# Common fixed point theorems for multi-valued mappings in complex-valued metric spaces

- Akbar Azam
^{1}, - Jamshaid Ahmad
^{1}Email author and - Poom Kumam
^{2}

**2013**:578

https://doi.org/10.1186/1029-242X-2013-578

© Azam et al.; licensee Springer. 2013

**Received: **26 July 2013

**Accepted: **14 November 2013

**Published: **11 December 2013

## Abstract

Azam *et al.* (Numer. Funct. Anal. Optim. 33(5):590-600, 2012) introduced the notion of complex-valued metric spaces and established a common fixed point result in the context of complex-valued metric spaces. In this paper, the existence of common fixed points is established for multi-valued mappings on the complex-valued metric spaces. Our results unify, generalize and complement the comparable results from the current literature.

**MSC:**46S40, 47H10, 54H25.

## Keywords

## 1 Introduction

It is a well-known fact that the mathematical results regarding fixed points of contraction-type mappings are very useful for determining the existence and uniqueness of solutions to various mathematical models. Over the last 40 years, the theory of fixed points has been developed regarding the results that are related to finding the fixed points of self and nonself nonlinear mappings in a metric space.

The study of fixed points for multi-valued contraction mappings was initiated by Nadler [1] and Markin [2]. Several authors proved fixed point results in different types of generalized metric spaces [3–17].

Azam *et al.* [9] introduced the concept of complex-valued metric space and obtained sufficient conditions for the existence of common fixed points of a pair of mappings satisfying a contractive-type condition. Subsequently, Rouzkard and Imdad [18] established some common fixed point theorems satisfying certain rational expressions in complex-valued metric spaces to generalize the results of [9]. In the same way, Sintunavarat and Kumam [19, 20] obtained common fixed point results by replacing the constant of contractive condition to control functions. Recently, Sitthikul and Saejung [21] and Klin-eam and Suanoom [22] established some fixed point results by generalizing the contractive conditions in the context of complex-valued metric spaces. Very recently, Ahmad *et al.* [5] obtained some new fixed point results for multi-valued mappings in the setting of complex-valued metric spaces.

The purpose of this paper is to study common fixed points of two multi-valued mappings satisfying a rational inequality without exploiting any type of commutativity condition in the framework of a complex-valued metric space. The results presented in this paper substantially extend and strengthen the results given in [5, 9] for the multi-valued mappings.

## 2 Preliminaries

- (i)
$Re({z}_{1})=Re({z}_{2})$, $Im({z}_{1})<Im({z}_{2})$,

- (ii)
$Re({z}_{1})<Re({z}_{2})$, $Im({z}_{1})=Im({z}_{2})$,

- (iii)
$Re({z}_{1})<Re({z}_{2})$, $Im({z}_{1})<Im({z}_{2})$,

- (iv)
$Re({z}_{1})=Re({z}_{2})$, $Im({z}_{1})=Im({z}_{2})$.

**Definition 1**Let

*X*be a nonempty set. Suppose that the mapping

- 1.
$0\precsim d(x,y)$ for all $x,y\in X$ and $d(x,y)=0$ if and only if $x=y$;

- 2.
$d(x,y)=d(y,x)$ for all $x,y\in X$;

- 3.
$d(x,y)\precsim d(x,z)+d(z,y)$ for all $x,y,z\in X$.

*d*is called a complex-valued metric on

*X*, and $(X,d)$ is called a complex-valued metric space. A point $x\in X$ is called an interior point of a set $A\subseteq X$ whenever there exists $0\prec r\in \mathbb{C}$ such that

*A*whenever, for every $0\prec r\in \mathbb{C}$,

*A*is called open whenever each element of

*A*is an interior point of

*A*. Moreover, a subset $B\subseteq X$ is called closed whenever each limit point of

*B*belongs to

*B*. The family

is a sub-basis for a Hausdorff topology *τ* on *X*.

