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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010, Article ID 123524, 17 pages doi:10.1155/2010/123524 Research Article A System of Random Nonlinear Variational Inclusions Involving Random Fuzzy Mappings and H·, ·-Monotone Set-Valued Mappings Xin-kun Wu and Yun-zhi Zou College of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China Correspondence should be addressed to Yun-zhi Zou, zouyz@scu.edu.cn Received 8 June 2010; Accepted 24 July 2010 Academic Editor: Qamrul Hasan Ansari Copyright q 2010 X k. Wu and Y z. Zou. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We introduce and study a new system of random nonlinear generalized variational inclusions involving random fuzzy mappings and set-valued mappings with H·, ·-monotonicity in two Hilbert spaces and develop a new algorithm which produces four random iterative sequences. We also discuss the existence of the random solutions to this new kind of system of variational inclusions and the convergence of the random iterative sequences generated by the algorithm. 1. Introduction The classic variational inequality problem VIF, K is to determine a vector x ∗ ∈ K ⊂ R n , such that F x ∗ T ,x− x ∗ ≥ 0, ∀x ∈ K, 1.1 where F is a given continuous function from K to R n and K is a given closed convex subset of the n-dimensional Euclidean space R n . This is equivalent to find an x ∗ ∈ K, such that 0 ∈ F x ∗ N ⊥ x ∗ , 1.2 where N ⊥ is normal cone operator. Due to its enormous applications in solving problems arising from the fields of eco- nomics, mechanics, physical equilibrium analysis, optimization and control, transportation 2 Journal of Inequalities and Applications equilibrium, and linear or nonlinear programming etcetera, variational inequality and its generalizations have been extensively studied during the past 40 years. For details, we refer readers to 1–7 and the references therein. It is not a surprise that many practical situations occur by chance and so variational inequalities with random variables/mappings have also been widely studied in the past decade. For instance, some random variational inequalities and random quasivariational inequalities problems have been introduced and studied by Chang 8, Chang and Huang 9, 10, Chang and Zhu 11, Huang 12, 13, Husain et al. 14, Tan et al. 15,Tan16,and Yuan 7. It is well known that one of the most important and interesting problems in the theory of variational inequalities is to develop efficient and implementable algorithms for solving variational inequalities and its generalizations. The monotonic properties of associated operators play essential roles in proving the existence of solutions and the convergence of sequences generated by iterative algorithms. In 2001, Huang and Fang 17 were t he first to introduce the generalized m-accretive mapping and give the definition of the resolvent operator for generalized m-accretive mappings in Banach spaces. They also showed some properties of the resolvent operator for generalized m-accretive mappings. Recently, Fang and Huang, Verma, and Cho and Lan investigated many generalized operators such as H- monotone, H-accretive, H, η-monotone, H, η-accretive, and A, η-accretive mappings. For details, we refer to 6, 17–22 and the references therein. In 2008, Zou and Huang 23 introduced the H ·, ·-accretive operator in