#### Abstract

The aim of this paper is to present common fixed point results of quasi-weak commutative mappings on a closed ball in the framework of multiplicative metric spaces. Example is presented to support the result proved herein. We also study sufficient conditions for the existence of a common solution of multiplicative boundary value problem. Our results extend and improve various recent results in the existing literature.

#### 1. Introduction and Preliminaries

The letters , , and will denote the set of all real numbers, the set of all nonnegative real numbers, and the set of all natural numbers, respectively.

Consistent with [1, 2], the following definitions and results will be needed in the sequel.

Definition 1 (see ). The multiplicative absolute value function is defined as

Using above definition of multiplicative absolute value function, we deduce the following proposition.

Proposition 2. For arbitrary , the following hold: (1);(2);(3);(4) if and only if ;(5).

Bashirov et al.  studied the concept of multiplicative calculus and proved the fundamental theorem of multiplicative calculus. Florack and Assen  displayed the use of the concept of multiplicative calculus in biomedical image analysis. Bashirov et al.  exploit the efficiency of multiplicative calculus over the Newtonian calculus. They demonstrated that the multiplicative differential equations are more suitable than the ordinary differential equations in investigating some problems in various fields. Furthermore, Bashirov et al.  illustrated the usefulness of multiplicative calculus with some interesting applications. With the help of multiplicative absolute value function, they defined the multiplicative distance between two nonnegative real numbers as well as between two positive square matrices. This provides the basis for multiplicative metric spaces.

Definition 3 (see ). Let be a nonempty set. A function is said to be a multiplicative metric on if for any , the following conditions hold:) and if and only if ;(); (). The pair (, ) is called a multiplicative metric space.

Example 4 (see ). Let be the collection of all -tuples of positive real numbers. Then defines a multiplicative metric on .

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

Definition 6. Let and be multiplicative metric spaces and an arbitrary but fixed point in . A map is said to be multiplicative continuous at if and only if in implies in for every multiplicative convergent sequence in . That is, given arbitrary , there exists such that whenever for .

Example 7. Let be the collection of all real-valued multiplicative continuous functions over . Then is a multiplicative metric space with defined by
For more examples of multiplicative metric spaces, we refer to [1, 2].

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

A sequence in is multiplicative Cauchy if and only if as .

For sake of brevity we skip the proof of the following lemma.

Lemma 9. Let be the collection of all real-valued multiplicative continuous functions over with the metric defined by Then is complete.

Definition 10 (see ). Let be maps. A point is called(1)fixed point of if ;(2)coincidence point of the pair if ;(3)common fixed point of the pair if . The sets of all fixed points of , coincidence points of the pair , and all common fixed points of the pair are denoted by , and , respectively.

One of the simplest and most useful results in fixed point theory is the Banach-Caccioppoli contraction mapping principle, a powerful tool in analysis for establishing existence and uniqueness of solution of problems in different fields. Over the years, this principle has been generalized in numerous directions in different spaces. These generalizations have been obtained either by extending the domain of the mapping or by considering a more general contractive condition on the mappings.

Recently, Ozavsar and Cervikel  generalized the celebrated Banach contraction mapping principle in the setup of multiplicative metric spaces.

Definition 11 (see ). Let be a multiplicative metric space. A mapping is said to be multiplicative contractive if there exists such that

Theorem 12 (see ). Let be a complete multiplicative metric space and a multiplicative contractive mapping. Then has a unique fixed point.

Definition 13. Let be a multiplicative metric space and let . The mapping is said to be -multiplicative contraction if there exists such that

Definition 14 (see ). Let be a multiplicative metric space and . The pair is said to be(a)commutative if for all in (b)weakly commutative if for all in .

He et al.  extended the results in  to two pairs of self-mappings satisfying certain commutative conditions on a multiplicative metric space. They actually proved the following result.

Theorem 15. Let , , , and be self-maps of a complete multiplicative metric space and () and weakly commuting pair with , , and one of the mappings , , , and is continuous. If where for holds. Then , , , and have a unique common fixed point.

The study of contractive conditions on the entire domain has been at the center of vigorous research activity (see  and references therein) and it has a wide range of applications in different areas such as nonlinear and adoptive control systems, parameterize estimation problems, fractal image decoding, computing magnetostatic fields in a nonlinear medium, and convergence of recurrent networks (see, e.g., ).

If a mapping does not satisfy a contractive condition on the entire space , a natural question arising in that direction is, whether it is still possible to guarantee the existence of a fixed point. An affirmative answer to this question is provided by (a) the restriction of the domain to the subset of , where the mapping is contractive (b) the suitable choice of a point in which force the Picard sequence to stay within the set . Recently Azam et al.  (see also, ) proved a significant result concerning the existence of fixed point of a mapping satisfying contractive conditions on a closed ball of a complete metric space.

The purpose of this paper is to establish the existence and uniqueness of common fixed point of quasi-weak commutative contractive mappings defined on a closed ball in a multiplicative metric space. Our results extend and improve the results of He et al. , Arshad et al. , and many others.