Let ${x}_{n}$ be a sequence in *X* and $x\in X$. If for every $c\in \mathbb{C}$ with $0\prec c$ there is ${n}_{0}\in \mathbb{N}$ such that for all $n>{n}_{0}$, $d({x}_{n},x)\prec c$, then $\{{x}_{n}\}$ is said to be convergent, $\{{x}_{n}\}$ converges to *x* and *x* is the limit point of $\{{x}_{n}\}$. We denote this by ${lim}_{n\to \mathrm{\infty}}{x}_{n}=x$, or ${x}_{n}\to x$, as $n\to \mathrm{\infty}$. If for every $c\in \mathbb{C}$ with $0\prec c$ there is ${n}_{0}\in \mathbb{N}$ such that for all $n>{n}_{0}$, $d({x}_{n},{x}_{n+m})\prec c$, where $m\in \mathbb{N}$, then $\{{x}_{n}\}$ is called a Cauchy sequence in $(X,d)$. If every Cauchy sequence is convergent in $(X,d)$, then $(X,d)$ is called a complete complex-valued metric space. We require the following lemmas.

**Lemma 2** [9]

*Let* $(X,d)$ *be a complex*-*valued metric space and let* $\{{x}_{n}\}$ *be a sequence in* *X*. *Then* $\{{x}_{n}\}$ *converges to* *x* *if and only if* $|d({x}_{n},x)|\to 0$ *as* $n\to \mathrm{\infty}$.

**Lemma 3** [9]

*Let* $(X,d)$ *be a complex*-*valued metric space and let* $\{{x}_{n}\}$ *be a sequence in* *X*. *Then* $\{{x}_{n}\}$ *is a Cauchy sequence if and only if* $|d({x}_{n},{x}_{n+m})|\to 0$ *as* $n\to \mathrm{\infty}$, *where* $m\in \mathbb{N}$.

## 3 Main result

Let $(X,d)$ be a complex-valued metric space.

We denote the family of nonempty, closed and bounded subsets of a complex valued metric space by $CB(X)$.

From now on, we denote $s({z}_{1})=\{{z}_{2}\in \mathbb{C}:{z}_{1}\u2aaf{z}_{2}\}$ for ${z}_{1}\in \mathbb{C}$, and $s(a,B)={\bigcup}_{b\in B}s(d(a,b))={\bigcup}_{b\in B}\{z\in \mathbb{C}:d(a,b)\u2aafz\}$ for $a\in X$ and $B\in CB(X)$.

**Remark 4** [5]

Let $(X,d)$ be a complex-valued metric space. If $\mathbb{C}=R$, then $(X,d)$ is a metric space. Moreover, for $A,B\in CB(X)$, $H(A,B)=infs(A,B)$ is the Hausdorff distance induced by *d*.

**Definition 5** [5]

**Definition 6** [5]

Let $(X,d)$ be a complex-valued metric space. A subset *A* of *X* is called bounded from below if there exists some $z\in X$ such that $z\u2aafa$ for all $a\in A$.

**Definition 7** [5]

for all $u\in Fx$.

**Definition 8** [5]

for all $u\in {W}_{x}(Ty)$, where ${l}_{x}(Ty)$ is called a lower bound of *T* associated with $(x,y)$.

**Definition 9** [5]

**Theorem 10**

*Let*$(X,d)$

*be a complete complex*-

*valued metric space and let*$S,T:X\to CB(X)$

*be multi*-

*valued mappings with g*.

*l*.

*b property such that*

*for all* $x,y\in X$ *and* $a+b+c<1$. *Then* *S* *and* *T* *have a common fixed point*.

