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Some new results for singlevalued and multivalued mixed monotone operators of Rhoades type
 Farshid Khojasteh^{1}Email author
https://doi.org/10.1186/16871812201373
© Khojasteh; licensee Springer. 2013
 Received: 11 May 2012
 Accepted: 1 March 2013
 Published: 28 March 2013
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 singlevalued and multivalued operators of Rhoades type are defined and two fixed point theorems are proved.
MSC:47H10, 47H07.
Keywords
 mixed monotone operator
 Rhoades type
 multivalued
 increasing inward mappings
 ${\mathcal{L}}^{\u2033}$function
1 Introduction and preliminaries
In (1987), mixed monotone operators were introduced by Guo and Lakshmikantham [1]. Then many authors studied them in Banach spaces and obtained lots of interesting results (see [2, 3] and [4–8]).
On the other hand, in (2001), Rhoades [9] introduced a new fixed point theorem as a generalization of Banach fixed point theorem.
Theorem 1.1 (Rhoades [9])
for each $x,y\in X$, where $\psi :[0,+\mathrm{\infty})\to [0,+\mathrm{\infty})$ is continuous, nondecreasing and ${\psi}^{1}(0)=\{0\}$ (i.e., weakly contractive mappings). Then T has a fixed point.
In this paper, a weak mixed monotone singlevalued and multivalued 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 $P\ne \{\theta \}$,

$a,b\in \mathsf{R}$, $a,b\ge 0$ 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 $yx\in P$. We write $x<y$ to indicate that $x\le y$ but $x\ne y$, while $x\ll y$ stands for $yx\in intP$, 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 $\{x\in E:{u}_{0}\le x\le {v}_{0}\}$ is denoted by $[{u}_{0},{v}_{0}]$.
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 $\alpha \in (\mathrm{\infty},+\mathrm{\infty})$. An operator $A:D\to D$ is said to be αconvex (αconcave) if it satisfies $A(tx)\le {t}^{\alpha}Ax$ ($A(tx)\ge {t}^{\alpha}Ax$) for $(t,x)\in (0,1)\times D$.
Definition 1.2 Let E be an ordered Banach space and $D\subset E$. An operator is called mixed monotone on $D\times D$ if $A:D\times D\to E$ and $A({x}_{1},{y}_{1})\le A({x}_{2},{y}_{2})$ 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 $A({x}^{\ast},{x}^{\ast})={x}^{\ast}$.
Let $\mathcal{C}(E)$ be a collection of all closed subsets of E.
Definition 1.3 For two subsets X, Y of E, we write

$X\u2aafY$ 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. $T:D\to \mathcal{C}(E)$ is called increasing (decreasing) upward if $u,v\in D$, $u\le v$ and $x\in T(u)$ imply there exists $y\in T(v)$ such that $x\le y$ ($x\ge y$). Similarly, $T:D\to \mathcal{C}(E)$ is called increasing (decreasing) downward if $u,v\in D$, $u\le v$ and $y\in T(v)$ imply there exists $x\in T(u)$ 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 multivalued operator $T:D\times D\to \mathcal{C}(E)$ is said to be mixed monotone upward if $T(x,y)$ is increasing upward in x and decreasing upward in y, i.e.,
(A_{1}) for each $y\in D$ and any ${x}_{1},{x}_{2}\in D$ with ${x}_{1}\le {x}_{2}$, if ${u}_{1}\in T({x}_{1},y)$, then there exists a ${u}_{2}\in T({x}_{2},y)$ such that ${u}_{1}\le {u}_{2}$;
(A_{2}) for each $x\in D$ and any ${y}_{1},{y}_{2}\in D$ with ${y}_{1}\le {y}_{2}$, if ${v}_{1}\in T(x,{y}_{1})$, then there exists a ${v}_{2}\in T(x,{y}_{2})$ such that ${v}_{1}\ge {v}_{2}$.
Definition 1.6 ${x}^{\ast}\in D$ is called a fixed point of T if ${x}^{\ast}\in T({x}^{\ast},{x}^{\ast})$.
