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Some new results for single-valued and multi-valued mixed monotone operators of Rhoades type

Fixed Point Theory and Applications20132013:73

https://doi.org/10.1186/1687-1812-2013-73

• Accepted: 1 March 2013
• Published:

Abstract

In (2008), Zhang proved the existence of fixed points of mixed monotone operators along with certain convexity and concavity conditions. In this paper, mixed monotone single-valued and multi-valued operators of Rhoades type are defined and two fixed point theorems are proved.

MSC:47H10, 47H07.

Keywords

• mixed monotone operator
• multi-valued
• increasing inward mappings
• ${\mathcal{L}}^{″}$-function

1 Introduction and preliminaries

In (1987), mixed monotone operators were introduced by Guo and Lakshmikantham . Then many authors studied them in Banach spaces and obtained lots of interesting results (see [2, 3] and ).

On the other hand, in (2001), Rhoades  introduced a new fixed point theorem as a generalization of Banach fixed point theorem.

Let $\left(X,d\right)$ be a complete metric space. Suppose that $T:X\to X$ is a single-valued mapping that satisfies
$d\left(Tx,Ty\right)\le d\left(x,y\right)-\psi \left(d\left(x,y\right)\right)$
(1)

for each $x,y\in X$, where $\psi :\left[0,+\mathrm{\infty }\right)\to \left[0,+\mathrm{\infty }\right)$ is continuous, nondecreasing and ${\psi }^{-1}\left(0\right)=\left\{0\right\}$ (i.e., weakly contractive mappings). Then T has a fixed point.

In this paper, a weak mixed monotone single-valued and multi-valued operator of Rhoades type is defined. Then two fixed point theorems for this kind of operators are proved.

Let E be a real Banach space. The zero element of E is denoted by θ. A subset P of E is called a cone if and only if:

• P is closed, nonempty and $P\ne \left\{\theta \right\}$,

• $a,b\in \mathsf{R}$, $a,b\ge 0$ and $x,y\in P$ imply that $ax+by\in P$,

• $x\in P$ and $-x\in P$ imply that $x=\theta$.

Given a cone $P\subset E$, a partial ordering ≤ with respect to P is defined by $x\le y$ if and only if $y-x\in P$. We write $x to indicate that $x\le y$ but $x\ne y$, while $x\ll y$ stands for $y-x\in intP$, where intP denotes the interior of P. The cone P is called normal if there exists a number $K>0$ such that $\theta \le x\le y$ implies $\parallel x\parallel \le K\parallel y\parallel$ for every $x,y\in E$. The least positive number satisfying this is called the normal constant of P.

Assume that ${u}_{0},{v}_{0}\in E$ and ${u}_{0}\le {v}_{0}$. The set $\left\{x\in E:{u}_{0}\le x\le {v}_{0}\right\}$ is denoted by $\left[{u}_{0},{v}_{0}\right]$.

Now, we recall the following definitions from [2, 3].

Definition 1.1 Let P be a cone of a real Banach space E. Suppose that $D\subset P$ and $\alpha \in \left(-\mathrm{\infty },+\mathrm{\infty }\right)$. An operator $A:D\to D$ is said to be α-convex (α-concave) if it satisfies $A\left(tx\right)\le {t}^{\alpha }Ax$ ($A\left(tx\right)\ge {t}^{\alpha }Ax$) for $\left(t,x\right)\in \left(0,1\right)×D$.

Definition 1.2 Let E be an ordered Banach space and $D\subset E$. An operator is called mixed monotone on $D×D$ if $A:D×D\to E$ and $A\left({x}_{1},{y}_{1}\right)\le A\left({x}_{2},{y}_{2}\right)$ for any ${x}_{1},{x}_{2},{y}_{1},{y}_{2}\in D$, where ${x}_{1}\le {x}_{2}$ and ${y}_{2}\ge {y}_{1}$. Also, ${x}^{\ast }\in D$ is called a fixed point of A if $A\left({x}^{\ast },{x}^{\ast }\right)={x}^{\ast }$.

Let $\mathcal{C}\left(E\right)$ be a collection of all closed subsets of E.

Definition 1.3 For two subsets X, Y of E, we write

• $X⪯Y$ if for all $x\in X$, there exists $y\in Y$ such that $x\le y$,

• $x\prec X$ if there exists $z\in X$ such that $x\ll z$,

• $X\prec x$ if for all $z\in X$, $z\ll x$.

Definition 1.4 Let D be a nonempty subset of E. $T:D\to \mathcal{C}\left(E\right)$ is called increasing (decreasing) upward if $u,v\in D$, $u\le v$ and $x\in T\left(u\right)$ imply there exists $y\in T\left(v\right)$ such that $x\le y$ ($x\ge y$). Similarly, $T:D\to \mathcal{C}\left(E\right)$ is called increasing (decreasing) downward if $u,v\in D$, $u\le v$ and $y\in T\left(v\right)$ imply there exists $x\in T\left(u\right)$ such that $x\le y$ ($x\ge y$). T is called increasing (decreasing) if T is an increasing (decreasing) upward and downward.

