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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 233, pp. 1--16.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/233\hfil Existence and uniqueness of fixed points]
{Existence and uniqueness of fixed points for mixed monotone
operators with perturbations}

\author[Y. Sang \hfil EJDE-2013/233\hfilneg]
{Yanbin Sang}  % in alphabetical order

\address{Yanbin Sang \newline
Department of Mathematics, North University of China,
Taiyuan, Shanxi, 030051, China}
\email{syb6662009@yahoo.com.cn,  Tel +86 351 3923592}

\thanks{Submitted April 10, 2013. Published October 18, 2013.}
\subjclass[2000]{47H07, 47H10, 34B10, 34B15}
\keywords{Sublinear; Mixed monotone operator;
 normal cone; time scales; \hfill\break\indent
nonlinear integral equation}

\begin{abstract}
 In this article, we study a class of mixed monotone
 operators with perturbations. Using a monotone iterative technique and
 the properties of cones, we show the existence and uniqueness 
 for fixed points for such operators.
 As applications, we prove the existence and uniqueness of positive
 solutions for nonlinear integral equations of second-order on time
 scales. In particular, we do not assume the existence of upper-lower
 solutions or compactness or continuity conditions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks


\section {Introduction}

Mixed monotone operators were introduced by Guo and Lakshmikantham
in \cite{g1}. Their study has wide applications in the applied sciences
such as engineering, biological chemistry technology, nuclear
physics and in mathematics (see \cite{g2,g3,z5} and references
therein). Various existence (and uniqueness) theorems of fixed
points for mixed monotone operators have been discussed extensively, see
for example \cite{b1,b4,d1,h2,l5,w2,z1,z6,z7}.
Bhaskar and Lakshmikantham \cite{b1}
established some coupled fixed point theorems for mixed monotone
operators in partially ordered metric spaces and discussed the
existence and uniqueness of a solution for a periodic boundary value
problem. Instead of using a direct proof as in \cite{b1},
 Drici, McRae and Devi \cite{d1} employed the notion of a reflection operator,
and investigated fixed point theorems for mixed monotone operators by
weakening the requirements in the contractive assumption and
strengthening the metric space utilized with a partial order. These
theorems are generalizations of the results of \cite{b1}. Moreover, in
\cite{h2}, Harjani, L\'{o}pez and Sadarangani generalized the main results
of \cite{b1} using the altering distance functions.


On the other hand, in recent years, there is much attention paid to
various existence and uniqueness theorems of fixed points for
monotone operators with perturbation. We would like to mention the
results of Li \cite{l1}, Li, Liang and Xiao \cite{l2,l3}, Liu, Zhang and Wang
\cite{l6}, and Zhai and Anderson \cite{z2}.  Li \cite{l1} proved the existence,
uniqueness and iteration of the positive fixed points for operator
$A=B+C$, where $B$ is a positive linear operator with the spectral
radius $r(B)<1$, and $C$ is a $\varphi$-concave increasing operator.
 Li, Liang and Xiao \cite{l2} obtained the existence and uniqueness of
positive fixed points for operator $C=A+B$, where $A$ is a
decreasing operator and $B$ is sublinear. Furthermore,  Li,
Liang and Xiao \cite{l3} used partial ordering methods, cone theory and
iterative technique to investigate the existence and uniqueness of
positive solutions of operator equation $A(x,x)+Bx=x$ in a real
ordered Banach space $E$, where $A$ is a mixed monotone operator
with convexity and concavity, and $B$ is affine. Liu, Zhang and Wang \cite{l6}
discussed the existence and uniqueness of positive solutions of
operator equation $A(x,x)+Bx=x$ on ordered Banach spaces, where $A$
is a mixed monotone operator, and $B$ is a sublinear operator.
Without any compactness and continuity of the operators, some new
fixed point theorems were obtained. Very recently,  Zhai and
Anderson \cite{z2} considered the existence and uniqueness of positive
solutions to the following operator equation on ordered Banach
spaces
$$Ax+Bx+Cx=x,$$
where $A$ is an increasing $\alpha$-concave operator, $B$ is an
increasing sub-homogeneous operator and $C$ is a homogeneous
operator.

However, we note that the upper-lower solutions conditions play a
fundamental role in the main results of \cite{b1,d1,h2,z6,l1,l2,l3,l6,l7},
 as we know, which are not easy to verify for some concrete nonlinear
equations. Thus, how to remove these conditions is an important and
interesting question, much effort has been devoted to this topic. In
\cite{z3}, without demanding the existence of upper and lower solutions
conditions, Zhai and Cao proved the existence, uniqueness and
monotone iterative techniques of fixed points for
$\tau$-$\varphi$-concave operators. Moreover,  Zhai, Yang
and Zhang \cite{z4} studied a class of nonlinear operator equations $x=Ax+x_0$
on ordered Banach spaces, where $A$ is a monotone generalized
concave operator. In particular, the authors did not suppose the
existence of upper-lower solutions conditions.

In this article, $E$ is a real Banach space with norm
$\|\cdot\|$, $P$ is a cone in $E$, $\theta$ is the zero element in
$E$. A partially ordered relation in $E$ is given by $x\leq y$ if and
only if $y-x\in P$. $A: P\times P\to P$ is said to be a mixed
monotone operator if $A(x,y)$ is nondecreasing in $x$ and
non-increasing in $y$; i.e., $u_i$, $v_i$ $(i=1,2)\in P$,
$u_1\leq u_2$, $v_1\geq v_2$ implies $A(u_1,v_1)\leq A(u_2,v_2)$. An element
$x\in P$ is called a fixed point of $A$ if $A(x,x)=x$.

Recall that a cone $P$ is said to be solid if the interior $P^\circ$
is nonempty and we denote $x\gg \theta$ if $x\in P^\circ$. $P$ is
said to be normal if there exists a positive constant $N$, such that
$\theta\leq x\leq y\Longrightarrow \|x\|\leq N\|y\|$, the smallest
$N$ is called the normal constant of $P$. For all $x, y\in E$, the
notation $x\sim y$ means that there exist $\lambda>0$ and $\mu>0$
such that $\lambda x\leq y\leq \mu x$. Clearly, $\sim$ is an
equivalence relation. Given $h>\theta$ (i.e., $h\geq \theta$ and
$h\neq \theta$), we denote by $P_h$ the set $P_h=\{x\in E\mid x\sim
h\}$. It is easy to see that $P_h\subset P$ is convex and $\lambda
P_h=P_h$ for all $\lambda>0$. If $P^\circ\neq \emptyset$ and $h\in
P^\circ$, it is clear that $P_h=P^\circ$.

An operator $B: E\to E$ is called a sublinear operator
if $B(sx)\leq sBx$, for $x\in P$ and $s\geq 1$.

All the concepts discussed above can be found in \cite{g1,g2,g3}. For more
results about mixed monotone operators and other related
concepts, the reader is referred to \cite{b4,d1,h2,l7,z1,z6,z6} and  the
references therein.


