{proof}


\begin{remark} \label{rmk4.1} \rm
  Corollary \ref{coro4.1} improves the results in Ma \cite[Theorem 4.1]{m2}.
Since assertion (b) in \cite[Theorem 4.1]{m2} fails to the case
$\lim_{x\to0^+}f(x)=+\infty$, which is due to the definition of
$M(r)=\max\{f(x): 0\leq x\leq r\}$.
However, Corollary \ref{coro4.1} is valid to the case
$\lim_{x\to0^+}f(x)=+\infty$ and provides more desirable intervals
of $\lambda$.

If $a(n, x)$ of \eqref{e1.2} is replaced with $a(n)g(x(n))x(n)$
of \eqref{e1.4}, then Corollary \ref{coro4.2} is exactly the same as
\cite[Theorem 4.3]{m2}.
\end{remark}

The following results are  direct consequences of Theorems \ref{thm3.3}
and \ref{thm3.4}.

\begin{corollary} \label{coro4.3}
 Assume that {\rm (H1), (H2)} hold and $c\in(0, \infty)$ is a
fixed constant, then
\begin{itemize}
\item[(i)] If $i_0=1$ or $2$, then \eqref{e1.3} has $i_0$
positive $T$-periodic solutions for
$$
\lambda>\frac{\bar{M}}{\bar{m}^2B
\min_{x\in[\frac{\bar{m}}{\bar{M}}c,c]}f(x)/x}.
$$

\item[(ii)] If $i_\infty=1$ or $2$, then \eqref{e1.3}
has $i_\infty$ positive $T$-periodic solutions for
$$
0<\lambda<\frac{1}{\bar{M}B\max_{x\in[c,\frac{\bar{M}}{\bar{m}}c]}
f(x)/x}.
$$
\end{itemize}
\end{corollary}

\begin{corollary} \label{coro4.4}
Assume that {\rm (H1), (H2)} hold and $i_0=i_\infty=0$, then
\begin{itemize}
\item[(1)] If $\bar{m}^2f_0>\bar{M}^2f_\infty$, Equation
 \eqref{e1.3} has a positive $T$-periodic solution for
$$
\frac{\bar{M}}{\bar{m}^2Bf_0}<\lambda<\frac{1}{\bar{M}Bf_{\infty}}.
$$

\item[(2)] If $\bar{m}^2f_\infty>\bar{M}^2f_0$, Equation
 \eqref{e1.3} has a positive $T$-periodic solution for
$$
\frac{\bar{M}}{\bar{m}^2Bf_\infty}<\lambda<\frac{1}{\bar{M}Bf_0}.
$$
\end{itemize}
\end{corollary}

\begin{remark} \label{rmk4.2}\rm
 Corollary \ref{coro4.3} improves the results in \cite[Theorem 4.4]{m2}.
Since assertion (b) in \cite[Theorem 4.4]{m2}
fails to the case $\lim_{x\to0^+}f(x)=+\infty$, which is due to
the definition of $M(r)=\max\{f(x): 0\leq x\leq r\}$.
However, Corollary \ref{coro4.3} is valid to the case
$\lim_{x\to0^+}f(x)=+\infty$ and provides more desirable
intervals of $\lambda$.

If $a(n, x)$ of \eqref{e1.3} is replaced with $a(n)g(x(n))x(n)$
of \eqref{e1.5}, then Corollary \ref{coro4.4} is exactly the same
as \cite[Theorem 4.6]{m2}.
\end{remark}


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\end{document}