Definition 1.5 Let D be a nonempty subset of E. A multi-valued operator $T:D×D\to \mathcal{C}\left(E\right)$ is said to be mixed monotone upward if $T\left(x,y\right)$ is increasing upward in x and decreasing upward in y, i.e.,

(A1) for each $y\in D$ and any ${x}_{1},{x}_{2}\in D$ with ${x}_{1}\le {x}_{2}$, if ${u}_{1}\in T\left({x}_{1},y\right)$, then there exists a ${u}_{2}\in T\left({x}_{2},y\right)$ such that ${u}_{1}\le {u}_{2}$;

(A2) for each $x\in D$ and any ${y}_{1},{y}_{2}\in D$ with ${y}_{1}\le {y}_{2}$, if ${v}_{1}\in T\left(x,{y}_{1}\right)$, then there exists a ${v}_{2}\in T\left(x,{y}_{2}\right)$ such that ${v}_{1}\ge {v}_{2}$.

Definition 1.6 ${x}^{\ast }\in D$ is called a fixed point of T if ${x}^{\ast }\in T\left({x}^{\ast },{x}^{\ast }\right)$.

Definition 1.7 

A function $\mathrm{\Psi }:\left[0,1\right)×P×P×E\to E$ is called an ${\mathcal{L}}^{″}$-function if $\mathrm{\Psi }\left(t,x,y,0\right)=0$, $\mathrm{\Psi }\left(t,x,y,s\right)\gg 0$ for $s\gg 0$, and $\mathrm{\Psi }\left(t,x,y,z\right) for all $\left(t,x,y,z\right)\in \left[0,1\right)×P×P×E$.

In 2011, Khojasteh and Razani  extended the results given by Zhang . Also, in 2011 Khojasteh and Razani  introduced the concept of integral with respect to a cone. We recall the following definitions and lemmas of cone integration and refer to [11, 12] for their proofs.

Definition 1.8 

Suppose that P is a cone in E. Let $a,b\in E$ and $a. Define
(2)
and
(3)

Definition 1.9 

The set $\left\{a={x}_{0},{x}_{1},\dots ,{x}_{n}=b\right\}$ is called a partition for $\left[a,b\right]$ if and only if the intervals ${\left\{\left[{x}_{i-1},{x}_{i}\right)\right\}}_{i=1}^{n}$ are pairwise disjoint and $\left[a,b\right]=\left\{{\bigcup }_{i=1}^{n}\left[{x}_{i-1},{x}_{i}\right)\right\}\cup \left\{b\right\}$. Denote $\mathcal{P}\left[a,b\right]$ as the collection of all partitions of $\left[a,b\right]$.

Definition 1.10 

For each partition Q of $\left[a,b\right]$ and each increasing function $\varphi :\left[a,b\right]\to E$, we define cone lower summation and cone upper summation as
${L}_{n}^{\mathrm{Con}}\left(\varphi ,Q\right)=\sum _{i=0}^{n-1}\varphi \left({x}_{i}\right)\parallel {x}_{i}-{x}_{i+1}\parallel$
(4)
and
${U}_{n}^{\mathrm{Con}}\left(\varphi ,Q\right)=\sum _{i=0}^{n-1}\varphi \left({x}_{i+1}\right)\parallel {x}_{i}-{x}_{i+1}\parallel ,$
(5)

respectively. Also, we denote $\parallel \mathrm{\Delta }\left(Q\right)\parallel =sup\left\{\parallel {x}_{i}-{x}_{i-1}\parallel ,{x}_{i}\in Q\right\}$.

Definition 1.11 

Suppose that P is a cone in E. $\varphi :\left[a,b\right]\to E$ is called an integrable function on $\left[a,b\right]$ with respect to a cone P or, to put it simply, a cone integrable function if and only if for all partition Q of $\left[a,b\right]$,
$\underset{\parallel \mathrm{\Delta }\left(Q\right)\parallel \to 0}{lim}{L}_{n}^{\mathrm{Con}}\left(\varphi ,Q\right)={S}^{\mathrm{Con}}=\underset{\parallel \mathrm{\Delta }\left(Q\right)\parallel \to 0}{lim}{U}_{n}^{\mathrm{Con}}\left(\varphi ,Q\right),$

where ${S}^{\mathrm{Con}}$ must be unique.

We show the common value ${S}^{\mathrm{Con}}$ by
${\int }_{a}^{b}\varphi \left(x\right)\phantom{\rule{0.2em}{0ex}}{d}_{P}\left(x\right)\phantom{\rule{1em}{0ex}}\text{or to simplicity}\phantom{\rule{1em}{0ex}}{\int }_{a}^{b}\varphi \phantom{\rule{0.2em}{0ex}}{d}_{p}.$

We denote the set of all cone integrable functions $\varphi :\left[a,b\right]\to E$ by ${\mathcal{L}}^{1}\left(\left[a,b\right],E\right)$.

Lemma 1.1 

Let M be a subset of P. The following conditions hold:
1. (1)

If $\left[a,b\right]\subseteq \left[a,c\right]\subset M$, then ${\int }_{a}^{b}f\phantom{\rule{0.2em}{0ex}}{d}_{p}\le {\int }_{a}^{c}f\phantom{\rule{0.2em}{0ex}}{d}_{p}$ for $f\in {\mathcal{L}}^{1}\left(M,P\right)$.

2. (2)

${\int }_{a}^{b}\left(\alpha f+\beta g\right)\phantom{\rule{0.2em}{0ex}}{d}_{p}=\alpha {\int }_{a}^{b}f\phantom{\rule{0.2em}{0ex}}{d}_{p}+\beta {\int }_{a}^{b}g\phantom{\rule{0.2em}{0ex}}{d}_{p}$ for $f,g\in {\mathcal{L}}^{1}\left(M,P\right)$ and $\alpha ,\beta \in \mathsf{R}$.

Remark 1.1 [, Remark 1.2]

Let P be a cone of E, and let $u\in P$. If for each $ϵ\in int\left(P\right)$, $0\le u\ll ϵ$, then $u=0$.

2 Main results

In this section, we introduce some new fixed point theorems in the class of mixed monotone operators. Due to this, the following definition is presented.

Definition 2.1 A mixed monotone operator $A:D×D\to E$ is said to be a Weak Mixed Monotone single-valued operator of Rhoades type (WM2R property for short) if
$A\left(tx,y\right)\le A\left(x,ty\right)-\mathrm{\Psi }\left(t,x,y,A\left(x,ty\right)\right)$
(6)

for all $\left(x,y\right)\in D×D$, where $\mathrm{\Psi }:\left[0,1\right)×P×P×E\to E$ is an ${\mathcal{L}}^{″}$-function.

Theorem 2.1 Let P be a cone of E, let S be a completely ordered closed subset of E with ${S}_{0}=S\mathrm{\setminus }\left\{\theta \right\}\subset intP$ and let $\lambda S\subset S$ for all $\lambda \in \left[0,1\right]$. Let ${u}_{0},{v}_{0}\in {S}_{0}$, $A:P×P\to E$ be a weak mixed monotone operator of Rhoades type with $A\left(\left(\left[\theta ,{v}_{0}\right]\cap S\right)×\left(\left[\theta ,{v}_{0}\right]\cap S\right)\right)\subset S$ satisfying the following conditions:
1. (I)

there exists ${r}_{0}>0$ such that ${u}_{0}\ge {r}_{0}{v}_{0}$,

2. (II)

$A\left({u}_{0},{v}_{0}\right)\ll {u}_{0}\ll {v}_{0}\ll A\left({v}_{0},{u}_{0}\right)$,

3. (III)

for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $A\left(u,v\right)\ll u\ll v$, there exists ${u}^{\prime }\in S$ such that $u\le A\left({u}^{\prime },v\right)\ll {u}^{\prime }\ll v$; similarly, for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $u\ll v\ll A\left(v,u\right)$, there exists ${v}^{\prime }\in S$ such that $u\ll {v}^{\prime }\ll A\left({v}^{\prime },u\right)\le v$.

Then A has at least one fixed point ${x}^{\ast }\in \left[{u}_{0},{v}_{0}\right]\cap S$.

Proof By the above condition (III), there exists ${u}_{1}\in S$ such that ${u}_{0}\le A\left({u}_{1},{v}_{0}\right)\ll {u}_{1}\ll {v}_{0}$. Then there exists ${v}_{1}\in S$ such that ${u}_{1}\ll {v}_{1}\ll A\left({v}_{1},{u}_{1}\right)\le {v}_{0}$. Likewise, there exists ${u}_{2}\in S$ such that ${u}_{1}\le A\left({u}_{2},{v}_{1}\right)\ll {u}_{2}\ll {v}_{1}$. Then there exists ${v}_{2}\in S$ such that ${u}_{2}\ll {v}_{2}\ll A\left({v}_{2},{u}_{2}\right)\le {v}_{1}$. In general, there exists ${u}_{n}\in S$ such that ${u}_{n-1}\le A\left({u}_{n},{v}_{n-1}\right)\ll {u}_{n}\ll {v}_{n-1}$. Then there exists ${v}_{n}\in S$ such that ${u}_{n}\ll {v}_{n}\ll A\left({v}_{n},{u}_{n}\right)\le {v}_{n-1}$ ($n=1,2,\dots$).

Take ${r}_{n}=sup\left\{r\in \left(0,1\right):{u}_{n}\ge r{v}_{n}\right\}$, thus $0<{r}_{0}<{r}_{1}<\cdots <{r}_{n}<{r}_{n+1}<\cdots <1$ and ${lim}_{n\to \mathrm{\infty }}{r}_{n}=sup\left\{{r}_{n}:n=0,1,2,\dots \right\}={r}^{\ast }\in \left(0,1\right]$. Since ${r}_{n+1}>{r}_{n}=sup\left\{r\in \left(0,1\right):{u}_{n}\ge r{v}_{n}\right\}$, thus ${u}_{n}\ngeqq {r}_{n+1}{v}_{n}$. In addition, S is completely ordered and $\lambda S\subset S$ for all $\lambda \in \left[0,1\right]$, then ${u}_{n}<{r}_{n+1}{v}_{n}$. Now, one can prove ${r}^{\ast }=1$. Otherwise, ${r}^{\ast }\in \left(0,1\right)$.

Since ${u}_{n}<{r}_{n+1}{v}_{n}$ and ${r}_{n+1}<{r}^{\ast }$, hence ${u}_{n}<{r}^{\ast }{v}_{n}$, and we have
$\begin{array}{rl}A\left({u}_{n+1},{v}_{n+1}\right)& \le A\left(\frac{1}{{r}^{\ast }}{u}_{n+1},{r}^{\ast }{v}_{n+1}\right)\\ \le A\left({u}_{n+1},{v}_{n+1}\right)-\mathrm{\Psi }\left({r}^{\ast },\frac{1}{{r}^{\ast }}{u}_{n+1},{v}_{n+1},A\left({u}_{n+1},{v}_{n+1}\right)\right)\\
(7)
which is a contradiction. Thus, ${r}^{\ast }=1$. Let $ϵ\gg 0$ be given. Choose $\delta >0$ such that $ϵ+{N}_{\delta }\left(0\right)\subseteq P$, where ${N}_{\delta }\left(0\right)=\left\{y\in E:\parallel y\parallel <\delta \right\}$. Since ${r}_{n}\to 1$, one can choose a natural number ${N}_{1}$ such that $\left(1-{r}_{n}\right){v}_{1}\in {N}_{\delta }\left(0\right)$ for all $n\ge {N}_{1}$. Therefore $\left(1-{r}_{n}\right){v}_{1}\ll ϵ$. Also, ${v}_{n}\le {v}_{1}$ and
$0<{v}_{n}-{u}_{n}\le \left(1-{r}_{n}\right){v}_{n}\le \left(1-{r}_{n}\right){v}_{1}\ll ϵ.$
(8)

By Remark 1.1, ${lim}_{n\to \mathrm{\infty }}{u}_{n}={lim}_{n\to \mathrm{\infty }}{v}_{n}$.

For all $n,p\ge 1$, applying the same argument, we have
$0<{v}_{n}-{v}_{n+p}\le {v}_{n}-{u}_{n}\ll ϵ.$
(9)
Also,
$0<{u}_{n+p}-{u}_{n}\le {v}_{n}-{u}_{n}\ll ϵ.$
(10)

Hence, $\left\{{u}_{n}\right\}$ and $\left\{{v}_{n}\right\}$ are Cauchy sequences in E, then there exist ${u}^{\ast },{v}^{\ast }\in E$ such that ${u}_{n}\to {u}^{\ast }$, ${v}_{n}\to {v}^{\ast }$ ($n\to \mathrm{\infty }$) and ${u}^{\ast }={v}^{\ast }$. Write ${x}^{\ast }={u}^{\ast }={v}^{\ast }$.

It is easy to see ${u}_{0}\le {u}_{n}\le {u}^{\ast }\le {v}_{n}\le {v}_{0}$ for all $n=1,2,\dots$ . In addition, S is closed, then ${u}^{\ast }\in \left[{u}_{n},{v}_{n}\right]\cap S\subset \left[{u}_{0},{v}_{0}\right]\cap S$ ($n=0,1,2,\dots$).

Finally, by the mixed monotone property of A,
${u}_{n-1}\le A\left({u}_{n},{v}_{n}\right)\le A\left({x}^{\ast },{x}^{\ast }\right)\le A\left({u}_{n},{v}_{n}\right)\le {u}_{n-1}.$
(11)

On taking limit on both sides of (11), when $n\to \mathrm{\infty }$, we have $A\left({x}^{\ast },{x}^{\ast }\right)={x}^{\ast }$. This means ${x}^{\ast }$ is a fixed point of A in $\left[{u}_{0},{v}_{0}\right]\cap S$. □

Corollary 2.1 Let P be a cone of E, let S be a completely ordered closed subset of E with ${S}_{0}=S\mathrm{\setminus }\left\{\theta \right\}\subset intP$ and let $\lambda S\subset S$ for all $\lambda \in \left[0,1\right]$. Let ${u}_{0},{v}_{0}\in {S}_{0}$, $A:P×P\to E$ satisfy
${\int }_{y}^{tx}\varphi \phantom{\rule{0.2em}{0ex}}{d}_{P}\le {\int }_{ty}^{x}\varphi \phantom{\rule{0.2em}{0ex}}{d}_{P}-\mathrm{\Psi }\left(t,x,y,{\int }_{ty}^{x}\varphi \phantom{\rule{0.2em}{0ex}}{d}_{P}\right)$
(12)
for all $\left(x,y\right)\in D×D$, where $\mathrm{\Psi }:\left[0,1\right)×P×P×E\to E$ is an ${\mathcal{L}}^{″}$-function, and let $\varphi :P\to P$ be a non-vanishing, cone integrable mapping on each $\left[a,b\right]\subset P$ such that for each $ϵ\gg 0$, ${\int }_{0}^{ϵ}\varphi \phantom{\rule{0.2em}{0ex}}{d}_{p}\gg 0$ and the mapping $\theta \left(x\right)={\int }_{0}^{x}\varphi \phantom{\rule{0.2em}{0ex}}{d}_{P}$ for $\left(x\ge 0\right)$ has a continuous inverse at zero. Also, $A\left(\left(\left[\theta ,{v}_{0}\right]\cap S\right)×\left(\left[\theta ,{v}_{0}\right]\cap S\right)\right)\subset S$ satisfies the following conditions:
1. (I)

there exists ${r}_{0}>0$ such that ${u}_{0}\ge {r}_{0}{v}_{0}$,

2. (II)

$A\left({u}_{0},{v}_{0}\right)\ll {u}_{0}\ll {v}_{0}\ll A\left({v}_{0},{u}_{0}\right)$,

3. (III)

for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $A\left(u,v\right)\ll u\ll v$, there exists ${u}^{\prime }\in S$ such that $u\le A\left({u}^{\prime },v\right)\ll {u}^{\prime }\ll v$; similarly, for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $u\ll v\ll A\left(v,u\right)$, there exists ${v}^{\prime }\in S$ such that $u\ll {v}^{\prime }\ll A\left({v}^{\prime },u\right)\le v$.

Then A has at least one fixed point ${x}^{\ast }\in \left[{u}_{0},{v}_{0}\right]\cap S$.

Proof

Define
$A\left(x,y\right)={\int }_{y}^{x}\varphi \phantom{\rule{0.2em}{0ex}}{d}_{P}.$

A is a mixed monotone operator, and one can easily see that all conditions of Theorem 2.1 hold. Thus we obtain the desired result. □

3 M3R property

In this section, we introduce a new fixed point theorem in the class of multi-valued mixed monotone operators. Due to this, the following definition is given.

Definition 3.1 A mixed monotone operator $T:D×D\to \mathcal{C}\left(E\right)$ is said to be a Mixed Monotone Multi-valued operator of Rhoades type (M3R property for short) if
$T\left(tx,y\right)⪯T\left(x,ty\right)-\mathrm{\Psi }\left(t,x,y,T\left(tx,y\right)\right)$
(13)

for each $\left(x,y\right)\in D×D$, where $\mathrm{\Psi }:\left[0,1\right)×P×P×E\to E$ is an ${\mathcal{L}}^{″}$-function.

Theorem 3.1 Let P be a cone of E, let S be a completely ordered closed subset of E with ${S}_{0}=S\mathrm{\setminus }\left\{\theta \right\}\subset intP$ and let $\lambda S\subset S$ for all $\lambda \in \left[0,1\right]$. Let ${u}_{0},{v}_{0}\in {S}_{0}$, $T:P×P\to \mathcal{C}\left(E\right)$ be a mixed monotone multi-valued operator of Rhoades type with $T\left(\left(\left[\theta ,{v}_{0}\right]\cap S\right)×\left(\left[\theta ,{v}_{0}\right]\cap S\right)\right)\subset S$ satisfying the following conditions:
1. (I)

there exists ${r}_{0}>0$ such that ${u}_{0}\ge {r}_{0}{v}_{0}$,

2. (II)

$T\left({u}_{0},{v}_{0}\right)\prec {u}_{0}\ll {v}_{0}\prec T\left({v}_{0},{u}_{0}\right)$,

3. (III)

for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $T\left(u,v\right)\prec u\ll v$, there exists ${u}^{\prime }\in S$ such that $u⪯T\left({u}^{\prime },v\right)\prec {u}^{\prime }\ll v$; similarly, for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $u\ll v\prec T\left(v,u\right)$, there exists ${v}^{\prime }\in S$ such that $u\ll {v}^{\prime }\prec T\left({v}^{\prime },u\right)⪯v$.

Then T has at least one fixed point ${x}^{\ast }\in \left[{u}_{0},{v}_{0}\right]\cap S$.

