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Discrete Dynamics in Nature and Society
Volume 2015, Article ID 532725, 10 pages
http://dx.doi.org/10.1155/2015/532725
Research Article

Common Fixed Points of Generalized Rational Type Cocyclic Mappings in Multiplicative Metric Spaces

1Department of Mathematics and Applied Mathematics, University Pretoria, Lynnwood Road, Pretoria 0002, South Africa
2Institute of Research and Development of Process IIDP, Faculty of Science and Technology, University of the Basque Country, Spain
3Division of Applied Mathematics, School of Education, Culture & Communication, Mälardalen University, 72123 Västerås, Sweden
4Department of Mathematics, COMSATS Institute of Information Technology, Abbottabad 22060, Pakistan

Received 5 May 2015; Accepted 24 June 2015

Academic Editor: Irena Rachůnková

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

The aim of this paper is to present fixed point result of mappings satisfying a generalized rational contractive condition in the setup of multiplicative metric spaces. As an application, we obtain a common fixed point of a pair of weakly compatible mappings. Some common fixed point results of pair of rational contractive types mappings involved in cocyclic representation of a nonempty subset of a multiplicative metric space are also obtained. Some examples are presented to support the results proved herein. Our results generalize and extend various results in the existing literature.

1. Introduction and Preliminaries

Fixed point theory provides the most important and traditional tools for proving the existence of solutions of many problems in both pure and applied 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. The study of fixed points of mappings which satisfy certain contractive conditions has primary applications in the solution of differential and integral equations (see, e.g., [15]). Theorems dealing with fixed point of certain mappings inspired and motivated the development of many other important kinds of points like coincidence points, intersection points, sectional points, and so forth.

One of the basic and most widely applied fixed point theorems in mathematical analysis is “Banach Contraction Mapping Principle” (or Banach’s Fixed Point Theorem). Kirk et al. [4] obtained some fixed point results for mappings satisfying cyclical contractive conditions. The Banach contraction mappings are continuous mappings, while cyclic contraction mappings need not be continuous. Păcurar and Rus [6] studied some fixed point results for cyclic weak contractions. Piątek [7] obtained some results on cyclic Meir-Keeler contractions in metric spaces. Using fixed point result of weakly contractive map, Karapinar [3] established some interesting fixed point results for cyclic -weak contraction mappings. Recently, Derafshpour et al. [8] obtained results on the existence of best proximity points of cyclic contractions. For more results in this direction, we refer to [915].

Banach contraction principle has been generalized either by extending the domain of the mapping or by considering a more general contractive condition on the mappings. Ozavsar and Cevikel [16] proved an analogous result to Banach contraction principle in the framework of multiplicative metric spaces. They also studied some topological properties of the relevant multiplicative metric space. Bashirov et al. [17] studied the concept of multiplicative calculus and proved a fundamental theorem of multiplicative calculus. They also illustrated the usefulness of multiplicative calculus with some interesting applications. Multiplicative calculus provides natural and straightforward way to compute the derivative of product and quotient of two functions [18]. It was shown that the multiplicative differential equations are more suitable than the ordinary differential equations in investigating some problems in economics and finance. Due to its operational simplicity and support to Newtonian calculus, it has attracted the attention of several researchers in the recent years. Furthermore, based on the definition of multiplicative absolute value function, they defined the multiplicative distance between two nonnegative real numbers and between two positive square matrices. This provided the basis for multiplicative metric spaces. Florack and van Assen [19] gave applications of multiplicative calculus in biomedical image analysis. He et al. [20] studied common fixed points for weak commutative mappings on a multiplicative metric space (see also [21]). Recently, Yamaod and Sintunavarat [22] obtained some fixed point results for generalized contraction mappings with cyclic ()-admissible mapping in multiplicative metric spaces.

In this paper, we obtain fixed point result of a generalized rational contractive mapping in the framework of multiplicative metric spaces. 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 types mappings involved in cocyclic representation of a nonempty subset of a multiplicative metric space. Our results generalize and extend comparable results in [3].

By , , , and we denote the set of all real numbers, the set of all nonnegative real numbers, the set of all -tuples of positive real numbers, and the set of all natural numbers, respectively.

