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Abstract and Applied Analysis

Volume 2013 (2013), Article ID 854965, 12 pages

http://dx.doi.org/10.1155/2013/854965

## Common Fixed Points for Multivalued Mappings in Complex Valued Metric Spaces with Applications

^{1}Department of Mathematics, COMSATS Institute of Information Technology, Chak Shahzad, Islamabad 44000, Pakistan^{2}Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand

Received 31 July 2013; Revised 21 November 2013; Accepted 24 November 2013

Academic Editor: Irena Rachůnková

Copyright © 2013 Jamshaid Ahmad et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this paper, we introduce the notion of multivalued contractive mappings in complex valued metric space and prove common fixed point theorems for two multivalued contractive mappings in complex valued metric spaces without using the notion of continuity. Our results improve and extend the results of Azam et al. (2011).

#### 1. Introduction

The Banach fixed point theorem was used to establish the existence of a unique solution for a nonlinear integral equation [1]. Moreover, this theorem plays an important role in several branches of mathematics. For instance, it has been used to show the existence of solutions of nonlinear Volterra integral equations and nonlinear integrodifferential equations in the Banach spaces and to show the convergence of algorithms in computational mathematics. Because of its importance and usefulness for mathematical theory, it has become a very popular tool of mathematical analysis in many directions. Nadler [2] introduced the concept of multivalued contraction mappings and obtained the fixed points results for multivalued mappings. Huang and Zhang [3] introduced the notion of cone metric space which is a generalization of metric spaces. They extended Banach contraction principle to cone metric spaces. Since then, Arshad et al. [4], Azam and Arshad [5], Latif and Shaddad [6], Karapınar [7], and many others obtained fixed point theorems in cone metric spaces (see [8]).

The fixed point results regarding rational contractive conditions cannot be extended in cone metric spaces. Azam et al. [9] introduced the concept of complex valued metric spaces and obtained sufficient conditions for the existence of common fixed points of a pair of mappings satisfying contractive type condition involving rational inequalities. In the same way, Rouzkard and Imdad [10] established some common fixed point theorems satisfying certain rational expressions in complex valued metric spaces which generalize, unify, and complement the results of Azam et al. [9]. Recently, Sintunavarat and Kumam [11] obtained common fixed point results by replacing constant of contractive condition with control functions. For more details in the subject, we refer to [12–19].

The aim of this paper is to extend the results of Azam et al. [9] to multivalued mappings in complex valued metric spaces.

#### 2. Preliminaries

Let be the set of complex numbers and . Define a partial order on as follows: It follows that if one of the following conditions is satisfied: In particular, we will write if and one of (i), (ii), and (iii) is satisfied and we will write if only (iii) is satisfied. Note that

*Definition 1. *Let be a nonempty set. Suppose that a mapping satisfies(1), for all and if and only if ;(2) for all ;(3), for all .

Then is called a *complex valued metric* on , and is called a *complex valued metric space*. A point is called * interior point* of a set whenever there exists such that

A point is called a *limit point* of whenever for every ,

is called *open* whenever each element of is an interior point of . Moreover, a subset is called closed whenever each limit point of belongs to . The family
is a subbasis for a Hausdorff topology on .

Let be a sequence in and . If for every with there is such that for all , , then is said to be *convergent*, converges to , and is thelimit point of . We denote this by , or , as . If for every with there is such that for all , , then is called a *Cauchy sequence* in . If every Cauchy sequence is convergent in , then is called a *complete *complex valued metric space. We require the following lemmas.

Lemma 2 (see [9]). *Let be a complex valued metric space and let be a sequence in . Then converges to if and only if as .*

Lemma 3 (see [9]). *Let be a complex valued metric space and let be a sequence in . Then is a Cauchy sequence if and only if as .*

#### 3. Main Results

Let be a complex valued metric space.

We denote by (resp., ) the set of nonempty (resp., closed and bounded) subsets of a complex valued metric space. Now, we denote for , and for and . For , we denote

*Remark 4. *Let be a complex valued metric space. If , then is a metric space. Moreover, for , is the Hausdorff distance induced by .

Let be a complex valued metric space and let be a collection of nonempty closed subsets of . Let be a multivalued map. For and , define Thus, for

*Definition 5. *Let be acomplex valued metric space. A nonempty subset of is called bounded from below if there exists some , such that for all .

*Definition 6. *Let be acomplex valued metric space. A multivalued mapping is called bounded from below if for there exists such that

*Definition 7. *Let be a complex valued metric space. The multivalued mapping is said to have lower bound property (l.b. property) on if for any , the multivalued mapping defined by
is bounded from below. That is, for there exists an element such that
for all , where is called lower bound of associated with .

*Definition 8. *Let be a complex valued metric space. The multivalued mapping is said to have the greatest lower bound property (g.l.b. property) on if the greatest lower bound of exists in for all . We denote by the g.l.b of . That is,

Theorem 9. *Let be a complete complex valued metric space and let be multivalued mappings with g.l.b. property such that
**
for all and , , and are nonnegative real numbers with . Then have a common fixed point.*

*Proof. *Let be an arbitrary point in and . From (15), we have
This implies that
Since , we have
So there exists some such that
Therefore,
By using the greatest lower bound property (g.l.b property) of and , we get
which implies that
Inductively, we can construct a sequence in such that for
with , for and .