 In 2010, Zhao \cite{z7} introduced the the following $h$-concave-convex
operator.


\begin{definition} \label{def1.1}\rm
 Let $A: P_h\times P_h  \to P_h$ and
$h\in P\setminus\{\theta\}$. If there exists an
 $\eta(u,v,t)>0$ such that
 $$
A(tu,t^{-1}v)\geq t(1+\eta(u,v,t))A(u,v),\quad \forall u, v\in
 P_h \text{ and } 0<t<1,
$$
Then $A$ is called an $h$-concave-convex operator.
\end{definition}

 Without assumptions on the coupled upper-lower
solutions, Zhao \cite{z7}  proved the following theorem.


\begin{theorem} \label{thm1.1}
 Suppose $P$ is a normal cone of $E$, $h\in P\setminus\{\theta\}$,
$A: P_h\times P_h\to P_h$ is a mixed
 monotone and $h$-concave-convex operator. Assume that one of the following
conditions is  satisfied
\begin{itemize}
\item[(A1)] for any $t\in (0,1)$, $\eta(u,v,t)$ is non-increasing with
respect to $u\in  P_h$, non-decreasing with respect to $v\in P_h$;

\item[(A2)] for any $t\in (0,1)$, $\eta(u,v,t)$ is non-decreasing
with respect to $u\in  P_h$, non-increasing with respect to $v\in P_h$,
and there exist $x_0, y_0\in P_h$, $x_0\leq y_0$ such that
$\limsup_{t\to 0^+} \eta(x_0,y_0,t)=+\infty$.
\end{itemize}
Then $A$ has exactly one fixed point $x^*$ in $P_h$. Moreover,
constructing successively the sequences
$$
x_n=A(x_{n-1},y_{n-1}),\quad
y_n=A(y_{n-1},x_{n-1}),\quad n=1,2,\dots,
$$
for any initial $x_0,y_0\in P_h$, we have $\|x_n-x^*\|\to 0$ and
$\|y_n-x^*\|\to 0$ as $n\to \infty$.
\end{theorem}

Recently, Zhai and Zhang \cite{z1}  proved a new
existence-uniqueness result of positive fixed points for mixed
monotone operators. They obtained the following result.


\begin{theorem} \label{thm1.2}
 Suppose $P$ is a normal cone of $E$,
and $A: P\times P\to P$ is a mixed
 monotone operator. Assume that the following
 conditions are satisfied
\begin{itemize}
\item[(B1)] there exists $h\in P$ with $h\neq \theta$ such that
$A(h,h)\in P_h$;

\item[(B2)] for any $u,v\in P$ and $t\in (0,1)$, there exists
$\varphi(t)\in (t,1]$ such that 
$$
A(tu,t^{-1}v)\geq \varphi(t)A(u,v).
$$
\end{itemize}
Then operator $A$ has a unique fixed point $x^*$ in $P_h$.
Moreover, for any initial $x_0,\ y_0\in P_h$, constructing
successively the sequences
$$
x_n=A(x_{n-1},y_{n-1}),\quad
y_n=A(y_{n-1},x_{n-1}),\quad  n=1,2,\dots,
$$
we have $\|x_n-x^*\|\to 0$ and $\|y_n-x^*\|\to 0$ as $n\to \infty$.
\end{theorem}

In this article, we use the partial ordering theory and monotone
iterative technique to obtain the existence and uniqueness of
solutions of the following operator equation
\begin{equation}
A(x,x)+Bx=x,\quad  x\in E,\label{e1.1}
\end{equation}
where $A$ is a mixed monotone operator, $B$ is sublinear, and $E$ is
a real ordered Banach space, we do not require the operator
discussed in this paper to have upper-lower solutions. In addition,
we show some applications to nonlinear integral equation and
second-order boundary-value problem on time scales.


\section{Abstract theorems}

Our main results are the following theorems.

\begin{theorem} \label{thm2.1}
 Let $P$ be a normal cone in $E$, and $A: P\times P\to P$ a mixed monotone
 operator. Let $B: E\to E$ be sublinear.
Assume that for all $a<t<b$, there exist two positive-valued
functions $\tau(t), \varphi(t,x,y)$ on an interval  $(a,b)$ such that
\begin{itemize}
\item[(H1)]  $\tau: (a,b)\to (0,1)$ is a surjection;
\item[(H2)] $\varphi(t,x,y)>\tau(t)$ for all $t\in (a,b)$, $x,y\in P$;
\item[(H3)] $A\big(\tau(t)x,\frac{1}{\tau(t)}y\big)
\geq \varphi(t,x,y)A(x,y)$ for all $t\in (a,b)$,  $x,y\in P$;
item[(H4)]  $(I-B)^{-1}: E\to E$ exists and is an
 increasing operator.
\end{itemize}
For any $t\in (a,b)$, $\varphi(t,x,y)$ is nondecreasing in $x$
for fixed $y$ and non-increasing in $y$ for fixed $x$.
 In addition,  suppose that there exist
 $h\in P\setminus \{\theta\}$ and $t_0\in (a,b)$ such that
\begin{equation}
\frac{\tau(t_0)}{\varphi(t_0,h,h)}h\leq (I-B)^{-1}A(h,h)
\leq \frac{1}{\tau(t_0)}h.\label{e2.1}
\end{equation}
 Then
\begin{itemize}
\item[(i)]  there are $u_0,\ v_0\in P_h$ and $r\in (0,1)$ such that
$rv_0\leq u_0\leq v_0$,
$u_0\leq (I-B)^{-1}A(u_0, v_0)\leq (I-B)^{-1}A(v_0, u_0)\leq v_0$;

\item[(ii)]  equation \eqref{e1.1}  has a unique solution $x^*$ in
$[u_0,v_0]$;

\item[(iii)]  for any initial $x_0,\ y_0\in P_h$, constructing
successively the sequences
$$
x_n=(I-B)^{-1}A(x_{n-1},y_{n-1}),\quad
y_n=(I-B)^{-1}A(y_{n-1},x_{n-1}),\quad n=1,2,\dots,
$$
we have $\|x_n-x^*\|\to 0$ and $\|y_n-x^*\|\to 0$ as
$n\to \infty$.
\end{itemize}
\end{theorem}


\begin{theorem} \label{thm2.2}
 Let $P$ be a normal cone in $E$, and $A: P\times P\to P$ a mixed monotone
 operator. Let $B: E\to E$ be sublinear.
Assume that for all $a<t<b$, there exist two positive-valued
functions $\tau(t), \varphi(t,x,y)$ on interval
 $(a,b)$ such that the properties
{\rm  (H1)--(H4)} in Theorem \ref{thm2.1}  are satisfied.
Furthermore, for any $t\in (a,b)$, $\varphi(t,x,y)$ is non-increasing
in $x$ for fixed $y$, and nondecreasing in $y$ for fixed $x$. In
addition,
 suppose that there exist
 $h\in P\setminus \{\theta\}$ and $t_0\in (a,b)$ such that
\begin{equation}
\tau(t_{0})h\leq (I-B)^{-1}A(h,h)
\leq \frac{\varphi\big(t_0,\frac{h}{\tau(t_0)},\tau(t_{0})h\big)}
{\tau(t_0)}h.\label{e2.2}
\end{equation}
Then the conclusions (i), (ii), (iii)  in Theorem \ref{thm2.1}
hold.
\end{theorem}


\begin{proof}[Proof of Theorem \ref{thm2.1}]
 For  convenience, we denote $C=(I-B)^{-1}A$. By the fact that operator
 $B$ is sublinear, we have $B\theta \geq \theta$, which together
 with (H4) imply
 $$
\theta\leq (I-B)^{-1}\theta \leq (I-B)^{-1}x,\quad x\in P.
$$
 Consequently, $(I-B)^{-1}$ is a positive operator. Hence, we have
that $C: P\times P\to P$. According to (H4), we know
that  $C$ is mixed monotone.

