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

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.

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])

Let $(X,d)$ be a complete metric space. Suppose that $T:X\to X$ is a single-valued mapping that satisfies

$$d(Tx,Ty)\le d(x,y)-\psi (d(x,y))$$

(1)

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 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 $P\ne \{\theta \}$,

• $a,b\in \mathsf{R}$, $a,b\ge 0$ 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 $y-x\in P$. We write $x<y$ 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 $\{x\in E:u_0\le x\le v_0\}$ 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 α-convex (α-concave) 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

• $X⪯Y$ 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. $T:D\to \mathcal{C}(E)$ 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 multi-valued 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.,

(A₁) for each $y\in D$ and any $x_1,x_2\in D$ with $x_1\le x_2$, if $u_1\in T(x_1,y)$, then there exists a $u_2\in T(x_2,y)$ 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 $v_1\in T(x,y_1)$, 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 ${\mathcal{L}}^{″}$-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]

Suppose that P is a cone in E. Let $a,b\in E$ and $a<b$. Define

$$[a,b]:=\{x\in E:x=tb+(1-t)a\text{ for some }t\in [0,1]\}$$

(2)

and

$$[a,b):=\{x\in E:x=tb+(1-t)a\text{ for some }t\in [0,1)\}.$$

(3)

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 ${\{[x_{i-1},x_i)\}}_{i=1}^n$ are pairwise disjoint and $[a,b]=\{{\bigcup }_{i=1}^n[x_{i-1},x_i)\}\cup \{b\}$. Denote $\mathcal{P}[a,b]$ as the collection of all partitions of $[a,b]$.

Definition 1.10 [12]

For each partition Q of $[a,b]$ and each increasing function $\varphi :[a,b]\to E$, we define cone lower summation and cone upper summation as

$$L_n^{\mathrm{Con}}(\varphi ,Q)=\sum _{i=0}^{n-1}\varphi (x_i)\parallel x_i-x_{i+1}\parallel$$

(4)

and

$$U_n^{\mathrm{Con}}(\varphi ,Q)=\sum _{i=0}^{n-1}\varphi (x_{i+1})\parallel x_i-x_{i+1}\parallel ,$$

(5)

respectively. Also, we denote $\parallel \mathrm{\Delta }(Q)\parallel =sup\{\parallel x_i-x_{i-1}\parallel ,x_i\in Q\}$.

Definition 1.11 [12]

Suppose that P is a cone in E. $\varphi :[a,b]\to E$ is called an integrable function on $[a,b]$ with respect to a cone P or, to put it simply, a cone integrable function if and only if for all partition Q of $[a,b]$,

$$\underset{\parallel \mathrm{\Delta }(Q)\parallel \to 0}{lim}{L}_n^{\mathrm{Con}}(\varphi ,Q)=S^{\mathrm{Con}}=\underset{\parallel \mathrm{\Delta }(Q)\parallel \to 0}{lim}{U}_n^{\mathrm{Con}}(\varphi ,Q),$$

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

We show the common value $S^{\mathrm{Con}}$ by

$${\int }_a^b\varphi (x)d_P(x)\text{or to simplicity}{\int }_a^b\varphi d_p.$$

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]

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

• (1) If $[a,b]\subseteq [a,c]\subset M$, then ${\int }_a^{b}f{d}_p\le {\int }_a^{c}f{d}_p$ for $f\in {\mathcal{L}}^1(M,P)$.

• (2) ${\int }_a^b(\alpha f+\beta g)d_p=\alpha {\int }_a^{b}f{d}_p+\beta {\int }_a^{b}g{d}_p$ 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 $ϵ\in int(P)$, $0\le u\ll ϵ$, 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 $A:D\times D\to E$ is said to be a Weak Mixed Monotone single-valued operator of Rhoades type (WM₂R property for short) if

$$A(tx,y)\le A(x,ty)-\mathrm{\Psi }(t,x,y,A(x,ty))$$

(6)

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

Theorem 2.1 Let P be a cone of E, let S be a completely ordered closed subset of E with $S_0=S\mathrm{\setminus }\{\theta \}\subset intP$ and let $\lambda S\subset S$ for all $\lambda \in [0,1]$. Let $u_0,v_0\in S_0$, $A:P\times P\to E$ be a weak mixed monotone operator of Rhoades type with $A(([\theta ,v_0]\cap S)\times ([\theta ,v_0]\cap S))\subset S$ satisfying the following conditions:

• (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 $u_{n-1}\le A(u_n,v_{n-1})\ll u_n\ll v_{n-1}$. Then there exists $v_n\in S$ such that $u_n\ll v_n\ll A(v_n,u_n)\le v_{n-1}$ ($n=1,2,\dots$).