Now for , we get
Therefore,
This implies that is a Cauchy sequence in . Since is complete, there exists such that as . We now show that and . From (15), we have
This implies that
Since , we have
By definition,
There exists some such that
that is,
By using the greatest lower bound property (g.l.b. property) of and , we have
Since
using (32), we get
Taking the limit as , we get as . By [9, lemma 2], we have as . Since is closed, . Similarly, it follows that . Thus, and have a common fixed point.

By setting in Theorem 9, we get the following corollary.

Corollary 10. *Let be a complete complex valued metric space and let be multivalued mappings with g.l.b property such that**
for all and are nonnegative real numbers with . Then have a common fixed point.*

By setting in Theorem 9, we get the following Corollary.

Corollary 11. *Let be a complete complex valued metric space and let be multivalued mapping with g.l.b property such that**
for all and , , and are nonnegative real numbers with . Then has a fixed point in .*

By Remark 4, we have the following corollaries from Theorem 9.

Corollary 12. *Let be a complete metric space and let be multivalued mappings such that
**
for all and , , and are nonnegative real numbers with . Then have a common fixed point.*

*Remark 13. *By equating , , to in all possible combinations, one can derive a host of corollaries which include the Banach fixed point theorem for multivalued mappings in complete metric space.

*Example 14. *Let . Define by
Then is a complex valued metric space. Consider the mappings defined by
The contractive condition of main theorem is trivial for the case when . Suppose without any loss of generality that all are nonzero and . Then
Consider,
Clearly, for any value of and and , we have
Thus,
Hence, all the conditions of our main theorem are satisfied and is a common fixed point of and .

In the following results, we considered the Kannan type contractive condition involving rational expressions.

Theorem 15. *Let be a complete complex valued metric space and let be multivalued mappings with g.l.b property such that
**
for all and . Then and have a common fixed point.*

*Proof. *Let and . From (44), we get
This implies that
Since , we have
So there exists some , such that
That is,
By using the greatest lower bound property (g.l.b property) of and , we get
which implies that
Since , we have
Thus,
Inductively, we can construct a sequence in such that for
with , for and . Now for , we get
and so
This implies that is a Cauchy sequence in . Since is complete, there exists such that as . We now show that and . So from (44), we get
This implies that
Since , we have
By definition,
There exists some such that
that is,
By using the greatest lower bound property (g.l.b property) of and , we get
Now by using the triangular inequality, we get
That is,
It follows that
By letting in the above inequality, we get

By [9, Lemma 2], we have as . Since is closed, . Similarly, it follows that . Thus, and have a common fixed point.

Corollary 16. *Let be a complete complex valued metric space and let be multivalued mappings with g.l.b property such that
**
for all and . Then and have a common fixed point.*

By setting in Theorem 15, we get the following corollary.

Corollary 17. *
for all and . Then has a fixed point.*

By Remark 4 we have the following corollaries.

Corollary 18. *Let be a complete metric space and let be multivalued mappings such that
**
for all and , , and are nonnegative real numbers with . Then have a common fixed point.*

*Remark 19. *By writing , , in all possible combinations, one can derive a host of corollaries which include the Kannan fixed point theorem for multivalued mappings in complete metric space.

In the following results, we considered the Chatterjea type locally contractive condition involving rational expressions.

Theorem 20. *Let be multivalued mappings with the g.l.b property on complete complex valued metric space , , and . If and satisfy
**
for all and
**
where , and are nonnegative real numbers with , then and have a common fixed point in .*

*Proof. *Let be an arbitrary point in . From (72), one can easily prove that
Thus, we have . From (71), we get
This implies that
By the definition, we can take such that
By the definition, we get
By using the g.l.b property of and , we get
Hence, we have
which implies that
where . From (73), we get
Note that
Thus, we have . From (71), we get
This implies that
By the definition, there exists such that
By the definition, we get
By using the g.l.b property of and , we get
Hence, we have
which implies that
where . Consider that
So . Continuing in this way, we can construct a sequence in such that for each ,
where , , and . Now, inductively, we can construct a sequence in such that for each ,
Thus, as in the proof of Theorem 15, is a Cauchy sequence in . Since is complete and is a closed subspace of , there exists such that as . Now, we show that and . From (71), we get
This implies that
By the definition, there exists such that
By the definition, we get
By using the g.l.b property of and , we get
Hence, we have
Now, by using the triangular inequality, we get
and so
By letting in the above inequality, we get as . By [9, Lemma 2.2], it follows that as . Since is closed, . Similarly, it follows that . Thus, and have a common fixed point in . This completes the proof.

By taking in Theorem 20, we get the following Corollary.

Corollary 21. *Let be multivalued mappings with the g.l.b property in complete complex valued metric space , , and . If and satisfy
**
for all and
**
where and are nonnegative real numbers with , then and have a common fixed point in .*

By taking in Theorem 20, we get the following.

Corollary 22. *Let be multivalued mapping with the g.l.b property in complete complex valued metric space , , and . If satisfies
**
for all and
**
where , and are nonnegative real numbers with , then has a fixed point in .*

#### 4. Applications

As an application of the main result (Corollary 22), we prove the following homotopy result.

Theorem 23. *Let be a complete complex valued metric space and let be an open subset of . Let be multivalued mapping with the g.l.b property. Suppose that there exist and such that*(a)* for all and ;*(b)* is a multivalued mapping satisfying
where ;*(c)*there exists a continuous increasing function such that
for all and , where .** Then has a fixed point if and only if has a fixed point.*

*Proof. *Suppose that has a fixed point ; so . From (a), . Define the following set: