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., [1–5]). 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 [9–15].

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 .