- Research
- Open Access

# Some new generalizations of Mizoguchi-Takahashi type fixed point theorem

- Gülhan Mınak
^{1}and - Ishak Altun
^{1}Email author

**2013**:493

https://doi.org/10.1186/1029-242X-2013-493

© Mınak and Altun; licensee Springer. 2013

**Received:**19 July 2013**Accepted:**10 October 2013**Published:**7 November 2013

## Abstract

In the light of the paper of Hasanzade Asl *et al.* (Fixed Point Theory Appl. 2012:212, 2012, doi:10.1186/1687-1812-2012-212), we obtain a fixed point theorem for multivalued mappings on a complete metric space. Our result is a generalized version of some results in the literature, including the famous result of Mizoguchi-Takahashi (J. Math. Anal. Appl. 141:177-188, 1989). Also, we give some examples to illustrate our result.

**MSC:**54H25, 47H10.

## Keywords

- fixed point
- multivalued mappings
- Mizoguchi-Takahashi’s fixed point theorem
- Nadler’s fixed point theorem

## 1 Introduction and preliminaries

*X*. It is well known that $H:\mathit{CB}(X)\times \mathit{CB}(X)\to \mathbb{R}$ defined by

*T*is called a multivalued contraction if for all $x,y\in X$, there exists $\lambda \in [0,1)$ such that

In 1969, Nadler [1] proved a fundamental fixed point theorem for multivalued maps: Every multivalued contraction on a complete metric space has a fixed point.

Then, a lot of generalizations of the result of Nadler have been given (see, for example, [2–5]). One of the most important generalizations of it was given by Mizoguchi and Takahashi [6]. We can find both a simple proof of Mizoguchi-Takahashi fixed point theorem and an example showing that it is a real generalization of Nadler’s result in [7]. We can also find some important results about this direction in [8–12].

**Definition 1** [2]

A function $k:[0,\mathrm{\infty})\to [0,1)$ is said to be an $\mathcal{MT}$-function if it satisfies $lim{sup}_{s\to {t}^{+}}k(s)<1$ for all $t\in [0,\mathrm{\infty})$ (Mizoguchi-Takahashi’s condition).

**Lemma 1** [9]

*Let* $k:[0,\mathrm{\infty})\to [0,1)$ *be an* $\mathcal{MT}$-*function*, *then the function* $h:[0,\mathrm{\infty})\to [0,1)$ *defined as* $h(t)=\frac{1+k(t)}{2}$ *is also an* $\mathcal{MT}$-*function*.

**Lemma 2** [9]

$k:[0,\mathrm{\infty})\to [0,1)$ *is an* $\mathcal{MT}$-*function if and only if for each* $t\in [0,\mathrm{\infty})$, *there exist* ${r}_{t}\in [0,1)$ *and* ${\epsilon}_{t}>0$ *such that* $k(s)\le {r}_{t}$ *for all* $s\in [t,t+{\epsilon}_{t})$.

**Theorem 1** [6]

*Let*$(X,d)$

*be a complete metric space*,

*and let*$T:X\to \mathit{CB}(X)$

*be a multivalued map*.

*Assume*

*for all* $x,y\in X$, *where* *k* *is an* $\mathcal{MT}$-*function*. *Then* *T* *has a fixed point*.

Recently, Samet *et al.* [13] introduced the notion of *α*-*ψ*-contractive mappings and gave some fixed point results for such mappings. Their results are closely related to some ordered fixed point results. Then, using their idea, some authors presented fixed point results for single and multivalued mappings (see, for example, [13–17]). First, we recall these results. Denote by Ψ the family of nondecreasing functions $\psi :[0,\mathrm{\infty})\to [0,\mathrm{\infty})$ such that ${\sum}_{n=1}^{\mathrm{\infty}}{\psi}^{n}(t)<\mathrm{\infty}$ for all $t>0$.

**Definition 2** [13]

*T*be a self-map on $X,\psi \in \mathrm{\Psi}$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ be a function. Then

*T*is called

*α*-

*ψ*-contractive whenever

for all $x,y\in X$.

Note that every Banach contraction mapping is an *α*-*ψ*-contractive mapping with $\alpha (x,y)=1$ and $\psi (t)=\lambda t$ for some $\lambda \in [0,1)$.

