# A New System of Random Generalized Variational Inclusions with Random Fuzzy Mappings and Random(H(·,·),ϕ)-η-Accretive Mappings in Banach Spaces

Authors: Sayyedeh Zahra Nazemi
Journal: ISRN Applied Mathematics
 (2012)

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## Abstract

We introduce a new notion of random(H(·,·),ϕ)-η-accretive mappings and prove the Lipschitz continuity of the random resolvent operator associated with the random (H(·,·),ϕ)-η-accretive mappings. We introduce and study a new system of random generalized
variational inclusions with random (H(·,·),ϕ)-η-accretive mappings and random fuzzy mappings in Banach spaces. By using the random resolvent operator, an iterative algorithm for solving such
system of random generalized variational inclusions is constructed in Banach spaces. Under some
suitable conditions, we prove the convergence of the iterative sequences generated by the algorithm.


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## Body

## 1. Introduction

Variational inclusions and variational inequalities have wide applications to many fields including, for example, economics, optimization and control theory, operators research, transportation network modeling, and mathematical programming. For these reasons, various variational inclusions and variational inequalities have been intensively studied in recent years. For details, we refer the reader to [1–3] and references therein.Recently, Wang and Ding [4], and Nazemi [5] introduced the cocepts of (H(·,·),η)-accretive mappings,  and (H,ϕ)-η-accretive mappings, respectively, which generalize the notion of H-accretive mappings, (H,η)-accretive mappings, H(·,·)-accretive mappings, (A,η)-accretive mappings and other existing accretive mappings as special cases. They also defined the resolvent operators associated with this mappings, and shown their Lipschitz continuity.On the other hand, it is well known that the study of the random equations involving the random operators in view of their need for dealing with probabilistic models in applied sciences is very important. In 1989, Chang and Zhu [6] and Chang [7] first introduced the concepts of variational inequalities for fuzzy mappings. Since then several classes of variational inequalities for fuzzy mappings were considered by Chang and Huang [8], Noor [9, 10], Noor and Al-Said [11], Ding [12], Ding and Park [13], and Park and Jeong [14, 15] in the setting of Hilbert spaces. Recently, the random variational inequalities and random variational inclusion problems have been introduced and studied by Dai [16], Cho and Lan [17], Lan [18], Ahmad and Bazán [19], Chang and Huang [20, 21], Zang and Zhu [22], Huang [23, 24], Husain et al. [25], etc.Motivated and inspired by the recent research works in this fascinating area, in this paper, we introduce a new class of random(H(·,·),ϕ)-η-accretive mappings and study a new system of random generalized variational inclusions with random (H(·,·),ϕ)-η-accretive mappings and random fuzzy mappings in Banach spaces. An iterative algorithm is defined to compute the approximate solutions of system of random generalized variational inclusions. The convergence of iterative sequences generated by the algorithm is also shown. Our results improve and generalize many known corresponding results.


## 2. Preliminaries

LetE be a real Banach space with dual space E* and 〈·,·〉  the dual pair between E* and E. Let F(E) be a collection of all 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 (2.1)(N)q={x∈E:N(x)≥q}
is called a q-cut set of N.Let(Ω,Σ) be a measurable space, where Ω is a set and Σ is a σ-algebra of subsets of Ω. Let β(E), 2E, CB(E), and H̃(·,·) be the class of Borel σ-fields in E, the family of all nonempty subsets of E, the family of all nonempty, closed and bounded subsets of E, and the Hausdorff metric on CB(E), respectively. Recall that the generalized duality mapping Jq:E→2E* is defined by(2.2)Jq(x)={f∈E*:〈x,f〉=‖x‖q,‖x‖q-1=∥f∥},∀x∈E,
where q>1 is a constant. In particular, J2 is the usual normalized duality mapping. It is known that, in general, Jq(x)=∥x∥q-1J2(x) for all x≠0, and Jq is single-valued if E* is strictly convex. In the sequel, unless otherwise specified, we always suppose that E is a real Banach space such that Jq is single-valued. If E=ℋ is a Hilbert space, then J2 becomes the identity mapping on H.The modulus of smoothness ofE is the function ρE:[0,∞)→[0,∞) defined by(2.3)ρE(t)=sup⁡{12(∥x+y∥+∥x-y∥)-1:∥x∥≤1,∥y∥≤t}.
A Banach space E is called uniformly smooth if(2.4)lim⁡t→0ρE(t)t=0.E is called q-uniformly smooth if there exists a constant c>0 such that(2.5)ρE(t)≤ctq,q>1.
Note that Jq is single-valued if E is uniformly smooth. In the following, we give an important inequality due to Xu [26] in q-uniformly smooth Banach spaces, which will play a crucial role in constructing the iterative algorithm to approximate the solution of system of variational inclusions.Lemma 2.1 (see [26]).
LetE be a real uniformly smooth Banach space. Then E is q-uniformly smooth if and only if there exists a constant cq>0 such that
(2.6)‖x+y‖q≤‖x‖q+q〈y,Jq(x)〉+cq‖y‖q.Definition 2.2.
A mappingx:Ω→E is said to be measurable if for any B∈β(E), {t∈Ω:x(t)∈B}∈Σ.Definition 2.3.
A mappingf:Ω×E→E is called a random single-valued mapping if for any x∈E, f(t,x)=y(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. Similarly, we can define a random mapping b:Ω×E×E→E. We will write ft(x)=f(t,x(t)) and bt(x,y)=b(t,x(t),y(t)) for all t∈Ω and x(t),y(t)∈E.
It is well known that a measurable mapping is necessarily a random mapping.Definition 2.4.
A multivalued mappingG:Ω→2E is said to be measurable if for any B∈β(E), G-1(B)={t∈Ω:G(t)∩B≠∅}∈Σ.Definition 2.5.
A mappingu:Ω→E is called a measurable section of a multivalued mapping Γ:Ω→2E if u is measurable and for any t∈Ω, u(t)∈Γ(t).Definition 2.6.
A mappingT: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 H̃-continuous if for any t∈Ω, T(t,·) is continuous in H̃(·,·), where H̃(·,·) is the Hausdorff metric on CB(E) defined as follows: for any given A,B∈CB(E),
(2.7)H̃(A,B)=max⁡{sup⁡x∈Ainf⁡y∈B∥x-y∥,sup⁡y∈Binf⁡x∈A∥x-y∥}.Definition 2.7.
A fuzzy mappingF:Ω→F(E) is called measurable  if for any α∈(0,1], (F(·))α:Ω→2E is a measurable multivalued mapping.Definition 2.8.
A fuzzy mappingF: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.Lemma 2.9 (see [27]).
LetT:Ω×E→CB(E)  be a H̃-continuous random multivalued mapping. Then for any measurable mapping w:Ω→E the multivalued mapping T(·,w(·)):Ω→CB(E) is measurable.Lemma 2.10 (see [27]).
LetS,T:Ω→CB(E) be two measurable multivalued mappings, ϵ>0   a constant, and v:Ω→E   a measurable selection of S. Then there exists a measurable selection w:Ω→E of T such that for all t∈Ω,
(2.8)∥v(t)-w(t)∥≤(1+ϵ)H̃(S(t),T(t)).Definition 2.11.
A random mappingP:Ω×E→E is said to be(i)
accretive if(2.9)〈Pt(x1)-Pt(x2),Jq(x1(t)-x2(t))〉≥0,∀x1(t),x2(t)∈E,t∈Ω;(ii)
r-strongly accretive if there exists a measurable function r:Ω→(0,∞) such that(2.10)〈Pt(x1)-Pt(x2),Jq(x1(t)-x2(t))〉≥r(t)‖x1(t)-x2(t)‖q,∀x1(t),x2(t)∈E,t∈Ω;(iii)
m-relaxed accretive if there exists a measurable function m:Ω→(0,∞) such that(2.11)〈Pt(x1)-Pt(x2),Jq(x1(t)-x2(t))〉≥-m(t)‖x1(t)-x2(t)‖q,∀x1(t),x2(t)∈E,t∈Ω;(iv)
(α,ξ)-relaxed cocoercive if there exist measurable functions α:Ω→(0,∞), ξ:Ω→(0,∞) such that(2.12)〈Pt(x1)-Pt(x2),Jq(x1(t)-x2(t))〉≥-α(t)‖Pt(x1)-Pt(x2)‖q+ξ(t)‖x1(t)-x2(t)‖q,∀x1(t),x2(t)∈E,t∈Ω;(v)
λ-Lipschitz continuous  if there exists a measurable function λ:Ω→(0,∞) such that(2.13)∥pt(x1)-pt(x2)∥≤λ(t)∥x1(t)-x2(t)∥,∀x1(t),x2(t)∈E,t∈Ω.Definition 2.12.
A random multivalued mappingM:Ω×E→2E is said to be(i)
accretive if(2.14)〈u(t)-v(t),Jq(x1(t)-x2(t))〉≥0,∀u(t)∈Mt(x1),v(t)∈Mt(x2),x1(t),x2(t)∈E,t∈Ω;(ii)
m-accretive if M is accretive and (It+ρ(t)Mt)(E)=E holds for all t∈Ω and ρ(t)>0.Remark 2.13.
