eu COMMON RANDOM FIXED POINT THEOREMS OF COMPATIBLE MAPPINGS FOR RANDOM MAPPINGS IN MULTIPLICATIVE METRIC SPACES

In this paper, we introduce the notions of compatible mappings in framework of random fixed points in multiplicative metric spaces and using these notions we prove common random fixed point theorems for these mappings. AMS Subject Classification: 47H10, 54H25


Introduction and Preliminaries
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 et al. [1] studied the multiplicative calculus and defined a new distance so called multiplicative distance.By using this idea, Özavsar and C ¸evikel [5] introduced the concept of multiplicative metric spaces and studied some topological properties in such spaces.Definition 1.1.( [1]) Let X be a nonempty set.A multiplicative metric is a mapping d : X × X → R + satisfying the following conditions: (i) d(x, y) ≥ 1 for all x, y ∈ X and d(x, y) = 1 if and only if x = y; (ii) d(x, y) = d(y, x) for all x, y ∈ X; (iii) d(x, y) ≤ d(x, z) • d(z, y) for all x, y, z ∈ X (multiplicative triangle inequality).
Then the mapping d together with X, that is, (X, d) is a multiplicative metric space.
Example 1.2.( [5]) Let R n + be the collection of all n-tuples of positive real numbers.Let d * : R n + × R n + → R be defined as follows: , where x = (x 1 , . . ., x n ), y = (y 1 , . . ., y n ) ∈ R n + and | • | * : R + → R + is defined by Then it is obvious that all conditions of a multiplicative metric are satisfied.Therefore (R n + , d * ) is a multiplicative metric space.Remark 1.3.( [7]) We note that multiplicative metrics and metric spaces are independent.
One can refer to [5] for detailed multiplicative metric topology.Definition 1.4.Let (X, d) be a multiplicative metric space.Then a sequence {x n } in X said to be (1) a multiplicative convergent to x if for every multiplicative open ball (2) a multiplicative Cauchy sequence if for all ǫ > 1, there exists (3) We call a multiplicative metric space complete if every multiplicative Cauchy sequence in it is multiplicative convergent to x ∈ X.
In 2012, Özavsar and C ¸evikel [5] gave the concept of multiplicative contraction mapping and proved some fixed point theorem of such mappings in a complete multiplicative metric spaces.Definition 1.5.Let f be a mapping of a multiplicative metric space (X, d) into itself.Then f is said to be a multiplicative contraction if there exists a real number λ ∈ [0, 1) such that d(f x, f y) ≤ d λ (x, y) for all x, y ∈ X.

Properties of Compatible Mappings for Random Mappings
Now we introduce the following.Definition 2.1.Let (X, d) be a multiplicative metric space and F : R × X → X be a mapping, where X is a nonempty set.Then a mapping g : R → X is said to be a random fixed point of the function 6 and a mapping g : R → X defined by g(t) = t for every t ∈ R. Then T has a unique random fixed point in X Next we introduce the notions of (weakly) commuting and compatible mappings for random mappings in multiplicative metric spaces as follows: Definition 2.3.Let (X, d) be a multiplicative metric space.Two mappings A and B : R × X → X are called (1)  Remark 2.4.Notice that commuting mappings are obviously weakly commuting.However, converse need not be true, in general, as follows.
Example 2.5.Let X = R be the usual multiplicative metric d defined as d(x, y) = 3 |x−y| .Define mappings A and B : R × X → X by for all x ∈ X, t ∈ R and g : R → X defined by g(t) = √ t.Then A and B are weakly commuting but not commuting.
Remark 2.6.Every weakly commuting mappings is compatible but converse need not be true, in general, as follows.
Example 2.7.Let X = R be the usual multiplicative metric d defined as d(x, y) = 3 |x−y| .Define mappings A and B : R × X → X by Then A and B are compatible but not weakly commuting mappings.Definition 2.8.Let (X, d) be a multiplicative metric space and A and B : R × X → X be mappings.Then the mapping A and B are called jointly continuous Next we give the properties of compatible mappings for random mappings.Proposition 2.9.Let (X, d) be a multiplicative metric space and A and B : R × X → X be mappings.Assume that A and B are compatible mappings.
Proof.Suppose that {g n } is a sequence in X defined by g n (t) = g(t), n = 1, 2, . .., for each t ∈ X and A(t, g(t)) = B(t, g(t)).Then we have A(t, g n (t)) and B(t, g n (t)) → A(t, g(t)) as n → ∞.Since A and B are compatible, we have Hence we have A(t, B(t, g(t)) = B(t, A(t, g(t)).Therefore, we have This completes the proof.
From Proposition 2.9 we have Proposition 2.10.Let (X, d) be a multiplicative metric space and A and B : R × X → X be mappings.
Suppose that Since A and B are compatible, we have This completes the proof.(b) The proof of lim n→∞ A(t, B(t, g n (t))) = B(t, g(t)) follows by similar arguments as in (a).(c) Suppose that A and B are jointly continuous.Since B(t, g n (t)) → g(t) as n → ∞ and A is jointly continuous, by (a), B(t, A(t, g n (t))) → A(t, g(t)) as n → ∞.On the other hand, B is also jointly continuous, B(t, B(t, g n (t))) → B(t, g(t)).Thus, we have A(t, g(t)) = B(t, g(t)) by the uniqueness of limit and so by Proposition 2.9, A(t, B(t, g(t))) = B(t, A(t, g(t))).This completes the proof.