**Definition 3** [13]

*T* is called *α*-admissible whenever $\alpha (x,y)\ge 1$ implies $\alpha (Tx,Ty)\ge 1$.

There exist some examples for *α*-admissible mappings in [13]. For convenience, we mention in here one of them. Let $X=[0,\mathrm{\infty})$. Define $T:X\to X$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ by $Tx=\sqrt{x}$ for all $x\in X$ and $\alpha (x,y)={e}^{x-y}$ for $x\ge y$ and $\alpha (x,y)=0$ for $x<y$. Then *T* is *α*-admissible.

**Definition 4** [14]

*α* is said to have (B) property whenever $\{{x}_{n}\}$ is a sequence in *X* such that $\alpha ({x}_{n},{x}_{n+1})\ge 1$ for all $n\in \mathbb{N}$ and ${x}_{n}\to x$, then $\alpha ({x}_{n},x)\ge 1$ for all $n\in \mathbb{N}$.

**Theorem 2** (Theorem 2.1 of [13])

*Let* $(X,d)$ *be a complete metric space and* $T:X\to X$ *be an* *α*-*admissible and* *α*-*ψ*-*contractive mapping*. *If there exists* ${x}_{0}\in X$ *such that* $\alpha ({x}_{0},T{x}_{0})\ge 1$ *and* *T* *is continuous*, *then* *T* *has a fixed point*.

**Remark 1** If we assume that *α* has (B) property instead of the continuity of *T*, then again *T* has a fixed point (Theorem 2.2 of [13]). If for each $x,y\in X$ there exists $z\in X$ such that $\alpha (x,z)\ge 1$ and $\alpha (y,z)\ge 1$, then *X* is said to have (H) property. Therefore, if *X* has (H) property in Theorem 2.1 and Theorem 2.2 in [13], then the fixed point of *T* is unique (Theorem 2.3 of [13]).

Then some generalizations of *α*-*ψ*-contractive mappings are given as follows.

**Definition 5** [14]

*T*is called a Ćirić type

*α*-

*ψ*-generalized contractive mapping whenever

Note that every Ćirić type generalized contraction mapping is a Ćirić type *α*-*ψ*-generalized contractive mapping with $\alpha (x,y)=1$ and $\psi (t)=\lambda t$ for some $\lambda \in [0,1)$.

**Theorem 3** (Theorem 2.3 of [14])

*Let* $(X,d)$ *be a complete metric space and* $T:X\to X$ *be an* *α*-*admissible and Ćirić type* *α*-*ψ*-*generalized contractive mapping*. *If there exists* ${x}_{0}\in X$ *such that* $\alpha ({x}_{0},T{x}_{0})\ge 1$ *and* *T* *is continuous or* *α* *has* (B) *property*, *then* *T* *has a fixed point*. *If* *X* *has* (H) *property*, *then the fixed point of* *T* *is unique*.

We can find some fixed point results for single-valued mappings in these directions in [15, 17]. Now we recall some multivalued case.

*T*is called multivalued

*α*-

*ψ*-contractive whenever

*T*is called multivalued ${\alpha}_{\ast}$-

*ψ*-contractive whenever

where ${\alpha}_{\ast}(Tx,Ty)=inf\{\alpha (a,b):a\in Tx,b\in Ty\}$. Similarly, if we replace $d(x,y)$ with $m(x,y)$, we can obtain Ćirić type multivalued *α*-*ψ*-generalized contractive and Ćirić type multivalued ${\alpha}_{\ast}$-*ψ*-generalized contractive mappings on *X*.

- (a)
*T*is said to be*α*-admissible whenever for each $x\in X$ and $y\in Tx$ with $\alpha (x,y)\ge 1$ implies $\alpha (y,z)\ge 1$ for all $z\in Ty$. - (b)
*T*is said to be ${\alpha}_{\ast}$-admissible whenever for each $x\in X$ and $y\in Tx$ with $\alpha (x,y)\ge 1$ implies ${\alpha}_{\ast}(Tx,Ty)\ge 1$.

**Remark 2** It is clear that ${\alpha}_{\ast}$-admissible maps are also *α*-admissible, but the converse may not be true as shown in the following example.

**Example 1**Let $X=[-1,1]$ and $\alpha :X\times X\to [0,\mathrm{\infty})$ be defined by $\alpha (x,x)=0$ and $\alpha (x,y)=1$ for $x\ne y$. Define $T:X\to \mathit{CB}(X)$ by

Let $x=-1$ and $y=0\in Tx=\{0,1\}$, then $\alpha (x,y)\ge 1$, but ${\alpha}_{\ast}(Tx,Ty)={\alpha}_{\ast}(\{0,1\},\{1\})=0$. Thus *T* is not ${\alpha}_{\ast}$-admissible. Now we show that *T* is *α*-admissible with the following cases:

Case 1. If $x=0$, then $y=1$ and $\alpha (x,y)\ge 1$. Also, $\alpha (y,z)\ge 1$ since $z=-1\in Ty=\{-1\}$.

Case 2. If $x=-1$, then $y\in \{0,1\}$ and $\alpha (x,y)\ge 1$. Also, $\alpha (y,z)\ge 1$ for all $z\in Ty$.

Case 3. If $x\notin \{-1,0\}$, then $y=-x$ and $\alpha (x,y)\ge 1$. Also, $\alpha (y,z)\ge 1$ since $z=x\in Ty=\{x\}$.

The purpose of this work is to present some generalizations of Mizoguchi-Takahashi’s fixed point theorem using this new idea.

## 2 Main results

**Theorem 4**

*Let*$(X,d)$

*be a complete metric space*,

*and let*$T:X\to \mathit{CB}(X)$

*be an*

*α*-

*admissible multivalued mapping such that*

*for all* $x,y\in X$, *where* *k* *is an* $\mathcal{MT}$-*function*. *Suppose that there exist* ${x}_{0}\in X$ *and* ${x}_{1}\in T{x}_{0}$ *such that* $\alpha ({x}_{0},{x}_{1})\ge 1$. *If* *T* *is continuous or* *α* *has* (B) *property*, *then* *T* *has a fixed point*.

*Proof*Define $h(t)=\frac{1+k(t)}{2}$, then from Lemma 1, $h:[0,\mathrm{\infty})\to [0,1)$ is an $\mathcal{MT}$-function. Let ${x}_{0}$ and ${x}_{1}$ be as mentioned in the hypothesis. If ${x}_{0}={x}_{1}$, then ${x}_{0}$ is a fixed point of

*T*. Assume ${x}_{0}\ne {x}_{1}$, then $\frac{1-k(d({x}_{0},{x}_{1}))}{2}d({x}_{0},{x}_{1})>0$. Therefore there exists ${x}_{2}\in T{x}_{1}$ such that

*T*is

*α*-admissible, ${x}_{1}\in T{x}_{0}$ and $\alpha ({x}_{0},{x}_{1})\ge 1$, then $\alpha ({x}_{1},u)\ge 1$ for all $u\in T{x}_{1}$. Thus $\alpha ({x}_{1},{x}_{2})\ge 1$ since ${x}_{2}\in T{x}_{1}$. If ${x}_{1}={x}_{2}$, then ${x}_{1}$ is a fixed point of

*T*. Assume ${x}_{1}\ne {x}_{2}$, then $\frac{1-k(d({x}_{1},{x}_{2}))}{2}d({x}_{1},{x}_{2})>0$. Therefore there exists ${x}_{3}\in T{x}_{2}$ such that

*T*is

*α*-admissible, then $\alpha ({x}_{2},{x}_{3})\ge 1$. In this way, we can construct a sequence $\{{x}_{n}\}$ in

*X*such that ${x}_{n+1}\in T{x}_{n}$, $\alpha ({x}_{n},{x}_{n+1})\ge 1$ and

*h*is an $\mathcal{MT}$-function, then $lim{sup}_{s\to {\lambda}^{+}}h(s)<1$ and $h(\lambda )<1$. Therefore from Lemma 2 there exist $r\in [0,1)$ and $\epsilon >0$ such that $h(s)\le r$ for all $s\in [\lambda ,\lambda +\epsilon )$. Since ${lim}_{n\to \mathrm{\infty}}d({x}_{n},{x}_{n+1})=\lambda $, then there exists ${n}_{0}\in \mathbb{N}$ such that $\lambda \le d({x}_{n},{x}_{n+1})<\lambda +\epsilon $ for all $n\ge {n}_{0}$ and so

and so $\{{x}_{n}\}$ is a Cauchy sequence. Since *X* is complete, there exists $z\in X$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}=z$.