IfE=ℋ is a Hilbert space, then we can obtain corresponding definitions of monotonicity, strongly monotonicity, relaxed monotonicity, and maximal monotonicity from Definitions 2.11 and 2.12.Definition 2.14.
A random mappingη:Ω×E×E→E is said to be τ-Lipschitz continuous if there exists a measurable function τ:Ω→(0,∞) such that
(2.15)∥ηt(x1,x2)∥≤τ(t)∥x1(t)-x2(t)∥,∀x1(t),x2(t)∈E,t∈Ω.Definition 2.15.
LetH,A:Ω×E→E and η:Ω×E×E→E be random single-valued mappings. Then a random multivalued mapping T:Ω×E→2E is said to be(i)
η-accretive if(2.16)〈u(t)-v(t),Jq(ηt(x1,x2))〉≥0,∀u(t)∈Tt(x1),v(t)∈Tt(x2),x1(t),x2(t)∈E,t∈Ω;(ii)
strictlyη-accretive if(2.17)〈u(t)-v(t),Jq(ηt(x1,x2))〉≥0,∀u(t)∈Tt(x1),v(t)∈Tt(x2),x1(t),x2(t)∈E,t∈Ω,
and equality holds if and only if x1(t)=x2(t);(iii)
γ-strongly η-accretive if there exists a measurable function γ:Ω→(0,∞) such that(2.18)〈u(t)-v(t),Jq(ηt(x1,x2))〉≥γ(t)‖x1(t)-x2(t)‖q,∀u(t)∈Tt(x1),v(t)∈Tt(x2),x1(t),x2(t)∈E,t∈Ω;(iv)
m-relaxed η-accretive if there exists a measurable function m:Ω→(0,∞) such that(2.19)〈u(t)-v(t),Jq(ηt(x1,x2))〉≥-m(t)‖x1(t)-x2(t)‖q,∀u(t)∈Tt(x1),v(t)∈Tt(x2),x1(t),x2(t)∈E,t∈Ω;(v)
η-m-accretive if T is η-accretive and (It+ρ(t)Tt)(E)=E holds for all t∈Ω and ρ(t)>0;(vi)
H-accretive if T is accretive and (Ht+ρ(t)Tt)(E)=E holds for all t∈Ω and ρ(t)>0;(vii)
(H,η)-accretive if T is η-accretive and (Ht+ρ(t)Tt)(E)=E holds for all t∈Ω and ρ(t)>0;(viii)
(A,η)-accretive if T is m-relaxed η-accretive and (At+ρ(t)Tt)(E)=E holds for all t∈Ω and ρ(t)>0.Definition 2.16.
LetH:E×E→E be a single-valued mapping and η:Ω×E×E→E and A,B:Ω×E→E   random single-valued mappings. Then(i)
H(A,·) is said to be α-strongly η-accretive with respect to A if there exists a measurable function α:Ω→(0,∞) such that
(2.20)〈H(At(x1),u(t))-H(At(x2),u(t)),Jq(ηt(x1,x2))〉≥α(t)‖x1(t)-x2(t)‖q,∀x1(t),x2(t),u(t)∈E,t∈Ω;(ii)
H(·,B) is said to be β-relaxed η-accretive with respect to B if there exists a measurable function β:Ω→(0,∞) such that
(2.21)〈H(u(t),Bt(x1))-H(u(t),Bt(x2)),Jq(ηt(x1,x2))〉≥-β(t)‖x1(t)-x2(t)‖q,∀x1(t),x2(t),u(t)∈E,t∈Ω;(iii)
H is said to be λ-Lipschitz continuous with respect to A in the first argument if there exists a measurable function λ:Ω→(0,∞) such that
(2.22)∥H(At(x1),u(t))-H(At(x2),u(t))∥≤λ(t)∥x1(t)-x2(t)∥,∀x1(t),x2(t),u(t)∈E,t∈Ω.Example 2.17.
Letℋ be a Hilbert space and S:Ω×ℋ→ℋ  a s-Lipschitz continuous and accretive mapping, ηt(x,y)=x(t)-y(t), H(x(t),y(t))=x(t)+y(t), At(x)=(1+ck/16)x(t), and Bt(x)=-ckSt(x)-(ck/8)x(t), for all (x(t),y(t))∈ℋ×ℋ, t∈Ω and ck>0. Then H(A,·) is (1+ck/16)-strongly η-accretive with respect to A and H(·,B) is ck(s+1/8)-relaxed η-accretive with respect to B.Definition 2.18.
LetE1,E2,…,Ek be real Banach spaces and for each i=1,2,…,k, let Ni:Ω×∏j=1kEj→Ei be a random single-valued mapping. Then Ni is said to be ξij-Lipschitz continuous in the (j+1)th argument if there exists a measurable function ξij:Ω→(0,∞) such that
(2.23)∥Nit(x1,…,xj-1,yj1,xj+1,…,xk)-Nit(x1,…,xj-1,yj2,xj+1,…,xk)∥≤ξij(t)∥yj1(t)-yj2(t)∥,∀yj1(t),yj2(t)∈Ej,xi(t)∈Ei,t∈Ω(i,j∈{1,2,…,k},i≠j).Definition 2.19.
Letℋ be a Hilbert space, H:ℋ×ℋ→ℋ   a single-valued mapping, and A,B:Ω×ℋ→ℋ   random single-valued mappings. H is said to be coercive with respect to A and B if
(2.24)lim⁡∥x(t)∥→∞〈H(At(x),Bt(x)),x(t)〉∥x(t)∥=+∞.Definition 2.20.
Letℋ be a Hilbert space, H:ℋ×ℋ→ℋ a single-valued mapping and A,B:Ω×ℋ→ℋ random single-valued mappings. H is said to be bounded with respect to A and B if H(At(P),Bt(P)) is bounded for every bounded subset P of ℋ and t∈Ω. H is said to be hemicontinuous with respect to A and B if for any fixed x(t),y(t),z(t)∈ℋ, the function k↦〈H(At(x+ky),Bt(x+ky)),z(t)〉 is continuous at 0+.


## 3. Random(H(·,·),ϕ)-η-Accretive Mappings

In this section, we will introduce a new class of random accretive mappings-(H(·,·),ϕ)-η-accretive mappings and discuss some properties of random (H(·,·),ϕ)-η-accretive mappings.Definition 3.1.
LetH:E×E→E be a single-valued mapping and η:Ω×E×E→E and ϕ,A,B:Ω×E→E   random single-valued mappings. Then the random multivalued mapping M:Ω×E→2E is said to be (H(·,·),ϕ)-η-accretive with respect to A and B if ϕ∘M is m-relaxed η-accretive and
(3.1)(H(At,Bt)+ϕt∘Mt)(E)=E,∀t∈Ω.Remark 3.2.
(i) IfH(At,Bt)=Ht and ϕ∘M is η-accretive, then the random (H(·,·),ϕ)-η-accretive mapping reduces to the random (H,ϕ)-η-accretive mapping.
(ii) Ifϕt(x)=λ(t)x(t), for all x(t)∈E, λ(t)>0, then the random (H(·,·),ϕ)-η-accretive mapping reduces to the random (H(·,·),η)-accretive mapping.Remark 3.3.
(i) IfM is η-accretive, H(At,Bt)=Ht, and ϕt(x)=λ(t)x(t), for all x(t)∈E, λ(t)>0, then the random (H(·,·),ϕ)-η-accretive mapping reduces to the random (H,η)-accretive mapping studied by Uea and Kumam [28].
(ii) IfM is m-relaxed η-accretive, H(At,Bt)=Ht, and ϕt(x)=λ(t)x(t), for all x(t)∈E, λ(t)>0, then the random (H(·,·),ϕ)-η-accretive mapping reduces to the random (A,η)-accretive mapping studied by Cho and Lan [17].
(iii) IfM is η-accretive, ηt(x,y)=x(t)-y(t), ϕt(x)=λ(t)x(t), for all x(t),y(t)∈E, λ(t)>0, then the random (H(·,·),ϕ)-η-accretive mapping reduces to the random H(·,·)-accretive mapping studied by Zhang [29].Example 3.4.
Letℋ   be a Hilbert space, λ:Ω→(0,∞) a measurable function; for every (x(t),y(t))∈ℋ×ℋ, ηt(x,y)=x(t)-y(t), ϕt(x)=λ(t)x(t);  M:Ω×ℋ→2H   a random maximal monotone mapping,   H:ℋ×ℋ→ℋ   a bounded, coercive, and hemicontinuous mapping, with respect to A and B, and α-strongly η-accretive with respect to A in the first argument and β-relaxed η-accretive with respect to B in second argument. Then M is a random (H(·,·),ϕ)-η-accretive mapping with respect to A and B.Proof.