Definition 1.7 [10]
A function $\mathrm{\Psi}:[0,1)\times P\times P\times E\to E$ is called an ${\mathcal{L}}^{\u2033}$function if $\mathrm{\Psi}(t,x,y,0)=0$, $\mathrm{\Psi}(t,x,y,s)\gg 0$ for $s\gg 0$, and $\mathrm{\Psi}(t,x,y,z)<z$ for all $(t,x,y,z)\in [0,1)\times P\times P\times E$.
In 2011, Khojasteh and Razani [10] extended the results given by Zhang [6]. Also, in 2011 Khojasteh and Razani [11] 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 [11]
Definition 1.9 [11]
The set $\{a={x}_{0},{x}_{1},\dots ,{x}_{n}=b\}$ is called a partition for $[a,b]$ if and only if the intervals ${\{[{x}_{i1},{x}_{i})\}}_{i=1}^{n}$ are pairwise disjoint and $[a,b]=\{{\bigcup}_{i=1}^{n}[{x}_{i1},{x}_{i})\}\cup \{b\}$. Denote $\mathcal{P}[a,b]$ as the collection of all partitions of $[a,b]$.
Definition 1.10 [12]
respectively. Also, we denote $\parallel \mathrm{\Delta}(Q)\parallel =sup\{\parallel {x}_{i}{x}_{i1}\parallel ,{x}_{i}\in Q\}$.
Definition 1.11 [12]
where ${S}^{\mathrm{Con}}$ must be unique.
We denote the set of all cone integrable functions $\varphi :[a,b]\to E$ by ${\mathcal{L}}^{1}([a,b],E)$.
Lemma 1.1 [11]
 (1)
If $[a,b]\subseteq [a,c]\subset M$, 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 $f\in {\mathcal{L}}^{1}(M,P)$.
 (2)
${\int}_{a}^{b}(\alpha f+\beta g)\phantom{\rule{0.2em}{0ex}}{d}_{p}=\alpha {\int}_{a}^{b}f\phantom{\rule{0.2em}{0ex}}{d}_{p}+\beta {\int}_{a}^{b}g\phantom{\rule{0.2em}{0ex}}{d}_{p}$ for $f,g\in {\mathcal{L}}^{1}(M,P)$ and $\alpha ,\beta \in \mathsf{R}$.
Remark 1.1 [[13], Remark 1.2]
Let P be a cone of E, and let $u\in P$. If for each $\u03f5\in int(P)$, $0\le u\ll \u03f5$, 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.
for all $(x,y)\in D\times D$, where $\mathrm{\Psi}:[0,1)\times P\times P\times E\to E$ is an ${\mathcal{L}}^{\u2033}$function.
 (I)
there exists ${r}_{0}>0$ such that ${u}_{0}\ge {r}_{0}{v}_{0}$,
 (II)
$A({u}_{0},{v}_{0})\ll {u}_{0}\ll {v}_{0}\ll A({v}_{0},{u}_{0})$,
 (III)
for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $A(u,v)\ll u\ll v$, there exists ${u}^{\prime}\in S$ such that $u\le A({u}^{\prime},v)\ll {u}^{\prime}\ll v$; similarly, for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $u\ll v\ll A(v,u)$, there exists ${v}^{\prime}\in S$ such that $u\ll {v}^{\prime}\ll A({v}^{\prime},u)\le v$.
Then A has at least one fixed point ${x}^{\ast}\in [{u}_{0},{v}_{0}]\cap S$.
Proof By the above condition (III), there exists ${u}_{1}\in S$ such that ${u}_{0}\le A({u}_{1},{v}_{0})\ll {u}_{1}\ll {v}_{0}$. Then there exists ${v}_{1}\in S$ such that ${u}_{1}\ll {v}_{1}\ll A({v}_{1},{u}_{1})\le {v}_{0}$. Likewise, there exists ${u}_{2}\in S$ such that ${u}_{1}\le A({u}_{2},{v}_{1})\ll {u}_{2}\ll {v}_{1}$. Then there exists ${v}_{2}\in S$ such that ${u}_{2}\ll {v}_{2}\ll A({v}_{2},{u}_{2})\le {v}_{1}$. In general, there exists ${u}_{n}\in S$ such that ${u}_{n1}\le A({u}_{n},{v}_{n1})\ll {u}_{n}\ll {v}_{n1}$. Then there exists ${v}_{n}\in S$ such that ${u}_{n}\ll {v}_{n}\ll A({v}_{n},{u}_{n})\le {v}_{n1}$ ($n=1,2,\dots $).
