RANDOM COMMON FIXED POINT THEOREM FOR RANDOM WEAKLY SUBSEQUENTIALLY CONTINUOUS GENERALIZED CONTRACTIONS WITH APPLICATION

Abstract: In this paper, we prove random common fixed point theorem for two pairs of random self mappings under a generalized contractive condition using subsequential continuity with compatibility of type (E). An example is given to justify our theorem. Our results in randomness extend and improve the results of S. Beloul [4]. Finally, we give an application to discuss the existence of a solution of random Hammerstein integral equations.


Introduction
The notion of commuting mappings is given by Jungck [13] to establish a common fixed point theorem.Two self mappings F and T of a metric space (X, d) are commuting if F T x = T F x for all x ∈ X.Later, weakly commuting mappings are defined by Sessa [19] as a generalization of commuting mappings.A pair of self mappings F and T are weakly commuting if d(F T x, T F x) ≤ d(F x, T x), for all x ∈ X.In [14,15,20] the concepts of compatible, compatible of type (A), compatible of type (B), compatible of type (C), compatible of type (P ) and weakly compatible mappings are presented as a generalization of weakly commuting as follows: Two self mappings F and T are: (1) compatible if lim One can note that, commuting ⇒ weakly commuting ⇒ compatible and compatibility of type (A) ⇒ compatibility of type (C), however compatibility (compatibility of type (A), compatibility of type (B) and compatibility of type (C)) are equivalent under the continuity of F and T .
In 2009, Bouhadjera and G. Thobie [6] introduced the concept of subsequential continuity as follows: Definition 1. Two self maps F and T of a metric space (X, d) is said to be subsequentially continuous if there exists a sequence {x n } such that lim Motivated by Definition 1, S. Beloul [4] give the following definitions Definition 2. Let F and T be two self mappings of a metric space (X, d), the pair {F, T } is said to be weakly subsequentially continuous, if there exists a sequence {x n } such that lim Definition 4. The pair {F, T } is said to be F −subsequentially continuous if there exists a sequence {x n } such that lim Remark 1 If the pair {F, T } is F −subsequentially continuous (or T − subsequentially continuous), then it is weakly subsequentially continuous.
In 1950, by the Prague school of probabilistic, random fixed point theory was initiated in the works of Hans [11] and spacek [21] as stochastic extension of classical fixed point theorem in separable Banach spaces, it plays an important role in the theory of random integral and random differential equations, also random fixed point theory can applied in various areas as optimization, approximation theory, variational inequalities and others.After paper of Bharucha-Ried [5], the field of random fixed point became most widely specially in functional analysis.A large number of researchers focussing on various aspects of random fixed point theory, some of these works see [3,1,12,[16][17][18].