#### 2. Main Result

In this section, we obtain several common fixed point results of mappings on multiplicative closed balls in the framework of multiplicative metric spaces.

Theorem 16. Let , , , and be self-maps of a complete multiplicative metric space and () and weakly commutative with , , and one of , , , and is continuous. If for some given point in and there exists with such that holds, where . Then there exists a unique common fixed point of , , , and in provided that for some in .

Proof. Let be a given point in . Since , we can choose a point in such that . Similarly, there exists a point such that . Indeed, it follows from the assumption that . Thus we can construct sequences and in such that Now we show that is a sequence in . Note that . Hence .
Assume for some . Then, if , it follows from (8) that Thus for all , where .
Similarly, , and we obtain Hence Therefore for all . Now Thus Since , we have that for all . This implies . By induction on , we conclude that for all . We claim that the sequence satisfies the multiplicative Cauchy criterion for convergence in . To see this let be such that ; then Consequently as . Hence the sequence is a multiplicative Cauchy sequence.
As is complete so is . Hence has a limit, say , in .
The fact that and are subsequences of makes .
Suppose is continuous; then By weak commutativity of a pair , we have Taking limit as on both sides of (18), we get which further implies that . Now by condition (8), we have Taking limit as on both sides of (20), we obtain That is, . Hence and is a fixed point of in . In similar way, by condition (8) we have Taking limit as on both sides of (22), we get Hence and is a fixed point of in .
Because of the fact that , let in be such that . So it follows from (8) that which implies and . Since the pair is weakly commutative from our assumptions, thus Hence . By (8), we obtain which implies . Hence is a common fixed point of , and in . If is continuous, then following arguments similar to those given above, we obtain that . Now suppose that is continuous. Thus As the pair is weakly commuting, we have Taking limit as on both sides of (28), we have and .
By contractive condition (8) we get Taking limit as on both sides of (29) implies that .
Hence and is a fixed point of in .
Since , let in be such that . It follows from condition (8) again that Taking limit as on both sides of (30) implies that . Thus . Since the pair is weakly commutative from our hypothesis, then which implies that . From (8), we have Taking limit as on both sides of (32) gives and . However , so let be such that . It follows from (8) also that which implies that . Hence . Since and are weakly commutative, gives . Applying condition (8), we obtain which implies that . Hence is a common fixed point of , , , and in . If is continuous, then using arguments similar to those given above, the result follows.
We proceed to show the uniqueness of the common fixed point of the maps , , , and . So, let be another common fixed point of , , , and . By (8), we have That is, . Hence and this implies that the common fixed point of , , , and is unique.

The following result generalizes Theorem 12. We obtain a common fixed point result in the setup of multiplicative metric spaces without the assumption of continuity and weak commutativity.

Theorem 17. Let and be two maps on a complete multiplicative metric space and an arbitrary point in . Suppose that there exists in such that for any and is satisfied. Then, there exists a unique common fixed point of and in .

Proof. Let be a given point in . Define a sequence in such that and for all . We show that for all . Note that This implies that . Let for some . Clearly, if , then Similarly, if , then Hence for any in , we have . Now implies that Thus we have Since , then for all .
Hence . By induction on , we conclude that for all . Now we show that is Cauchy in . Therefore for each such that , Note that as we get . Hence is a multiplicative Cauchy sequence. By the completeness of , it follows that for a point . Also, we have which on taking limit as tends to infinity gives Thus, is a fixed point of . In a similar manner, we can observe that is a fixed point of . Thus and have a common fixed point in . Note also that is a unique common fixed point of and in . Indeed, if is another fixed point of and , then which implies that .

Corollary 18. Let and be two maps on a complete multiplicative metric space . Suppose that there exists in such that Then, there exists a unique common fixed point of and in .

Proof. Put in Theorem 17.

Example 19. Let and be a multiplicative metric defined by . Note that is a complete multiplicative metric space. Define mappings  and  by Obviously, maps are continuous, and are weak commutative with , and . Choose ; then there exists such that and gives . Also, . Thus Moreover, for and with we have and so holds. Note also that That is, Thus all conditions of Theorem 16 are satisfied. Moreover is the unique common fixed point of , , , and in .

#### 3. Application to Multiplicative Boundary Value Problems

Consider a system of two multiplicative differential equations where for sufficiently small and and are multiplicative continuous functions defined from to , with and in . It is easy to see that solution of problem (53) is equivalently a solution of the following multiplicative integral equations Suppose and satisfy the following multiplicative Lipschitz type condition with respect to second coordinate. That is for some constant and . Let ; then there exists a unique solution of (54) on some closed interval , for sufficiently small such that .

Proof. Define and as and . Assume and are arbitrary members of ; then This implies that and satisfy all the hypothesis of Corollary 18; we conclude that and admit a unique common fixed point which is clearly the (unique) common solution to the multiplicative integral equations (54) and hence to (53).

#### Conflict of Interests

The authors declare that they have no competing interests.