Discrete Dynamics in Nature and Society
Volume 2015 (2015), Article ID 147303, 11 pages
http://dx.doi.org/10.1155/2015/147303

Research Article

Common Fixed Points of Generalized Cocyclic Mappings in Complex Valued Metric Spaces

Mujahid Abbas,1,2 Manuel De la Sen,³ and Talat Nazir⁴

¹Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia
²Department of Mathematics and Applied Mathematics, University of Pretoria, Lynnwood Road, Pretoria 0002, South Africa
³Institute of Research and Development of Process (IIDP), Faculty of Science and Technology, University of the Basque Country, 48940 Leioa, Spain
⁴Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan

Received 19 August 2015; Accepted 18 October 2015

Academic Editor: Victor S. Kozyakin

Copyright © 2015 Mujahid Abbas 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.

We present fixed point results of mappings satisfying generalized contractive conditions in complex valued metric spaces. As an application, we obtain a common fixed point of a pair of weakly compatible mappings. Some common fixed point results of generalized contractive-type mappings involved in cocyclic representation of a nonempty subset of a complex valued metric space are also obtained. Some examples are also presented to support the results proved herein. These results extend and generalize many results in the existing literature.

1. Introduction and Preliminaries

Fixed point theory is one of the famous and traditional theories in mathematics that provide important and traditional tools for proving the existence of solutions of many problems in both applied and pure mathematics. In metric fixed point theory, the interplay between contractive condition and the existence and uniqueness of a fixed point has been very strong and fruitful. There are many applications to the study of fixed points of mappings which satisfy contractive conditions related to the solutions of differential and integral equations [1–3]. Fixed point results for mappings inspired and motivated the development of many other important kinds of points like equilibrium points, coincidence points, periodic points, sectional points, intersection points, and so forth.

Banach’s fixed point theorem is one of the basic and most widely applied fixed point theorems in mathematical analysis. Kirk et al. [4] established fixed point theorems for mappings satisfying cyclical contractive conditions. The cyclic contraction mappings need not be continuous, while the Banach contraction mappings are continuous mappings. Păcurar and Rus [5] proved some fixed point results for cyclic weakly contractive mappings. Karapinar [6] obtained fixed point results for cyclic weakly contractive mappings. Derafshpour and Rezapour [7] proved some results on the existence of best proximity points of cyclic contractions. Recently, Abbas et al. [8] established some common fixed point results of rational-type cocyclic mappings in the setup of multiplicative metric spaces. For further interesting results in this direction, we refer readers to [9–17]. There are many generalizations of metric spaces such as partial metric spaces, generalized metric spaces, cone metric spaces, and quasimetric spaces. Recently, Azam et al. [18] obtained the generalization of Banach’s contraction principal by introducing the concept of complex valued metric space and established some common fixed point theorems for mappings involving rational expressions which are not meaningful in cone metric spaces. Later, some other useful results appeared in [19–24]. Abbas et al. [25] obtained the well-posedness and periodic point property of mappings satisfying a rational inequality in an ordered complex valued metric space. Recently, Nashine et al. [26] proved common fixed point theorems under rational contractions in complex valued metric spaces.

In this paper, we establish fixed point result of a generalized contractive mapping in the setup of complex valued metric spaces. Further, by employing this result, a common fixed point of a pair of weakly compatible mappings is obtained. We study the sufficient conditions for the existence of common fixed points of pair of rational contractive-type mappings involved in cocyclic representation of a nonempty subset of a complex valued metric space. Our results generalize and extend comparable results.

By , and we denote the set of all real numbers, the set of all nonnegative real numbers, the set of all complex numbers, the set of all nonnegative complex numbers, and the set of all natural numbers, respectively.

Consistent with Azam et al. [18], the following definitions and results will be needed in the sequel.

Let . Define a partial order on as follows:

if and only if , .

It follows that if one of the following conditions is satisfied:

(1).(2).(3).(4).

In particular, we will write if one of (1), (2), and (3) is satisfied and we will write if only (3) is satisfied.

The partial order on satisfies the following properties:

(i)If , then , where is the zero of complex number.(ii) is equivalent to (iii)If and is a real number, then (iv)If and with , then (v) and do not imply (vi) does not imply . Moreover, if and , then

Definition 1. Let be a nonempty set. Suppose that the mapping satisfies the conditions

(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 an interior point of a set whenever there exists such that . A subset in is called open whenever each point of is an interior point of . The family is a subbasis for a Hausdorff topology on .
A point is called a limit point of whenever, for every . A subset is called closed whenever each limit point of belongs to .
Let be a sequence in and If, for every , with there is such that, for all , , then is called the limit of and we write 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.

Lemma 2 (see [18]). Let be a complex valued metric space and let be a sequence in . Then converges to if and only if as .