*Proof*Let ${x}_{0}$ be an arbitrary point in

*X*and ${x}_{1}\in S{x}_{0}$. From (3.1), we have

*S*and

*T*, we get

*a*,

*b*,

*c*are nonnegative reals and $a+b+c<1$, so $\rho =\frac{a}{1-a}<1$, so we have

*X*such that for $n=0,1,2,\dots $ ,

*X*. Since

*X*is complete, there exists $\nu \in X$ such that ${x}_{n}\to \nu $ as $n\to \mathrm{\infty}$. We now show that $\nu \in T\nu $ and $\nu \in S\nu $. From (3.1), we get

*S*and

*T*, we have

By letting $n\to \mathrm{\infty}$ in the above inequality, we get $|d(\nu ,{\nu}_{n})|\to 0$ as $n\to \mathrm{\infty}$. By Lemma 2 [9], we have ${\nu}_{n}\to \nu $ as $n\to \mathrm{\infty}$. Since *Tν* is closed, so $\nu \in T\nu $. Similarly, it follows that $\nu \in S\nu $. Thus *S* and *T* have a common fixed point. □

**Corollary 11**

*Let*$(X,d)$

*be a complete complex*-

*valued metric space and let*$S,T:X\to CB(X)$

*be multi*-

*valued mappings with g*.

*l*.

*b property such that*

*for all* $x,y\in X$ *and* $0\le \alpha <1$. *Then* *S* *and* *T* *have a common fixed point*.

*Proof* By taking $a=b$ and $c=0$ in Theorem 10. □

**Corollary 12**

*Let*$(X,d)$

*be a complete complex*-

*valued metric space and let*$T:X\to CB(X)$

*be a multi*-

*valued mapping with g*.

*l*.

*b property such that*

*for all* $x,y\in X$ *and* $a+b+c<1$. *Then* *T* *has a fixed point*.

*Proof* By taking $S=T$ in Theorem 10. □

Now we obtain a common fixed point result discussed by Khan [23] in the setting of complex-valued metric spaces.

**Theorem 13**

*Let*$(X,d)$

*be a complete complex*-

*valued metric space and let*$S,T:X\to CB(X)$

*be multi*-

*valued mappings with g*.

*l*.

*b property such that*

*for all* $x,y\in X$ *and* $0\le \alpha <1$. *Then* *S* *and* *T* *have a common fixed point*.

*Proof*Let ${x}_{0}\in X$ and ${x}_{1}\in S{x}_{0}$. From (3.4), we get

*S*and

*T*, we get

*X*such that for $n=0,1,2\dots $ , $|d({x}_{n},{x}_{n+1})|\le {a}^{n}|d({x}_{0,}{x}_{1})|$ with $a<1$, ${x}_{2n+1}\in S{x}_{2n}$ and ${x}_{2n+2}\in T{x}_{2n+1}$. Now, for $m>n$, we get

*X*. Since

*X*is complete, so there exists $\nu \in X$ such that ${x}_{n}\to \nu $ as $n\to \mathrm{\infty}$. We now show that $\nu \in T\nu $ and $\nu \in S\nu $. From (3.4), we have

*S*and

*T*, we get

By letting $n\to \mathrm{\infty}$ in the above inequality, we get $|d(\nu ,{\nu}_{n})|\to 0$ as $n\to \mathrm{\infty}$. By Lemma 2 [9], we have ${\nu}_{n}\to \nu $ as $n\to \mathrm{\infty}$. Since *Tν* is closed, so $\nu \in T\nu $. Similarly, it follows that $\nu \in S\nu $. Thus *S* and *T* have a common fixed point. □

**Corollary 14**

*Let*$(X,d)$

*be a complete complex*-

*valued metric space and let*$T:X\to CB(X)$

*be a multi*-

*valued mapping with g*.

*l*.

*b property such that*

*for all* $x,y\in X$ *and* $0\le \alpha <1$. *Then* *T* *has a fixed point*.

*Proof* By setting $S=T$ in Theorem 13. □

Now we give an example which satisfies our main result.

**Example 15**Let $X=[0,1]$. Define $d:X\times X\to \mathbb{C}$ as follows:

*x*,

*y*are nonzero and $x<y$. Then

*a*and

*c*and $b=\frac{1}{3}$, we have

Hence all the conditions of Theorem 10 are satisfied and 0 is a common fixed point of *S* and *T*.

## 4 Conclusion

In this paper, we have established common fixed point results for Chatterjea-type contractive mappings in the context of complex-valued metric spaces. Our results may be the motivation for other authors to extend and improve these results to be suitable tools for their applications.

## Declarations

## Authors’ Affiliations

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