Some Random Fixed-Point Theorems for Weakly Contractive Random Operators in a Separable Banach Space

,e aim of this paper is to prove a common random fixed-point and some random fixed-point theorems for random weakly contractive operators in separable Banach spaces. A randomMann iterative process is introduced to approximate the fixed point. Finally, the main result is supported by an example and used to prove the existence and the uniqueness of a solution of a nonlinear stochastic integral equation system.


Introduction
e fixed-point theory has been revealed as a very powerful and important tool in the study of different mathematical models expressed in the forms of differential equations [1], integral equations [2,3], fractional differential equations [4,5], matrix equations [6], etc. Also, its applications are very useful and interesting in economics, in game theory, in computer science, and in other domains.
Probabilistic functional analysis is one of the essential mathematical disciplines that are applied to solving problems, characterized with uncertainties, known as probabilistic models. e random fixed-point theorems are stochastic generalizations of classical fixed-point theorems which are known as deterministic results and are required for the theory of random equations, random matrices, random partial differential equations, and various classes of random operators. e theory of random fixed point was initiated by the Prague School of Probability in the 1950s. e random fixedpoint theory finds its roots in the work ofŠpaček [7] and Hanš [8,9]. ey established a stochastic generalization of Banach contraction principle (BCP), and they applied their results to study the existence of a solution of random linear Fredholm integral equations. In 1976, Bharucha-Reid published his review article [10] which has attracted the attention of several researchers and which has led to the development of random fixed-point theory. In 1979, Itoh [11] extendedŠpaček's and Hanš's theorems to multivalued contraction random mappings. e result obtained by Itoh in [11] was applied to solve a random differential equation in Banach space. In the recent past, random differential equations and random integral equations have been solved by random fixed-point theorems (see, for example, [12][13][14][15][16]). For some important contributions in the random fixedpoint theory, we invite the reader to consult [17][18][19][20][21][22][23][24][25] and the references therein.
It is necessary to mention that the BCP is the first fundamental deterministic fixed-point theorem in a metric space.
Theorem 1 (see [26]). If (X, d) is a complete metric space and T: X ⟶ X is a self-mapping such that d(Tx, Ty) ≤ k d(x, y), (1) for all x, y ∈ X and k ∈ ]0, 1[, then T has a unique fixed point.
Theorem 2 (see [27]). If (X, d) is a complete metric space and T: X ⟶ X is a self-mapping such that for all x, y ∈ X, and ϕ: Several interesting weak contractions were considered in various frameworks (see, for example, [28][29][30][31] and references therein). Among all these weak contractions, we are interested in the one studied by Eslamian and Abkar in [32]. We state the result in the following.
Theorem 3 (see [32]). If (X, d) is a complete metric space and T: X ⟶ X is a (ψ 1 , ψ 2 , φ)-weakly contractive selfmapping, i.e., then T has a unique fixed point.Here, the three functions ψ 1 , ψ 2 , φ: R + ⟶ R + , called control functions, satisfy the following conditions: In this study, we prove a common random fixed-point theorem and some random fixed-point theorems for random (ψ 1 , ψ 2 , φ)-weakly contractive operators in a separable Banach space, where the three control functions ψ 1 , ψ 2 , and φ satisfy all the conditions (a) − (f) except the condition (d) which is replaced by the weak condition: As an application, we show the existence and the uniqueness of a random solution for a system of nonlinear integral equations. To prove our main results, we need to recall the following concepts and results. For more details, the reader may consult [33,34].