Preliminaries
Let (Ω, Σ) denotes a measurable space consisting of a set Ω and sigma algebra Σ of subsets of Ω, X stands for a separable Banach space and ξ n : Ω → X is a measurable sequence Motivated by definitions of S. Beloul [4] in deterministic, we state the following definitions in stochastic which are used in prove of our result: Definition 6.Let X be a Polish space (separable complete metric space) and T, F : Ω × X −→ X then {F, T } is said to be weakly random subsequentially continuous if there exists a sequence {x n (ω)} such that Definition 7. Let X be a Polish space (separable complete metric space) and T, F : Ω×X −→ X then {F, T } is said to be weakly random subsequentially continuous if there exists a sequence {x n (ω)} such that lim Definition 8.A random self mappings F and T are said to be: for some x ∈ X, ω ∈ Ω and there exists a measurable sequence Let Ω = [0, 1] and let be the sigma measurable algebra subset of [0,1].Define T, F : Ω × X → X be any random operators as follows: for all ω ∈ Ω, x ∈ X, also define the sequence of measurable mapping x n (ω) : By the simple calculations, we have Therefore: (i) the pair {F, T } is weakly random subsequentially continuous, (ii) A random mapping T is T −random subsequentially continuous and F is F −random subsequentially continuous, (iii) a random mappings F and T are random compatible of type (E).Let a function φ : [0, ∞) → [0, ∞) is said to be contractive modulus if φ(t) < t for all t > 0 (see [17]) and non decreasing such that φ(0) = 0.
In this paper, we will establish a stochastic version of random common fixed point theorem for the pairs of random operators using concept of random subsequentially continuous with compatible of type (E), and an example to satisfy our result.Also we use this result to find the existence and uniqueness of a solution of random integral equations.S. Beloul [4] studied the following general contractive type mappings and proved some fixed point theorems for this mappings under weakly subsequentially continuous and compatible of type (E): Let X be a metric space, A, B, S, T : X → X are four self mappings such that for all x, y ∈ X, We now prove the random analogue of S. Beloul [4] with ψ = 2φ.
Proof.Since the pair {T, Q} is weakly random subsequentially continuous, then it is T −random subsequentially continuous and there exists a sequence measurable mappings {y n (ω)} in X such that and lim Again, since F and P are weakly random subsequentially continuous, then there exists a sequence measurable mapping x n (ω), such that and lim n→∞ for all ω ∈ Ω, and x(ω) : Ω → X be measurable mapping.Also (3.9) Now, we claim that F (ω, x(ω)) = T (ω, y(ω)).If not, substituting from (3.5) and (3.9) in (3.1), we get Since φ is a contractive modulus, then again, we will show that x(ω) = F (ω, x(ω)).If not, then from (3.1) and the inequality (3.10), we get Putting n → ∞ in above inequality and using (3.2), we have From definition φ, we obtain that this is a contradiction again, therefore Taking the limit as n → ∞ in above inequality and using (3.2), we obtain From definition φ, we get d(x(ω), y(ω)) < d(x(ω), y(ω)) which is a contradiction again.Therefore x(ω) = y(ω) is a common random fixed point of the four random mappings F, T, P and Q.
Putting φ(t) = 1 2 t, T = F and Q = P in Theorem 3.1, we obtain the following corollary Corollary 10.Let X be a Polish space and T, Q : Ω×X → X are random mappings satisfy for all x, y ∈ X and ω ∈ Ω, If the pair {T, Q} is weakly random subsequentially continuous and random compatible of type (E).Then T and Q have a unique common random fixed point in X.Therefore, the two pairs {T, Q} and {F, P } are weakly random subsequentially continuous and random compatible of type (E).To satisfy the contractive condition (3.12), we consider the following cases: (i) If we take x, y ∈ C − {1}, then the inequality is satisfied, (ii) By taking x = y = 1 in C, we get (iii) If we take x = 1 and y ∈ C − {1}, we get (iv) If we take y = 1 and x ∈ C − {1}, we have From the four cases, the contractive (3.1) is satisfied.Consequently all conditions of Theorem 9 are satisfied, and x(ω) = 5 is a unique common random fixed point of the four mappings F, T, P and Q.

Application to Integral Equation
Using Theorem 9, we discuss the existence and uniqueness of a solution of random integral equations (see [22]) as the following: where (i) R = (−∞, ∞) is a compact with metric d on R × R, (ii) ω ∈ Ω is supporting set of measurable space (Ω, R) for each R ⊂ , (iii) x(ω, t) is the unknown vector-valued random variables for each t ∈ R, (iv) K(ω, s, t) is the stochastic kernel defined for t, s ∈ R, (v) x(ω, s) is vector-valued function of s ∈ R.
Let C(Ω, R) is the space of all continuous functions from (Ω × R) into R, and it is a Banach space with a metric d(x, y) = sup |x − y| .
Suppose that for all x, y ∈ C(Ω, R) there exists a δ ∈ (0, 1) such that compatible of type (A) if lim n→∞ d(F T x n , T 2 x n ) = 0 and lim n→∞ d(T F x n , F 2 x n ) = 0, (3) compatible of type (B) if