Banach spaces which provides a unified framework for the existing H-monotone, H, η-monotone, and A, η-monotone operators in Hilbert spaces and H-accretive, H, η-accretive, and A, η-accretive operators in Banach spaces. In 1965, Zadeh 24 introduced the concept of fuzzy sets, which became a cornerstone of modern fuzzy mathematics. To explore connections among VIs, fuzzy mapping and random mappings, in 1997, Huang 25 introduced the concept of random fuzzy mappings and studied the random nonlinear quasicomplementarity problem for random fuzzy mappings. Later, Huang 26 studied the random generalized nonlinear variational inclusions for random fuzzy mappings. In 2005, Ahmad and Baz ´ an 27 studied a class of random generalized nonlinear mixed variational inclusions for random fuzzy mappings and constructed an iterative algorithm for solving such random problems. For related work in this hot area, we refer to Ahmad and Farajzadeh 28, Ansari and Yao 29, Chang and Huang 9, 10, Cho and Huang 30, Cho and Lan 31, Huang 25, 26, 32, Huang et al. 33,andthe references therein. Motivated and inspired by recent research work mentioned above in this field, in this paper, we t ry to inject some new energy into this interesting field by studying on a new kind of random nonlinear variational inclusions in two Hilbert spaces. We will prove the existence of random solutions to the system of inclusions and propose an algorithm which produces a convergent iterative sequence. For a suitable choice of some mappings, we can obtain several known results 10, 11, 21, 23, 31, 34 as special cases of the main results of this paper. 2. Preliminaries Throughout this paper, let Ω, A be a measurable space, where Ω is a set and A is a σ-algebra over Ω.LetX 1 be a separable real Hilbert space endowed with a norm · X 1 and an inner product ·, · X 1 .LetX 2 be a separable real Hilbert space endowed with a norm · X 2 and an inner product ·, · X 2 . Journal of Inequalities and Applications 3 We denote by D·, · the Hausdorff metric between two nonempty closed bounded subsets, where the Hausdorff metric between A and B is defined by D A, B max sup a∈A inf b∈B d a, b , sup b∈B inf a∈A d a, b . 2.1 We denote by BX 1 ,2 X 1 ,andCBX 1 the class of Borel σ-fields in X 1 , and the family of all nonempty subsets of X 1 , the family of all nonempty closed bounded subsets of X 1 . In this paper, to make it self-contained, we start with the following basic definitions and similar definitions can also be found in 26, 32, 34. Definition 2.1. A mapping x 1 : Ω → X 1 is said to be measurable if for any B ∈BX 1 , { t ∈ Ω : x t ∈ B } ∈A. 2.2 Definition 2.2. A mapping T 1 : Ω × X 1 → X 1 is called a random mapping if for any x ∈ X 1 , z 1 tT 1 t, x is measurable. Definition 2.3. A random mapping T 1 : Ω × X 1 → X 1 is said to be continuous if for any t ∈ Ω, T 1 t, · : X 1 → X 1 is continuous. Definition 2.4. A set-valued mapping V 1 : Ω → 2 X 1 is said to be measurable if for any B ∈ BX 1 , V −1 1 B { v ∈ Ω : V 1 v ∩ B / ∅ } ∈A. 2.3 Definition 2.5. A mapping u : Ω → X 1 is called a measurable selection of a set-valued measurable mapping U : Ω → 2 X 1 if u is measurable and for any t ∈ Ω, ut ∈ Ut. Definition 2.6. A set-valued mapping W 1 : Ω × X 1 → 2 X 1 is called random set-valued if for any x 1 ∈ X 1 , W 1 ·,x 1 : Ω → 2 X 1 is a measurable set valued mapping. Definition 2.7. A random set-valued mapping W 1 : Ω × X 1 → CBX 1 is said to be ξ E t-D- continuous if there exists a measurable function ξ E : Ω → 0, ∞, such that D W 1 t, x 1 t ,W 1 t, x 2 t ≤ ξ E t x 1 t − x 2 t X 1 , 2.4 for all t ∈ Ω and x 1 t,x 2 t ∈ X 1 . Definition 2.8. A set-valued mapping A : X 1 → 2 X 1 is said to be monotone if for all x 1 ,y 1 ∈ X 1 and u 1 ∈ Ax 1 , v 1 ∈ Ay 1 , u 1 − v 1 ,x 1 − y 1 X 1 ≥ 0. 