A Common Random Fixed Point Theorem for Weakly Compatible Mappings in Cone Random Metric Spaces

In this paper, we prove a unique common random fixed point theorem in the f ramework of cone random metric spaces for four weakly random compatible mappings under strict contractive condition. Some corollaries of this theorem for three and two weakly random compatible mappings and for one random mapping are derived. Two examples to justify our theorem are given. Our results extend some previous work related to cone random metric spaces from the current existing literature.


Introduction
Fixed point theory has the diverse applications in different branches of mathematics, statistics, engineering, and economics in dealing with the problems arising in approximation theory, potential theory, game theory, theory of differential equations, theory of integral equations, and others. Developments in the investigation on fixed points of non-expansive mappings, contractive mappings in different spaces like metric spaces, Banach spaces, Fuzzy metric spaces and cone metric spaces have almost been saturated. The study of random fixed point theorems was initiated by the Prague school of probabilistic in 1950's [9,10,23]. The introduction of randomness leads to several new questions of measurability of solutions, probabilistic and statistical aspects of random solutions. Common random fixed point theorems are stochastic generalization of classical common fixed point theorems. Random methods have revolutionized the financial markets. The survey article by Bharucha-Reid [8] in 1976 attracted the attention of several mathematicians and gave wings to the theory. The results ofŠpaček and Hanš in multi-valued contractive mappings was extended by Itoh [14]. Now this theory has become the full fledged research area and various ideas associated with random fixed point theory are used to give the solution of nonlinear system see [5-7, 11, 20]. Common random fixed points and random coincidence points of a pair of compatible random operators and fixed point the-orems for contractive random operators in Polish spaces are obtained by Papageorgiou [15,16] and Beg [3,4].
In [12] Huang and Zhang generalized the concept of metric spaces, replacing the set of real numbers by an ordered Banach space, hence they have defined the cone metric spaces. They also described the convergence of sequences and introduced the notion of completeness in cone metric spaces. They have proved some fixed point theorems of contractive mappings on complete cone metric space with the assumption of normality of a cone. According to this concept, several other authors [1,13,19,22] studied the existence of fixed points and common fixed points of mappings satisfying contractive type condition on a normal cone metric space. In 2008, the assumption of normality in cone normal spaces is deleted by Rezapour and Hamlbarani [19], which is an important event in developing fixed point theory in cone metric spaces.
The aim of this paper is to extends the contractive condition (2.1) for four, three and two random mappings and establish a unique random fixed point results under this condition in random cone metric spaces using the concept of weakly random compatible mappings.

Definition [21]
Let (E, τ ) be a topological vector space. A subset p of E is called a cone if the following conditions satisfied: (c 1 ) p is closed, nonempty and p = {0}; (c 2 ) a, b ∈ R, a, b ≥ 0 and x, y ∈ p ⇒ ax + by ∈ p; (c 3 ) If x ∈ p and −x ∈ p ⇒ x = 0. For a given cone p ⊂ E, we define a partial ordering ≤ with respect to p by x ≤ y iff y − x ∈ p. We shall write x < y to indicate that x ≤ y but x = y, while x y will stand for y − x ∈ p • , where p • indicate to the interior of p. [12,24] Let X be a nonempty set. Assume that the mapping d :

Definition
Then d is called a cone metric [12] or K−metric [24] on X and (X, d) is called a cone metric space [12].
The concept of a cone metric space is more general than that of a metric space, because each metric space is a cone metric space where E = R and p = [0, +∞).

Example [12]
is a cone metric space with normal cone p where K = 1.

Example [18]
Let E = l 2 , p = {{x n } n≥1 ∈ E 2 : x n ≥ 0, for all n}, (X, ρ) a metric space and d : Then (X, d) is a cone metric space. Clearly, the above examples present that the class of cone metric spaces contains the class of metric spaces.

Definition [15]
Let (X, d) be a cone metric space. We say that {x n } is: (i) a Cauchy sequence if for every ε in E with 0 ε, then there is an N such that for all n, m > N, d(x n , x m ) ε; (ii) a convergent sequence if for every ε in E with 0 ε, then there is an N such that for all n > N, d(x n , x) ε for some fixed x in X.
A cone metric space X is said to be complete if every Cauchy sequence in X is convergent in X.
The following definitions are given in [21].

Definition (Measurable function)
Let (Ω, Σ) be a measurable space with Σ−a sigma algebra of subsets of Ω and M be a nonempty subset of a metric space X = (X, d). Let 2 M be the family of nonempty subsets of M and C(M ) the family of all nonempty closed subsets of

Definition (Random operator)
The mapping T : Ω × M → X is called a random operator iff for each fixed x ∈ M, the mapping T (., x) : Ω → X is measurable.

Definition (Continuous random mapping)
A random operator T : Ω × M → X is called continuous random operator if for each fixed x ∈ M and ω ∈ Ω, the mapping T (ω, .) : Ω → X is continuous.

Definition (Cone random metric space)
Let M be a nonempty set and the mapping d : Ω×M → p, where p is a cone, ω ∈ Ω be a selector, satisfy the following conditions: for all x, y, z ∈ M and ω ∈ Ω be a selector, (iv) for any x, y ∈ M, ω ∈ Ω, d(x(ω), y(ω)) is nonincreasing and left continuous. Then d is called cone random metric on M and (M, d) is called a cone random metric space.

Main Results
In this section we shall prove a common random fixed point theorem under a generalized contraction condition for four mappings satisfying some conditions in the setting of cone random metric spaces.
If we take P = Q in above theorem we obtain the following corollary.
Putting P = Q and S = T in above theorem we get the following corollary.
Letting P = I (where I is the identity mapping defined by I(ω, x) = x(ω) for all ω ∈ Ω) in Corollary 3.3, we have
Finally, we present some examples to verify the requirements of Theorem 3.1 as follows.
Hence α(ω) = 2 3 ∈ (0, 1), therefore all requirements of Theorem 3.1 are satisfied and 0 is a unique random fixed point of S, T, P and Q.