Take ${r}_{n}=sup\{r\in (0,1):{u}_{n}\ge r{v}_{n}\}$, thus $0<{r}_{0}<{r}_{1}<\cdots <{r}_{n}<{r}_{n+1}<\cdots <1$ and ${lim}_{n\to \mathrm{\infty}}{r}_{n}=sup\{{r}_{n}:n=0,1,2,\dots \}={r}^{\ast}\in (0,1]$. Since ${r}_{n+1}>{r}_{n}=sup\{r\in (0,1):{u}_{n}\ge r{v}_{n}\}$, thus ${u}_{n}\ngeqq {r}_{n+1}{v}_{n}$. In addition, S is completely ordered and $\lambda S\subset S$ for all $\lambda \in [0,1]$, then ${u}_{n}<{r}_{n+1}{v}_{n}$. Now, one can prove ${r}^{\ast}=1$. Otherwise, ${r}^{\ast}\in (0,1)$.
By Remark 1.1, ${lim}_{n\to \mathrm{\infty}}{u}_{n}={lim}_{n\to \mathrm{\infty}}{v}_{n}$.
Hence, $\{{u}_{n}\}$ and $\{{v}_{n}\}$ 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 ${u}^{\ast}\in [{u}_{n},{v}_{n}]\cap S\subset [{u}_{0},{v}_{0}]\cap S$ ($n=0,1,2,\dots $).
On taking limit on both sides of (11), when $n\to \mathrm{\infty}$, we have $A({x}^{\ast},{x}^{\ast})={x}^{\ast}$. This means ${x}^{\ast}$ is a fixed point of A in $[{u}_{0},{v}_{0}]\cap S$. □
 (I)
there exists ${r}_{0}>0$ such that ${u}_{0}\ge {r}_{0}{v}_{0}$,
 (II)
$A({u}_{0},{v}_{0})\ll {u}_{0}\ll {v}_{0}\ll A({v}_{0},{u}_{0})$,
 (III)
for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $A(u,v)\ll u\ll v$, there exists ${u}^{\prime}\in S$ such that $u\le A({u}^{\prime},v)\ll {u}^{\prime}\ll v$; similarly, for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $u\ll v\ll A(v,u)$, there exists ${v}^{\prime}\in S$ such that $u\ll {v}^{\prime}\ll A({v}^{\prime},u)\le v$.
Then A has at least one fixed point ${x}^{\ast}\in [{u}_{0},{v}_{0}]\cap S$.
Proof
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 M_{3}R property
In this section, we introduce a new fixed point theorem in the class of multivalued mixed monotone operators. Due to this, the following definition is given.
for each $(x,y)\in D\times D$, where $\mathrm{\Psi}:[0,1)\times P\times P\times E\to E$ is an ${\mathcal{L}}^{\u2033}$function.
 (I)
there exists ${r}_{0}>0$ such that ${u}_{0}\ge {r}_{0}{v}_{0}$,
 (II)
$T({u}_{0},{v}_{0})\prec {u}_{0}\ll {v}_{0}\prec T({v}_{0},{u}_{0})$,
 (III)
for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $T(u,v)\prec u\ll v$, there exists ${u}^{\prime}\in S$ such that $u\u2aafT({u}^{\prime},v)\prec {u}^{\prime}\ll v$; similarly, for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $u\ll v\prec T(v,u)$, there exists ${v}^{\prime}\in S$ such that $u\ll {v}^{\prime}\prec T({v}^{\prime},u)\u2aafv$.
Then T has at least one fixed point ${x}^{\ast}\in [{u}_{0},{v}_{0}]\cap S$.
Proof By the above condition (III), there exists ${u}_{1}\in S$ such that ${u}_{0}\u2aafT({u}_{1},{v}_{0})\prec {u}_{1}\ll {v}_{0}$. Then there exists ${v}_{1}\in S$ such that ${u}_{1}\ll {v}_{1}\prec T({v}_{1},{u}_{1})\u2aaf{v}_{0}$. Likewise, there exists ${u}_{2}\in S$ such that ${u}_{1}\u2aafT({u}_{2},{v}_{1})\prec {u}_{2}\ll {v}_{1}$. Then there exists ${v}_{2}\in S$ such that ${u}_{2}\ll {v}_{2}\prec T({v}_{2},{u}_{2})\le {v}_{1}$. In general, there exists ${u}_{n}\in S$ such that ${u}_{n1}\u2aafT({u}_{n},{v}_{n1})\prec {u}_{n}\ll {v}_{n1}$. Then there exists ${v}_{n}\in S$ such that ${u}_{n}\ll {v}_{n}\prec T({v}_{n},{u}_{n})\u2aaf{v}_{n1}$ ($n=1,2,\dots $).
Suppose that $x\in T({u}_{n+1},{v}_{n+1})$ is arbitrary. We have ${u}_{n+1}\le (1/{r}^{\ast}){u}_{n+1}$. If ${x}_{1}={u}_{n+1}$, ${x}_{2}=(1/{r}^{\ast}){u}_{n+1}$ and $y={v}_{n+1}$, then by (A_{1}) of Definition 1.5, there exists $z\in T((1/{r}^{\ast}){u}_{n+1},{v}_{n+1})$ such that $x\le z$. Thus, $T({u}_{n+1},{v}_{n+1})\u2aafT((1/{r}^{\ast}){u}_{n+1},{v}_{n+1})$.
By Remark 1.1, ${lim}_{n\to \mathrm{\infty}}{u}_{n}={lim}_{n\to \mathrm{\infty}}{v}_{n}$.
Hence, $\{{u}_{n}\}$ and $\{{v}_{n}\}$ 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 ${x}^{\ast}={u}^{\ast}={v}^{\ast}$.
□
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 $E=\mathsf{R}$, $P=[0,+\mathrm{\infty})$ and $S=P$. Then ${S}_{0}=int(P)=(0,+\mathrm{\infty})$.
 (I)
${u}_{0}\ge {r}_{0}{v}_{0}$,
 (II)
$A({u}_{0},{v}_{0})=\frac{1}{3}\ll {u}_{0}\ll {v}_{0}\ll A({v}_{0},{u}_{0})=3$,
 (III)
for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $A(u,v)\ll u\ll v$, there exists ${u}^{\prime}\in S$ such that $u\le A({u}^{\prime},v)\ll {u}^{\prime}\ll v$; similarly, for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $u\ll v\ll A(v,u)$, there exists ${v}^{\prime}\in S$ such that $u\ll {v}^{\prime}\ll A({v}^{\prime},u)\le v$.
4 Application
The following result is given by Zhang [6] and is obtained by our main result.
 (I)
for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $A(u,v)\ll u\ll v$, there exists ${u}^{\prime}\in S$ such that $u\le A({u}^{\prime},v)\ll {u}^{\prime}\ll v$; similarly, for $u,v\in [{u}_{0},{v}_{0}]\cap S$ with $u\ll v\ll A(v,u)$, there exists ${v}^{\prime}\in S$ such that $u\ll {v}^{\prime}\ll A({v}^{\prime},u)\le v$.
 (II)
there exists an element ${w}_{0}\in [{u}_{0},{v}_{0}]\cap S$ such that $\varphi (t,x,x)\le \varphi (t,{w}_{0},{w}_{0})$ for all $(t,x)\in (0,1)\times ([{u}_{0},{v}_{0}]\cap S)$, and ${lim}_{s\to {t}^{}}\varphi (s,{w}_{0},{w}_{0})<t$ for all $t\in (0,1)$.
Then A has at least one fixed point ${x}^{\ast}\in [{u}_{0},{v}_{0}]\cap S$.
Thus, by Theorem 2.1 the desired result is obtained. □
Declarations
Authors’ Affiliations
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