Main Results
Now, we prove the random fixed point theorems of compatible mappings for random mappings in a multiplicative metric space.Theorem 3.1.Let (X, d) be a complete multiplicative metric space and A, B, S and T : R × X → X be mappings satisfying the following conditions; S(t, X) ⊂ B(t, X) and T (t, X) ⊂ A(t, X); for all x, y ∈ X and t ∈ R, where λ ∈ (0, 1/2); (C 3 ) one of A, B, S and T is jointly continuous.
Assume that the pairs A, S and B, T are compatible.Then A, B, S and T have a unique common random fixed point.
Hence, in general, Taking n → ∞, we have {y n (t)} is a multiplicative Cauchy sequence and since X is complete so {y n (t)} converges to a point z(t) ∈ X as n → ∞.Also subsequences of {y n (t)} also converges to a point z(t) ∈ X, that is Now suppose that A is jointly continuous, so This implies that d(z(t), T (t, u(t))) = 1 and hence T (t, u(t)) = z(t).Since B and T are compatible and B(t, u(t)) = T (t, u(t)) = z(t), by Proposition 2.9, we have ))) = 1 and hence Also, on putting x(t) = z(t) and y(t) = z(t) in (C 2 ), we have This implies that z(t) = B(t, z(t)).Hence Therefore A, B, S and T have a common random fixed point z(t) in X.
Similarly, we can also complete the proof when B is jointly continuous.
On This implies that z(t) = v(t).Therefore, A, B, S and T have a unique common fixed point in X.This completes the proof.
From Theorem 3.1, Remarks 2.4 and 2.6, we follow the following corollary.
Corollary 3.2.Let (X, d) be a complete multiplicative metric space and A, B, S and T : R × X → X be mappings satisfying the conditions (C 1 )-(C 3 ).
Assume that the pairs A, S and B, T are commuting.Then A, B, S and T have a unique common random fixed point.
From Theorem 3.1 and Remark 2.6, we follow the following corollary.Corollary 3.3.Let (X, d) be a complete multiplicative metric space and A, B, S and T : R × X → X be mappings satisfying the conditions (C 1 )-(C 3 ).
Assume that the pairs A, S and B, T are weakly commuting.Then A, B, S and T have a unique common random fixed point.Remark 3.4.1.In Theorem 3.1, if we take R to be a singleton set, then the result can reduce into the result of Kang et al. [4].Also Theorem 3.1 generalize the result of Kumar et al. [6,Theorem 3.1].
2. In Corollary 3.2, if we take R to be a singleton set, then the result can reduce into the result of Gu et al. [2].
3. In Corollary 3.3, if we take R to be a singleton set, then the result can reduce into the result of He et al. [3].