2.5 Definition 2.9. Let f 1 ,g 1 : X 1 → X 1 and H 1 : X 1 × X 1 → X 1 be three single-valued mappings and A : X 1 → 2 X 1 be a set-valued mapping. A is said to be H 1 ·, ·-monotone with respect to operators f 1 and g 1 if A is monotone and H 1 f 1 ,g 1 λAX 1 X 1 , for every λ>0. 4 Journal of Inequalities and Applications Definition 2.10. The inverses of A : X 1 → 2 X 1 and B : X 2 → 2 X 2 are defined as follows, respectively, A −1 y x ∈ X 1 : y ∈ A x , ∀y ∈ X 1 , B −1 y x ∈ X 2 : y ∈ B x , ∀y ∈ X 2 . 2.6 Definition 2.11. p : Ω × X 1 → X 1 is said to be 1 monotone if p t, x 1 t − p t, x 2 t ,x 1 t − x 2 t X 1 ≥ 0, ∀t ∈ Ω, ∀x 1 t ,x 2 t ∈ X 1 , 2.7 2 strictly monotone if p is monotone and p t, x 1 t − p t, x 2 t ,x 1 t − x 2 t X 1 0 ⇐⇒ x 1 t x 2 t , ∀t ∈ Ω, ∀x 1 t ,x 2 t ∈ X 1 , 2.8 3 δ p t-strongly monotone if there exists some measurable function δ p : Ω → 0, ∞, such that p t, x 1 t − p t, x 2 t ,x 1 t − x 2 t X 1 ≥ δ p t x 1 t − x 2 t 2 x 1 , ∀t ∈ Ω, ∀x 1 t ,x 2 t ∈ X 1 , 2.9 4 σ p t-Lipschitz continuous if there exists some measurable function σ p : Ω → 0, ∞, such that pt, x 1 t − p t, x 2 t X 1 ≤ σ p t x 1 t − x 2 t X 1 , ∀t ∈ Ω, ∀x 1 t ,x 2 t ∈ X 1 . 2.10 Definition 2.12. A single-valued mapping M : X 1 × X 1 × X 2 → X 1 is said to be 1 ζ A t-strongly monotone with respect to the random single-valued mapping s M : Ω × X 1 → X 1 in the first argument if there exists some measurable function ζ A : Ω → 0, ∞, such that M s M t, u 1 t , ·, · − M s M t, u 2 t , ·, · ,u 1 t − u 2 t X 1 ≥ ζ A t u 1 t − u 2 t 2 X 1 , 2.11 for all t ∈ Ω and u 1 t,u 2 t ∈ X 1 , 2 ξ M t-Lipschitz continuous with respect to the random single-valued mapping s M : Ω × X 1 → X 1 in its fi rst argument if there exists some measurable function ξ M : Ω → 0, ∞, such that Ms M t, u 1 t, ·, · − Ms M t, u 2 t, ·, · X 1 ≤ ξ M t u 1 t − u 2 t X 1 , 2.12 for all t ∈ Ω and u 1 t,u 2 t ∈ X 1 , Journal of Inequalities and Applications 5 3 β M t-Lipschitz continuous with respect to its second argument if there exists some measurable function β M : Ω → 0, ∞, such that M·,x 1 t, · − M·,x 2 t, · X 1 ≤ β M t x 1 t − x 2 t X 1 , 2.13 for all t ∈ Ω and x 1 t,x 2 t ∈ X 1 , 4 η M t-Lipschitz continuous with respect to its third argument if there exists some measurable function η M : Ω → 0, ∞ such that M·, ·,y 1 t − M ·, ·,y 2 t X 1 ≤ η M t y 1 t − y 2 t X 2 , 2.14 for all t ∈ Ω and y 1 t,y 2 t ∈ X 2 ; Definition 2.13. Assume that p : Ω × X 1 → X 1 is a random single-valued mapping, f 1 : X 1 → X 1 , g 1 : X 1 → X 1 ,andH 1 f 1 ,g 1 : X 1 → X 1 are three single-valued mappings, H 1 f 1 ,g 1 is said to be 1 μ A t-strongly monotone with respect to the mapping p if there exists some measurable function μ A : Ω → 0, ∞ such that H 1 f 1 p t, x 1 t ,g 1 p t, x 1 t − H 1 f 1 p t, y 1 t ,g 1 p t, y 1 t ,x 1 t − y 1 t X 1 ≥ μ A t x 1 t − y 1 t 2 X 1 , 2.15 for all t ∈ Ω and x 1 t,y 1 t ∈ X 1 , 2 a A t-Lipschitz continuous with respect to the mapping p if there exists some measurable function a A : Ω → 0, ∞ such that H 1 f 1 p t, x 1 t ,g 1 p t, x 1 t − H 1 f 1 p t, y 1 t ,g 1 p t, y 1 t X 1 ≤ a A t x 1 t − y 1 t X 1 , 2.16 for all t ∈ Ω and x 1 t,y 1 t ∈ X 1 . 