For everyt∈Ω, λ(t)M is maximal monotone since M is maximal monotone. Since H is bounded, coercive, and hemicontinuous with respect to A and B,α-strongly η-accretive with respect to A in the first argument and β-relaxed η-accretive with respect to B in the second argument, it follows from Corollary 32.26 of [30] that (H(At,Bt)+ϕt∘Mt)(ℋ)=ℋ holds for every t∈Ω. Thus M is a random (H(·,·),ϕ)-η-accretive mapping with respect to A and B.Example 3.5.
Letℋ=ℛ=(-∞,+∞), Mt(x)=1/(1+x(t)2), At(x)=2x(t), Bt(y)=-y(t), H(At(x),Bt(y))=At(x)+Bt(y), ηt(x,y)=arctan(x(t))-arctan(y(t)), and ϕt∘Mt(x)=∫-∞x(t)Mt(s)ds, for all x(t),y(t)∈ℛ. Then
(3.2)〈ϕt∘Mt(x)-ϕt∘Mt(y),ηt(x,y)〉=〈(arctan(x(t))+π2)-(arctan(y(t))+π2),arctan(x(t))-arctan(y(t))〉=(arctan(x(t))-arctan(y(t)))2≥0,(H(At,Bt)+ϕt∘Mt)(x)=x(t)+arctan(x(t))+π2.
This implies that (H(At,Bt)+ϕt∘Mt) is surjective. Thus M is a random (H(·,·),ϕ)-η-accretive mapping with respect to A and B.Definition 3.6.
Letη:Ω×E×E→E, ϕ:Ω×E→E be random single-valued mappings, H(A,B) be α-strongly η-accretive with respect to A and β-relaxed η-accretive with respect to B and M:Ω×E→2E be a random (H(·,·),ϕ)-η-accretive mapping with respect to A and B. Then the general resolvent operator RMt,ϕtH(·,·),ηt:Ω×E→2E is defined by
(3.3)RMt,ϕtH(⋅,⋅),ηt(u)=(H(At,Bt)+ϕt∘Mt)-1(u),∀u(t)∈E,t∈Ω.Remark 3.7.
(i) IfM is η-accretive, H(At,Bt)=Ht, and ϕt(x)=λ(t)x(t), for all x(t)∈E, λ(t)>0, then the resolvent operator RMt,ϕtH(·,·),ηt reduces to the resolvent operator JHtλ(t),Mt introduced by Uea and Kumam [28].
(ii) IfM is m-relaxed η-accretive, H(At,Bt)=Ht and ϕt(x)=λ(t)x(t), for all x(t)∈E, λ(t)>0, then the resolvent operator RMt,ϕtH(·,·),ηt reduces to the resolvent operator Rηt,Mtλ(t),At introduced by Cho and Lan [17].
(iii) IfM is η-accretive, ηt(x,y)=x(t)-y(t), ϕt(x)=λ(t)x(t), for all x(t),y(t)∈E, λ(t)>0, then the resolvent operator RMt,ϕtH(·,·),ηt reduces to the resolvent operator RM,λH(·,·) introduced by Zhang [29].Theorem 3.8.
Letϕ:Ω×E→E be a random single-valued mapping, η:Ω×E×E→E be τ-Lipschitz continuous and H(A,B) be α-strongly η-accretive with respect to A and β-relaxed η-accretive with respect to B and α(t)≥β(t), ∀t∈Ω. Let M:Ω×E→2E be a random (H(·,·),ϕ)-η-accretive mapping with respect to A and B. Then the resolvent operator RMt,ϕtH(·,·),ηt is τ(t)q-1/(r(t)-m(t))-Lipschitz continuous for r(t)>m(t), where r(t)=α(t)-β(t), ∀t∈Ω, that is,
(3.4)∥RMt,ϕtH(⋅,⋅),ηt(u)-RMt,ϕtH(⋅,⋅),ηt(v)∥≤τ(t)q-1r(t)-m(t)∥u(t)-v(t)∥,∀u(t),v(t)∈E,t∈Ω.Proof.
Letu(t),v(t)∈E be any given points; it follows from Definition 3.6 that
(3.5)RMt,ϕtH(⋅,⋅),ηt(u)=(H(At,Bt)+ϕt∘Mt)-1(u),RMt,ϕtH(⋅,⋅),ηt(v)=(H(At,Bt)+ϕt∘Mt)-1(v).
This implies that
(3.6)(u(t)-H(At(RMt,ϕtH(⋅,⋅),ηt(u)),Bt(RMt,ϕtH(⋅,⋅),ηt(u))))∈ϕt∘Mt(RMt,ϕtH(⋅,⋅),ηt(u)),(v(t)-H(At(RMt,ϕtH(⋅,⋅),ηt(v)),Bt(RMt,ϕtH(⋅,⋅),ηt(v))))∈ϕt∘Mt(RMt,ϕtH(⋅,⋅),ηt(v)).
Since ϕ∘M is m-relaxed η-accretive, we get
(3.7)-m(t)‖RMt,ϕtH(⋅,⋅),ηt(u)-RMt,ϕtH(⋅,⋅),ηt(v)‖q≤〈u(t)-H(At(RMt,ϕtH(⋅,⋅),ηt(u)),Bt(RMt,ϕtH(⋅,⋅),ηt(u)))-(v(t)-H(At(RMt,ϕtH(⋅,⋅),ηt(v)),Bt(RMt,ϕtH(⋅,⋅),ηt(v)))),Jq(ηt(RMt,ϕtH(⋅,⋅),ηt(u),RMt,ϕtH(⋅,⋅),ηt(v)))〉=〈u(t)-v(t)-(H(At(RMt,ϕtH(⋅,⋅),ηt(u)),Bt(RMt,ϕtH(⋅,⋅),ηt(u)))-H(At(RMt,ϕtH(⋅,⋅),ηt(v)),Bt(RMt,ϕtH(⋅,⋅),ηt(v)))),Jq(ηt(RMt,ϕtH(⋅,⋅),ηt(u),RMt,ϕtH(⋅,⋅),ηt(v)))〉.
From (3.7) and Lipschitz continuity of η, we have
(3.8)τ(t)q-1∥u(t)-v(t)∥‖RMt,ϕtH(⋅,⋅),ηt(u)-RMt,ϕtH(⋅,⋅),ηt(v)‖q-1≥∥u(t)-v(t)∥‖ηt(RMt,ϕtH(⋅,⋅),ηt(u),RMt,ϕtH(⋅,⋅),ηt(v))‖q-1≥〈u(t)-v(t),Jq(ηt(RMt,ϕtH(⋅,⋅),ηt(u),RMt,ϕtH(⋅,⋅),ηt(v)))〉≥〈H(At(RMt,ϕtH(⋅,⋅),ηt(u)),Bt(RMt,ϕtH(⋅,⋅),ηt(u)))-H(At(RMt,ϕtH(⋅,⋅),ηt(v)),Bt(RMt,ϕtH(⋅,⋅),ηt(v))),Jq(ηt(RMt,ϕtH(⋅,⋅),ηt(u),RMt,ϕtH(⋅,⋅),ηt(v)))〉-m(t)‖RMt,ϕtH(⋅,⋅),ηt(u)-RMt,ϕtH(⋅,⋅),ηt(v)‖q≥(α(t)-β(t)-m(t))‖RMt,ϕtH(⋅,⋅),ηt(u)-RMt,ϕtH(⋅,⋅),ηt(v)‖q=(r(t)-m(t))‖RMt,ϕtH(⋅,⋅),ηt(u)-RMt,ϕtH(⋅,⋅),ηt(v)‖q.
Hence
(3.9)∥RMt,ϕtH(⋅,⋅),ηt(u)-RMt,ϕtH(⋅,⋅),ηt(v)∥≤τ(t)q-1r(t)-m(t)∥u(t)-v(t)∥,∀u(t),v(t)∈E,t∈Ω.
This completes the proof.