Since $B$ is sublinear, we know that for any $x\in P$ and
$\beta\in (0,1)$, we obtain
$$
(I-B)(\beta x)\leq \beta (I-B)x.$$
Thus
$$
(I-B)(\beta (I-B)^{-1}x)\leq \beta (I-B)(I-B)^{-1}x=\beta x;
$$
i.e.,
$(I-B)(\beta (I-B)^{-1}x)\leq\beta x$.
Therefore, we have
\begin{equation}
\beta(I-B)^{-1}x\leq (I-B)^{-1}(\beta x).\label{e2.3}
\end{equation}
 For any $t\in (a,b)$, it follows from (H1)-(H4) and \eqref{e2.3} that
\begin{equation}
\begin{aligned}
C\big(\tau(t)x,\frac{1}{\tau(t)}y\big)
&=(I-B)^{-1}A\big(\tau(t)x,\frac{1}{\tau(t)}y\big)\\
&\geq (I-B)^{-1}\varphi(t,x,y)A(x,y)\\
&\geq \varphi(t,x,y)(I-B)^{-1}A(x,y)\\
&=\varphi(t,x,y)C(x,y).
\end{aligned}\label{e2.4}
\end{equation}
Since
 $\tau(t_0)<\varphi(t_0,h,h)$, we can take a positive integer $k$
 such that
\begin{equation}
\big(\frac{\varphi(t_0,h,h)}{\tau(t_0)}\big)^k\geq
 \frac{1}{\tau(t_0)}.\label{e2.5}
\end{equation}
Let $u_0=[\tau(t_0)]^{k}h,\ v_0=\frac{1}{[\tau(t_0)]^k}h$, and
 construct successively the sequences
 $$
u_n=C(u_{n-1},v_{n-1}),\quad
v_n=C(v_{n-1},u_{n-1}),\quad n=1,2,\dots.
$$
It is  clear that $u_0, v_0\in P_h$ and $u_0<v_0$,
$u_1=C(u_0,v_0)\leq  C(v_0,u_0)=v_1$. In general, we obtain
$u_n\leq v_n$, $n=1,2,\dots$. Note that $\varphi(t,x,y)>\tau(t)$ for
all $t\in (a,b)$, $x,y\in P$. Combining \eqref{e2.1} with \eqref{e2.4},
we have
\begin{align*}
u_1
&=C(u_0,v_0)=C\Big([\tau(t_0)]^k
h,\frac{h}{[\tau(t_0)]^k}\Big)\\
& =C\Big(\tau(t_0)[\tau(t_0)]^{k-1}
h,\frac{1}{\tau(t_0)}\frac{h}{[\tau(t_0)]^{k-1}}\Big)\\
&\geq \varphi \Big(t_0,[\tau(t_0)]^{k-1}h,\frac{h}{[\tau(t_0)]^{k-1}}\Big)
 C\Big([\tau(t_0)]^{k-1} h,\frac{h}{[\tau(t_0)]^{k-1}}\Big)\\
&=\varphi\Big(t_0,[\tau(t_0)]^{k-1}h,\frac{h}{[\tau(t_0)]^{k-1}}\Big)
 C\Big(\tau(t_0)[\tau(t_0)]^{k-2}
h,\frac{1}{\tau(t_0)}\frac{h}{[\tau(t_0)]^{k-2}}\Big)\\
&\geq\varphi \Big(t_0,[\tau(t_0)]^{k-1}h,\frac{h}{[\tau(t_0)]^{k-1}}\Big)
\varphi \Big(t_0,[\tau(t_0)]^{k-2}h,\frac{h}{[\tau(t_0)]^{k-2}}\Big)\\
&\quad\times C\Big([\tau(t_0)]^{k-2} h,\frac{h}{[\tau(t_0)]^{k-2}}\Big)\\
&\geq \dots\geq \varphi
\Big(t_0,[\tau(t_0)]^{k-1}h,\frac{h}{[\tau(t_0)]^{k-1}}\Big)
\varphi \Big(t_0,[\tau(t_0)]^{k-2}h,\frac{h}{[\tau(t_0)]^{k-2}}\Big)\\
&\quad\times \dots \varphi(t_0,h,h)C(h,h) \\
&\geq [\tau(t_0)]^{k-1}\varphi(t_0,h,h)C(h,h) \\
&\geq [\tau(t_0)]^{k}h=u_0.
\end{align*}
From \eqref{e2.4}, we have
\begin{equation}
C\big(\frac{x}{\tau(t)},\tau(t)y\big)
\leq \frac{1}{\varphi\big(t,\frac{x}{\tau(t)},\tau(t)y\big)}C(x,y),\quad
\forall t\in (a,b),\quad x, y\in P.\label{e2.6}
\end{equation}
Note that $\varphi(t,x,y)$ is nondecreasing in $x$
 and nonincreasing in $y$, it follows from \eqref{e2.1},
 \eqref{e2.5} and \eqref{e2.6} that
\begin{align*}
v_1
&=C(v_0,u_0)=C\Big(\frac{h}{[\tau(t_0)]^k},[\tau(t_0)]^k
h\Big)\\
& =C\Big(\frac{1}{\tau(t_0)}\frac{h}{[\tau(t_0)]^{k-1}},
 \tau(t_0)[\tau(t_0)]^{k-1} h\Big) \\
&\leq\frac{1}{\varphi
\big(t_0,\frac{h}{[\tau(t_0)]^{k}},[\tau(t_0)]^{k}h\big)}
 C\Big(\frac{h}{[\tau(t_0)]^{k-1}},[\tau(t_0)]^{k-1}h\Big) \\
&=\frac{1}{\varphi \big(t_0,\frac{h}{[\tau(t_0)]^{k}},[\tau(t_0)]^{k}h\big)}
 C\Big(\frac{1}{\tau(t_0)}\frac{h}{[\tau(t_0)]^{k-2}},\tau(t_0)
 [\tau(t_0)]^{k-2}h\Big) \\
 &\leq\frac{1}{\varphi\big(t_0,\frac{h}{[\tau(t_0)]^{k}},[\tau(t_0)]^{k}h\big)}
\frac{1}{\varphi \big(t_0,\frac{h}{[\tau(t_0)]^{k-1}},[\tau(t_0)]^{k-1}h\big)}\\
&\quad \times C\Big(\frac{h}{[\tau(t_0)]^{k-2}},[\tau(t_0)]^{k-2} h\Big) 
\leq\dots\\
&\leq\frac{1}{\varphi\big(t_0,\frac{h}{[\tau(t_0)]^{k}},[\tau(t_0)]^{k}h\big)}
\frac{1}{\varphi\big(t_0,\frac{h}{[\tau(t_0)]^{k-1}},[\tau(t_0)]^{k-1}h\big)}\\
&\quad\times  \dots \frac{1}{\varphi\big(t_0,\frac{h}{\tau(t_0)},\tau(t_0)h\big)}
C(h,h) \\
&<\frac{1}{[\varphi(t_0,h,h)]^{k}}\frac{h}{\tau(t_0)}\\
&\leq\frac{1}{[\tau(t_0)]^{k}}h=v_0.
\end{align*}
Thus, we obtain
\begin{equation}
u_0\leq u_1\leq v_1\leq v_0. \label{e2.7}
\end{equation}
By induction, it is easy to obtain that
$$
u_0\leq u_1\leq \dots \leq u_n\leq \dots \leq v_n\leq \dots \leq v_1\leq v_0.
$$
Take any
$r\in (0,[\tau(t_0)]^{2k})$, then $r\in (0,1)$ and $u_0\geq r v_0$.
So we can know that
$$
u_n\geq u_0\geq r v_0\geq r v_n,\quad n=1,2,\dots.
$$
Let
$$
r_n=\sup\{r>0|u_n\geq r v_n\},\quad n=1,2,\dots.
$$
Thus, we have $u_n\geq r_n v_n,\ n=1,2,\dots $, and then
$$
u_{n+1}\geq u_n\geq r_n v_n\geq r_n v_{n+1},\quad n=1,2,\dots.
$$
Therefore, $r_{n+1}\geq r_n$; i.e.,
$$
0<r_0\leq r_1\leq\dots \leq r_n\leq \dots \leq 1.
$$
Set $r^*=\lim_{n\to \infty}r_n$, we will show that
$r^*=1$. In fact, if $0<r^*<1$, by (H1), there exists $t_1\in
(a,b)$ such that $\tau(t_1)=r^*$. Consider the following two cases:

