Turkish Journal of Analysis and Number Theory, 2016, Vol. 4, No. 2, 39-43 Available online at http://pubs.sciepub.com/tjant/4/2/3 ©Science and Education Publishing DOI:10.12691/tjant-4-2-3 Weak Compatibility and Related Fixed Point Theorem for Six Maps in Multiplicative Metric Space Kamal Kumar 1 , Nisha Sharma 2 , Rajeev Jha 3 , Arti Mishra 2 , Manoj Kumar 4,* 1 Department of Mathematics, Pt. JLN Govt. College Faridabad, Sunrise University, Alwar (Rajasthan), India 2 Department of Mathematics, Manav Rachna International University, Faridabad, Haryana, India 3 Department of Mathematics, Teerthankar Mahaveer University, Moradabad (U.P), India 4 Departtment of Mathematics, Lovely Professional University, Punjab, India *Corresponding author: manojantil18@gmail.com Abstract We consider six self-maps satisfying the condition of commuting and weak compatibility of mappings and the purpose of this paper is to give some common fixed points theorems for complete multiplicative metric space. Keywords: commuting mapping, complete multiplicative metric spaces, weakly compatible maps and common fixed points Cite This Article: Kamal Kumar, Nisha Sharma, Rajeev Jha, Arti Mishra, and Manoj Kumar, “Weak Compatibility and Related Fixed Point Theorem for Six Maps in Multiplicative Metric Space.” Turkish Journal of Analysis and Number Theory, vol. 4, no. 2 (2016): 29-43. doi: 10.12691/tjant-4-2-3. 1. Introduction The concept of multiplicative metric spaces is introduced by M. Özavsar [8].They also gave some topological properties of the relevant multiplicative metric space and now it’s more general than well-known metric space. Fixed point theorems are admirable tool for Existence and uniqueness of the solutions to various mathematical models like differential, integral and partial differential equations and vibrational inequalities etc. The study of metric space plays very important role to many fields both in pure and applied science [4]. Abounding researchers extended the notion of a metric space such as vector valued metric space of Perov [3], a cone metric spaces of Huang and Zhang [7], a modular metric spaces of Chistyakov [17], for details about multiplicative metric space and related concepts, we refer the reader to[8] etc. It is well know that the set of positive real numbers ℝ + is not complete according to the usual metric. To overcome this problem, In 2008, Bashirov [2] Introduced the concept of multiplicative metric spaces as follows: Definition 1.1. [8] Let X be a nonempty set. Multiplicative metric is a mapping d: X×X → ℝ + satisfying the following conditions (1.1) d(x, y) ≥ 1 for all x, y ∈ X and d(x, y) = 1 if and only if x = y, (1.2) d(x, y) = d(y, x) for all x, y ∈ X, (1.3) d(x, z) ≤ d(x, y) ∙d(y, z) for all x, y, z ∈ X (multiplicative triangle inequality) To articulate the importance of this study, we should first note that ℝ + is a complete multiplicative metric space with respect to the multiplicative metric. Furthermore, we introduce concept of multiplicative contraction mapping and prove some fixed point theorems of multiplicative contraction mappings on multiplicative metric spaces. Definition 1.2. [8] (Multiplicative ball) Let (X, d) be a multiplicative metric space, x ∈ X and ε > 1. We now define a set B ε (x) = {y ∈ X | d(x, y) < ε}, which is called multiplicative open ball of radius ε with centre x. Similarly, one can describe multiplicative closed ball as ( ) { } y X|d Bε(x x, ) . y ε = ∈ ≤ Definition 1.3. [8] (Multiplicative interior point): Let (X, d) be a multiplicative metric space and A⊂ X. Then we call x ∈ A a multiplicative interior point of A if there exists an ε > 1 such that B ε (x) ⊂ A. The collection of all interior points of A is called multiplicative interior of A and denoted by int(A). Definition 1.4. [8] (Multiplicative open set): Let (X, d) be a multiplicative metric space and A⊂ X. If every point of A is a multiplicative interior point of A, i.e., A = int(A), then A is called a multiplicative open set. Definition 1.5. [8] Let (X, d) be a multiplicative metric space. A point x ∈ X is said to be multiplicative limit point of S ⊂ X if and only if (B ε (x) \ {x}) ∩ S ≠∅ for every ε > 1. The set of all multiplicative limit points of the set S is denoted by S ′. Definition 1.6. [8] Let (X, d) be a multiplicative metric space. We call a set S ⊂ X multiplicative closed in (X, d) if S contains all of its multiplicative limit points. Definition 1.7. [8] (Multiplicative convergence): Let (X, d) be a multiplicative metric space, {x n } be a sequence in X and x ∈ X. If for every multiplicative open ball B ε (x), there exists a natural number N such that n ≥ N⇒x n ∈B ε (x), then the sequence {x n } is said to be multiplicative convergent to x, denoted by x n → x (n → ∞).