Proof By the above condition (III), there exists ${u}_{1}\in S$ such that ${u}_{0}⪯T\left({u}_{1},{v}_{0}\right)\prec {u}_{1}\ll {v}_{0}$. Then there exists ${v}_{1}\in S$ such that ${u}_{1}\ll {v}_{1}\prec T\left({v}_{1},{u}_{1}\right)⪯{v}_{0}$. Likewise, there exists ${u}_{2}\in S$ such that ${u}_{1}⪯T\left({u}_{2},{v}_{1}\right)\prec {u}_{2}\ll {v}_{1}$. Then there exists ${v}_{2}\in S$ such that ${u}_{2}\ll {v}_{2}\prec T\left({v}_{2},{u}_{2}\right)\le {v}_{1}$. In general, there exists ${u}_{n}\in S$ such that ${u}_{n-1}⪯T\left({u}_{n},{v}_{n-1}\right)\prec {u}_{n}\ll {v}_{n-1}$. Then there exists ${v}_{n}\in S$ such that ${u}_{n}\ll {v}_{n}\prec T\left({v}_{n},{u}_{n}\right)⪯{v}_{n-1}$ ($n=1,2,\dots$).

Take ${r}_{n}=sup\left\{r\in \left(0,1\right):{u}_{n}\ge r{v}_{n}\right\}$, thus $0<{r}_{0}<{r}_{1}<\cdots <{r}_{n}<{r}_{n+1}<\cdots <1$, and ${lim}_{n\to \mathrm{\infty }}{r}_{n}=sup\left\{{r}_{n}:n=0,1,2,\dots \right\}={r}^{\ast }\in \left(0,1\right]$. Since ${r}_{n+1}>{r}_{n}=sup\left\{r\in \left(0,1\right):{u}_{n}\ge r{v}_{n}\right\}$, thus ${u}_{n}\ngeqq {r}_{n+1}{v}_{n}$. In addition, S is completely ordered and $\lambda S\subset S$ for all $\lambda \in \left[0,1\right]$, then ${u}_{n}<{r}_{n+1}{v}_{n}$. Now, one can prove ${r}^{\ast }=1$. Otherwise, ${r}^{\ast }\in \left(0,1\right)$. We claim
$T\left({u}_{n+1},{v}_{n+1}\right)⪯T\left(\left(1/{r}^{\ast }\right){u}_{n+1},{r}^{\ast }{v}_{n+1}\right).$
(14)

Suppose that $x\in T\left({u}_{n+1},{v}_{n+1}\right)$ is arbitrary. We have ${u}_{n+1}\le \left(1/{r}^{\ast }\right){u}_{n+1}$. If ${x}_{1}={u}_{n+1}$, ${x}_{2}=\left(1/{r}^{\ast }\right){u}_{n+1}$ and $y={v}_{n+1}$, then by (A1) of Definition 1.5, there exists $z\in T\left(\left(1/{r}^{\ast }\right){u}_{n+1},{v}_{n+1}\right)$ such that $x\le z$. Thus, $T\left({u}_{n+1},{v}_{n+1}\right)⪯T\left(\left(1/{r}^{\ast }\right){u}_{n+1},{v}_{n+1}\right)$.

Also, if ${y}_{1}={r}^{\ast }{v}_{n+1}$, ${y}_{2}={v}_{n+1}$ and $x=\left(1/{r}^{\ast }\right){u}_{n+1}$, then for $w\in T\left(\left(1/{r}^{\ast }\right){u}_{n+1},{r}^{\ast }{v}_{n+1}\right)$, there exists $h\in T\left(\left(1/{r}^{\ast }\right){u}_{n+1},{v}_{n+1}\right)$ such that $w\ge h$. It means that
$T\left(\left(1/{r}^{\ast }\right){u}_{n+1},{v}_{n+1}\right)⪯T\left(\left(1/{r}^{\ast }\right){u}_{n+1},{r}^{\ast }{v}_{n+1}\right).$
(15)
Thus,
$\begin{array}{rl}T\left({u}_{n+1},{v}_{n+1}\right)& ⪯T\left(\left(1/{r}^{\ast }\right){u}_{n+1},{r}^{\ast }{v}_{n+1}\right)\\ ⪯T\left({u}_{n+1},{v}_{n+1}\right)-\mathrm{\Psi }\left(\frac{1}{{r}^{\ast }},{u}_{n+1},{r}^{\ast }{v}_{n+1},T\left({u}_{n+1},{v}_{n+1}\right)\right)\\ \prec T\left({u}_{n+1},{v}_{n+1}\right),\end{array}$
(16)
and this is a contradiction. Therefore, ${r}^{\ast }=1$. Let $ϵ\gg 0$ be given. Choose $\delta >0$ such that $ϵ+{N}_{\delta }\left(0\right)\subseteq P$, where ${N}_{\delta }\left(0\right)=\left\{y\in E:\parallel y\parallel <\delta \right\}$. Since ${r}_{n}\to 1$, one can choose a natural number ${N}_{1}$ such that $\left(1-{r}_{n}\right){v}_{1}\in {N}_{\delta }\left(0\right)$ for all $n\ge {N}_{1}$. Therefore $\left(1-{r}_{n}\right){v}_{1}\ll ϵ$. Also, ${v}_{n}\le {v}_{1}$ and
$0<{v}_{n}-{u}_{n}\le \left(1-{r}_{n}\right){v}_{n}\le \left(1-{r}_{n}\right){v}_{1}\ll ϵ.$
(17)

By Remark 1.1, ${lim}_{n\to \mathrm{\infty }}{u}_{n}={lim}_{n\to \mathrm{\infty }}{v}_{n}$.

For all $n,p\ge 1$, applying the same argument, we have
$0<{v}_{n}-{v}_{n+p}\le {v}_{n}-{u}_{n}\ll ϵ.$
(18)
Also,
$0<{u}_{n+p}-{u}_{n}\le {v}_{n}-{u}_{n}\ll ϵ.$
(19)

Hence, $\left\{{u}_{n}\right\}$ and $\left\{{v}_{n}\right\}$ are Cauchy sequences in E, then there exist ${u}^{\ast },{v}^{\ast }\in E$ such that ${u}_{n}\to {u}^{\ast }$, ${v}_{n}\to {v}^{\ast }$ ($n\to \mathrm{\infty }$) and ${u}^{\ast }={v}^{\ast }$. Write ${x}^{\ast }={u}^{\ast }={v}^{\ast }$.

It is easy to see that ${u}_{n}⪯T\left({u}_{n+1},{v}_{n+1}\right)⪯T\left({x}^{\ast },{x}^{\ast }\right)⪯T\left({v}_{n+1},{u}_{n+1}\right)⪯{v}_{n}$ for all $n=1,2,\dots$ . Thus, there exists ${z}_{n}\in T\left({x}^{\ast },{x}^{\ast }\right)$ such that ${u}_{n}\le {z}_{n}\le {v}_{n}$. By taking limit on both sides of (17),
$0<{z}_{n}-{u}_{n}\le \left(1-{r}_{n}\right){v}_{n}\le \left(1-{r}_{n}\right){v}_{1}\ll ϵ.$
(20)
So, ${z}_{n}\to {x}^{\ast }$. Since T has closed values, then ${x}^{\ast }\in T\left({x}^{\ast },{x}^{\ast }\right)$ and
${x}^{\ast }\in \left[{u}_{n},{v}_{n}\right]\cap S\subset \left[{u}_{0},{v}_{0}\right]\cap S.$

□

Remark 3.1 One can see easily that Theorem 2.1 should be included as a corollary of Theorem 3.1.

Example 3.1 Let $E=\mathsf{R}$, $P=\left[0,+\mathrm{\infty }\right)$ and $S=P$. Then ${S}_{0}=int\left(P\right)=\left(0,+\mathrm{\infty }\right)$.

Define $A:\left[0,+\mathrm{\infty }\right)×\left[0,+\mathrm{\infty }\right)\to \mathsf{R}$ as
$A\left(x,y\right)=\left\{\begin{array}{ll}\frac{x}{y},& \left(x,y\right)\ne \left(0,0\right),\\ 0,& \left(x,y\right)=\left(0,0\right).\end{array}$
A is a mixed monotone operator. Now suppose that $\mathrm{\Psi }:\left[0,1\right)×P×P×E\to E$ is as $\mathrm{\Psi }\left(t,x,y,s\right)=\left(1-{t}^{2}\right)s$. Then Ψ is an ${\mathcal{L}}^{″}$-function. Moreover,
$A\left(tx,y\right)\le A\left(x,ty\right)-\mathrm{\Psi }\left(t,x,y,A\left(x,ty\right)\right)$
for each $x,y\in {S}_{0}$. Also, by taking ${u}_{0}=\frac{1}{2}$, ${v}_{0}=\frac{3}{2}$ and ${r}_{0}=\frac{1}{4}$, we have
1. (I)

${u}_{0}\ge {r}_{0}{v}_{0}$,

2. (II)

$A\left({u}_{0},{v}_{0}\right)=\frac{1}{3}\ll {u}_{0}\ll {v}_{0}\ll A\left({v}_{0},{u}_{0}\right)=3$,

3. (III)

for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $A\left(u,v\right)\ll u\ll v$, there exists ${u}^{\prime }\in S$ such that $u\le A\left({u}^{\prime },v\right)\ll {u}^{\prime }\ll v$; similarly, for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $u\ll v\ll A\left(v,u\right)$, there exists ${v}^{\prime }\in S$ such that $u\ll {v}^{\prime }\ll A\left({v}^{\prime },u\right)\le v$.

For further explanation on (III), since $A\left({u}_{0},{v}_{0}\right)=\frac{1}{3}\ll {u}_{0}\ll {v}_{0}$, by (III) there exists ${u}_{1}\in S$ such that ${u}_{0}\ll A\left({u}_{1},{v}_{0}\right)\ll {u}_{1}\ll {v}_{0}$. It means that $\frac{1}{2}\ll \frac{{u}_{1}}{\frac{3}{2}}\ll {u}_{1}\ll \frac{3}{2}$. Thus ${u}_{1}$ must be greater than $\frac{3}{4}$. Therefore we can set ${u}_{1}=\frac{\frac{3}{4}+1}{2}$. Similarly, since $\frac{7}{8}={u}_{1}\ll {v}_{0}=\frac{3}{2}\ll A\left({v}_{0},{u}_{1}\right)=\frac{12}{7}$, thus by (III) there exists ${v}_{1}\in S$ such that ${u}_{1}\ll {v}_{1}\ll A\left({v}_{1},{u}_{1}\right)\le {v}_{0}$. It means that ${v}_{1}$ must be less than $\frac{21}{16}$. We can set ${v}_{1}=\frac{\frac{21}{16}+1}{2}$. By the continuity of such ways, we can consider the following reflexive sequences:
${u}_{0}=\frac{1}{2},\phantom{\rule{2em}{0ex}}{v}_{0}=\frac{3}{2},\phantom{\rule{2em}{0ex}}{u}_{n}=\frac{{u}_{n-1}{v}_{n-1}+1}{2}\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}{v}_{n}=\frac{{v}_{n-1}{u}_{n}+1}{2},$
which satisfy (I), (II) and (III) (see Figure 1). Moreover, ${u}_{n}\to 1$ and ${v}_{n}\to 1$ and $A\left(1,1\right)=1$. Figure 1 A ( u 0 , v 0 ) ≪ u 0 ≪ u 1 ≪ ⋯ ≪ u n ≪ ⋯ ≪ 1 ≪ ⋯ ≪ v n ≪ ⋯ ≪ v 1 ≪ v 0 ≪ A ( v 0 , u 0 ) .

4 Application

The following result is given by Zhang  and is obtained by our main result.

Corollary 4.1 Let P be a normal cone of E, let S be a completely ordered closed subset of E with ${S}_{0}=S\mathrm{\setminus }\left\{\theta \right\}\subset intP$ and let $\lambda S\subset S$ for all $\lambda \in \left[0,1\right]$. Let ${u}_{0},{v}_{0}\in {S}_{0}$, $A:P×P\to E$ be a mixed monotone operator with $A\left(\left(\left[\theta ,{v}_{0}\right]\cap S\right)×\left(\left[\theta ,{v}_{0}\right]\cap S\right)\right)\subset S$ and $A\left({u}_{0},{v}_{0}\right)\ll {u}_{0}\ll {v}_{0}\ll A\left({v}_{0},{u}_{0}\right)$. Assume that there exists a function $\varphi :\left(0,1\right)×\left(\left[{u}_{0},{v}_{0}\right]\cap S\right)×\left(\left[{u}_{0},{v}_{0}\right]\cap S\right)\to \left(0,+\mathrm{\infty }\right)$ such that $A\left(tx,y\right)\le \varphi \left(t,x,y\right)A\left(x,ty\right)$, where $0<\varphi \left(t,x,x\right) for all $\left(t,x,y\right)\in \left(0,1\right)×\left(\left[{u}_{0},{v}_{0}\right]\cap S\right)×\left(\left[{u}_{0},{v}_{0}\right]\cap S\right)$. Suppose that
1. (I)

for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $A\left(u,v\right)\ll u\ll v$, there exists ${u}^{\prime }\in S$ such that $u\le A\left({u}^{\prime },v\right)\ll {u}^{\prime }\ll v$; similarly, for $u,v\in \left[{u}_{0},{v}_{0}\right]\cap S$ with $u\ll v\ll A\left(v,u\right)$, there exists ${v}^{\prime }\in S$ such that $u\ll {v}^{\prime }\ll A\left({v}^{\prime },u\right)\le v$.

2. (II)

there exists an element ${w}_{0}\in \left[{u}_{0},{v}_{0}\right]\cap S$ such that $\varphi \left(t,x,x\right)\le \varphi \left(t,{w}_{0},{w}_{0}\right)$ for all $\left(t,x\right)\in \left(0,1\right)×\left(\left[{u}_{0},{v}_{0}\right]\cap S\right)$, and ${lim}_{s\to {t}^{-}}\varphi \left(s,{w}_{0},{w}_{0}\right) for all $t\in \left(0,1\right)$.

Then A has at least one fixed point ${x}^{\ast }\in \left[{u}_{0},{v}_{0}\right]\cap S$.

Proof Set $\mathrm{\Psi }\left(t,x,y,z\right)=\left(1-\varphi \left(t,x,y\right)\right)z$. Then Ψ is an ${\mathcal{L}}^{″}$-function, and we have
$A\left(tx,y\right)\le \varphi \left(t,x,y\right)A\left(x,ty\right)=A\left(x,ty\right)-\mathrm{\Psi }\left(t,x,y,A\left(x,ty\right)\right).$

Thus, by Theorem 2.1 the desired result is obtained. □

Authors’ Affiliations

(1)
Department of Mathematics, Arak Branch, Islamic Azad University, Arak, Iran

References 