Preliminaries
Let (X, ‖.‖) be a separable Banach space, β X be the σ-algebra of all Borel subsets of X, and (Ω, β, μ) be a complete probability measure space with the measure μ and β be the σ-algebra of subsets of Ω. Let C be a nonempty subset of X.
(i) A mapping x: Ω ⟶ X is said to be a random variable with values in X if the inverse image under the mapping x of every Borel set B of X belongs to β, that is, Definition 2 (see [33,34]).
(i) A mapping x: Ω ⟶ X is said to be a finitely-valued random variable if it is constant on each of a finite number of disjoint sets A i ∈ β and equal to 0 on Ω − ( ∪ n i�1 A i ). (ii) A mapping x: Ω ⟶ X is said to be a simple random variable if it is finitely valued. (iii) A mapping x: Ω ⟶ X is said to be a strong (or Bochner) random variable if there exists a sequence (x n ) n of simple random variables which converges to x almost surely, that is, there exists a set A 0 ∈ β with μ(A 0 ) � 0 such that Next, we introduce the notion of a weak random variable.
Definition 3 (see [33]). A mapping x: Ω ⟶ X is said to be a weak (or Pettis) random variable if the functions x * (x) are real-valued random variables for each x * ∈ X * , where X * denotes the first normed dual space of X.
Remark 1 (see [33]). In this work, we restrict our attention to the case where X is a separable Banach space. In this setting, the concept of weak and strong random variables is equivalent. e following definition of the mode of convergence for Banach space-valued random variables, which we use in the sequel, is borrowed from [33]. e sequence (x n ) n converges to x in Ω strongly almost surely if there exists a set A 0 ∈ β with μ(A 0 ) � 0 such that We recall the following results from the study by Joshi and Bose ( [34], eorem 6.1.2). Theorem 4. Let x and y be two strong random variables and α and β be two constants. en, the following statements hold: (i) αx + βy is a strong random variable (ii) If f is a real-valued random variable and x is a strong random variable, then fx is a strong random variable (iii) If (x n ) n is a sequence of strong random variables converging strongly to x almost surely, then x is a strong random variable 2 International Journal of Mathematics and Mathematical Sciences Definition 5 (see [35]).
(i) A mapping T: Ω × C ⟶ C is said to be random operator if for each x ∈ C, the mapping T(., x): Ω ⟶ C is measurable (ii) A random operator T: Ω × C ⟶ C is continuous if the set of all ω ∈ Ω for which T(ω, .) is continuous has measure one roughout this paper, we denote RV(X) as the set of all X− valued random variables and we adopt the following definition of the random fixed point given by Joshi and Bose in [34].

Main Results
In this section, we prove a common random fixed-point theorem and some random fixed-point theorems for (ψ 1 , ψ 2 , φ)− weakly contractive mappings in a separable Banach space.

Theorem 5.
Let (X, ‖.‖) be a separable Banach space, C be a nonempty closed subset of X, and (Ω, β, μ) be a complete probability measure space. Let T, S: Ω × C ⟶ C be two continuous random operators satisfying almost surely, for all x 1 , x 2 ∈ RV(C). en, there exists a unique common random fixed point of S and T.
For all y, z ∈ RV(C), we denote by E y,z the set of elements ω ∈ Ω such that For all n ∈ N * , we have is implies that International Journal of Mathematics and Mathematical Sciences Indeed, let us assume that there exists n 0 ∈ N * such that Since ψ 1 is increasing, we get en, which is a contradiction. Hence, for all n ∈ N * , we have It follows that the sequence (‖x n+1 (ω) − x n (ω)‖) n is decreasing and consequently there exists l ≥ 0 such that Since and by using the continuity of ψ 1 and ψ 2 and the lower semicontinuity of φ, we obtain that ψ 1 (l) ≤ ψ 2 (l) − φ(l), which is a contradiction unless l � 0. Hence, Now, fix ω in M and let us prove that (x n (ω)) n is a Cauchy sequence in C. For this, it is sufficient to show that the subsequence (x 2n (ω)) n is a Cauchy sequence. If we assume the contrary, then Furthermore, corresponding to n k , we can choose m k in such a way that it is the smallest integer with m k > n k satisfying Consequently, By using the triangular inequality, we obtain