Case i: There exists an integer $N$ such that $r_N=r^*$. In this case,
we have $r_n=r^*$ and $u_n\geq r^* v_n$ for all $n\geq N$ hold.
Hence
\begin{align*}
u_{n+1}
&=C(u_n,v_n)\geq C\Big(r^* v_n,\frac{1}{r^*}u_n\Big) \\
&=C\Big(\tau(t_1) v_n,\frac{1}{\tau(t_1)}u_n\Big)\\
&\geq \varphi(t_1,v_n,u_n)C(v_n,u_n) \\
&\geq \varphi(t_1,u_0,v_0)C(v_n,u_n)=\varphi(t_1,u_0,v_0)v_{n+1},\quad
n\geq N.
\end{align*}
By the definition of $r_n$, we have
$$
r_{n+1}=r^*\geq \varphi(t_1,u_0,v_0)>\tau(t_1)=r^*,\quad n\geq N,
$$
which is a contradiction.

Case ii: For all integers $n,\ r_n<r^*$. Then we obtain
$0<\frac{r_n}{r^*}<1$. By (H1), there exist $z_n\in (a,b)$ such
that $\tau(z_n)=\frac{r_n}{r^*}$. Hence
\begin{align*}
u_{n+1}
&=C(u_n,v_n)\geq C\Big(r_n v_n,\frac{1}{r_n}u_n\Big)\\
&=C\Big(\frac{r_n}{r^*}r^* v_n,\frac{1}{\frac{r_n}{r^*}r^* }u_n\Big)
=C\Big(\tau(z_n)r^* v_n,\frac{1}{\tau(z_n)r^*}u_n\Big)\\
&\geq \varphi\Big(z_n,r^* v_n, \frac{1}{r^*}u_n\Big)
C\Big(r^* v_n, \frac{1}{r^*}u_n\Big)\\
& \geq \varphi\Big(z_n,r^* u_0, \frac{1}{r^*}
v_0\Big)C\Big(\tau(t_1)v_n,\frac{1}{\tau(t_1)}u_n\Big)\\
&\geq \varphi\Big(z_n,r^* u_0, \frac{1}{r^*}
v_0\Big)\varphi(t_1,v_n,u_n)C(v_n,u_n) \\
&\geq \varphi\Big(z_n,r^* u_0, \frac{1}{r^*} v_0\Big)
\varphi(t_1,u_0,v_0)v_{n+1}.
\end{align*}
By the definition of $r_n$, we have
$$
r_{n+1}\geq \varphi\big(z_n,r^* u_0,\frac{1}{r^*}v_0\big)
\varphi(t_1,u_0,v_0)>\tau(z_n)\varphi(t_1,u_0,v_0)
=\frac{r_n}{r^*}\varphi(t_1,u_0,v_0).
$$
Let $n\to \infty$, we have
$$
r^* \geq \varphi(t_1,u_0,v_0)>\tau(t_1)=r^*,
$$
which is also a contradiction. Thus, $\lim_{n\to \infty}r_n=1$.


Furthermore, as in the proof of \cite[Theorem 2.1]{l2}, there
exists $x^{*}\in [u_0,v_0]$ such that
$\lim_{n\to \infty}u_n=\lim_{n\to \infty}v_n=x^*$, and $x^*$ is
the fixed point of operator $C$.

In the following, we prove that $x^*$ is the unique fixed point of
$C$ in $P_h$. In fact, suppose that $x_*\in P_h$ is another fixed
point of operator $C$. Let
$$
c_1=\sup\big\{0<c\leq 1| c x_*\leq x^*\leq \frac{1}{c}x_*\big\}.
$$
Clearly, $0<c_1\leq 1$ and $c_1 x_* \leq x^* \leq \frac{1}{c_1}x_*$.
If $0<c_1<1$, according to (H1), there exists $t_2\in (a,b)$ such
that $\tau(t_2)=c_1$. Then
\begin{align*}
x^*&=C(x^*,x^*)\geq C\Big(c_1 x_*,\frac{1}{c_1}x_*\Big)\\
&=C\Big(\tau(t_2)x_*,\frac{1}{\tau(t_2)}x_*\Big)\\
&\geq\varphi(t_2,x_*,x_*)C(x_*,x_*)=\varphi(t_2,x_*,x_*)x_*,
\end{align*}
and
\begin{align*}
x^*&=C(x^*,x^*)\leq C\Big(\frac{1}{c_1}x_*,c_1 x_*\Big)\\
&=C\Big(\frac{1}{\tau(t_2)}x_*,\tau(t_2)x_*\Big)
\\
&\leq\frac{1}{\varphi\big(t_2,\frac{x_*}{\tau(t_2)},\tau(t_2)x_*\big)}
C(x_*,x_*)\\
&=\frac{1}{\varphi\big(t_2,\frac{x_*}{\tau(t_2)},\tau(t_2)x_*\big)}x_*.
\end{align*}
Since
$$
\varphi(t_2,x_*,x_*)\leq \varphi\Big(t_2,\frac{x_*}{\tau(t_2)},\tau(t_2)x_*\Big),
$$
we have
$$
\varphi(t_2,x_*,x_*)x_*\leq x^*\leq
\frac{1}{\varphi(t_2,x_*, x_*)}x_*.
$$
Hence, $c_1\geq \varphi(t_2,x_*,x_*)>\tau(t_2)=c_1$, which is a contradiction.
Thus we have $c_1=1$; i.e., $x_*=x^*$. Therefore, $C$ has a unique fixed
point $x^*$ in $P_h$. Note that $[u_0,v_0]\subset P_h$, so we know
that $x^*$ is the unique fixed point of $C$ in $[u_0,v_0]$. For any
initial $x_0,\ y_0\in P_h$, we can choose a small number
$\overline{e}\in (0,1)$ such that
$$
\overline{e}h\leq x_0\leq \frac{1}{\overline{e}}h,\quad
\overline{e}h\leq y_0\leq \frac{1}{\overline{e}}h.
$$
From (H1), there is $t_3\in (a,b)$ such that $\tau(t_3)=\overline{e}$, thus
$$
\tau(t_3)h\leq x_0\leq \frac{1}{\tau(t_3)}h,\ \ \ \ \
\tau(t_3)h\leq y_0\leq \frac{1}{\tau(t_3)}h.$$ We can choose a
sufficiently large positive integer $q$ such that
$$
\Big(\frac{\varphi(t_3,h,h)}{\tau(t_3)}\Big)^q
\geq \frac{1}{\tau(t_3)}.
$$
Take $\hat{u}_0=[\tau(t_3)]^q h$,
$\hat{v}_0=\frac{1}{[\tau(t_3)]^q}h$. We can find that
$$
\hat{u}_0\leq x_0\leq \hat{v}_0,\quad
\hat{u}_0\leq y_0\leq \hat{v}_0.
$$
Constructing successively the sequences
\begin{gather*}
x_n=C(x_{n-1},y_{n-1}),\quad
 y_n=C(y_{n-1},x_{n-1}),\quad n=1,2,\dots, \\
\hat{u}_n=C(\hat{u}_{n-1},\hat{v}_{n-1}),\quad
\hat{v}_n=C(\hat{v}_{n-1},\hat{u}_{n-1}),\quad n=1,2,\dots.
\end{gather*}
By using the mixed monotone properties of operator
$C$, we have
$$
\hat{u}_n\leq x_n\leq \hat{v}_n,\quad
\hat{u}_n\leq y_n\leq \hat{v}_n,\quad n=1,2,\dots.
$$
Similarly to the above proof, we
can know that there exists $y^*\in P_h$ such that
$$
C(y^*,y^*)=y^*,\quad
\lim_{n\to \infty}\hat{u}_n=\lim_{n\to \infty}\hat{v}_n=y^*.
$$
By the uniqueness of fixed points of
operator $C$ in $P_h$, we have $y^*=x^*$. Taking into account that
$P$ is normal, we deduce that
$\lim_{n\to \infty}x_n=\lim_{n\to \infty}y_n=x^*$. This completes
the proof.
\end{proof}

\begin{proof}[Proof of Theorem \ref{thm2.2}]
As in the proof of  Theorem \ref{thm2.1}, it suffices to verify that \eqref{e2.7} holds.
For any $t\in (a,b)$, note that
$\varphi(t,x,y)$ is non-increasing in $x$
 and nondecreasing in $y$, it follows from \eqref{e2.2},
 \eqref{e2.4} and \eqref{e2.5} that
\begin{align*}
&u_1\\
&=C(u_0,v_0)=C\Big([\tau(t_0)]^k
h,\frac{h}{[\tau(t_0)]^k}\Big)\\
&\geq\varphi \Big(t_0,[\tau(t_0)]^{k-1}h,\frac{h}{[\tau(t_0)]^{k-1}}\Big)
 \varphi\Big(t_0,[\tau(t_0)]^{k-2}h,\frac{h}{[\tau(t_0)]^{k-2}}\Big)
 \dots \varphi(t_0,h,h)C(h,h) \\
&\geq [\varphi(t_0,h,h)]^k \tau(t_0)h \\
&\geq [\tau(t_0)]^{k}h=u_0.
\end{align*}
Note that $\varphi(t,x,y)>\tau(t)$ for all $t\in  (a,b)$, $x,y\in P$.
Combining \eqref{e2.2} with \eqref{e2.6}, we obtain
\begin{align*}
v_1
&=C(v_0,u_0)=C\Big(\frac{h}{[\tau(t_0)]^k},[\tau(t_0)]^kh\Big)\\
&\leq\frac{1}{\varphi \big(t_0,\frac{h}{[\tau(t_0)]^{k}},[\tau(t_0)]^{k}h\big)}
\frac{1}{\varphi\big(t_0,\frac{h}{[\tau(t_0)]^{k-1}},[\tau(t_0)]^{k-1}h\big)}
\dots \\
&\quad\times \frac{1}{\varphi\big(t_0,\frac{h}{\tau(t_0)},\tau(t_0)h\big)}C(h,h)
\\
&<\frac{1}{[\tau(t_0)]^{k-1}}\frac{1}{\varphi
 \big(t_0,\frac{h}{\tau(t_{0})},\tau(t_0)h\big)}C(h,h)\\
 &\leq\frac{1}{[\tau(t_0)]^{k}}h=v_0.
\end{align*}
Thus, we know that \eqref{e2.7} holds. The rest proof is similar to
that of Theorem \ref{thm2.1}, we omit it here.
\end{proof}



\begin{remark} \label{rmk2.1}\rm
 Compared with Theorem \ref{thm1.1}, the main
 contribution in this paper to weaken the restriction on operator $A$;
 i.e., the condition $A: P_h \times P_h \to P_h$
 in Theorem \ref{thm1.1} is replaced by \eqref{e2.1} in
 Theorem \ref{thm2.1} and \eqref{e2.2} in
 Theorem \ref{thm2.2}. We also remove the condition
 ``there exist $x_0,y_0\in P_h$, $x_0\leq y_0$ such that
$\limsup_{t\to 0^+} \eta(x_0,y_0,t)=+\infty$'' in
Theorem \ref{thm1.1}.
In Theorems \ref{thm1.2}, and \ref{thm2.2} we consider more general operators;
 i.e., concave and convex mixed monotone operators with perturbations.
\end{remark}

By the proof of \cite[Corollary 2.5]{h2}, we can obtain the following
corollary.


\begin{corollary} \label{coro2.1}
Let $P$ be a normal cone in $E$, and $A: P\times P\to P$ a mixed monotone
 operator. Let $B$ be a linear operator in $E$, such that
\begin{itemize}
\item[(C1)]  $\|B\|<1$, there exists some number $b\geq 0$ such that
$B+bI\geq 0$;

\item[(C2)] $A$ satisfies the conditions of Theorem \ref{thm2.1}
 or Theorem \ref{thm2.2}.
\end{itemize}
Then equation \eqref{e1.1} has a unique solution $x^*$ in
$[u_0,v_0]$.
\end{corollary}


\section{An application to integral equations}

We consider nonlinear integral equation
\begin{equation}
\int_{a_1}^{a_2}G(t,s)[f(x(s))+g(x(s))]ds
=[1+G_1(t)]x(t)-G_2(t)x(t+\tau),\quad
t\in \mathbb{R},\label{e3.1}
\end{equation}
where $a_1,a_2,\tau$ are constants.

Let $E=C(\mathbb{R})$ denote the real Banach space of all bounded
and continuous functions on $\mathbb{R}$ with the supremum norm.
Define a cone 
$$
P=\{x\in E: x(t)\geq 0,\,\forall t\in \mathbb{R}\}.
$$

\begin{theorem} \label{thm3.1}
Assume that
\begin{itemize}
\item[(D1)] Let $G_1, G_2\in E$, $G(t,s)$ be
 uniformly continuous on $\mathbb{R}\times [a_1,a_2]$, $f(x)$ be
 increasing, $g(x)$ be decreasing and $f(x)\geq 0,\ g(x)\geq 0$ for
 $x\geq 0$;

 item[(D2)] there exist $g_1, g_2\in [0,+\infty)$ such that
 $0\leq G_1(t)\leq g_1$, $0\leq G_2(t)\leq g_2$, and $g_1+g_2<1$,
 where $t\in \mathbb{R}$;

\item[(D3)]  there exist $\tau(t), \varphi(t,x_1,x_2)$ on an interval
 $t\in\mathbb{R}$  such that $\tau:\mathbb{R}\to (0,1)$ is a
 surjection and $\varphi(t,x_1,x_2)>\tau(t)$ for all 
$t\in\mathbb{R}$, $x_1, x_2\in P$ which satisfy
\begin{align*}
&\int_{a_1}^{a_2}G(t,s)\big[f(\tau(\mu)x_1(s))
+g\big(\frac{1}{\tau(\mu)}x_2(s)\big)\big]ds\\  
& \geq \varphi(\mu,x_1,x_2)
\int_{a_1}^{a_2}G(t,s)[f(x_1(s))+g(x_2(s))]ds,\quad \forall
\mu\in\mathbb{R},\; x_1, x_2\in P;
 \end{align*}