3 α A -strongly monotone with respect to f 1 in the first argument if there exists a positive constant α A , such that H 1 f 1 x 1 ,u 1 − H 1 f 1 y 1 ,u 1 ,x 1 − y 1 X 1 ≥ α A x 1 − y 1 2 X 1 , 2.17 for all x 1 ,y 1 ,u 1 ∈ X 1 , 6 Journal of Inequalities and Applications 4 β A -relaxed monotone with respect to g 1 in the second argument if there exists a positive constant β A , such that H 1 u 1 ,g 1 x 1 − H 1 u 1 ,g 1 y 1 ,x 1 − y 1 X 1 ≥−β A x 1 − y 1 2 X 1 , 2.18 for all x 1 ,y 1 ,u 1 ∈ X 1 . Let FX 1 be a collection of all fuzzy sets over X 1 . A mapping F from Ω into FX 1 is called a fuzzy mapping. If F is a f uzzy mapping on X 1 , then for any given t ∈ Ω, Ftdenote it by F t in the sequel is a fuzzy set on X 1 and F t y is the membership function of y in F t . Let A ∈FX 1 , α ∈ 0, 1, then the set A α { x ∈ X 1 : A x ≥ α } 2.19 is called an α-cut set of fuzzy set A. Definition 2.14. A random fuzzy mapping F : Ω →FX 1 is said to be measurable if for any given α ∈ 0, 1, F· α : Ω → 2 X 1 is a measurable set-valued mapping. Definition 2.15. A fuzzy mapping E : Ω × X 1 →FX 1 is called a random fuzzy mapping if for any given x 1 ∈ X 1 , E·,x 1 : Ω →FX 1 is a measurable fuzzy mapping. Remark 2.16. The above is mainly about some definitions in X 1 . There are similar definitions and notations for operators in X 2 . Let E : Ω × X 1 →FX 1 and F : Ω × X 2 →FX 2 be two random fuzzy mappings satisfying the following condition ∗∗: ∗∗ there exist two mappings α : X 1 → 0, 1 and β : X 2 → 0, 1, such that E t,x 1 αx 1 ∈ CB X 1 , ∀ t, x 1 ∈ Ω × X 1 , F t,x 2 βx 2 ∈ CB X 2 , ∀ t, x 2 ∈ Ω × X 2 . 2.20 By using the random fuzzy mappings E and F, we can define the two set-valued mappings E ∗ and F ∗ as follows, respectively, E ∗ : Ω × X 1 −→ CB X 1 , t, x 1 −→ E t,x 1 αx 1 , ∀ t, x 1 ∈ Ω × X 1 , F ∗ : Ω × X 2 −→ CB X 2 , t, x 2 −→ E t,x 2 αx 2 , ∀ t, x 2 ∈ Ω × X 2 . 2.21 It follows that E ∗ t, x 1 E t,x 1 αx 1 { z 1 ∈ X 1 : E t,x 1 z 1 ≥ α x 1 } , F ∗ t, x 2 F t,x 2 βx 2 z 2 ∈ X 2 : F t,x 2 z 2 ≥ β x 2 . 2.22 It is easy to see that E ∗ and F ∗ are two random set-valued mappings. We call E ∗ and F ∗ the random set-valued mappings induced by the fuzzy mappings E and F, respectively. Journal of Inequalities and Applications 7 Problem 1. Let f 1 ,g 1 : X 1 → X 1 be two single-valued mappings and s M ,p : Ω × X 1 → X 1 be two random single-valued mappings. Let f 2 ,g 2 : X 2 → X 2 be two single-valued mappings and s N ,q : Ω × X 2 → X 2 be two random single-valued mappings. Let H 1 : X 1 × X 1 → X 1 , H 2 : X 2 × X 2 → X 2 , M : X 1 × X 1 × X 2 → X 1 and N : X 2 × X 1 × X 2 → X 2 be four single- valued mappings. Suppose that A : X 1 → 2 X 1 is an H 1 ·, ·-monotone mapping with respect to f 1 and g 1 and B : X 2 → 2 X 2 is an H 2 ·, ·-monotone mapping with respect to f 2 and g 2 . E : Ω × X 1 →FX 1 and F : Ω × X 2 →FX 2 are two random fuzzy mappings, α, β, E ∗ ,and F ∗ are the same as the above. Assume that pt, ut ∩ domA / ∅ and qt, vt ∩ domB / ∅ for all t ∈ Ω. We consider the following problem. Find four measurable mappings u, x : Ω → X 1 and v,y : Ω → X 2 , such that E t,ut x t ≥ α u t , F t,vt y t ≥ β v t , 0 ∈ M s M t, u t ,x t ,y t A p t, u t , 0 ∈ N s N t, v t ,x t ,y t B q t, v t , 2.23 for all t ∈ Ω. Problem 1 is called a system of generalized random nonlinear variational inclusions involving random fuzzy mappings and set-valued mappings with H·, ·-monotonicity in two Hilbert spaces. A set of the four measurable mappings x, y, u, and v is called one solution of Problem 1. 