If *T* is continuous, then from the inequality $d({x}_{n+1},Tz)\le H(T{x}_{n},Tz)$, we have $d(z,Tz)=0$ and so $z\in Tz$.

*α*has (B) property. Then $\alpha ({x}_{n},z)\ge 1$ for all $n\in \mathbb{N}$. Therefore

and, taking limit $n\to \mathrm{\infty}$, we have $d(z,Tz)=0$ and so $z\in Tz$. □

Although ${\alpha}_{\ast}$-admissibility implies *α*-admissibility of *T*, we will give the following theorem. However, the contractive condition is slightly different from (2.1).

**Theorem 5**

*Let*$(X,d)$

*be a complete metric space*,

*and let*$T:X\to \mathit{CB}(X)$

*be an*${\alpha}_{\ast}$-

*admissible multivalued mapping such that*

*for all* $x,y\in X$, *where* *k* *is an* $\mathcal{MT}$-*function*. *Suppose that there exist* ${x}_{0}\in X$ *and* ${x}_{1}\in T{x}_{0}$ *such that* $\alpha ({x}_{0},{x}_{1})\ge 1$. *If* *T* *is continuous or* *α* *has* (B) *property*, *then* *T* *has a fixed point*.

*Proof*Define $h(t)=\frac{1+k(t)}{2}$, then from Lemma 1, $h:[0,\mathrm{\infty})\to [0,1)$ is an $\mathcal{MT}$-function. Let ${x}_{0}$ and ${x}_{1}$ be as mentioned in the hypothesis. If ${x}_{0}\in T{x}_{0}$, then ${x}_{0}$ is a fixed point of

*T*. Let ${x}_{0}\notin T{x}_{0}$. Since ${x}_{0}\ne {x}_{1}$, then $\frac{1-k(d({x}_{0},{x}_{1}))}{2}d({x}_{0},{x}_{1})>0$. If ${x}_{1}\in T{x}_{1}$, ${x}_{1}$ is a fixed point of

*T*. Let ${x}_{1}\notin T{x}_{1}$. Also, since

*T*is ${\alpha}_{\ast}$-admissible, ${\alpha}_{\ast}(T{x}_{0},T{x}_{1})\ge 1$. Therefore, there exists ${x}_{2}\in T{x}_{1}$ such that

*T*. Let ${x}_{2}\notin T{x}_{2}$. Since $\alpha ({x}_{2},{x}_{3})\ge {\alpha}_{\ast}(T{x}_{1},T{x}_{2})\ge 1$, then ${\alpha}_{\ast}(T{x}_{2},T{x}_{3})\ge 1$. In this way, we can construct a sequence $\{{x}_{n}\}$ in

*X*such that ${x}_{n+1}\in T{x}_{n}$, $\alpha ({x}_{n},{x}_{n+1})\ge 1$ and

for all $n\in \mathbb{N}$. As in the proof of Theorem 4, we can show that $\{{x}_{n}\}$ is a Cauchy sequence in *X*. Since *X* is complete, there exists $z\in X$ such that ${lim}_{n\to \mathrm{\infty}}{x}_{n}=z$.

If *T* is continuous, then from the inequality $d({x}_{n+1},Tz)\le H(T{x}_{n},Tz)$, we have $d(z,Tz)=0$ and so $z\in Tz$.

*α*has (B) property. Then $\alpha ({x}_{n},z)\ge 1$ for all $n\in \mathbb{N}$. Since

*T*is ${\alpha}_{\ast}$-admissible, ${\alpha}_{\ast}(T{x}_{n},Tz)\ge 1$. Therefore

and, taking limit $n\to \mathrm{\infty}$, we have $d(z,Tz)=0$ and so $z\in Tz$. □

Now we give an example to illustrate our main theorems. Note that Theorem 1 cannot be applied to this example.

**Example 2**Let $X=[-1,1]$ and $d(x,y)=|x-y|$. Define $T:X\to \mathit{CB}(X)$ by

*T*is ${\alpha}_{\ast}$-admissible and

*k*is any $\mathcal{MT}$-function. Indeed, first we show that

*T*is ${\alpha}_{\ast}$-admissible. If $\alpha (x,y)\ge 1$, then $x,y\in [-\frac{1}{2},\frac{1}{2}]$ and hence

Therefore *T* is ${\alpha}_{\ast}$-admissible.