\item[(D4)] for fixed $t\in\mathbb{R}$, $\varphi(t,x_1,x_2)$
 is non-increasing in $x_1$ and non-decreasing in $x_2$;

\item[(D5)]  there exist $e\in P\setminus \{0\}$ and
 $t_0\in\mathbb{R}$ such that
\begin{align*}
\tau(t_0)e(t)
&\leq \int_{a_1}^{a_2}G(t,s)[f(e(s))+g(e(s))]ds+G_2(t)e(t+\tau)-G_1(t)e(t)\\
&\leq \frac{\varphi\big(t_0,\frac{e(t)}{\tau(t_0)},\tau(t_0)e(t)\big)}
{\tau(t_0)}e(t), \quad \forall t\in\mathbb{R}.
\end{align*}
\end{itemize}
Then  \eqref{e3.1}  has a unique positive
solution $x^*$ in $P_e$.
\end{theorem}

\begin{proof} We rewrite  \eqref{e3.1} as
$$
x(t)=\int_{a_1}^{a_2}G(t,s)[f(x(s))+g(x(s))]ds+G_2(t)x(t+\tau)-G_1(t)x(t),\quad
t\in \mathbb{R}.
$$ 
Define
\begin{gather*}
A(x_1,x_2)(t)=\int_{a_1}^{a_2}G(t,s)[f(x_1(s))+g(x_2(s))]ds,\quad
t\in \mathbb{R},\\
Bx(t)=G_2(t)x(t+\tau)-G_1(t)x(t),\quad t\in \mathbb{R}.
\end{gather*}
According to (D1), we have that $A: P\times P\to P$
is a mixed monotone operator.

For the linear operator $B$, we have $\|B\|\leq g_1+g_2<1$, and
$B+bI\geq 0$ for $b\geq g_1$.

On the other hand, for any $\mu\in\mathbb{R}$ and $x_1,\ x_2\in P$,
according to (D3), we obtain
\begin{align*}
A\big(\tau(\mu)x_1,\frac{1}{\tau(\mu)}x_2\big)
&=\int_{a_1}^{a_2}G(t,s)
\big[f(\tau(\mu)x_1(s))+g\big(\frac{1}{\tau(\mu)}x_2(s)\big)\big]ds\\ 
&\geq \varphi(\mu,x_1,x_2)\int_{a_1}^{a_2}G(t,s)[f(x_1(s))+g(x_2(s))]ds\\ 
&=\varphi(\mu,x_1,x_2)A(x_1,x_2).
\end{align*}
That is,
$$
A\Big(\tau(\mu)x_1,\frac{1}{\tau(\mu)}x_2\Big)
\geq \varphi(\mu,x_1,x_2)A(x_1,x_2),\quad\text{for }
 \mu\in\mathbb{R},\;x_1,x_2\in P.
$$
 In addition, from (D5), for any $t\in \mathbb{R}$, we have
\begin{align*}
\tau(t_0)e(t)
&\leq A(e,e)+B(e)\\
&= \int_{a_1}^{a_2}G(t,s)[f(e(s))+g(e(s))]ds+G_2(t)e(t+\tau)-G_1(t)e(t)\\
&\leq \frac{\varphi\big(t_0,\frac{e(t)}{\tau(t_0)},\tau(t_0)e(t)\big)}{\tau(t_0)}
e(t),  \quad \forall t\in\mathbb{R}.
\end{align*}
Consequently, all conditions in Corollary \ref{coro2.1} are satisfied;
therefore,   \eqref{e3.1} has a unique positive
solution in $P_e$.
\end{proof}

\section{An application to a boundary-value problem on time scales}

 In this section, we will apply Theorem \ref{thm2.2} to the
following second-order boundary value problem with Sturm-Liouville
boundary conditions on time scales
\begin{gather}
(py^{\Delta})^{\nabla}(t)+[f(t,y(t))+g(t,y(t))]=0,\quad
  t\in (a,b]_{\mathbb{T}}, \label{e4.1}\\
\alpha y(a)-\beta (py^{\Delta})(a)=0,\quad
 \gamma y^{\sigma}(b)+\delta (py^{\Delta})(b)=0,\label{e4.2}
\end{gather}
where
\begin{gather}
p: [a,\sigma(b)]_{\mathbb{T}}\to (0,+\infty),\quad  
p\in C[a,\sigma(b)]_{\mathbb{T}},\label{e4.3}\\
\beta,\delta\in (0,+\infty),\quad \alpha,\gamma\in [0,+\infty),\quad
\beta\gamma+\alpha\delta+\alpha\gamma\int_{a}^{\sigma(b)}\frac{\Delta
\tau}{p(\tau)}>0.\label{e4.4}
\end{gather}
Some definitions and theorems on time scales can be found in \cite{a4,b2,b3,h3}
which are excellent
references for the calculus on time scales.
The study of dynamic equations on time scales goes back to its
founder Hilger \cite{h3}, and is a new area of still fairly theoretical
exploration in mathematics. In recent years, there has been a great
deal of research work on the existence of positive solutions of
second-order boundary value problems on time scales, we refer the
reader to \cite{a1,a2,a3, a4,b2,b3,c1,e1,h1,j1,l4,s1,w1}
for some recent results. Such
investigations can provide accurate information of phenomena that
manifest themselves partly in continuous time and partly in discrete
time. We would like to mention some results of Anderson and Wong
\cite{a2}, Jankowski \cite{j1} and Wang, Wu and Wu \cite{w1},
 which motivated us to
consider problem \eqref{e4.1} and \eqref{e4.2}.

Anderson and Wong \cite{a2} studied the second-order time scale
semipositone boundary value problem
$$
(py^{\Delta})^{\nabla}(t)+\lambda f(t, u(t))=0,\quad  t\in (a,b]_{\mathbb{T}}
$$
with Sturm-Liouville boundary conditions \eqref{e4.2}.

The methods of lower and upper solutions have
been applied extensively in proving the existence results for
dynamic equations on time scales.  Jankowski \cite{j1} investigated
second order dynamic equations with deviating arguments on time
scales of the form
\begin{gather*}
 -x^{\Delta\Delta}(t)=f(t,x(t),x(\alpha(t)))\equiv (Fx)(t),\quad
  t\in[0,T]_{\mathbb{T}}, \\
x(0)=k_{1}\in \mathbb{R},\quad x(T)=k_{2}\in \mathbb{R}.
\end{gather*}
They formulated sufficient conditions, under which such problems
have a minimal and a maximal solution in a corresponding region
bounded by upper-lower solutions.

 Wang, Wu and Wu \cite{w1} considered a method of generalized
quasilinearization, with even-order $k$ $(k\geq 2)$ convergence, for
the problem
\begin{gather*}
-(p(t)x^{\Delta})^{\nabla}+q(t)x^{\sigma}=f(t,x^{\sigma})+g(t,x^{\sigma}),\quad
  t\in [a,b]_{\mathbb{T}}, \\
\tau_{1}x(\rho(a))-\tau_{2}x^{\Delta}(\rho(a))=0,\quad
 x(\sigma(b))-\tau _{3}x(\eta)=0. 
\end{gather*}
The main contribution in \cite{w1} is to relaxed the monotone conditions on
$f^{(i)}(t,x)$ and $g^{(i)}(t,x)$ for $1<i<k$, including a more general
concept of upper and lower solution in mathematical biology,  so
that the high-order convergence of the iterations is ensured for a
larger class of nonlinear functions on time scales.