3. Random Iterative Algorithm In order to prove the main results, we need the following lemmas. Lemma 3.1 see 23. Let H 1 , f 1 , g 1 , and A be defined as in Problem 1.LetH 1 f 1 ,g 1 be α A - strongly monotone with respect to f 1 , β A -relaxed monotone with respect to g 1 ,whereα A >β A . Suppose that A : X 1 → 2 X 1 is an H 1 ·, ·-monotone set-valued mapping with respect to f 1 and g 1 , then the resolvent operator R H 1 ·,· A,λ Hf, gλA −1 is a single-valued mapping. Lemma 3.2 see 23. Let H 1 , f 1 , g 1 , A be defined as in Problem 1.LetH 1 f 1 ,g 1 be α A -strongly monotone with respect to f 1 , β A -relaxed monotone with respect to g 1 ,whereα A >β A . Suppose that A : X 1 → 2 X 1 is an H 1 ·, ·-monotone set-valued mapping with respect to f 1 and g 1 . Then, the resolvent operator R H 1 ·,· A,λ is 1/α A − β A -Lipschitz continuous. Remark 3.3. Some interesting examples concerned with the H 1 ·, ·-monotone mapping and the resolvent operator R H 1 ·,· A,λ can be found in 23. Lemma 3.4 see Chang 8. Let V : Ω × X 1 → CBX 1 be a D-continuous random set-valued mapping. Then for any given measurable mapping u : Ω → X 1 , the set-valued mapping V ·,u· : Ω → CBX 1 is measurable. 8 Journal of Inequalities and Applications Lemma 3.5 see Chang 8. Let V, W : Ω → CBX 1 be two measurable set-valued mappings, and let ε>0 be a constant and u : Ω → X 1 a measurable selection of V . Then there exists a measurable selection v : Ω → X 1 of W, such that for all t ∈ Ω, u t − v t ≤ 1 ε D V t ,W t . 3.1 Lemma 3.6. The four measurable mappings x, u : Ω → X 1 and y, v : Ω → X 2 are solution of Problem 1 if and only if, for all t ∈ Ω, x t ∈ E ∗ t, u t , x t ∈ F ∗ t, v t , p t, u t R H 1 ·,· A,λ H 1 f 1 p t, u t ,g 1 p t, u t − λM s M t, u t ,x t ,y t , q t, u t R H 2 ·,· B,ρ H 2 f 2 q t, v t ,g 2 q t, v t − ρN s N t, v t ,x t ,y t , 3.2 where R H 1 ·,· A,λ H 1 f 1 ,g 1 λA −1 and R H 2 ·,· B,ρ H 2 f 2 ,g 2 ρB −1 are two resolvent operators. Proof. From the definitions of R H 1 ·,· A,λ and R H 2 ·,· B,ρ , one has H 1 f 1 p t, u t ,g 1 p t, u t − λM s M t, u t ,x t ,y t ∈ H 1 f 1 p t, u t ,g 1 p t, u t λA p t, u t , ∀t ∈ Ω, H 2 f 2 q t, v t ,g 2 q t, v t − ρN s N t, v t ,x t ,y t ∈ H 2 f 2 q t, v t ,g 2 q t, v t ρB q t, v t , ∀t ∈ Ω. 3.3 Hence, 0 ∈ M s M t, u t ,x t ,y t A p t, u t , ∀t ∈ Ω, 0 ∈ N s N t, v t ,x t ,y t B q t, v t , ∀t ∈ Ω. 3.4 Thus, x, y, u, v is a set of solution of Problem 1. This completes the proof. Now we use Lemma 3.6 to construct the following algorithm. Let u 0 : Ω → X 1 and v 0 : Ω → X 2 be two measurable mappings, then by Himmelberg 35, there exist x 0 : Ω → X 1 , a measurable selection of E ∗ ·,u 0 · : Ω → CBX 1 and y 0 : Ω → X 2 , a measurable selection of F ∗ ·,v 0 · : Ω → CBX 2 . We now propose the following algorithm. Journal of Inequalities and Applications 9 Algorithm 3.7. For any given measurable mappings u 0 : Ω → X 1 and v 0 : Ω → X 2 , iterative sequences that attempt to solve Problem 1 are defined as follows: u n1 t u n t − p t, u n t R H 1 ·,· A,λ H 1 f 1 p t, u n t ,g 1 p t, u n t − λM s M t, u n t ,x n t ,y n t , v n1 t v n t − q t, v n t R H 2 ·,· B,ρ H 2 f 2 q t, v n t ,g 2 q t, v n t − ρN s N t, v n t ,x n t ,y n t . 