Now we consider the following cases:

Case 1. Let $x,y\in X$ with $\{x,y\}\cap \{[-1,-\frac{3}{4})\cup (\frac{3}{4},1]\}\ne \mathrm{\varnothing}$, then ${\alpha}_{\ast}(Tx,Ty)=0$. Thus (2.2) is satisfied.

and so again (2.2) is satisfied.

Therefore there is no $\mathcal{MT}$-function satisfying (1.1).

**Remark 3** If we take $\alpha :X\times X\to [0,\mathrm{\infty})$ by $\alpha (x,y)=1$, then any multivalued mappings $T:X\to \mathit{CB}(X)$ are *α*-admissible as well as ${\alpha}_{\ast}$-admissible. Therefore, Mizoguchi-Takahashi’s fixed point theorem is a special case of Theorem 4 and Theorem 5.

We can obtain some ordered fixed point results from our theorems as follows. First we recall some ordered notions. Let *X* be a nonempty set and ⪯ be a partial order on *X*.

**Definition 8** [18]

Let *A*, *B* be two nonempty subsets of *X*, the relations between *A* and *B* are defined as follows:

(r_{1}) If for every $a\in A$ there exists $b\in B$ such that $a\u2aafb$, then $A{\prec}_{1}B$.

(r_{2}) If for every $b\in B$ there exists $a\in A$ such that $a\u2aafb$, then $A{\prec}_{2}B$.

(r_{3}) If $A{\prec}_{1}B$ and $A{\prec}_{2}B$, then $A\prec B$.

**Remark 4** [18]

≺_{1} and ≺_{2} are different relations between *A* and *B*. For example, let $X=\mathbb{R}$, $A=[\frac{1}{2},1]$, $B=[0,1]$, ⪯ be the usual order on *X*, then $A{\prec}_{1}B$ but $A{\nprec}_{2}B$; if $A=[0,1]$, $B=[0,\frac{1}{2}]$, then $A{\prec}_{2}B$ while $A{\nprec}_{1}B$.

**Remark 5** [18]

≺_{1}, ≺_{2} and ≺ are reflexive and transitive, but are not antisymmetric. For instance, let $X=\mathbb{R}$, $A=[0,3]$, $B=[0,1]\cup [2,3]$, ⪯ be the usual order on *X*, then $A\prec B$ and $B\prec A$, but $A\ne B$. Hence, they are not partial orders.

**Corollary 1**

*Let*$(X,\u2aaf)$

*be a partially ordered set and suppose that there exists a metric*

*d*

*in*

*X*

*such that*$(X,d)$

*is a complete metric space*.

*Let*$T:X\to \mathit{CB}(X)$

*be a multivalued mapping such that*

*for all*$x,y\in X$

*with*$x\u2aafy$,

*where*

*k*

*is an*$\mathcal{MT}$-

*function*.

*Suppose that there exists*${x}_{0}\in X$

*such that*$\{{x}_{0}\}{\prec}_{1}T{x}_{0}$.

*Assume that for each*$x\in X$

*and*$y\in Tx$

*with*$x\u2aafy$,

*we have*$y\u2aafz$

*for all*$z\in Ty$.

*If*

*T*

*is continuous or*

*X*

*satisfies the following condition*:

*then* *T* *has a fixed point*.

*Proof*Define the mapping $\alpha :X\times X\to [0,\mathrm{\infty})$ by

for all $x,y\in X$. Also, since $\{{x}_{0}\}{\prec}_{1}T{x}_{0}$, then there exists ${x}_{1}\in T{x}_{0}$ such that ${x}_{0}\u2aaf{x}_{1}$ and so $\alpha ({x}_{0},{x}_{1})\ge 1$. Now let $x\in X$ and $y\in Tx$ with $\alpha (x,y)\ge 1$, then $x\u2aafy$ and so, by the hypotheses, we have $y\u2aafz$ for all $z\in Ty$. Therefore, $\alpha (y,z)\ge 1$ for all $z\in Ty$. This shows that *T* is *α*-admissible. Finally, if *T* is continuous or *X* satisfies (2.3), then *T* is continuous or *α* has (B) property. Therefore, from Theorem 4, *T* has a fixed point. □

**Remark 6** We can give a similar corollary using ≺_{2} instead of ≺_{1}.

## Declarations

### Acknowledgements

The authors are grateful to the referees for their suggestions that contributed to the improvement of the paper.

