An Iterative Algorithm for Random Quasi

Applied Mathematical Sciences, Vol. 8, 2014, no. 97, 4799 - 4814 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.46495

An Iterative Algorithm for Random Quasi-Variational-Like Inclusions in Banach Spaces Syed Shakaib Irfan College of Engineering, P. O. Box 6677 Qassim University, Buraidah 51452 Al-Qassim, Kingdom of Saudi Arabia Zeid Ibrahim Al-Muhimied Department of Mathematics College of Science, P.O. Box 6644 Qassim University, Buraidah 51452 Al-Qassim, Kingdom of Saudi Arabia c 2014 Syed Shakaib Irfan and Zeid Ibrahim Al-Muhimied. This is an open access article disCopyright tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract In this paper, we consider the completely generalized nonlinear random quasivariational-like inclusions problem for random fuzzy mappings and define. By using ηt -proximal mapping due to Ahmad [6]. We propose an iterative algorithm for Jρ(t) computing the approximate solutions of completely generalized nonlinear random quasi-variational-like inclusions problem for random fuzzy mappings. Our results improved and generalize many known corresponding results.

Mathematics Subject Classification: 47J22, 47H06, 49J40 Keywords: Random quasi-variational-like inclusions, Algorithm, Random fuzzy mapηt pings, η-subdifferential, Jρ(t) -proximal mapping, Banach space

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Syed Shakaib Irfan and Zeid Ibrahim Al-Muhimied

1. Introduction and Preliminaries The theory of variational inequality has become a very powerful tool for studying a wide range of problems arising in pure and applied sciences. Due to its applications in optimization, economics, physics and engineering sciences this theory was extended and generalized in many directions. Variational and quasi-variational inclusions are the useful and important generalization of the variational inequalities. In 1994, Hassouni and Moudafi [14] used the resolvent operator technique for maximal monotone mappings to study mixed type variational inequalities with single valued mappings which are called variational inclusions and developed a perturbed algorithm for finding the approximate solutions of mixed variational inequalities. Since then this class of variational inclusions have been extended and generalized by Adly [1], Ahmad and Ansari [3], Ansari and Siddiqi [8], Jin and Liou [19], and Huang [24] etc. In 1965, L. Zadeh [22] at the University of California introduced fuzzy set theory. This theory has gained importance in analysis from both theoretical and practical point of view. In 1989, Chang and Zhu [11] first introduced and studied the concept of a class of variational inequalities for fuzzy mappings. Since then several class of variational inequalities with fuzzy mappings were considered by Ding and Park [13], Huang [16] and Lan [20] etc. On the other hand, the random variational inequality, random quasi-variational inequality, random variational inclusion and random complementarity problems have been introduced and studied by Ahmad and Farajzadeh [5], J. Balooee [9], Chang [10], Huang [17], Kumam [21] and Zhang [23] etc. In this paper, we consider the completely generalized nonlinear random quasi-variationalηt -proximal like inclusions problem for random fuzzy mappings and define. By using Jρ(t) mapping due to Ahmad [6]. We propose an iterative algorithm for computing the approximate solutions of completely generalized nonlinear random quasi-variational-like inclusions problem for random fuzzy mappings. Our results improved and generalize many known corresponding results. In this paper, let (Ω, Σ) be a measurable space, where Ω is a set and Σ is a σ-algebra of subsets of Ω. Let E be a Banach space with the topological dual space E ∗ and u, x be the pair between u ∈ E ∗ and x ∈ E. We denote B(E), 2E , CB(E) and H(., .) the class of Borel σ-fields in E, the family of all nonempty subsets of E, the family of all nonempty closed bounded subsets of E and the Housdorff metric H(A, B) = max{supu∈A infv∈B d(u, v), supv∈B infu∈A d(u, v)} ∗

on CB(E) respectively. The normalized duality mapping J : E → 2E is defined by J (x) = {f ∈ E ∗ : f, x = x.f = f = x}, ∀ x ∈ E.