Lemma 3. Let be a complex valued metric space and let be a sequence in . Then is a Cauchy sequence if and only if as .

Recall that if and are two self-maps on a set and for some in , then is called a coincidence point of and and is called a point of coincidence of and

Definition 4 (see [27]). Two self-maps, and , on a nonempty set are called weakly compatible if they commute at their coincidence point.

We will also need the following proposition from [27].

Proposition 5. Let and be weakly compatible self-maps on a set If and have a unique point of coincidence——then is the unique common fixed point of and

Definition 6 (see [12]). Let be a finite collection of nonempty subsets of a set , where is some positive integer and . The set is said to have a cocyclic representation with respect to the collection and a pair if

(1),(2)

Recall that a function is said to be lower semicontinuous at in if for every there exists a neighborhood of such that for all in . This can be expressed as Also is upper semicontinuous at in if for every there exists a neighborhood of such that for all in . This can be expressed as .

Definition 7. The control functions and are defined as follows:

(i) is a continuous nondecreasing function with if and only if .(ii) is a lower semicontinuous function with if and only if

2. Main Results

Now, we prove fixed and common fixed point theorems in the setup of complex valued metric space. Our first result is the following.

Theorem 8. Let be a complete complex valued metric space and . Suppose that there exist control functions and such that, for ,

holds, where

Then has almost one fixed point.

Proof. Let be an arbitrary element in Construct a sequence in as or equivalently as for If, for some holds, then we have and is the fixed point of . So we assume that , for all ; that is, for all . From (1), we have

where

If , then

and, by the property of , we obtain
In case , then

implies , a contradiction.
Finally if , then gives a contradiction.
Thus

for all and is the decreasing of positive real numbers. Consequently, there exists such that converges to . Suppose that . Now

where
If , then lower semicontinuity of gives that

which implies , a contradiction as .
In case , then we obtain

which implies that , and so which implies , a contradiction.
If , then we have , a contradiction.
Therefore ; that is, .
Now, we claim that . If not, then there exist and sequences , in , with , such that for all Without any loss of generality, we can assume that Since is a subsequence of convergent sequence as , therefore, as . Now

implies that

From (12) and inequality , it follows that . Also, the inequality and (12) give that , and hence we have

From (13) and inequality , it is implied that , while inequality and (13) imply that , and hence we have

From (12) and , we have , and the inequality and (12) give that . So

As

so we have

Now, if , then, from (1), it follows that

and taking upper limit as implies that , a contradiction as .
In case , then, from (1), it follows that , a contradiction.
When , then, from (1), it follows that

which implies ; that is, , a contradiction.
Thus, we obtain that , and hence is a Cauchy sequence in the complex valued metric space
Next, we assume there exists a point such that .
Thus,

Taking limit as , we conclude that Hence

and taking upper limit as implies that

which further implies that
To prove the uniqueness of fixed point of , assume on the contrary that and with . Note that

where

Now, if , then from (23) it follows that

a contradiction as Hence .
In case , then (23) gives a contradiction.
Finally, if , then, from (23),

which gives , a contradiction. Hence for all cases we have

Corollary 9. Let be a complete complex valued metric space and . Suppose that there exist control functions and such that for

holds for , where

Then has almost one fixed point in and .

Proof. Taking It follows from Theorem 8 that has a unique fixed point . Also implies that is also the fixed point of . From the uniqueness of fixed point of , it is implied that

Recently, Haghi et al. [28] proved the following lemma.

Lemma 10. Let be a nonempty set and let be a self-map on . Then there exists a subset such that is one-to-one and .

Theorem 11. Let be a complex valued metric space and . Suppose that there exist control functions and such that for any

holds, where

If is a complete subspace of , then and have at most one coincidence point in . Furthermore, if and are weakly compatible, then and have a unique common fixed point in .

Proof. It follows from Lemma 10 that there exists such that is one-to-one and . Define a map by . Since is one-to-one on , is well-defined. Note that

for all , where

Since is complete, by Theorem 8, there exists such that . Hence, and have at most one point of coincidence in . By Proposition 5, and have at most one common fixed point.

3. Cocyclic Contraction Mappings

In this section, we prove common fixed point results for self-maps satisfying cocyclic contractions defined on a complex valued metric space. Our first result is the following.

Theorem 12. Let be a complex valued metric space, let be nonempty closed subsets of , and let . Suppose that are such that

(a) has a cocyclic representation with respect to pair and to the collection ;(b)there exist control functions and such that, for any ,  ,

 holds where

 with .

If is complete subspace of for each , then and have a unique coincidence point provided that and is closed. Furthermore, if and are weakly compatible, then and have a unique common fixed point.