## Authors’ Affiliations

## References

- Nadler SB: Multivalued contraction mappings.
*Pac. J. Math.*1969, 30: 475–488. 10.2140/pjm.1969.30.475MathSciNetView ArticleGoogle Scholar - Du W-S: On coincidence point and fixed point theorems for nonlinear multivalued maps.
*Topol. Appl.*2012, 159: 49–56. 10.1016/j.topol.2011.07.021View ArticleGoogle Scholar - Kamran T, Kiran Q: Fixed point theorems for multi-valued mappings obtained by altering distances.
*Math. Comput. Model.*2011, 54: 2772–2777. 10.1016/j.mcm.2011.06.065MathSciNetView ArticleGoogle Scholar - Reich S: Some problems and results in fixed point theory.
*Contemp. Math.*1983, 21: 179–187.View ArticleGoogle Scholar - Reich S: Fixed points of contractive functions.
*Boll. Unione Mat. Ital.*1972, 5: 26–42.Google Scholar - Mizoguchi N, Takahashi W: Fixed point theorems for multivalued mappings on complete metric spaces.
*J. Math. Anal. Appl.*1989, 141: 177–188. 10.1016/0022-247X(89)90214-XMathSciNetView ArticleGoogle Scholar - Suzuki T: Mizoguchi-Takahashi’s fixed point theorem is a real generalization of Nadler’s.
*J. Math. Anal. Appl.*2008, 340: 752–755. 10.1016/j.jmaa.2007.08.022MathSciNetView ArticleGoogle Scholar - Berinde M, Berinde V: On a general class of multi-valued weakly Picard mappings.
*J. Math. Anal. Appl.*2007, 326: 772–782. 10.1016/j.jmaa.2006.03.016MathSciNetView ArticleGoogle Scholar - Du W-S: Some new results and generalizations in metric fixed point theory.
*Nonlinear Anal., Theory Methods Appl.*2010, 73(5):1439–1446. 10.1016/j.na.2010.05.007View ArticleGoogle Scholar - Du W-S, Karapınar E, Shahzad N: The study of fixed point theory for various multivalued non-self-maps.
*Abstr. Appl. Anal.*2013., 2013: Article ID 938724 10.1155/2013/938724Google Scholar - He Z, Du W-S, Lin IJ: The existence of fixed points for new nonlinear multivalued maps and their applications.
*Fixed Point Theory Appl.*2011., 2011: Article ID 84Google Scholar - Shatanavi W, Ćojbašić Rajić V, Radenović S, Al-Rawashden A: Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces.
*Fixed Point Theory Appl.*2012., 2012: Article ID 106Google Scholar - Samet B, Vetro C, Vetro P: Fixed point theorems for
*α*-*ψ*-contractive type mappings.*Nonlinear Anal.*2012, 75: 2154–2165. 10.1016/j.na.2011.10.014MathSciNetView ArticleGoogle Scholar - Hasanzade Asl J, Rezapour S, Shahzad N: On fixed points of
*α*-*ψ*-contractive multifunctions.*Fixed Point Theory Appl.*2012., 2012: Article ID 212 10.1186/1687-1812-2012-212Google Scholar - Karapınar E, Samet B: Generalized
*α*-*ψ*-contractive type mappings and related fixed point theorems with applications.*Abstr. Appl. Anal.*2012., 2012: Article ID 793486 10.1155/2012/793486Google Scholar - Mohammadi B, Rezapour S, Shahzad N: Some results on fixed points of
*α*-*ψ*-Ćirić generalized multifunctions.*Fixed Point Theory Appl.*2013., 2013: Article ID 24Google Scholar - Nawab H, Karapınar E, Salimi P, Akbar F:
*α*-admissible mappings and related fixed point theorems.*J. Inequal. Appl.*2013., 2013: Article ID 114Google Scholar - Feng Y, Liu S: Fixed point theorems for multi-valued increasing operators in partially ordered spaces.
*Soochow J. Math.*2004, 30(4):461–469.MathSciNetGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.