Consistent with [16, 17], the following definitions and results will be needed in the sequel.

Definition 1 (see [17]). Let be a nonempty set. A mapping is said to be a multiplicative metric on if for any , the following conditions hold:(i) and if and only if .(ii).(iii).
The pair is called a multiplicative metric space.

We define absolute valued function that includes the negative real numbers in its domain.

Definition 2. A multiplicative absolute value function is defined as

Using the definition of multiplicative absolute value function, we can prove the following proposition.

Proposition 3. For arbitrary , the multiplicative absolute value function satisfies the following:(1).(2).(3) if and if .(4).

Example 4 (see [16]). Let . Thendefine multiplicative metrices on , where and .

Definition 5 (see [16]). Let be a multiplicative metric space, an arbitrary point in , and . A multiplicative open ball of radius centered at is the set .
A sequence in a multiplicative metric space is said to be multiplicative convergent to some point if, for any given , there exists such that for all . If converges to , we write as .

Definition 6 (see [16]). A sequence in a multiplicative metric space is multiplicative convergent to in if and only if as .

Definition 7. Let and be two multiplicative metric spaces and an arbitrary but fixed element of . A mapping is said to be multiplicative continuous at if and only if in implies that in . That is, given arbitrary , there exists which depends on and such that for all those in for which .

Definition 8 (see [16]). A sequence in a multiplicative metric space is said to be multiplicative Cauchy sequence if, for any , there exists such that for all .
A multiplicative metric space is said to be complete if every multiplicative Cauchy sequence in is multiplicative convergent in .

Theorem 9 (see [16]). A sequence in a multiplicative metric space is multiplicative Cauchy if and only if as .

Example 10. Let be the collection of all real-valued multiplicative continuous functions over with the multiplicative metric defined bywhere is a multiplicative absolute value function. Then is complete.

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 11 (see [23]). 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 [23].

Proposition 12. 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 13 (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), .

Definition 14. 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

In this section, we obtain several fixed and common fixed point results of self-maps satisfying certain generalized contractive conditions in the framework of multiplicative metric space.

We start with the following result.

Theorem 15. Let be a complete multiplicative metric space and . Suppose that there exist control functions and such thatfor any , whereThen has a unique fixed point.

Proof. Let be a given point in . Define a sequence in as or equivalently as for . If for some , then we have and the result follows. Assume that , for all ; that is, for all . From (4), we havewhereIf for some , then implies , a contradiction. Hence for all . Also, for all and hence for all . Thus, we havewhich implies thatfor all . Thus is (strictly) decreasing of positive real numbers. Consequently, there exists such that converges to . Suppose that . Nowand lower semicontinuity of gives thatwhich implies that , a contradiction as . 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 , then as . Nowimplies thatFrom (13) and inequality , it follows that . Also, the inequality and (13) give that , and hence we haveEquation (14) and inequality imply that , while inequality and (14) imply that , and hence we haveFrom (13) and , we have , and the inequality and (13) give that . SoAswe have . From (4), it follows thatTaking upper limit as implies that , a contradiction as . Thus, we obtain that , and hence is a multiplicative Cauchy sequence in . Next, we assume there exists a point such that ; equivalently,Note thatTaking limit as , we conclude that . Henceon taking upper limit as implies thatwhich further implies that .
To prove the uniqueness of fixed point of , assume on the contrary that and with . Note thatwhereFrom (23) it follows thata contradiction as . Hence .

Corollary 16. Let be a complete multiplicative metric space and . Suppose that there exist control functions and such thatfor any and whereThen has a unique fixed point and .

Proof. Set . From Theorem 15, has a unique fixed point . Now which implies that is also a fixed point of . By the uniqueness of fixed point of , we have .

Now, we recall the following lemma from [24].

Lemma 17. Let be a nonempty set and . Then there exists a subset such that and is one-to-one.

Theorem 18. Let be a multiplicative metric space and . Suppose that there exist control functions and such thatholds for any , whereIf is a complete subspace of , then and have a unique coincidence point in . Moreover, if and are weakly compatible, then and have at most one common fixed point.