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, $r^{\ast }\in (0,1)$.

Since $u_n<r_{n+1}{v}_n$ and $r_{n+1}<r^{\ast }$, hence $u_n<r^{\ast }{v}_n$, and we have

$$\begin{array}{rl}A(u_{n+1},v_{n+1})& \le A(\frac{1}{r^{\ast }}{u}_{n+1},r^{\ast }{v}_{n+1})\\ \le A(u_{n+1},v_{n+1})-\mathrm{\Psi }(r^{\ast },\frac{1}{r^{\ast }}{u}_{n+1},v_{n+1},A(u_{n+1},v_{n+1}))\\ <A(u_{n+1},v_{n+1}),\end{array}$$

(7)

which is a contradiction. Thus, $r^{\ast }=1$. Let $ϵ\gg 0$ be given. Choose $\delta >0$ such that $ϵ+N_{\delta }(0)\subseteq P$, where $N_{\delta }(0)=\{y\in E:\parallel y\parallel <\delta \}$. Since $r_n\to 1$, one can choose a natural number $N_1$ such that $(1-r_n)v_1\in N_{\delta }(0)$ for all $n\ge N_1$. Therefore $(1-r_n)v_1\ll ϵ$. Also, $v_n\le v_1$ and

$$0<v_n-u_n\le (1-r_n)v_n\le (1-r_n)v_1\ll ϵ.$$

(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, $\{u_n\}$ 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 $u^{\ast }\in [u_n,v_n]\cap S\subset [u_0,v_0]\cap S$ ($n=0,1,2,\dots$).

Finally, by the mixed monotone property of A,

$$u_{n-1}\le A(u_n,v_n)\le A(x^{\ast },x^{\ast })\le A(u_n,v_n)\le u_{n-1}.$$

(11)

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 $[u_0,v_0]\cap S$. □

Corollary 2.1 Let P be a cone of E, let S be a completely ordered closed subset of E with $S_0=S\mathrm{\setminus }\{\theta \}\subset intP$ and let $\lambda S\subset S$ for all $\lambda \in [0,1]$. Let $u_0,v_0\in S_0$, $A:P\times P\to E$ satisfy

$${\int }_y^{tx}\varphi d_P\le {\int }_{ty}^x\varphi d_P-\mathrm{\Psi }(t,x,y,{\int }_{ty}^x\varphi d_P)$$

(12)

for all $(x,y)\in D\times D$, where $\mathrm{\Psi }:[0,1)\times P\times P\times E\to E$ is an ${\mathcal{L}}^{″}$-function, and let $\varphi :P\to P$ be a non-vanishing, cone integrable mapping on each $[a,b]\subset P$ such that for each $ϵ\gg 0$, ${\int }_0^{ϵ}\varphi d_p\gg 0$ and the mapping $\theta (x)={\int }_0^x\varphi d_P$ for $(x\ge 0)$ has a continuous inverse at zero. Also, $A(([\theta ,v_0]\cap S)\times ([\theta ,v_0]\cap S))\subset S$ satisfies the following conditions:

• (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

Define

$$A(x,y)={\int }_y^x\varphi d_P.$$

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 $T:D\times D\to \mathcal{C}(E)$ is said to be a Mixed Monotone Multi-valued operator of Rhoades type (M₃R property for short) if

$$T(tx,y)⪯T(x,ty)-\mathrm{\Psi }(t,x,y,T(tx,y))$$

(13)

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

Theorem 3.1 Let P be a cone of E, let S be a completely ordered closed subset of E with $S_0=S\mathrm{\setminus }\{\theta \}\subset intP$ and let $\lambda S\subset S$ for all $\lambda \in [0,1]$. Let $u_0,v_0\in S_0$, $T:P\times P\to \mathcal{C}(E)$ be a mixed monotone multi-valued operator of Rhoades type with $T(([\theta ,v_0]\cap S)\times ([\theta ,v_0]\cap S))\subset S$ satisfying the following conditions:

• (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 $u⪯T(u^{\prime },v)\prec u^{\prime }\ll v$; 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 $u\ll v^{\prime }\prec T(v^{\prime },u)⪯v$.