We would also like to mention the results of Sang \cite{s1}. There, we
considered the existence of positive solutions and established the
corresponding iterative schemes for the  $m$-point boundary
value problem on time scales
\begin{gather}
u^{\Delta\nabla}(t)+f(t, u(t))=0,\quad t\in [0,1]\subset\mathbb{T}, \label{e4.5}\\
\beta u(0)-\gamma u^{\Delta}(0)=0,\quad
 u(1)=\sum_{i=1}^{m-2}\alpha_{i}u(\xi_{i}),\quad m\geq 3.\label{e4.6}
\end{gather}
By considering the ``heights'' of the nonlinear term $f$ on some
bounded sets and applying monotone iterative techniques on a Banach
space, we did not only obtain the existence of positive solutions
for problem \eqref{e4.5} and \eqref{e4.6}, but also established the iterative
schemes for approximating the solutions. In essence, we combined the
method of lower and upper solutions with the cone expansion and
compression fixed point theorem of norm type.

We note that the upper and lower solutions conditions are both
required in \cite{j1,w1}, and the researchers mentioned above
\cite{j1,s1,w1}  only studied the existence of positive solutions. 
Therefore, it is natural to discuss the uniqueness and iteration of positive
solutions to problem \eqref{e4.1} and \eqref{e4.2}.

To obtain our main result, we will employ several lemmas.
These lemmas are based on the linear equation
\begin{equation}
-(py^{\Delta})^{\nabla}(t)=u(t), \quad t\in (a,b]_{\mathbb{T}}, \label{e4.7}
\end{equation}
with boundary conditions \eqref{e4.2}. Define the constant
\begin{equation}
d:=\beta\gamma+\alpha\delta+\alpha\gamma\int_{a}^{\sigma(b)}\frac{\Delta
\tau}{p(\tau)}.\label{e4.8}
\end{equation}


\begin{lemma}[\cite{a2}] \label{lem4.1}
 Assume \eqref{e4.3}  and \eqref{e4.4}. Then  
problem \eqref{e4.7}-\eqref{e4.2}  has a unique solution
 $y$ such that
$$
y(t)=\int_{a}^{b}G(t,s)u(s)\nabla s,\quad t\in [a,\sigma(b)]_{\mathbb{T}},
$$ 
where the Green function is
\begin{equation}
G(t,s)=\frac{1}{d} \begin{cases}
\big(\beta+\alpha\int_{a}^{s}\frac{\Delta\tau}{p(\tau)}\big)
\big(\delta+\gamma\int_{t}^{\sigma(b)}\frac{\Delta \tau}{p(\tau)}\big)
 & a\leq s\leq t\leq \sigma(b), \\[4pt]
\big(\beta +\alpha\int_{a}^{t}\frac{\Delta \tau}{p(\tau)}\big)
\big(\delta+\gamma\int_{s}^{\sigma(b)}\frac{\Delta\tau}{p(\tau)}\big)
 & a\leq t\leq s\leq \sigma(b),
\end{cases} \label{e4.9}
\end{equation}
for all $t,s\in [a,\sigma(b)]_{\mathbb{T}}$, where $d$ is given by
\eqref{e4.8}.
\end{lemma}

\begin{lemma}[\cite{a2}] \label{lem4.2}  Assume
\eqref{e4.3}  and \eqref{e4.4}.Then the Green function \eqref{e4.9} satisfies
$$
g(t)G(s,s)\leq G(t,s)\leq G(s,s),\quad t,s\in [a,\sigma(b)]_{\mathbb{T}},
$$ 
where
\begin{equation}
g(t)=\min_{t\in[a,\sigma(b)]_{\mathbb{T}}}
\Big\{\frac{\delta+\gamma\int_{t}^{\sigma(b)}\frac{\Delta
\tau}{p(\tau)}}{\delta+\gamma\int_{a}^{\sigma(b)}\frac{\Delta
\tau}{p(\tau)}},\frac{\beta+\alpha\int_{a}^{t}\frac{\Delta
\tau}{p(\tau)}}{\beta+\alpha\int_{a}^{\sigma(b)}\frac{\Delta
\tau}{p(\tau)}}\Big\}\in [0,1].
\label{e4.10}
\end{equation}
\end{lemma}

Let the Banach space $\mathbb{B}=C[a,\sigma(b)]_{\mathbb{T}}$ be
equipped with norm
$$
\|u\|=\sup_{t\in [a,\sigma(b)]_{\mathbb{T}}}|u(t)|.
$$
In this space define a cone 
$$
K=\{u\in \mathbb{B}:\min_{t\in [a,\sigma(b)]_{\mathbb{T}}}u(t)\geq 0\}.
$$
Our main result is the following theorem.

\begin{theorem} \label{thm4.1} Assume that
\begin{itemize}
\item[(E1)]  $f,g: (a,b]_{\mathbb{T}}\times [0,+\infty)\to [0,+\infty)$
 are continuous;

\item[(E2)]  for fixed $t$, $f(t,u)$ is nondecreasing in $u$,
 $g(t,u)$ is non-increasing in $u$;

\item[(E3)]  there exist $\tau(t), \varphi(t,y_1,y_2)$ on the interval 
$(a,b)_{\mathbb{T}}$  such that $\tau: (a,b)_{\mathbb{T}}\to (0,1)$ is a
 surjection and $\varphi(t,y_1,y_2)>\tau(t)$  for all 
$t\in(a,b)_{\mathbb{T}}$, $y_1,y_2\in K$ which satisfy
\begin{align*}
&\int_{a}^{b}G(t,s)\big[f(s,\tau(\lambda)y_1(s))
+g\big(s,\frac{1}{\tau(\lambda)}y_2(s)\big)\big]\nabla s \\  
& \geq \varphi(\lambda,y_1,y_2)
\int_{a}^{b}G(t,s)[f(s,y_1(s))+g(s,y_2(s))]\nabla s,\quad
\forall \lambda\in(a,b)_{\mathbb{T}},\; y_1, y_2\in K;
\end{align*}

\item[(E4)]  for any $t\in(a,b)_{\mathbb{T}}$, $\varphi(t,y_1,y_2)$
 is non-increasing in $y_1$ for fixed $y_2$ and nondecreasing in $y_2$ 
for fixed $y_1$;