## 4. A New System of Random Generalized Variational Inclusions

In this section, we will introduce a new system of random generalized variational inclusions with random(H(·,·),ϕ)-η-accretive mappings and construct an iterative algorithm for solving this system of random generalized variational inclusions.Let fori,j=1,2,…,k, Tji:Ω×Ej→F(Ei),  and Si:Ω×Ei→F(Ei) be random fuzzy mappings satisfying the following condition (C).(C)
Fori,j=1,2,…,k, there exist mappings aj:Ej→[0,1], bi:Ei→[0,1], such that(4.1)(Tjit,yj)aj(yj)∈CB(Ei),∀(t,yj)∈Ω×Ej(i,j=1,2,…,k),(Sit,zi)bi(zi)∈CB(Ei),∀(t,zi)∈Ω×Ei(i=1,2,…,k).By using the random fuzzy mappingsTji, Si(i,j=1,2,…,k), we can define random multivalued mappings T̃ji, S̃i as follows.(4.2)T̃ji:Ω×Ej⟶CB(Ei),yj⟼(Tjit,yj)aj(yj)(i,j=1,2,…,k),S̃i:Ω×Ei⟶CB(Ei),zi⟼(Sit,zi)bi(zi)(i=1,2,…,k),
where (Sit,zi)bi(zi)=Si(t,zi(t)).In the sequel, fori,j=1,2,…,k, T̃ji and S̃i are called the random multivalued mappings induced by the random fuzzy mappings Tji and Si, respectively.Fori=1,2,…,k, given mappings ai:Ei→[0,1], bi:Ei→[0,1], let T1i:Ω×E1→F(Ei), T2i:Ω×E2→F(Ei),…,Tki:Ω×Ek→F(Ei), Si:Ω×Ei→F(Ei) be random fuzzy mappings, Hi:Ei×Ei→Ei be a single-valued mapping and ϕi,Ai,Bi,fi,pi:Ω×Ei→Ei, ηi:Ω×Ei×Ei→Ei, Fi,Ni:Ω×∏j=1kEj→Ei be random single-valued mappings and Mi:Ω×Ei×Ei→2Ei be a random (Hi(·,·),ϕi)-ηi-accretive mapping with respect to Ai and Bi in the second argument with Imfi∩domMit(·,wi)≠∅, for each wi(t)∈Ei and t∈Ω. Now we consider the following system of random generalized variational inclusions:Fori=1,2,…,n, find measurable mappings xi,u1i,u2i,…,uki,vi:Ω→Ei such that for all t∈Ω and for each i=1,2,…,k,xi(t)∈Ei, T1it,x1(t)(u1i(t))≥a1(x1(t)), T2it,x2(t)(u2i(t))≥a2(x2(t)), …, Tkit,xk(t)(uki(t))≥ak(xk(t)), Sit,xi(t)(vi(t))≥bi(xi(t)) such that(4.3)0∈Fit(x1-p1t(x1),x2-p2t(x2),…,xk-pkt(xk))+Nit(ui1,ui2,…,uik)+Mit(fit(xi),vi).If fori=1,2,…,k, Ei=ℋi is a Hilbert space and Mit(·,vi)=∂φit(·), for all t∈Ω, xi(t)∈ℋi and vi(t)∈S̃it(xi), where ∂φi denote the subdifferential of a proper, convex, and lower semicontinuous function φi:Ω×ℋi→R∪{+∞}, then system of random generalized variational inclusions (4.3) reduces to the following system of random generalized mixed variational inequalities.Fori=1,2,…,k, find measurable mappings xi,u1i,u2i,…,uki:Ω→ℋi such that for all t∈Ω and for each i=1,2,…,k, xi(t)∈ℋi, T1it,x1(t)(u1i(t))≥a1(x1(t)), T2it,x2(t)(u2i(t))≥a2(x2(t)), …, Tkit,xk(t)(uki(t))≥ak(xk(t)) such that(4.4)〈Fit(x1-p1t(x1),x2-p2t(x2),…,xk-pkt(xk))+Nit(ui1,ui2,…,uik),yi(t)-fit(xi)〉≥φit(fit(xi))-φit(yi),∀t∈Ω,yi(t)∈Hi.We remark that system of random generalized variational inclusions (4.3) and system of random generalized mixed variational inequalities (4.4) include as special cases, many kinds of random variational inequalities and random variational inclusions of [17, 19–21], etc.Lemma 4.1.
Fori=1,2,…,k, let Hi:Ei×Ei→Ei be a single-valued mapping and ηi:Ω×Ei×Ei→Ei, ϕi,Ai,Bi:Ω×Ei→Ei  random single-valued mappings such that Hi(Ai,Bi) be αi-strongly ηi-accretive with respect to Ai and βi-relaxed ηi-accretive with respect to Bi. Then the set of measurable mappings xi,u1i,u2i,…,uki:Ω→Ei(i=1,2,…,k) is a random solution of problem (4.3) if and only if for all t∈Ω, xi(t)∈Ei, u1i(t)∈T̃1it(x1), u2i(t)∈T̃2it(x2),…,uki(t)∈T̃kit(xk), vi(t)∈S̃it(xi), and
(4.5)fit(xi)=RMit(⋅,vi),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xi)),Bit(fit(xi)))-ϕit∘Fit(x1-p1t(x1),x2-p2t(x2),…,xk-pkt(xk))-ϕit∘Nit(ui1,ui2,…,uik)(fit(xi))],
where RMit(·,vi),ϕitHi(·,·),ηit=(Hi(Ait,Bit)+ϕit∘Mit(·,vi))-1.Algorithm 4.2.
Suppose that fori,j=1,2,…,k, Tji:Ω×Ej→F(Ei), Si:Ω×Ei→F(Ei)  be random fuzzy mappings satisfying the condition (C). Let for i=1,2,…,k, T̃1i:Ω×E1→CB(Ei), T̃2i:Ω×E2→CB(Ei),…,T̃ki:Ω×Ek→CB(Ei), S̃i:Ω×Ei→CB(Ei) be H̃-continuous random multivalued mappings induced by T1i,T2i,…,Tki and Si, respectively. Let Hi:Ei×Ei→Ei be a single-valued mapping; ηi:Ω×Ei×Ei→Ei, ϕi,Ai,Bi:Ω×Ei→Ei, Fi,Ni:Ω×∏j=1kEj→Ei   random single-valued mappings; the random multivalued mapping Mi:Ω×Ei×Ei→2Ei be (Hi(·,·),ϕi)-ηi-accretive with respect to Ai and Bi in the second argument. For any given measurable mapping xi:Ω→Ei(i=1,2,…,k), the multivalued mappings T̃1i(·,x10(·)), T̃2i(·,x20(·)),…,T̃ki(·,xk0(·)), S̃i(·,xi0(·))(i=1,2,…,k) are measurable by Lemma 2.9. Hence for i=1,2,…,k, there exist measurable selections u1i0:Ω→Ei of T̃1i(·,x10(·)), u2i0:Ω→Ei of T̃2i(·,x20(·)),…,uki0:Ω→Ei of T̃ki(·,xk0(·)) and vi0:Ω→Ei of S̃i(·,xi0(·)), by Himmelberg [31]. Let for i=1,2,…,k,
(4.6)xi1(t)=xi0(t)-fit(xi0)+RMit(⋅,vi0),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xi0)),Bit(fit(xi0)))-ϕit∘Fit(x10-p1t(x10),x20-p2t(x20),…,xk0-pkt(xk0))-ϕit∘Nit(ui10,ui20,…,uik0)].
It is easy to see that for i=1,2,…,k, xi1:Ω→Ei is measurable. By Lemma 2.10, there exist measurable selections u1i1:Ω→Ei of T̃1i(·,x11(·)), u2i1:Ω→Ei of T̃2i(·,x21(·)),…,uki1:Ω→Ei of T̃ki(·,xk1(·)) and vi1:Ω→Ei of S̃i(·,xi1(·))(i=1,2,…,k) such that for all t∈Ω and i=1,2,…,k,
(4.7)∥u1i0(t)-u1i1(t)∥≤(1+1)H̃(T̃1it(x10),T̃1it(x11)),∥u2i0(t)-u2i1(t)∥≤(1+1)H̃(T̃2it(x20),T̃2it(x21)),⋮∥uki0(t)-uki1(t)∥≤(1+1)H̃(T̃kit(xk0),T̃kit(xk1)),∥vi0(t)-vi1(t)∥≤(1+1)H̃(S̃it(xi0),S̃it(xi1)).
Let for i=1,2,…,k,
(4.8)xi2(t)=xi1(t)-fit(xi1)+RMit(⋅,vi1),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xi1)),Bit(fit(xi1)))-ϕit∘Fit(x11-p1t(x11),x21-p2t(x21),…,xk1-pkt(xk1))-ϕit∘Nit(ui11,ui21,…,uik1)],
then xi2:Ω→Ei is measurable. Continuing the above process inductively, we can define the following random iterative sequences {xin(t)}, {u1in(t)}, {u2in(t)},…,{ukin(t)} and {vin(t)}(i=1,2,…,k) for solving problem (4.3) as follows:
(4.9)xin+1(t)=xin(t)-fit(xin)+RMit(⋅,vin),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xin)),Bit(fit(xin)))-ϕit∘Fit(x1n-p1t(x1n),x2n-p2t(x2n),…,xkn-pkt(xkn))-ϕit∘Nit(ui1n,ui2n,…,uikn)(Ait(fit(xin)),Bit(fit(xin)))],u1in(t)∈T̃1it(x1n),u2in(t)∈T̃2it(x2n),…,ukin(t)∈T̃kit(xkn),vin(t)∈S̃it(xin),∥u1in(t)-u1in+1(t)∥≤(1+1n+1)H̃(T̃1it(x1n),T̃1it(x1n+1)),∥u2in(t)-u2in+1(t)∥≤(1+1n+1)H̃(T̃2it(x2n),T̃2it(x2n+1)),⋮∥ukin(t)-ukin+1(t)∥≤(1+1n+1)H̃(T̃kit(xkn),T̃kit(xkn+1)),∥vin(t)-vin+1(t)∥≤(1+1n+1)H̃(S̃it(xin),S̃it(xin+1)),
for any t∈Ω, i=1,2,…,k and n=0,1,2,….Theorem 4.3.