3.5 Choose x n1 t ∈ E ∗ t, u n1 t and y n1 t ∈ F ∗ t, v n1 t, such that x n1 t − x n t X 1 ≤ 1 ε n1 D E ∗ t, u n1 t ,E ∗ t, u n t , y n1 t − y n t X 2 ≤ 1 ε n1 D F ∗ t, v n1 t ,F ∗ t, v n t , 3.6 for any t ∈ Ω and n 0, 1, 2, 3, Remark 3.8. The existence of x n and y n is guaranteed by Lemmas 3.4 and 3.5. 4. Existence and Convergence Theorem 4.1. Let X 1 and X 2 be two separable real Hilbert spaces. Suppose that s M ,p: Ω×X 1 → X 1 and s N ,q : Ω × X 2 → X 2 are four random mappings. Suppose that f 1 ,g 1 : X 1 → X 1 , H 1 : X 1 × X 1 → X 1 , f 2 ,g 2 : X 2 → X 2 , H 2 : X 2 × X 2 → X 2 are six single-valued mappings. Assume that 1 A : X 1 → 2 X 1 is an H 1 ·, ·-monotone with respect to operators f 1 and g 1 , 2 B : X 2 → 2 X 2 is an H 2 ·, ·-monotone with respect to operators f 2 and g 2 , 3 pt, ut ∩ domA / ∅ and qt, vt ∩ domB / ∅ for all t ∈ Ω, 4 M : X 1 × X 1 × X 2 → X 1 is ζ A t-monotone with respect to the mapping s M in the first argument, ξ M t-Lipschitz continuous with respect to mapping s M in the first argument, β M t-Lipschitz continuous with respect to the second argument and η M t-Lipschitz continuous with respect to the third argument, 5 N : X 2 × X 1 × X 2 → X 2 is ζ B t-monotone with respect to the mapping s N in the first argument, ξ N t-Lipschitz continuous with respect to mapping s N in the first argument, β N t-Lipschitz continuous with respect to the second argument and η N t-Lipschitz continuous with respect to the third argument, 6 Let E : Ω × X 1 →FX 1 and F : Ω × X 2 →FX 2 be two random fuzzy mappings satisfying the condition ∗∗, α, β, E ∗ and F ∗ are four mappings induced by E and F. E ∗ and F ∗ are ξ E t-D-Lipschitz and ξ F t-D-Lipschitz continuous, respectively; 7 p is δ p t-strongly monotone with respect to its second argument, σ p t-Lipschitz continuous with respect to its second argument, H 1 f 1 ,g 1 is μ A t-strongly monotone 10 Journal of Inequalities and Applications with respect to the mapping p and a A t-Lipschitz continuous with respect to the mapping p, 8 H 1 f 1 ,g 1 is α A -strongly monotone with respect to f 1 , and β A -relaxed monotone with respect to g 1 ,whereα A >β A , 9 q is δ q t-strongly monotone with respect to its second argument and σ q t-Lipschitz continuous with respect to its second argument, H 2 f 2 ,g 2 is μ B t-strongly monotone with respect to the mapping q, and a B t-Lipschitz continuous with respect to the mapping q, 10 H 2 f 2 ,g 2 is α B -strongly monotone with respect to f 2 and β B -relaxed monotone with respect to g 2 ,whereα B >β B , If A t λ α A − β A β M t ξ E t 2 1 − 2δ p t σ p t 2 1 α A − β A 2 1 − 2μ A t a A t 2 1 α A − β A 2 1 − 2λζ A t λ 2 ξ M t 2 , B t λ α A − β A η M t ξ F t , C t ρ α B − β B β N t ξ E t , D t ρ α B − β B η N t ξ F t 2 1 − 2δ q t σ q t 2 1 α B − β B 2 1 − 2μ B t a B t 2 1 α B − β B 2 1 − 2ρζ B t ρ 2 ξ N t 2 , 0 <A t C t < 1, ∀t ∈ Ω, 0 <B t D t < 1, ∀t ∈ Ω, 4.1 then there exist four measurable mappings x, u : Ω → X 1 and y,v : Ω → X 2 ,such that x, y, u, v is a set of solution of Problem 1. Moreover, lim n →∞ x n t x t , lim n →∞ y n t y t , lim n →∞ u n t u t , lim n →∞ v n t v t , 4.2 where x n t, y n t, u n t, and v n t are defined as in Algorithm 3.7. 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