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 $u_0⪯T(u_1,v_0)\prec u_1\ll v_0$. Then there exists $v_1\in S$ such that $u_1\ll v_1\prec T(v_1,u_1)⪯v_0$. Likewise, there exists $u_2\in S$ such that $u_1⪯T(u_2,v_1)\prec u_2\ll v_1$. 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 $u_{n-1}⪯T(u_n,v_{n-1})\prec u_n\ll v_{n-1}$. Then there exists $v_n\in S$ such that $u_n\ll v_n\prec T(v_n,u_n)⪯v_{n-1}$ ($n=1,2,\dots$).

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, $r^{\ast }\in (0,1)$. We claim

$$T(u_{n+1},v_{n+1})⪯T((1/r^{\ast })u_{n+1},r^{\ast }{v}_{n+1}).$$

(14)

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 (A₁) of Definition 1.5, there exists $z\in T((1/r^{\ast })u_{n+1},v_{n+1})$ such that $x\le z$. Thus, $T(u_{n+1},v_{n+1})⪯T((1/r^{\ast })u_{n+1},v_{n+1})$.

Also, if $y_1=r^{\ast }{v}_{n+1}$, $y_2=v_{n+1}$ and $x=(1/r^{\ast })u_{n+1}$, then for $w\in T((1/r^{\ast })u_{n+1},r^{\ast }{v}_{n+1})$, there exists $h\in T((1/r^{\ast })u_{n+1},v_{n+1})$ such that $w\ge h$. It means that

$$T((1/r^{\ast })u_{n+1},v_{n+1})⪯T((1/r^{\ast })u_{n+1},r^{\ast }{v}_{n+1}).$$

(15)

Thus,

$$\begin{array}{rl}T(u_{n+1},v_{n+1})& ⪯T((1/r^{\ast })u_{n+1},r^{\ast }{v}_{n+1})\\ ⪯T(u_{n+1},v_{n+1})-\mathrm{\Psi }(\frac{1}{r^{\ast }},u_{n+1},r^{\ast }{v}_{n+1},T(u_{n+1},v_{n+1}))\\ \prec T(u_{n+1},v_{n+1}),\end{array}$$

(16)

and this is a contradiction. Therefore, $r^{\ast }=1$. Let $ϵ\gg 0$ be given. Choose $\delta >0$ such that $ϵ+N_{\delta }(0)\subseteq P$, where $N_{\delta }(0)=\{y\in E:\parallel y\parallel <\delta \}$. Since $r_n\to 1$, one can choose a natural number $N_1$ such that $(1-r_n)v_1\in N_{\delta }(0)$ for all $n\ge N_1$. Therefore $(1-r_n)v_1\ll ϵ$. Also, $v_n\le v_1$ and

$$0<v_n-u_n\le (1-r_n)v_n\le (1-r_n)v_1\ll ϵ.$$

(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, $\{u_n\}$ 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 that $u_n⪯T(u_{n+1},v_{n+1})⪯T(x^{\ast },x^{\ast })⪯T(v_{n+1},u_{n+1})⪯v_n$ for all $n=1,2,\dots$ . Thus, there exists $z_n\in T(x^{\ast },x^{\ast })$ such that $u_n\le z_n\le v_n$. By taking limit on both sides of (17),

$$0<z_n-u_n\le (1-r_n)v_n\le (1-r_n)v_1\ll ϵ.$$

(20)

So, $z_n\to x^{\ast }$. Since T has closed values, then $x^{\ast }\in T(x^{\ast },x^{\ast })$ and

$$x^{\ast }\in [u_n,v_n]\cap S\subset [u_0,v_0]\cap S.$$

 □

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 $E=\mathsf{R}$, $P=[0,+\mathrm{\infty })$ and $S=P$. Then $S_0=int(P)=(0,+\mathrm{\infty })$.