International Journal of Mathematics and Mathematical Sciences
Hence, Also, Letting k ⟶ + ∞, we obtain By the same way, we have en, We have, for all k ∈ N, By passing to the upper limit, we get ψ 1 (ϵ) ≤ ψ 2 (ϵ) − φ(ϵ), which is a contradiction since ϵ > 0.
is shows that (x 2n (ω)) n is a Cauchy sequence in C, for each ω ∈ M. Using (20), it is easy to check that (x n (ω)) n is a Cauchy sequence in C, for each ω ∈ M.
Since C is a closed subset of the Banach space X, then C is complete, which implies that, for all ω ∈ M, the sequence (x n (ω)) n converges by norm in C. Let x: Ω ⟶ C be the mapping such that x(ω) � lim n⟶+∞ x n (ω), for each ω ∈ M. Since the sequence (x n ) n converges strongly almost surely to x, then, according to ([33], eorem 1.6), x is a C-valued random variable.
Since x(ω) � lim n⟶+∞ x n (ω) and by using the continuity of S and T, we get S(ω, x(ω)) � T(w, x(ω)) � x(ω), for each ω ∈ M. Hence, It means that x is a common random fixed point of S and T.
To prove the uniqueness of this common fixed point, let y be another common random fixed point of S and T. Consider the two sets is implies that ‖x(ω) − y(ω)‖ � 0.
is proves the uniqueness of the common random fixed point of S and T.
en, there exists a unique random fixed point of T.
Let ω ∈ Ω. We have en, 6 International Journal of Mathematics and Mathematical Sciences Consequently, for all x � (x 1 ; x 2 ), y � (y 1 ; y 2 ) ∈ RV (C) and for each ω ∈ Ω, (41) All conditions of Corollary 1 are satisfied and T has a random fixed point which is In Corollary 1, if ψ 1 (t) � ψ 2 (t) � t, for all t ≥ 0, we obtain the following corollary which is an improvement of ( [36], eorem 5.2) in a separable Banach space.

Corollary 2.
Let (X, ‖.‖) be a separable Banach space, C be a nonempty closed subset of X, and (Ω, β, μ) be a complete probability measure space. Let T: Ω × C ⟶ C be a continuous random operator satisfying almost surely, for all x 1 , x 2 ∈ RV(C). en, there exists a unique random fixed point of T.

Corollary 3.
Let (X, ‖.‖) be a separable Banach space, C be a nonempty closed subset of X, and (Ω, β, μ) be a complete probability measure space. Let T: Ω × C ⟶ C be a continuous random operator satisfying the following condition: almost surely, for all x 1 , x 2 ∈ RV(C). en, there exists a unique random fixed point of T.

Corollary 4.
Let (X, ‖.‖) be a separable Banach space, C be a nonempty closed subset of X, and (Ω, β, μ) be a complete probability measure space. Let T: Ω × C ⟶ C be a continuous random operator satisfying the following condition: almost surely, for all x 1 , x 2 ∈ RV(C) and k ∈ [0, 1[. en, there exists a unique random fixed point of T.

Random Mann Iteration Scheme
In the following, we investigate the convergence of random Mann iteration scheme applied to a (ψ 1 , ψ 2 , φ)-weakly contractive random operator.