\item[(E5)]  there exist $h\in K\setminus \{0\}$ and
 $t_0\in(a,b)_{\mathbb{T}}$ such that
 $$
\tau(t_0)h(t)\leq \int_{a}^{b}G(t,s)[f(h(s))+g(h(s))]\nabla s
 \leq \frac{\varphi\big(t_0,\frac{h(t)}{\tau(t_0)},\tau(t_0)h(t)\big)}
 {\tau(t_0)}h(t),
$$
for all $t\in[a,\sigma(b)]_{\mathbb{T}}$.
\end{itemize}
Then \eqref{e4.1}-\eqref{e4.2}  has a unique positive
solution $x^*$ in $K_h$. Moreover, for any initial condition
 $x_0, y_0\in K_h$, constructing successively the sequences
\begin{gather*}
x_n(t)=\int_a^{b}G(t,s)[f(s,x_{n-1}(s))+g(s,y_{n-1}(s))]\nabla
s,\quad t\in [a,\sigma(b)]_{\mathbb{T}},\; n=1,2,\dots,\\
y_n(t)=\int_a^{b}G(t,s)[f(s,y_{n-1}(s))+g(s,x_{n-1}(s))]\nabla
s,\quad t\in [a,\sigma(b)]_{\mathbb{T}},\; n=1,2,\dots,
\end{gather*}
we have $\|x_n-x^*\|\to 0$ and $\|y_n-x^*\|\to 0$ as
$n\to \infty$.
\end{theorem}

\begin{proof} Define
$$
F(y_1,y_2)(t)=\int_{a}^{b}G(t,s)[f(s,y_1(s))+g(s,y_2(s))]\nabla s,\quad
t\in[a,\sigma(b)]_{\mathbb{T}}.
$$ 
By Lemma \ref{thm4.1}, we can know that
problem \eqref{e4.1}-\eqref{e4.2} is equivalent to the fixed point equation
$$
y(t)=F(y,y)(t),\quad t\in[a,\sigma(b)]_{\mathbb{T}}.
$$
 By (E2), it is easy to check that $F: K\times
K\to K$ is mixed monotone.

For any $\lambda\in(a,b)_{\mathbb{T}}$ and $y_1, y_2\in K$, from (E3)
it follows that
\begin{align*}
F\big(\tau(\lambda)y_1,\frac{1}{\tau(\lambda)}y_2\big)
&=\int_{a}^{b}G(t,s) \big[f(s,\tau(\lambda)y_1(s))
+g\big(s,\frac{1}{\tau(\lambda)}y_2(s)\big)\big]\nabla s\\ 
&\geq \varphi(\lambda,y_1,y_2)\int_{a}^{b}G(t,s)[f(s,y_1(s))
 +g(s,y_2(s))]\nabla s \\ 
&=\varphi(\lambda,y_1,y_2)F(y_1,y_2);
\end{align*}
i.e.,
$$
F\big(\tau(\lambda)y_1,\frac{1}{\tau(\lambda)}y_2\big)
\geq \varphi(\lambda,y_1,y_2)F(y_1,y_2),\quad \text{for }
\lambda\in(a,b)_{\mathbb{T}},\; y_1, y_2\in K.
$$ 
Moreover, from (E5), we obtain 
\begin{align*}
\tau(t_0)h(t)
&\leq F(h,h)= \int_{a}^{b}G(t,s)[f(s,h(s))+g(s,h(s))]\nabla s\\ 
&\leq \frac{\varphi\big(t_0,\frac{h(t)}{\tau(t_0)},\tau(t_0)h(t)\big)}
 {\tau(t_0)}h(t),
 \quad \forall t\in [a,\sigma(b)]_{\mathbb{T}}.
\end{align*}
Note that all the conditions in Theorem \ref{thm2.2} hold, which implies the
the conclusions of Theorem \ref{thm4.1}.
\end{proof}

We conclude this article with the following example.


\begin{example} \label{examp4.1}\rm
 Let $\mathbb{T}=\{2^k\}_{k\in \mathbb{Z}}\cup \{0\}$,
 where $\mathbb{Z}$ denotes the set of integers. 
Consider the following problem on time scales $\mathbb{T}$:
\begin{gather}
(y^{\Delta})^{\nabla}(t)+[f(y(t))+g(y(t))]=0,\quad
   t\in (0,1]_{\mathbb{T}}, \label{e4.11}\\
y(0)-(y^{\Delta})(0)=0,\quad y^{\sigma}(1)+(y^{\Delta})(1)=0,\label{e4.12}
\end{gather}
where $f(y)=2+y^{1/2}$, $g(y)=1/(7+y)$. It is easy to
check that $f, g:[0,+\infty)\to [0,+\infty)$ are
continuous, and $f$ is non-decreasing in $y$, $g$ is non-increasing in
$y$.
For any $\lambda\in(0,1)$, $y_1,y_2 \in K$, we have
\begin{equation}
\begin{aligned}
&\int_0^1 G(t,s)\left[2+(\lambda y_1
 (s))^{1/2}+\frac{1}{7+\frac{1}{\lambda}y_2 (s)}\right]\nabla s\\
&\geq \lambda \int_0^1
 G(t,s)\big[2\lambda^{-1}+\lambda^{-\frac{1}{2}} y_1
 ^{1/2}(s)+\frac{1}{7+y_2 (s)}\big]\nabla s \\
&\geq\lambda \frac{2\lambda^{-1}+\lambda^{-\frac{1}{2}}y_1^{1/2}(s)
 +\frac{1}{7+y_2 (s)}}{2+y_1 ^{1/2}(s)+\frac{1}{7+y_2 (s)}}\int_0^1
 G(t,s)\big[2+y_1 ^{1/2}(s)+\frac{1}{7+y_2 (s)}\big]\nabla s.
\end{aligned}\label{e4.13}
\end{equation}
In \eqref{e4.13}, we note that
$$
\lambda<\varphi(\lambda,y_1,y_2)=\lambda
\frac{2\lambda^{-1}+\lambda^{-\frac{1}{2}}y_1^{1/2}+\frac{1}{7+y_2}}{2+y_1
^{1/2}+\frac{1}{7+y_2}}<1.
$$
 For any $\lambda\in (0,1)$, by
means of some calculations, we can obtain that $\varphi$ is
non-increasing in $y_1$ for fixed $y_2$ and nondecreasing in $y_2$
for fixed $y_1$.

In the following, it suffices to verify that the condition (E5)
of Theorem \ref{thm4.1} is satisfied. Some direct calculations show that
$g(t)=\frac{1}{1+\sigma(1)}=\frac{1}{3}$, and
\begin{align*}
\int_{0}^{1}G(s,s)\nabla s
&=4+\sum_{n=0}^{\infty}2^{-1-2n}-\sum_{n=0}^{\infty}2^{-3-3n}
 -\sum_{n=0}^{\infty}2^{-2n}+\sum_{n=0}^{\infty}2^{-1-3n}\\  
&=\frac{79}{21}>0.
\end{align*}
From Lemma \ref{lem4.2} it follows that that
$$
\frac{79}{63}\leq\int_0^1 G(t,s)\nabla s\leq \frac{79}{21},\quad
t\in [0,\sigma(1)]_{\mathbb{T}}.
$$ 
Since $f(1)+g(1)=25/8$, we choose $h=1$, $t_0=2^{-9}$, it is easy to 
check that
\begin{align*}
0.01
&<\frac{79}{63}\cdot\frac{25}{8}\leq \int_0^1
G(t,s)[f(1)+g(1)]\nabla s \\ 
&\leq \frac{79}{21}\cdot\frac{25}{8}
 <\frac{2(2^{-9})^{-1}+(2^{-9})^{-1/2}(2^9)^{1/2}
+\frac{1}{7+2^{-9}}}
{2+(2^{-9})^{-1/2}+\frac{1}{7+2^{-9}}}.
\end{align*}
This implies that (E5) of Theorem \ref{thm4.1} holds. The proof
is complete.
\end{example}

\subsection*{Acknowledgments}
The research was supported by the National Natural Science Foundation of
China, Tian Yuan Foundation (11226119), the Scientific and
Technological Innovation Programs of Higher Education Institutions
in Shanxi, and the Youth Science Foundation of Shanxi Province
(2013021002-1).

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\end{document}