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If E = H is a Hilbert space then J becomes the identity mapping on H. In this paper, we will use the following definitions and lemmas. Definition 1.1. A mapping x : Ω → E is said to be measurable if for any B ∈ B(E), {t ∈ Ω : x(t) ∈ B} ∈ Σ. Definition 1.2. A mapping f : Ω × E → E is called a random mapping if for any x ∈ E, f (t, x) = x(t) is measurable. A random mapping f is said to be continuous if for any t ∈ Ω, the mapping f (t, .) : E → E is continuous. Definition 1.3. A multivalued mapping T : Ω → 2E is said to be measurable if for any B ∈ B(E), T −1 (B) = {t ∈ Ω : T (t) ∩ B = φ} ∈ Σ. Definition 1.4. A mapping u : Ω → E is called a measurable selection of a multivalued measurable mapping T : Ω → 2E if u is a measurable and for any t ∈ Ω, u(t) ∈ T (t). Definition 1.5. A mapping T : Ω×E → 2E is called a random multivalued mapping if for any x ∈ E, T (., x) is measurable. A random multivalued mapping T : Ω × E → CB(E) is said to be H-continuous if for any t ∈ Ω, T (t, .) is continuous the Housdorff metric on CB(E) Let η : Ω × E × E → E and φ : Ω × E → R ∪ {+∞}. A vector w ∗ (t) ∈ E ∗ is called an η-subdiferential of φ at x(t) ∈ domφ if w ∗ (t), η(t, y(t), x(t)) ≤ φ(t, y(t)) − φ(t, x(t)) ∀ y(t) ∈ E. Each φ can be associated with the following η-subdifferential map ∂ηt φ defined by ⎧ ⎪ ⎪ ⎨

∂ηt φ(t, x(t)) =

w ∗ (t) ∈ E : w ∗(t), η(t, y(t), x(t)) ≤ φ(t, y(t)) − φ(t, x(t)) ∀ y(t) ∈ E, x(t) ∈ domφ ⎪ ⎪ ⎩ φ x(t) ∈ / domφ

Let F (E) be a collection of fuzzy sets over E. A mapping F from E into F (E) is called a fuzzy mapping on E. If F is a fuzzy mapping on E, then F (x) (denote it by Fx , in the sequel) is a fuzzy set on E and Fx (y) is the membership function of y in Fx . Let N ∈ F (E), q ∈ [0, 1]. Then the set (N)q = {x ∈ E : N(x) ≥ q} is called a q-cut set of N. Definition 1.6. A fuzzy mapping F : Ω → F (E) is called measurable, if for any α ∈ (0, 1], (F (.))α : Ω → 2E is a measurable multivalued mapping.

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Definition 1.7. A fuzzy mapping F : Ω × E → F (E) is called a random fuzzy mapping if for any x ∈ E, F (., x) : Ω → F (E) is a measurable fuzzy mapping. Clearly, the random fuzzy mappings include multivalued mappings, random multivalued mappings and fuzzy mappings as special cases. Definition 1.8. Let A : Ω×E → CB(E) be a random multivalued mapping, J : Ω×E → E ∗ , g : Ω × E → E, η : Ω × E × E → E be the single valued mappings (i) A is said to be Lipschitz continuous if there exists a measurable function λA : Ω → (0, ∞) such that H(A(t, x(t)), A(t, y(t))) ≤ λA (t)x(t) − y(t), ∀ x(t), y(t) ∈ E; (ii) J is said to be η-strongly accretive if there exist a measurable function α : Ω → (0, ∞) such that J(t, x(t)) − J(t, y(t)), η(t, x(t), y(t) ≥ α(t)x(t) − y(t)2, ∀ x(t), y(t) ∈ E; (iii) η is said to be Lipschitz continuous if there exist a measurable function τ : Ω → (0, ∞) such that η(t, x(t), y(t)) ≤ τ (t)x(t) − y(t), ∀ x(t), y(t) ∈ E; (iv) g is said to be k-strongly accretive (k(t) > 0) if for any x(t), y(t) ∈ E, there exists j(t, x(t) − y(t)) ∈ J (t, x(t) − y(t)) such that j(t, x(t) − y(t)), g(t, x(t)) − g(t, y(t)) ≥ k(t)x(t) − y(t)2 , where J is the normalized duality mapping.