Proof. By Lemma 17, there exists such that and is one-to-one. Define a map by . Since is one-to-one on ,   is well defined. Note thatfor all whereSince is complete, by Theorem 15, there exists such that . Hence, and have a unique point of coincidence. From Proposition 12, and have a unique common fixed point.

3. Cocyclic Contractions

Now we obtain common fixed point results for self-maps satisfying certain cocyclic contractions defined on a multiplicative metric space. We start with the following.

Theorem 19. Let be a multiplicative metric space, nonempty closed subsets of , and . Suppose that are such that(a) has a cocyclic representation with respect to pair and the collection ;(b)there exists control functions and such that, for any , , the following,holds wherewith .
If is 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 at most one common fixed point.

Proof. Let be a given point in . Then there exists such that . Choose a point in such that . This can be done because . Continuing this process, for , there exists such that having chosen in , we obtain in such that . If, for some , we have , then implies that is the coincidence point of and . Assume that for all . From (b), we havewhereIf, for some , , then andimply that , a contradiction as . Hence for all . Thus . On the other hand, we have . Hence .
Similarly we obtain that . Thus the sequence is nonincreasing. Consequently, there exists such that . Suppose that . Nowand lower semicontinuity of gives thatwhich implies that , a contradiction. Therefore . That is,Assume that is not a multiplicative Cauchy sequence. Then, there is , and there are even integers and with such thatand . Note thatFrom (39) and (40), it follows thatBy (42) andwe have Also, by (42) andwe obtain that . HenceFrom (39) andwe have . By (45) and the inequalitywe have . ThusNote thatConsequently, Since and lie in different but adjacent labelled sets and for some , we haveTaking upper limit as implies that , a contradiction as , and hence is a multiplicative Cauchy sequence in . Since is complete, there exists such thatConsequently, 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, . Similarly, we can choose a subsequence in out of the sequence . Obviously, . As is closed, . 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 havewhereOn taking upper limit as we obtain thatand hence . Thus is the coincidence point of and .
Note that being a finite intersection of closed sets is closed and hence complete. Consider the restrictions of and on ; that is, ,  . Obviously, Also is closed and hence complete. From Theorem 18, it follows that and have a unique coincidence point in . As and are weakly compatible, from Proposition 12, it follows that and have a unique common fixed point.

Corollary 20. Let be a multiplicative metric space, nonempty closed subsets of , and . Suppose that are such that(a) has a cocyclic representation with respect to pair and the collection for some ;(b)there exists two control functions and such that, for any , ,holds for some , where with .
If is 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 a unique common fixed point provided that and are commuting.

Proof. Set and . From Theorem 19, 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 uniqueness of common fixed point of and , we have .

Example 21. Let and let be a multiplicative metric on defined by , where is a real number. For some , set , , and . Define bywhere with and . Note that and . has a cocyclic representation with respect to pair and the collection .
Define byClearly is continuous and nondecreasing, is a lower semicontinuous, and if and only if .
We show that condition (b) is satisfied. Now, for , impliesWhen and , Thus, and satisfy all the conditions of Theorem 19. Moreover, and have at most one common fixed point.

Example 22. Let and be the multiplicative metric defined by . Suppose , , and and , where . Define byNote that , , and . Hence is a cocyclic representation between and .
Let be as defined in Example 21. To check condition (b), we consider the following cases:(1)Let and . If , then and hence (b) holds. If , then we have(2)Let and If and , then and hence (b) is satisfied. If and , then we obtain thatWhen and , we haveFor and , we obtain that(3)Let and . If , then and (b) is satisfied. If , then we haveHence and satisfy all the conditions of Theorem 19. Moreover, is a unique common fixed point of and in .

Denote ; is a Lebesgue integrable with finite integral on each compact subset of and for each , .

As a consequence of Theorem 19, we obtain following common fixed point result for mappings satisfying cocyclic contractive condition of integral type in a multiplicative metric space.

Corollary 23. Let be a multiplicative metric space, , nonempty closed subsets of , and . Suppose that are mappings such that(a) has a cocyclic representation with respect to pair and the collection ;(b)there exists control functions and such that, for any , , the following,is satisfied, where , and