International Journal of Mathematics and Mathematical Sciences
Definition 8. (random Mann iteration scheme [35]). Let T: Ω × C ⟶ C be a random operator, where C is a nonempty convex subset of a separable Banach space X. A random Mann iteration scheme is the sequence of C-valued random variables (x n ) n defined, for all ω ∈ Ω, by x n+1 (ω) � 1 − c n x n (ω) + c n T ω, x n (ω) , for all n ∈ N x 0 : Ω ⟶ C is an arbitrary measurable mapping , where 0 ≤ c n ≤ c n+1 ≤ 1 and 0 ≤ lim n⟶∞ c n � c ≤ 1, for all n ∈ N. In particular, if c n � 1, for all n ∈ N, the sequence (x n ) n is said to be a random Picard iteration scheme. Theorem 6. Let (X, ‖.‖) be a separable Banach space, C be a nonempty closed convex subset of X, and (Ω, β, μ) be a complete probability measure space.
Let T: Ω × C ⟶ C be a continuous (ψ 1 , ψ 2 , φ)-weakly contractive random operator. Assume that ψ 1 is convex. en, the following two statements hold: (i) ere exists a unique random fixed point x: Ω ⟶ C of T (ii) e random Mann iteration scheme converges strongly almost surely to the unique random fixed point x of T Proof (i) From Corollary 1, T has a unique random fixed point Let E x 1 ,x 2 be the set of elements ω ∈ Ω such that en, is is a contradiction, since for each t > 0, en, for all ω ∈ M, Consider the set Since M⊆L and μ(M) � 1, then μ(L) � 1. is shows that the sequence (x n ) n of the C-valued random variable converges strongly almost surely to the unique random fixed point x.

Applications to Nonlinear Stochastic Integral Equations System
In this section, we give an application of eorem 5 to show the existence and the uniqueness of a solution of a nonlinear stochastic integral equations system (NSIE) presented as follows: ; ω)f(s, y(s; ω))dλ(s), where we have the following: (a) R is the locally compact real space with the usual norm of reals and λ is the Lebesgue measure on R (b) ω ∈ Ω, where Ω is the supporting set of the probability measure space (Ω, β, μ) (c) For all t ∈ R, x(t; .) and y(t; .) are two unknown elements in RV(R) (t; ω) is the stochastic free term defined for t ∈ R (e) k(t, s; ω) is the stochastic kernel defined for t and s in R (f ) f and g are two real-valued functions Remark 2 (see [37]). e topological space R is the union of a countable family of compact subsets K n having the properties that K n ⊂ K n+1 and that for any other compact set in R, there is a K i which contains it.
Let B and D be two Banach spaces. e pair (B, D) is said to be admissible with respect to the linear operator T if T(ω)(B) ⊂ D a.s. Lemma 1 (see [39]).
Let ρ be a positive real number. Consider the ball is centered on 0 and the neutral element in D is (67) Theorem 7. e following conditions hold: (1) B and D are Banach spaces stronger than C(R, L 2 (Ω, β, μ)) so that (B, D) is admissible by respect to the integral operator imposed by (66). (2) e functions F and G maps from M(ρ) into B and there exists three control functions ψ 1 , ψ 2 , and φ such that for any x 1 , x 2 ∈ M(ρ), (68) then U(ω)x ∈ M(ρ) a.s. By the same argument, we prove that V(ω)x ∈ M(ρ) a.s. Let x 1 , x 2 ∈ M(ρ). We have en, 10 International Journal of Mathematics and Mathematical Sciences which shows that U(ω) and V(ω) are (ψ 1 , ψ 2 , φ)-weakly contractive mappings almost surely on M(ρ). en, the two operators U, V: Ω × M(ρ) ⟶ M(ρ) are (ψ 1 , ψ 2 , φ)-weakly contractive random operators. erefore, by eorem 5, there is a unique common random fixed point of random operators U and V, which is the unique stochastic solution of NSIE. is completes the proof.

Conclusion
In this paper, we have the following: eorem 5 is a random generalization of the main result of [32] in a separable Banach space Corollary 1 generalizes and improves ( [36], eorem 2.5) in the setting of Banach spaces Corollary 4 is a random version of Banach contraction principle in a separable Banach space eorem 6 extends ( [36], eorem 5.3) to (ψ 1 , ψ 2 , ϕ)-weakly contractive random operators eorem 7 shows the utility of our main result in solving a system of nonlinear stochastic integral equations is work will open the door for other deterministic results that can be randomized, for example, [29].

Data Availability
No data were used to support this study.

Conflicts of Interest
e authors declare that there are no conflicts of interest.