Definition 1.9. Let E be a Banach space with the dual space E ∗ , φ : Ω×E → R {+∞} be a proper, η-subdifferential random functional, η : Ω × E × E → E and J : Ω × E → E ∗ be the random mappings. If for any given point x(t) ∈ E ∗ and ρ(t) > 0, there is a unique fixed point x(t) ∈ E satisfying J(t, x(t)) − x∗ (t), η(t, y(t), x(t)) + ρ(t)φ(t, y(t)) − ρ(t)φ(t, x(t)) ≥ 0, ∀y(t) ∈ E ∂ φ

ηt t (x∗ (t)) is said to be J ηt -proximal mapping The mapping x∗ (t) → x(t), denoted by Jρ(t) of φt . We have x∗ (t) − J(t, x(t)) ∈ ρ(t)∂ηt φ(., ., x(t)), it follows that

∂ φ

ηt t (x∗ (t)) = (J + ρ(t)∂ηt φt )−1 (x∗ (t)). Jρ(t)

Let T, A, S : Ω × E → F (E) be three random fuzzy mappings satisfying the following condition (∗):

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(∗) : There exist three mappings a, b, c : E → (0, 1] such that (Tt,x )a(x) ∈ CB(E), (At,x )b(x) ∈ CB(E), (St,x )c(x) ∈ CB(E), ∀ (t, x) ∈ Ω × E. By using the random fuzzy mappings T, A and S, we can define three random multivalued mappings T˜, A˜ and S˜ respectively as follows: T˜ : Ω × E → CB(E), x → (Tt,x )a(x) , ∀ (t, x) ∈ Ω × E; A˜ : Ω × E → CB(E), x → (At,x )b(x) , ∀ (t, x) ∈ Ω × E; and S˜ : Ω × E → CB(E), x → (St,x )c(x) , ∀ (t, x) ∈ Ω × E. In the sequel, T˜ , A˜ and S˜ are called the random multivalued mappings induced by the random fuzzy mappings T, A, and S respectively. Given mappings a, b, c : E → (0, 1], random fuzzy mappings T, A, S : Ω × E → F (E), random mappings J : Ω × E → E ∗ , N : Ω × E × E × E → E, f, g, h, s : Ω × E → E, η : Ω × E × E → E and φ : Ω × E × E → R {+∞} be such that for each fixed x(t) ∈ E, φ(t, ., ) is a lower semi continuous, η-subdifferentialbe functional on E satisfying Img ∩ dom(∂ηt φ(t, ., .)) = φ. We consider the following completely generalized nonlinear random quasi-variational-like inclusion problem: Find measurable mappings x, u, v, w : Ω → E such that for all t ∈ Ω, x(t) ∈ E, y(t) ∈ E, Tt,x(t) u(t) ≥ a(x(t)), At,x(t) v(t) ≥ b(x(t)), St,x(t) w(t) ≥ c(x(t)) such that g(t, x(t)) ∩ dom(∂ηt φ(t, ., .)) = φ and J(t, (N (t, f (t, u(t)), h(t, v(t)), s(t, w(t))))), η(t, y(t), g(t, x(t))) ≥ φ(t, g(t, x(t)), x(t)) − φ(t, y(t), x(t))

(1.1)

The set of measurable mappings (x, u, v, w) is called a random solution of (1.1). Special Cases. (i) If E = H is a Hilbert space, N(t, f (t), h(t), s(t)) = f (t)−(h(t)−s(t)), η(t, u(t), v(t)) = u(t) − v(t) and φ(t, x(t), y(t)) = φ(t, x(t)) then (1.1) reduces to the problem of finding measurable mappings x, u, v, w : Ω → H such that ∀t ∈ Ω, x(t) ∈ H, y(t) ∈ H, Tt,x(t)) (u(t)) ≥ a(x(t)), At,x(t)) (v(t)) ≥ b(x(t)), St,x(t)) (w(t)) ≥ c(x(t)), g(t, x(t)) dom(∂φ) = φ, and f (t, u(t))−(h(t, v(t))−s(t, w(t)), y(t)−g(t, x(t)) ≥ φ(t, g(t, x(t))−φ(t, y(t)). (1.2) Problem (1.2) is introduced and studied by Ahmad and Bazán [4].

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(ii) If A, S, T are three random multivalued mappings and a(x) = b(x) = c(x) = 1, ∀x ∈ H, then problem (1.2) is equivalent to finding measurable mappings x, u, v, w : Ω → H such that ∀t ∈ Ω, x(t) ∈ H, y(t) ∈ H, u(t) ∈ T (t, x(t)), v(t) ∈ A(t, x(t)), w(t) ∈ S(t, x(t)), g(t, w(t)) dom(∂φ) = φ, and f (t, u(t))−(h(t, v(t))−s(t, w(t)), y(t)−g(t, x(t)) ≥ φ(t, g(t, x(t))−φ(t, y(t)). (1.3) A slight variant form of (1.3) is considered by S. S. Irfan [18]. (iii) If E = H is a Hilbert space, h, s = I are identity mappings N(t, f (t), h(t), s(t)) = f (t) + h(t) − s(t) and φ(t, x(t), y(t)) = φ(t, x(t)) then (1.1) reduces to the problem of finding measurable mapping x : Ω → H such that ∀t ∈ Ω, x(t) ∈ H, y(t) ∈ H, and f (t, u(t)) + v(t) − w(t), η(t, y(t), g(t, x(t))) ≥ φ(t, g(t, x(t)) − φ(t, y(t)).

(1.4)

Which is called the generalized nonlinear random variational inclusions for random multivalued operators in Hilbert spaces. The determinate form of the problem (1.4) was studied by Agarwal et al.[2]. (iv) If η(t, u(t), v(t)) = u(t) − v(t) ∀t ∈ Ω u(t), v(t) ∈ H, then problem (1.4) reduces to the problem of finding measurable mappings x, u : Ω → H such that u(t) ∈ T (t, x(t)), and f (t, u(t)) + v(t) − w(t), y(t) − g(t, x(t)) ≥ φ(t, g(t, x(t)) − φ(t, y(t)).

(1.5)

The determinate form is a generalization of the problem (1.4) considered in [12].

Definition 1.10. A random mapping f : Ω×E×E → R {+∞} is said to be 0-diagonally quasi-concave (0-DQCV) in y(t), if for any finite subset {x1 (t), ...xn (t)} ⊂ E and for any y(t) = Σni=1 λi (t)xi (t) with λi (t) ≥ 0 and Σni=1 λi (t) = 1 min1≤i≤n f (t, xi (t), y(t)) ≤ 0. Lemma 1.2.[6] Let D be a non empty convex subset of a topological vector space and f : Ω × D × D → R {+∞} be such that (i) for each x(t) ∈ D, y(t) → f (t, x(t), y(t)) is lower semicontinuous on each compact subset of D; (ii) for each finite set {x1 (t)....xn (t)} ∈ D and for each y(t) = Σni=1 λi (t)xi (t) with λi (t) ≥ 0 and Σm i=1 λi (t) = 1, min1≤i≤m f (t, xi (t), y(t)) ≤ 0;

Iterative algorithm

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(iii) there exist a nonempty compact convex subset D0 of D and a nonempty compact subset K of D such that for each y(t) ∈ D \ K, there is a x(t) ∈ c0 (D0 {y(t)}) satisfying f (t, x(t), y(t)) > 0. Then there exist yˆ(t) ∈ D such that f (t, x(t), yˆ(t)) ≤ 0, ∀ x(t) ∈ D. Now we give some sufficient conditions which guarantee the existence and Lipschitz continuity of the J ηt -proximal mapping of a proper functional on reflexive Banach space. Theorem 1.1.[6]. Let E be a reflexive Banach space with the dual space E ∗ and φ : Ω × E → R ∪ {+∞} be a lower semicontinuous, η-subdifferentiable, proper functional which may not be convex. Let J : Ω × E → E ∗ be η-strongly accretive with constant α(t) > 0. Let η : Ω × E × E → E be Lipschitz continuous with constant τ (t) > 0 such that η(t, x(t), y(t)) = −η(t, y(t), x(t)), for all x(t), y(t) ∈ E and for any x(t) ∈ E, the function h(t, y(t), x(t)) = x∗ (t) − J(t, x(t)), η(t, y(t), x(t)) is 0-DQCV in y(t). Then for any ρ(t) > 0, and any x∗ (t) ∈ E ∗ , there exists a unique x(t) ∈ E such that J(t, x(t)) − x∗ (t), η(t, y(t), x(t)) + ρ(t)φ(t, y(t)) − ρ(t)φ(t, x(t)) ≥ 0, ∀ y(t) ∈ E. ∂ φ

ηt t That is x(t) = Jρ(t) (x∗ (t)) and so the J ηt -proximal mapping of φ is well defined.

Theorem 1.2.[6]. Let E be a reflexive Banach space with the dual space E ∗ , J : Ω × E → E ∗ is η-strongly accretive continuous mapping with constant α(t) > 0, φ : Ω × E → R ∪ {+∞} be a lower semicontinuous, η-subdifferentiable proper functional. Let η : Ω × E × E → E be Lipschitz continuous with constant τ (t) > 0 such that η(t, x(t), y(t)) = −η(t, y(t), x(t)), for all x(t), y(t) ∈ E and for any given x(t) ∈ E, the functional h(t, y(t), x(t)) = x∗ (t) − J(t, x(t)), η(t, y(t), x(t)) is 0-DQCV in y(t), ρ(t) > 0 ∂ηt φt τ (t) is an arbitrary constant. Then the J ηt -proximal mapping Jρ(t) of φ is α(t) -Lipschitz continuous.

2. Random Iterative Algorithm Definition 2.1. The mapping N : Ω × E × E × E → E is said to be (i) Lipschitz continuous with respect to the first argument, if there exists a constant λN1 (t) > 0 such that N (u1 (t), v(t), w(t)) − N(u2 (t), v(t), w(t)) ≤ λN1 (t)u1(t) − u2 (t), ∀ u1 (t) ∈ T (t, x1 (t)), u2 (t) ∈ T (t, x2 (t)) and x1 (t), x2 (t) ∈ E; (ii) Lipschitz continuous with respect to the second argument, if there exists a constant λN2 (t) > 0 such that N (u(t), v1 (t), w(t)) − N(u(t), v2 (t), w(t)) ≤ λN2 (t)v1 (t) − v2 (t), ∀ v1 (t) ∈ A(t, x1 (t)), v2 (t) ∈ A(t, x2 (t)) and x1 (t), x2 (t) ∈ E;

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(iii) Lipschitz continuous with respect to the third argument, if there exists a constant λN3 (t) > 0 such that N (u(t), v(t), w1(t)) − N(u(t), v(t), w2(t)) ≤ λN3 (t)w1 (t) − w2 (t), ∀ w1 (t) ∈ S(t, x1 (t)), w2 (t) ∈ S(t, x2 (t)) and x1 (t), x2 (t) ∈ E. Lemma 2.1.[10]. Let T : Ω × E → CB(E) be a H-continuous random multivalued mapping. Then for any measurable mapping u : Ω → E, the multivalued mapping T (., u(t)) : Ω → CB(E) is measurable. Lemma 2.2.[10]. Let T, A : Ω × E → CB(E) be two mesurable multivalued mappings,

> 0 be a constant and u : Ω → E be measurable selection of T . Then there exist a measurable selection v : Ω → E of A such that for all t ∈ Ω u(t) − v(t) ≤ (1 + )H(T (t), A(t)). Lemma 2.3.[7] Let E be a real Banach space and J : E → E ∗ be the normalized duality mapping. Then for any x, y ∈ E, x + y2 ≤ x2 + 2y, j(x + y), ∀ j(x + y) ∈ J (x + y). We first transfer (1.1) into a fixed point problem. Theorem 2.1. The set of measurable mappings x, u, v, w : Ω → E is a random solution ˜ x(t)), w(t) ∈ of (1.1) if and only if for all t ∈ Ω, x(t) ∈ E, u(t) ∈ T˜ (t, x(t)), v(t) ∈ A(t, ˜ x(t)) and S(t, ∂ φ(.,.,x(t))

ηt g(t, x(t)) = Jρ(t)

{J(t, g(t, x(t))) − ρ(t)J(t, N(t, f (t, u(t)), h(t, v(t)), s(t, w(t)))))}. (3.1) ∂ηt (.,x(t)) −1 ηt Where Jρ(t) = (J + ρ(t)∂ηt φ(., ., x(t))) is the J -proximal mapping of φ(., ., x(t)) and ρ : Ω → (0, ∞) is a measurable function. ˜ x(t)), w(t) ∈ S(t, ˜ x(t)) Proof. Assume that x(t) ∈ E, u(t) ∈ T˜ (t, x(t)), v(t) ∈ A(t, satisfies relation (3.1), i.e., ∂ φ(.,.,x(t))

ηt g(t, x(t)) = Jρ(t)

∂ φ(.,.,x(t))

{J(t, g(t, x(t))) − ρ(t)J(t, N(t, f (t, u(t)), h(t, v(t)), s(t, w(t)))))}.

ηt Since Jρ(t) = (J + ρ(t)∂ηt φ(., ., x(t)))−1 , the above equality holds if and only if J(t, g(t, x(t))) − ρ(t)J(t, N(t, f (t, u(t)), h(t, v(t)), s(t, w(t))))) ∈ J(t, g(t, x(t))) + ρ(t)∂ηt φ(t, g(t, x(t)), x(t)) By the definition of η-subdifferential of φ(., ., x(t)), the above relation holds if and only if

Iterative algorithm

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φ(t, y(t), x(t)) − φ(t, g(t, x(t)), x(t)) ≥ −J(t, N(t, f (t, u(t)), h(t, v(t)), s(t, w(t)))), η(t, y(t), g(t, x(t))), ∀ y(t) ∈ E Hence we have J(t, N(t, f (t, u(t)), h(t, v(t)), s(t, w(t)))), η(t, y(t), g(t, x(t))) ≥ φ(t, g(t, x(t)), x(t)) − φ(t, y(t), x(t)), ∀ y(t) ∈ E. i.e. (x, u, v, w) is a solution of the (1.1). This fixed point formulation enables us to suggest the following proximal point algorithm. Algorithm 2.1. Let T, A, S : E → F (E) be random fuzzy mappings satisfying the condi˜ S˜ : Ω × E → CB(E) be H-continuous random multivalued mappings tion (∗). Let T˜, A, induced by T, A and S, respectively. Let f, g, h, s : Ω×E → E, N : Ω×E×E×E → E, η : Ω×E ×E → E, J : Ω×E → E ∗ be the single-valued random mappings with g(t, E) = E. Let φ : Ω × E × E → R ∪ {+∞} be a lower semicontinuous, η-subdifferentiable proper random functional on E satisfying g(t, E) ∩ dom∂ηt φ(., ., x(t)) = φ. For any given measur˜ x0 (.)), S(., ˜ x0 (.)) : able mappings x0 : Ω → E, the multivalued mappings T˜ (., x0 (.)), A(., Ω → CB(E) are measurable by Lemma 2.1. Hence there exist measurable selections ˜ x0 (.)) and w0 : Ω → E of S(., ˜ x0 (.)) by u0 : Ω → E of T˜ (., x0 (.)), v0 : Ω → E of A(., Himmelberg [15]. By g(t, E) = E, there exists a point x1 (t) ∈ E such that ∂ φ(.,.,x0 (t))

ηt g(t, x1 (t)) = Jρ(t)

{J(t, g(t, x0 (t)))−ρ(t)J(t, N(t, f (t, u0 (t)), h(t, v0 (t)), s(t, w0 (t)))))}

It is easy to see that x1 : Ω → E is measurable. By Lemma 2.2, there exist measurable ˜ x1 (.)) and w1 : Ω → E of S(., ˜ x1 (.)) selections u1 : Ω → E of T˜ (., x1 (.)), v1 : Ω → E of A(., such that for all t ∈ Ω, u1 (t) − u0 (t) ≤ (1 + 1)H(T (t, x1 (t)), T (t, x0 (t))), v1 (t) − v0 (t) ≤ (1 + 1)H(A(t, x1 (t)), A(t, x0 (t))), w1 (t) − w0 (t) ≤ (1 + 1)H(S(t, x1 (t)), S(t, x0 (t))). Let ∂ φ(.,.,x1 (t))

ηt g(t, x2 (t)) = Jρ(t)

{J(t, g(t, x1 (t)))−ρ(t)J(t, N(t, f (t, u1 (t)), h(t, v1 (t)), s(t, w1 (t)))))}

then x2 : Ω → E is measurable. Continuing the above process inductively, we can define the following random iterative sequences {xn (t)}, {un (t)}, {vn (t)} and {wn (t)} for solving (1.1) as follows: ∂ φ(.,.,xn (t))

ηt g(t, xn+1 (t)) = Jρ(t)

{J(t, g(t, xn (t)))−ρ(t)J(t, N(t, f (t, un (t)), h(t, vn (t)), s(t, wn (t)))))} (3.2)

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un (t) ∈ T˜ (t, xn (t)), un+1 (t) − un (t) ≤ (1 + (n + 1)−1 )H(T (t, xn+1 (t)), T˜ (t, xn (t))), ˜ xn (t)), vn+1 (t) − vn (t) ≤ (1 + (n + 1)−1 )H(A(t, xn+1 (t)), A(t, ˜ xn (t))), vn (t) ∈ A(t, ˜ xn (t)), wn+1 (t) − wn (t) ≤ (1 + (n + 1)−1 )H(S(t, xn+1 (t)), S(t, ˜ xn (t))), wn (t) ∈ S(t, where ρ(t) > 0 is a constant.

3. Convergence Result Theorem 3.1. Let T, A, S : Ω × E → F (E) be three random fuzzy mappings satisfying ˜ S˜ : Ω × E → CB(E) be three random multivalued mappings condition (∗). Let T˜ , A, ˜ and S˜ are H-Lipschitz contininduced by T, A and S respectively. Suppose that T˜, A, uous mappings with constants λT (t), λA (t) and λS (t), respectively. Let f, g, h, s : Ω × E → E are Lipschitz continuous random mappings with constants λf (t), λg (t), λh (t) and λs (t), respectively and g is k(t)-strongly accretive (k(t) ∈ (0, 1)) satisfying g(t, E) = E. Let η : Ω × E × E → E be Lipschitz continuous with constant τ (t) > 0 such that η(t, x(t), y(t)) = −η(t, y(t), x(t)) for all x(t), y(t) ∈ E and for each given x(t) ∈ E, the function h(t, y(t), x(t)) = x∗ (t) − J(t, x(t)), η(t, y(t), x(t)) is 0-DQCV in y(t). Let N : Ω × E × E × E → E be λN1 (t)-Lipschitz continuous, λN2 (t)-Lipschitz continuous and λN3 (t)-Lipschitz continuous with respect to the first argument, second argument and third argument respectively. Let φ : Ω × E × E → R ∪ {+∞} be such that for each x(t) ∈ E, φ(., ., x(t)) is a lower semicontinuous, η-subdifferentiable, proper functional satisfying g(t, x(t)) ∈ dom(∂ηt φ(., ., x(t))). Let J : Ω × E → E ∗ is η-strongly accretive with constant α(t) > 0 and λj (t)-Lipschitz continuous. Suppose that there exists a constant ρ(t) > 0 such that for each x(t), y(t) ∈ E, x∗ (t) ∈ E ∗ ∂ φ(.,.,xn (t))

ηt Jρ(t)

∂ φ(.,.,xn−1 (t))

ηt (x∗ ) − Jρ(t)

(x∗ (t)) ≤ μ(t)xn (t) − xn−1 (t)

(3.3)

and the following conditions holds 2τ 2 (t)λ2j (t)λ2g (t) 3 − μ2 (t) − < k(t) < 1 α2 (t) 2

0 < ρ(t)