Fori=1,2,…,k, let Ei be a q-uniformly smooth Banach space and ϕi:Ω×Ei→Ei be a ϱi-Lipschitz continuous mapping satisfying ϕit(x+y)=ϕit(x)+ϕit(y) and kerϕit={0}, for all t∈Ω. Let ηi:Ω×Ei×Ei→Ei be τi-Lipschitz continuous, Hi:Ei×Ei→Ei be a single-valued mapping, and Ai,Bi:Ω×Ei→Ei   random single-valued mappings such that Hi(Ai,Bi) be αi-strongly ηi-accretive with respect to Ai and βi-relaxed ηi-accretive with respect to Bi and Hi(Ai,Bi) be δi1-Lipschitz continuous with respect to Ai and δi2-Lipschitz continuous with respect to Bi. Let fi:Ω×Ei→Ei be (ϵi,μi)-relaxed cocoercive and ξi-Lipschitz continuous, and pi:Ω×Ei→Ei be γpi-strongly accretive and λpi-Lipschitz continuous. Suppose that Fi:Ω×∏j=1kEj→Ei be ζij-Lipschitz continuous in the (j+1) th argument for j=1,2,…,k, Ni:Ω×∏j=1kEj→Ei be νij-Lipschitz continuous in the (j+1) th argument for j=1,2,…,k. Let Mi:Ω×Ei×Ei→2Ei be a random multivalued mapping such that Mi  is a (Hi(·,·),ϕi)-ηi-accretive mapping with respect to Ai and Bi in the second argument. Let T1i:Ω×E1→F(Ei), T2i:Ω×E2→F(Ei),…,Tki:Ω×Ek→F(Ei), Si:Ω×Ei→F(Ei) be random fuzzy mappings satisfying the condition (C) and T̃1i:Ω×E1→CB(Ei), T̃2i:Ω×E2→CB(Ei),…,T̃ki:Ω×Ek→CB(Ei), and S̃i:Ω×Ei→CB(Ei)   random multivalued mappings induced by T1i,T2i,…,Tki and Si, respectively. Suppose that T̃1i, T̃2i,…,T̃ki, S̃i are H̃-continuous with constants t1i, t2i, …, tki, and λsi, respectively.In addition, if
(4.10)∥RMit(⋅,yi1),ϕitHi(⋅,⋅),ηit(zi)-RMit(⋅,yi2),ϕitHi(⋅,⋅),ηit(zi)∥≤λi(t)∥yi1(t)-yi2(t)∥,∀t∈Ω,yi1(t),yi2(t),zi(t)∈Ei(i=1,2,…,k),(4.11)0<(1+qϵ1(t)ξ1(t)q-qμ1(t)+cqξ1(t)q)1/q+λ1(t)λs1(t)+τ1(t)q-1r1(t)-m1(t)×(∑j=1k((δ11(t)+δ12(t))qξ1(t)q+qϱ1(t)(1-qγp1(t)+cqλp1(t)q)1/qζ11(t)(δ11(t)+δ12(t))q-1ξ1(t)q-1+cqϱ1(t)q(1-qγp1(t)+cqλp1(t)q)ζ11(t)q)1/q+∑j=1kϱ1(t)ν1j(t)t1j(t))+∑j=2kτj(t)q-1rj(t)-mj(t)ϱj(t)ζj1(t)(1-qγp1(t)+cqλp1(t)q)1/q<1,0<(1+qϵ2(t)ξ2(t)q-qμ2(t)+cqξ2(t)q)1/q+λ2(t)λs2(t)+τ2(t)q-1r2(t)-m2(t)×(∑j=1k((δ21(t)+δ22(t))qξ2(t)q+qϱ2(t)(1-qγp2(t)+cqλp2(t)q)1/qζ22(t)(δ21(t)+δ22(t))q-1ξ2(t)q-1+cqϱ2(t)q(1-qγp2(t)+cqλp2(t)q)ζ22(t)q)1/q+∑j=1kϱ2(t)ν2j(t)t2j(t))+τ1(t)q-1r1(t)-m1(t)ϱ1(t)ζ12(t)(1-qγp2(t)+cqλp2(t)q)1/q+∑j=3kτj(t)q-1rj(t)-mj(t)ϱj(t)ζj2(t)(1-qγp2(t)+cqλp2(t)q)1/q<1,…,0<(1+qϵk(t)ξk(t)q-qμk(t)+cqξk(t)q)1/q+λk(t)λsk(t)+τk(t)q-1rk(t)-mk(t)×(∑j=1k((δk1(t)+δk2(t))qξk(t)q+qϱk(t)(1-qγpk(t)+cqλpk(t)q)1/qζkk(t)(δk1(t)+δk2(t))q-1ξk(t)q-1+cqϱk(t)q(1-qγpk(t)+cqλpk(t)q)ζkk(t)q)1/q+∑j=1kϱk(t)νkj(t)tkj(t))+∑j=1k-1τj(t)q-1rj(t)-mj(t)ϱj(t)ζjk(t)(1-qγpk(t)+cqλpk(t)q)1/q<1,
then there exist measurable mappings xi, u1i,u2i,…,uki,vi:Ω→Ei such that (4.3) holds. Moreover for all t∈Ω,xin(t)→xi(t),u1in(t)→u1i(t), u2in(t)→u2i(t),…,ukin(t)→uki(t), vin(t)→vi(t), where {xin(t)}, and {u1in(t)},…,{ukin(t)}, {vin(t)} are random sequences obtained by Algorithm 4.2.Proof.
Fori=1,2,…,k, from (4.10), Lemma 4.1, and Algorithm 4.2, we have
(4.12)∥xin+1(t)-xin(t)∥=‖xin(t)-fit(xin)+RMit(⋅,vin),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xin)),Bit(fit(xin)))-ϕit∘Fit(x1n-p1t(x1n),x2n-p2t(x2n),…,xkn-pkt(xkn))-ϕit∘Nit(ui1n,ui2n,…,uikn)]-(xin-1(t)-fit(xin-1)+RMit(⋅,vin-1),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))-ϕit∘Fit(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),…,xkn-1-pkt(xkn-1))-ϕit∘Nit(ui1n-1,ui2n-1,…,uikn-1)](Ait(fit(xin-1)),Bit(fit(xin-1))))(Ait(fit(xin)),Bit(fit(xin)))‖≤‖xin(t)-xin-1(t)-(fit(xin)-fit(xin-1))‖+‖RMit(⋅,vin),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xin)),Bit(fit(xin)))-ϕit∘Fit(x1n-p1t(x1n),x2n-p2t(x2n),…,xkn-pkt(xkn))-ϕit∘Nit(ui1n,ui2n,…,uikn)(Ait(fit(xin)),Bit(fit(xin)))]-RMit(⋅,vin),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))-ϕit∘Fit(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),…,xkn-1-pkt(xkn-1))-ϕit∘Nit(ui1n-1,ui2n-1,…,uikn-1)(Ait(fit(xin-1)),Bit(fit(xin-1)))]RMit(⋅,vin),ϕitHi(⋅,⋅),ηit‖+‖RMit(⋅,vin),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))-ϕit∘Fit(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),…,xkn-1-pkt(xkn-1))-ϕit∘Nit(ui1n-1,ui2n-1,…,uikn-1)(Ait(fit(xin-1)),Bit(fit(xin-1)))]-RMit(⋅,vin-1),ϕitHi(⋅,⋅),ηit[Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))-ϕit∘Fit(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),…,xkn-1-pkt(xkn-1))-ϕit∘Nit(ui1n-1,ui2n-1,…,uikn-1)(Ait(fit(xin-1)),Bit(fit(xin-1)))]‖≤‖xin(t)-xin-1(t)-(fit(xin)-fit(xin-1))‖+τi(t)q-1ri(t)-mi(t)‖(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),…,xkn-1-pkt(xkn-1))Hi(Ait(fit(xin)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))-(ϕit∘Fit(x1n-p1t(x1n),x2n-p2t(x2n),…,xkn-pkt(xkn))-ϕit∘Fit(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),…,xkn-1-pkt(xkn-1)))‖+τi(t)q-1ri(t)-mi(t)ϱi(t)‖Nit(ui1n,ui2n,…,uikn)-Nit(ui1n-1,ui2n-1,…,uikn-1)‖+λi(t)‖vin(t)-vin-1(t)‖.
Since fi:Ω×Ei→Ei be (ϵi,μi)-relaxed cocoercive and ξi-Lipschitz continuous, we have
(4.13)‖xin(t)-xin-1(t)-(fit(xin)-fit(xin-1))‖q≤‖xin(t)-xin-1(t)‖q-q〈fit(xin)-fit(xin-1),Jq(xin(t)-xin-1(t))〉+cq‖fit(xin)-fit(xin-1)‖q≤(1+qϵi(t)ξi(t)q-qμi(t)+cqξi(t)q)‖xin(t)-xin-1(t)‖q.
Using γpi-strongly accretivity and λpi-Lipschitz continuity of pi, ϱi-Lipschitz continuity of φi we have
(4.14)‖xin(t)-xin-1(t)-(pit(xin)-pit(xin-1))‖q≤‖xin(t)-xin-1(t)‖q-q〈pit(xin)-pit(xin-1),Jq(xin(t)-xin-1(t))〉+cq‖pit(xin)-pit(xin-1)‖q≤(1-qγpi(t)+cqλpi(t)q)‖xin(t)-xin-1(t)‖q,‖Hi(Ait(fit(xin)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))-(ϕit∘Fit(x1n-p1t(x1n),x2n-p2t(x2n),…,xkn-pkt(xkn))-ϕit∘Fit(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),…,xkn-1-pkt(xkn-1)))‖≤‖Hi(Ait(fit(xin)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))-(ϕit∘Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-pit(xin),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))-ϕit∘Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-1-pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn)))‖+ϱi(t)‖Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-1-pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))-Fit(x1n-1-p1t(x1n-1),…,xi-1n-1-pi-1t(xi-1n-1),xin-1-pit(xin-1),xi+1n-1-pi+1t(xi+1n-1),…,xkn-1-pkt(xkn-1))‖.
Since Hi(Ai,Bi) is δi1-Lipschitz continuous with respect to Ai and δi2-Lipschitz continuous with respect to Bi, we have
(4.15)‖Hi(Ait(fit(xin)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))‖≤‖Hi(Ait(fit(xin)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin)))‖+‖Hi(Ait(fit(xin-1)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))‖≤(δi1(t)+δi2(t))‖fit(xin)-fit(xin-1)‖≤(δi1(t)+δi2(t))ξi(t)‖xin(t)-xin-1(t)‖.
Since Fi:Ω×∏j=1kEj→Ei is ζii-Lipschitz continuous in the (i+1)th argument, by (4.15), we have
(4.16)‖Hi(Ait(fit(xin)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))-(ϕit∘Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-pit(xin),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))-ϕit∘Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-1pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn)))(Ait(fit(xin)),Bit(fit(xin)))(Ait(fit(xin)),Bit(fit(xin)))‖q≤‖Hi(Ait(fit(xin)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin-1)))‖q-q〈(Hi(Ait(fit(xin)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin-1))))ϕit∘Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-pit(xin),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))-ϕit∘Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-1-pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn)(xin-1)(xin-1)),Jq(Hi(Ait(fit(xin)),Bit(fit(xin)))-Hi(Ait(fit(xin-1)),Bit(fit(xin-1))))〉+cqϱi(t)q‖xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-pit(xin),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))-Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-1-pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))‖q≤((δi1(t)+δi2(t))qξi(t)q+qϱi(t)(1-qγpi(t)+cqλpi(t)q)1/qζii(t)(δi1(t)+δi2(t))q-1ξi(t)q-1+cqϱi(t)q(1-qγpi(t)+cqλpi(t)q)ζii(t)q)‖xin(t)-xin-1(t)‖q,i=1,2,…,k.
Since Fi:Ω×∏j=1kEj→Ei is ζij-Lipschitz continuous in the (j+1)th argument, for j=1,…,i-1,i+1,…,k, we have
(4.17)‖Fit(x1n-p1t(x1n),…,xi-1n-pi-1t(xi-1n),xin-1-pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))-Fit(x1n-1-p1t(x1n-1),…,xi-1n-1-pi-1t(xi-1n-1),xin-1-pit(xin-1),xi+1n-1-pi+1t(xi+1n-1),…,xkn-1-pkt(xkn-1))‖≤‖Fit(x1n-p1t(x1n),x2n-p2t(x2n),…,xi-1n-pi-1t(xi-1n),xin-1-pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))-Fit(x1n-1-p1t(x1n-1),x2n-p2t(x2n),…,xi-1n-pi-1t(xi-1n),xin-1-pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))‖+‖Fit(x1n-1-p1t(x1n-1),x2n-p2t(x2n),x3n-p3t(x3n),…,xi-1n-pi-1t(xi-1n),xin-1-pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))-Fit(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),x3n-p3t(x3n),…,xi-1n-pi-1t(xi-1n),xin-1-pit(xin-1),xi+1n-pi+1t(xi+1n),…,xkn-pkt(xkn))‖+⋯+‖Fit(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),x3n-1-p3t(x3n-1),…,xi-1n-1-pi-1t(xi-1n-1),xin-1-pit(xin-1),xi+1n-1-pi+1t(xi+1n-1),…,xk-1n-1-pk-1t(xk-1n-1),xkn-pkt(xkn))-Fit(x1n-1-p1t(x1n-1),x2n-1-p2t(x2n-1),x3n-1-p3t(x3n-1),…,xi-1n-1-pi-1t(xi-1n-1),xin-1-pit(xin-1),xi+1n-1-pi+1t(xi+1n-1),…,xk-1n-1-pk-1t(xk-1n-1),xkn-1-pkt(xkn-1))‖≤ζi1(t)(1-qγp1(t)+cqλp1(t)q)1/q‖x1n(t)-x1n-1(t)‖+ζi2(t)(1-qγp2(t)+cqλp2(t)q)1/q‖x2n(t)-x2n-1(t)‖+⋯+ζi,i-1(t)(1-qγpi-1(t)+cqλpi-1(t)q)1/q‖xi-1n(t)-xi-1n-1(t)‖+ζi,i+1(t)(1-qγpi+1(t)+cqλpi+1(t)q)1/q‖xi+1n(t)-xi+1n-1(t)‖+⋯+ζik(t)(1-qγpk(t)+cqλpk(t)q)1/q‖xkn(t)-xkn-1(t)‖=∑j=1i-1ζij(t)(1-qγpj(t)+cqλpj(t)q)1/q‖xjn(t)-xjn-1(t)‖+∑j=i+1kζij(t)(1-qγpj(t)+cqλpj(t)q)1/q‖xjn(t)-xjn-1(t)‖,i=1,2,…,k.
It follows from the Lipschitz continuity of Ni, the H̃-continuity of T̃1i,T̃2i,…,T̃ki(i=1,2,…,k) that
(4.18)‖Nit(ui1n,ui2n,…,uikn)-Nit(ui1n-1,ui2n-1,…,uikn-1)‖≤‖Nit(ui1n,ui2n,…,uikn)-Nit(ui1n-1,ui2n,…,uikn)‖+‖Nit(ui1n-1,ui2n,ui3n,…,uikn)-Nit(ui1n-1,ui2n-1,ui3n,…,uikn)‖+⋯+‖Nit(ui1n-1,ui2n-1,ui3n-1,…,ui,k-1n-1,uikn)-Nit(ui1n-1,ui2n-1,ui3n-1,…,ui,k-1n-1,uikn-1)‖≤∑j=1kνij(t)‖uijn(t)-uijn-1(t)‖≤∑j=1kνij(t)(1+1n)H̃(T̃ijt(xin),T̃ijt(xin-1))≤∑j=1kνij(t)(1+1n)tij(t)‖xin(t)-xin-1(t)‖,i=1,2,…,k.
It follows from (4.12)–(4.18) that for each i=1,2,…,k,
(4.19)∥xin+1(t)-xin(t)∥≤[∑j=1k(1+qϵi(t)ξi(t)q-qμi(t)+cqξi(t)q)1/q+λi(t)λsi(t)(1+1n)+τi(t)q-1ri(t)-mi(t)×(∑j=1k((δi1(t)+δi2(t))qξi(t)q+qϱi(t)(1-qγpi(t)+cqλpi(t)q)1/qζii(t)(δi1(t)+δi2(t))q-1ξi(t)q-1+cqϱi(t)q(1-qγpi(t)+cqλpi(t)q)ζii(t)q)1/q+∑j=1kϱi(t)νij(t)(1+1n)tij(t))]∥xin(t)-xin-1(t)∥+∑j=1i-1τi(t)q-1ri(t)-mi(t)ϱi(t)ζij(t)(1-qγpj(t)+cqλpj(t)q)1/q∥xjn(t)-xjn-1(t)∥+∑j=i+1kτi(t)q-1ri(t)-mi(t)ϱi(t)ζij(t)(1-qγpj(t)+cqλpj(t)q)1/q∥xjn(t)-xjn-1(t)∥.
Therefore,
(4.20)∑i=1k‖xin+1(t)-xin(t)‖≤∑i=1k{[∑j=1k(1+qϵi(t)ξi(t)q-qμi(t)+cqξi(t)q)1/q+λi(t)λsi(t)(1+1n)+τi(t)q-1ri(t)-mi(t)×(∑j=1k((δi1(t)+δi2(t))qξi(t)q+qϱi(t)(1-qγpi(t)+cqλpi(t)q)1/q×ζii(t)(δi1(t)+δi2(t))q-1ξi(t)q-1+cqϱi(t)q(1-qγpi(t)+cqλpi(t)q)ζii(t)q)1/q+∑j=1kϱi(t)νij(t)(1+1n)tij(t))]‖xin(t)-xin-1(t)‖+∑j=1i-1τi(t)q-1ri(t)-mi(t)ϱi(t)ζij(t)(1-qγpj(t)+cqλpj(t)q)1/q‖xjn(t)-xjn-1(t)‖+∑j=i+1kτi(t)q-1ri(t)-mi(t)ϱi(t)ζij(t)(1-qγpj(t)+cqλpj(t)q)1/q∥xjn(t)-xjn-1(t)∥}=[∑j=1k(1+qϵ1(t)ξ1(t)q-qμ1(t)+cqξ1(t)q)1/q+λ1(t)λs1(t)(1+1n)+τ1(t)q-1r1(t)-m1(t)×(∑j=1k((δ11(t)+δ12(t))qξ1(t)q+qϱ1(t)(1-qγp1(t)+cqλp1(t)q)1/qζ11(t)(δ11(t)+δ12(t))q-1ξ1(t)q-1+cqϱ1(t)q(1-qγp1(t)+cqλp1(t)q)ζ11(t)q)1/q+∑j=1kϱ1(t)ν1j(t)(1+1n)t1j(t))+∑j=2kτj(t)q-1rj(t)-mj(t)ϱj(t)ζj1(t)(1-qγp1(t)+cqλp1(t)q)1/q]‖x1n(t)-x1n-1(t)‖+[∑j=3k(1+qϵ2(t)ξ2(t)q-qμ2(t)+cqξ2(t)q)1/q+λ2(t)λs2(t)(1+1n)+τ2(t)q-1r2(t)-m2(t)×(∑j=1k((δ21(t)+δ22(t))qξ2(t)q+qϱ2(t)(1-qγp2(t)+cqλp2(t)q)1/qζ22(t)(δ21(t)+δ22(t))q-1ξ2(t)q-1+cqϱ2(t)q(1-qγp2(t)+cqλp2(t)q)ζ22(t)q)1/q+∑j=1kϱ2(t)ν2j(t)(1+1n)t2j(t))+τ1(t)q-1r1(t)-m1(t)ϱ1(t)ζ12(t)(1-qγp2(t)+cqλp2(t)q)1/q+∑j=3kτj(t)q-1rj(t)-mj(t)ϱj(t)ζj2(t)(1-qγp2(t)+cqλp2(t)q)1/q]‖x2n(t)-x2n-1(t)‖+⋯+[+∑j=1k-1(1+qϵk(t)ξk(t)q-qμk(t)+cqξk(t)q)1/q+λk(t)λsk(t)(1+1n)+τk(t)q-1rk(t)-mk(t)×(∑j=1k((δk1(t)+δk2(t))qξk(t)q+qϱk(t)(1-qγpk(t)+cqλpk(t)q)1/qζkk(t)(δk1(t)+δk2(t))q-1ξk(t)q-1+cqϱk(t)q(1-qγpk(t)+cqλpk(t)q)ζkk(t)q)1/q+∑j=1kϱk(t)νkj(t)(1+1n)tkj(t))+∑j=1k-1τj(t)q-1rj(t)-mj(t)ϱj(t)ζjk(t)(1-qγpk(t)+cqλpk(t)q)1/q]‖xkn(t)-xkn-1(t)‖≤θn(t)(∑i=1k‖xin(t)-xin-1(t)‖),
where(4.21)θn(t)=max⁡⁡{(1+qϵ1(t)ξ1(t)q-qμ1(t)+cqξ1(t)q)1/q+λ1(t)λs1(t)(1+1n)+τ1(t)q-1r1(t)-m1(t)×(∑j=1k((δ11(t)+δ12(t))qξ1(t)q+qϱ1(t)(1-qγp1(t)+cqλp1(t)q)1/qζ11(t)(δ11(t)+δ12(t))q-1ξ1(t)q-1+cqϱ1(t)q(1-qγp1(t)+cqλp1(t)q)ζ11(t)q)1/q+∑j=1kϱ1(t)ν1j(t)(1+1n)t1j(t))+∑j=2kτj(t)q-1rj(t)-mj(t)ϱj(t)ζj1(t)(1-qγp1(t)+cqλp1(t)q)1/q,(1+qϵ2(t)ξ2(t)q-qμ2(t)+cqξ2(t)q)1/q+λ2(t)λs2(t)(1+1n)+τ2(t)q-1r2(t)-m2(t)×(∑j=1k((δ21(t)+δ22(t))qξ2(t)q+qϱ2(t)(1-qγp2(t)+cqλp2(t)q)1/qζ22(t)(δ21(t)+δ22(t))q-1ξ2(t)q-1+cqϱ2(t)q(1-qγp2(t)+cqλp2(t)q)ζ22(t)q)1/q+∑j=1kϱ2(t)ν2j(t)(1+1n)t2j(t))+τ1(t)q-1r1(t)-m1(t)ϱ1(t)ζ12(t)(1-qγp2(t)+cqλp2(t)q)1/q+∑j=3kτj(t)q-1rj(t)-mj(t)ϱj(t)ζj2(t)(1-qγp2(t)+cqλp2(t)q)1/q,…,(1+qϵk(t)ξk(t)q-qμk(t)+cqξk(t)q)1/q+λk(t)λsk(t)(1+1n)+τk(t)q-1rk(t)-mk(t)×(∑j=1k((δk1(t)+δk2(t))qξk(t)q+qϱk(t)(1-qγpk(t)+cqλpk(t)q)1/qζkk(t)(δk1(t)+δk2(t))q-1ξk(t)q-1+cqϱk(t)q(1-qγpk(t)+cqλpk(t)q)ζkk(t)q)1/q+∑j=1kϱk(t)νkj(t)(1+1n)tkj(t))+∑j=1k-1τj(t)q-1rj(t)-mj(t)ϱj(t)ζjk(t)(1-qγpk(t)+cqλpk(t)q)1/q}.
Let
(4.22)θ(t)=max{(1+qϵ1(t)ξ1(t)q-qμ1(t)+cqξ1(t)q)1/q+λ1(t)λs1(t)+τ1(t)q-1r1(t)-m1(t)×(∑j=1k((δ11(t)+δ12(t))qξ1(t)q+qϱ1(t)(1-qγp1(t)+cqλp1(t)q)1/qζ11(t)(δ11(t)+δ12(t))q-1ξ1(t)q-1+cqϱ1(t)q(1-qγp1(t)+cqλp1(t)q)ζ11(t)q)1/q+∑j=1kϱ1(t)ν1j(t)t1j(t))+∑j=2kτj(t)q-1rj(t)-mj(t)ϱj(t)ζj1(t)(1-qγp1(t)+cqλp1(t)q)1/q,(1+qϵ2(t)ξ2(t)q-qμ2(t)+cqξ2(t)q)1/q+λ2(t)λs2(t)+τ2(t)q-1r2(t)-m2(t)×(∑j=1k((δ21(t)+δ22(t))qξ2(t)q+qϱ2(t)(1-qγp2(t)+cqλp2(t)q)1/qζ22(t)(δ21(t)+δ22(t))q-1ξ2(t)q-1+cqϱ2(t)q(1-qγp2(t)+cqλp2(t)q)ζ22(t)q)1/q+∑j=1kϱ2(t)ν2j(t)t2j(t))+τ1(t)q-1r1(t)-m1(t)ϱ1(t)ζ12(t)(1-qγp2(t)+cqλp2(t)q)1/q+∑j=3kτj(t)q-1rj(t)-mj(t)ϱj(t)ζj2(t)(1-qγp2(t)+cqλp2(t)q)1/q,…,(1+qϵk(t)ξk(t)q-qμk(t)+cqξk(t)q)1/q+λk(t)λsk(t)+τk(t)q-1rk(t)-mk(t)×(∑j=1k((δk1(t)+δk2(t))qξk(t)q+qϱk(t)((1-qγpk(t)+cqλpk(t)q))1/qζkk(t)(δk1(t)+δk2(t))q-1ξk(t)q-1+cqϱk(t)q(1-qγpk(t)+cqλpk(t)q)ζkk(t)q)1/q+∑j=1kϱk(t)νkj(t)tkj(t))+∑j=1k-1τj(t)q-1rj(t)-mj(t)ϱj(t)ζjk(t)(1-qγpk(t)+cqλpk(t)q)1/q}.
Then θn(t)→θ(t), for all t∈Ω.
Define∥·∥* on E=∏i=1kEi by
(4.23)‖(z1(t),z2(t),…,zk(t))‖*=∥z1(t)∥+∥z2(t)∥+⋯+∥zk(t)∥,∀(z1(t),z2(t),…,zk(t))∈E.
It is easy to see that (E,∥·∥*) is a Banach space.
Definezn+1(t)=(x1n+1(t),x2n+1(t),…,xkn+1(t)). Then, we have
(4.24)‖zn+1(t)-zn(t)‖*=∥x1n+1(t)-x1n(t)∥+∥x2n+1(t)-x2n(t)∥+⋯+∥xkn+1(t)-xkn(t)∥.
From the condition (4.11), we know that for all t∈Ω, 0<θ(t)<1, and hence there exists an n0>0 and θ0(t)∈(0,1) such that θn(t)≤θ0(t), for all n≥n0. Therefore, by (4.20) and (4.24), we have
(4.25)‖zn+1(t)-zn(t)‖*≤θ0(t)‖zn(t)-zn-1(t)‖*,∀n≥n0.
It follows from (4.25) that
(4.26)‖zn+1(t)-zn(t)‖*≤θ0(t)n-n0‖zn0+1(t)-zn0(t)‖*.
Hence, for any m≥n≥n0, it follows that
(4.27)‖x1m(t)-x1n(t)‖≤‖zm(t)-zn(t)‖*≤∑i=nm-1‖zi+1(t)-zi(t)‖*≤∑i=nm-1θ0(t)i-n0‖zn0+1(t)-zn0(t)‖*.
Since 0<θ0(t)<1, for all t∈Ω, it follows from (4.27) that ∥x1m(t)-x1n(t)∥→0 as n→∞, and hence {x1n(t)} is a Cauchy sequence in E1. By the same argument, we also have that {xin(t)} is a Cauchy sequence in Ei(i=2,3,…,k). Thus there exist measurable mappings x1:Ω→E1, x2:Ω→E2,…,xk:Ω→Ek such that for all t∈Ω, x1n(t)→x1(t), x2n(t)→x2(t),…,xkn(t)→xk(t) as n→∞.
Now we prove thatu1in(t)→u1i(t)∈T̃1it(x1), u2in(t)→u2i(t)∈T̃2it(x2),…,ukin(t)→uki(t)∈T̃kit(xk), vin(t)→vi(t)∈S̃it(xi)(i=1,2,…,k). In fact, it follows from the Lipschitz continuity of T̃1i, T̃2i,…,T̃ki, S̃i and Algorithm 4.2 that for i=1,2,…,k,
(4.28)‖u1in(t)-u1in-1(t)‖≤(1+1n)t1i(t)‖x1n(t)-x1n-1(t)‖,‖u2in(t)-u2in-1(t)‖≤(1+1n)t2i(t)‖x2n(t)-x2n-1(t)‖,⋮‖ukin(t)-ukin-1(t)‖≤(1+1n)tki(t)‖xkn(t)-xkn-1(t)‖,‖vin(t)-vin-1(t)‖≤(1+1n)λsi(t)‖xin(t)-xin-1(t)‖.
From (4.28), we know that {u1in(t)}, {u2in(t)},…,{ukin(t)}, {vin(t)}(i=1,2,…,k) are also Cauchy sequences. Therefore, there exist u1i(t)∈Ei, u2i(t)∈Ei,…,uki(t)∈Ei, vi(t)∈Ei(i=1,2,…,k) such that u1in(t)→u1i(t), u2in(t)→u2i(t),…,ukin(t)→uki(t) as n→∞. Further, for i=1,2,…,k,
(4.29)d(u1i(t),T̃1it(x1))≤‖u1i(t)-u1in(t)‖+d(u1in(t),T̃1it(x1))≤‖u1i(t)-u1in(t)‖+H̃(T̃1it(x1n),T̃1it(x1))≤‖u1i(t)-u1in(t)‖+t1i(t)∥x1n(t)-x1(t)∥⟶0,∀t∈Ω.
Since T̃1it(x1) is closed, we have u1i(t)∈T̃1it(x1)(i=1,2,…,k). Similarly, u2i(t)∈T̃2it(x2),…,uki(t)∈T̃kit(xk), vi(t)∈S̃it(xi)(i=1,2,…,k). This completes the proof.Now we show the existence of solutions for problem (4.4).Lemma 4.4.
Problem (4.4) is equivalent to find measurable mappings xi,u1i,u2i,…,uki:Ω→ℋi such that for all t∈Ω and i=1,2,…,k, xi(t)∈ℋi, T1it,x1(t)(u1i(t))≥a1(x1(t)), T2it,x2(t)(u2i(t))≥a2(x2(t)),…,Tkit,xk(t)(uki(t))≥ak(xk(t)) such that
(4.30)fit(xi)=Jρi(t)φit[fit(xi)-ρi(t)(Fit(x1-p1t(x1),x2-p2t(x2),…,xk-pkt(xk))+Nit(ui1,ui2,…,uik))],
where ρi:Ω→(0,∞) is a measurable function and Jρi(t)φit=(It+ρi(t)∂φit)-1.
If fori=1,2,…,k, Ei=ℋi is a Hilbert space and Mit(·,vi)=∂φit(·), for all t∈Ω, xi(t)∈ℋi and vi(t)∈S̃it(xi), then Algorithm 4.2 reduces to Algorithm 4.5.Algorithm 4.5.
For any given measurable mappingxi0:Ω→ℋi(i=1,2,…,k), compute the following random iterative sequences {xin(t)}, {u1in(t)}, {u2in(t)}, …, {ukin(t)}(i=1,2,…,k) for solving problem (4.4) as follows:
(4.31)xin+1(t)=xin(t)-fit(xin)+Jρi(t)φit[fit(xin)-ρi(t)(Fit(x1n-p1t(x1n),x2n-p2t(x2n),…,xkn-pkt(xkn))+Nit(ui1n,ui2n,…,uikn))],u1in(t)∈T̃1it(x1n),u2in(t)∈T̃2it(x2n),…,ukin(t)∈T̃kit(xkn),∥u1in(t)-u1in+1(t)∥≤(1+1n+1)H̃(T̃1it(x1n),T̃1it(x1n+1)),∥u2in(t)-u2in+1(t)∥≤(1+1n+1)H̃(T̃2it(x2n),T̃2it(x2n+1)),⋮∥ukin(t)-ukin+1(t)∥≤(1+1n+1)H̃(T̃kit(xkn),T̃kit(xkn+1)),
for any t∈Ω, i=1,2,…,k, n=0,1,2,….From Theorem4.3, we have the following Theorem.Theorem 4.6.
Fori=1,2,…,k, let ℋi be a Hilbert space, fi:Ω×ℋi→ℋi be  (ϵi,μi)-relaxed cocoercive and ξi-Lipschitz continuous, and pi:Ω×ℋi→ℋi be γpi-strongly accretive and λpi-Lipschitz continuous. Let Fi:Ω×∏j=1kℋj→ℋi be ζij-Lipschitz continuous in the (j+1)th argument for j=1,2,…,k, and Ni:Ω×∏j=1kℋj→ℋi be νij-Lipschitz continuous in the (j+1)th argument for j=1,2,…,k. Let T1i:Ω×ℋ1→F(ℋi), T2i:Ω×ℋ2→F(ℋi),…,Tki:Ω×ℋk→F(ℋi) be random fuzzy mappings satisfying the condition (C); T̃1i:Ω×ℋ1→CB(ℋi), T̃2i:Ω×ℋ2→CB(ℋi),…,T̃ki:Ω×ℋk→CB(ℋi)   be random multivalued mappings induced by T1i,T2i,…,Tki, respectively. Suppose that T̃1i, T̃2i,…,T̃ki are H̃-continuous with constants t1i, t2i,…,tki, respectively. If there exists a constant ρi(t)>0(i=1,2,…,k) such that
(4.32)0<(1+2ϵ1(t)ξ1(t)2-2μ1(t)+ξ1(t)2)1/2+(ξ1(t)2+2ρ1(t)(1-2γp1(t)+λp1(t)2)1/2ζ11(t)ξ1(t)+ρ1(t)2(1-2γp1(t)+λp1(t)2)ζ11(t)2)1/2+∑j=1kρ1(t)ν1j(t)t1j(t)+∑j=2kρj(t)ζj1(t)(1-2γp1(t)+λp1(t)2)1/2<1,0<(1+2ϵ2(t)ξ2(t)2-2μ2(t)+ξ2(t)2)1/2+(ξ2(t)2+2ρ2(t)(1-2γp2(t)+λp2(t)2)1/2ζ22(t)ξ2(t)+ρ2(t)2(1-2γp2(t)+λp2(t)2)ζ22(t)2(1-2γp2(t)+λp2(t)2)1/2)1/2+∑j=1kρ2(t)ν2j(t)t2j(t)+ρ1(t)ζ12(t)(1-2γp2(t)+λp2(t)2)1/2+∑j=3kρj(t)ζj2(t)(1-2γp2(t)+λp2(t)2)1/2<1,…,0<(1+2ϵk(t)ξk(t)2-2μk(t)+ξk(t)2)1/2+(ξk(t)2+2ρk(t)(1-2γpk(t)+λpk(t)2)1/2ζkk(t)ξk(t)+ρk(t)2(1-2γpk(t)+λpk(t)2)ζkk(t)2)1/2+∑j=1kρk(t)νkj(t)tkj(t)+∑j=1k-1ρj(t)ζjk(t)(1-2γpk(t)+λpk(t)2)1/2<1.
Then there exist measurable mappings xi, u1i,u2i,…,uki:Ω→ℋi such that (4.4) holds. Moreover, for all t∈Ω, xin(t)→xi(t), u1in(t)→u1i(t), u2in(t)→u2i(t),…,ukin(t)→uki(t), where {xin(t)}, {u1in(t)},…,{ukin(t)} are random sequences obtained by Algorithm 4.5.



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*Source: 731058-2012-05-22.xml*