Define $A:[0,+\mathrm{\infty })\times [0,+\mathrm{\infty })\to \mathsf{R}$ as

$$A(x,y)=\{\begin{array}{ll}\frac{x}{y},& (x,y)\ne (0,0),\\ 0,& (x,y)=(0,0).\end{array}$$

A is a mixed monotone operator. Now suppose that $\mathrm{\Psi }:[0,1)\times P\times P\times E\to E$ is as $\mathrm{\Psi }(t,x,y,s)=(1-t^2)s$. Then Ψ is an ${\mathcal{L}}^{″}$-function. Moreover,

$$A(tx,y)\le A(x,ty)-\mathrm{\Psi }(t,x,y,A(x,ty))$$

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

• (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 $u\ll v^{\prime }\ll A(v^{\prime },u)\le v$.

For further explanation on (III), since $A(u_0,v_0)=\frac{1}{3}\ll u_0\ll v_0$, by (III) there exists $u_1\in S$ such that $u_0\ll A(u_1,v_0)\ll u_1\ll v_0$. 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 $\frac{7}{8}=u_1\ll v_0=\frac{3}{2}\ll A(v_0,u_1)=\frac{12}{7}$, thus by (III) there exists $v_1\in S$ such that $u_1\ll v_1\ll A(v_1,u_1)\le v_0$. 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:

$$u_0=\frac{1}{2},v_0=\frac{3}{2},u_n=\frac{u_{n-1}{v}_{n-1}+1}{2}\text{and}{v}_n=\frac{v_{n-1}{u}_n+1}{2},$$

which satisfy (I), (II) and (III) (see Figure 1). Moreover, $u_n\to 1$ and $v_n\to 1$ and $A(1,1)=1$.

Figure 1

$\mathit{A}\mathbf{(}{\mathit{u}}_{\mathbf{0}}\mathbf{,}{\mathit{v}}_{\mathbf{0}}\mathbf{)}\mathbf{\ll }{\mathit{u}}_{\mathbf{0}}\mathbf{\ll }{\mathit{u}}_{\mathbf{1}}\mathbf{\ll }\mathbf{\cdots }\mathbf{\ll }{\mathit{u}}_{\mathit{n}}\mathbf{\ll }\mathbf{\cdots }\mathbf{\ll }\mathbf{1}\mathbf{\ll }\mathbf{\cdots }\mathbf{\ll }{\mathit{v}}_{\mathit{n}}\mathbf{\ll }\mathbf{\cdots }\mathbf{\ll }{\mathit{v}}_{\mathbf{1}}\mathbf{\ll }{\mathit{v}}_{\mathbf{0}}\mathbf{\ll }\mathit{A}\mathbf{(}{\mathit{v}}_{\mathbf{0}}\mathbf{,}{\mathit{u}}_{\mathbf{0}}\mathbf{)}$ .

4 Application

The following result is given by Zhang [6] 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 $S_0=S\mathrm{\setminus }\{\theta \}\subset intP$ and let $\lambda S\subset S$ for all $\lambda \in [0,1]$. Let $u_0,v_0\in S_0$, $A:P\times P\to E$ be a mixed monotone operator with $A(([\theta ,v_0]\cap S)\times ([\theta ,v_0]\cap S))\subset S$ and $A(u_0,v_0)\ll u_0\ll v_0\ll A(v_0,u_0)$. Assume that there exists a function $\varphi :(0,1)\times ([u_0,v_0]\cap S)\times ([u_0,v_0]\cap S)\to (0,+\mathrm{\infty })$ such that $A(tx,y)\le \varphi (t,x,y)A(x,ty)$, where $0<\varphi (t,x,x)<t$ for all $(t,x,y)\in (0,1)\times ([u_0,v_0]\cap S)\times ([u_0,v_0]\cap S)$. Suppose that

• (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 $x^{\ast }\in [u_0,v_0]\cap S$.

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

$$A(tx,y)\le \varphi (t,x,y)A(x,ty)=A(x,ty)-\mathrm{\Psi }(t,x,y,A(x,ty)).$$

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