Proof. Let be a given element. By the given condition, there exists such that . Since , we can choose a point such that Continuing this process, for , there exists such that, having chosen , we obtain such that . If for some ,   ; that is, and is the coincidence point of and . Taking for all , we have

where

If, for some ,  , then

implies that
In case , then

implies , a contradiction.
When , then gives a contradiction.
Thus

for all and is the decreasing of positive real numbers. Consequently, there exists such that converges to Suppose that Now

where

If  , then lower semicontinuity of gives that

which implies that , a contradiction as .
In case , then we obtain

which implies that , and so which implies , a contradiction.
If , then we have , a contradiction. Therefore ; that is,

Now we claim that is a complex valued Cauchy sequence. If not, there are and even integers and with such that

and Note that

From (44) and (45), it follows that

By (47) and

we have . Also, by (47) and

we obtain that . Hence

From (44) and

we have . By (50) and the inequality

we have . Thus

As

so we have .
Now, if , then, from (b), it follows that

and taking upper limit as implies that , a contradiction as .
In case , then, from (b), it follows that , a contradiction.
When , then, from (b), it follows that

which implies ; that is, , a contradiction.
Hence is a Cauchy sequence in . Since is complete, there exists such that

Consequently, we can find a point in such that
Now we show that From condition (a) and for some , we can choose a subsequence in out of the sequence Obviously, . As is closed, so . Similarly, we can choose a subsequence in out of the sequence Obviously, . As is closed, so . Continuing this way, we obtain that and hence .
Now we show that . Since , there exists some in such that . Choose a subsequence of with . From (b), we have

where

which gives . Thus and hence Thus is the coincidence point of and .
Thus being a finite intersection of closed sets is closed and hence complete. Consider the restrictions mappings . Clearly, Also is closed and hence complete. From Theorem 11, it follows that and have at most one coincidence point in . As and are weakly compatible, from Proposition 5, it follows that and have at most one common fixed point.

Corollary 13. Let be a complex valued metric space, let be nonempty closed subsets of , and let . Suppose that are such that

(a) has a cocyclic representation with respect to pair and to the collection for some ;(b)there exist two control functions, and , such that, for any ,

 holds for some , where

 with .

If is a complete subspace of for each , then and have a unique coincidence point provided that and is closed. Moreover, if and are weakly compatible, then and have a unique common fixed point. Furthermore, and have at most one common fixed point provided that and commute.

Proof. Set and From Theorem 12, and have a unique common fixed point . Now and imply that is the common fixed point of and . Also, and imply that is also the common fixed point of and By the uniqueness of common fixed point of and , we have .

Example 14. Let and let be the complex valued metric defined by , where Suppose and and , where and . Define by

where with Note that ,  , and Hence is a cocyclic representation with respect to pair and the collection . Define by

Clearly is continuous and nondecreasing and is lower semicontinuous with if and only if
We show that condition (b) is satisfied:

(I)Let and If , then and hence (b) holds. If , then we have

(II)Let If and , then and hence (b) is satisfied. If and , then we obtain that

 When and , we have

 For and , we obtain that

(III)Let If , then and (b) is satisfied. If , then we have

Hence and satisfy all the conditions of Theorem 12. Moreover, is a unique common fixed point of and in .

We denote is Lebesgue integrable with finite integral on each compact subset of and, for each ,   . As a consequence of Theorem 12, we obtain the following common fixed point result for mappings satisfying cocyclic contractive condition of integral type in a complex valued metric space.

Corollary 15. Let be a complex valued metric space, let , be nonempty closed subsets of , and let . Suppose that are mappings such that

(a) has a cocyclic representation with respect to pair and to the collection ;(b)there exist control functions and such that, for any ,

 is satisfied, where and

 with .

If is a complete subspace of for each , then and have a unique coincidence point provided that and is closed. Moreover, if and are weakly compatible, then and have a unique common fixed point.

Proof. Define by From (69), it follows that

which can be written as

where and . Clearly, is continuous and nondecreasing, is lower semicontinuous, and if and only if Hence, by Theorem 12, and have a unique common fixed point.

Remark 16. It is worth mentioning that the results obtained in this paper are distinct from the results in [8] in the following sense:

(1)The results in this paper are proved in the setup of complex valued metric spaces in which the distance between any two elements is a complex number but the results proved in [8] are in the setup of multiplicative metric spaces, where the distance between any two elements is a real number greater than or equal to 1.(2)The contractive conditions imposed on the mappings established in this paper are substantially different from contractions used in [8].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The second author thanks the Spanish Ministry of Economy and Competitiveness for its partial support of this work through Grant DPI2012-30651. The authors also thank the Basque Government for its support through Grant IT378-10 and the University of Basque Country for its support through Grant UFI 11/07.

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