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Volume 2010, Article ID 123524, 17 pagesdoi:10.1155/2010/123524 Research Article A System of Random Nonlinear Variational Inclusions Involving Random Fuzzy Mappings and Xin-kun Wu and Yu

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

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

Copyrightq 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∗

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

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

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 to1 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 Chang8, Chang and Huang

9,10, Chang and Zhu 11, Huang 12,13, Husain et al 14, Tan et al 15, Tan 16, and Yuan7

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 the 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, Zadeh24 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´an27 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 Farajzadeh28, 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, and the references therein

Motivated and inspired by recent research work mentioned above in this field, in this paper, we try 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 results10,11,21,23,31,34 as special cases of the main results of this paper

2 Preliminaries

Throughout this paper, letΩ, A be a measurable space, where Ω is a set and A is a σ-algebra

overΩ Let X1 be a separable real Hilbert space endowed with a norm · X1 and an inner product X1 Let X2be a separable real Hilbert space endowed with a norm · X2and an inner product X

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

 sup

a∈Ainf

b∈B d a, b, sup

b∈B inf

a∈A d a, b



We denote byBX1, 2X1, and CBX1 the class of Borel σ-fields in X1, and the family

of all nonempty subsets of X1, the family of all nonempty closed bounded subsets of X1

In this paper, to make it self-contained, we start with the following basic definitions and similar definitions can also be found in26,32,34

z1 1t, x is measurable.

T1t, · : X1 → X1is continuous

BX1,

measurable mapping U :Ω → 2X1if u is measurable and for any t ∈ Ω, ut ∈ Ut.

any x1∈ X1, W1·, x1 : Ω → 2X1is a measurable set valued mapping

t-D-continuous if there exists a measurable function ξ E:Ω → 0, ∞, such that

D W1t, x1t, W1t, x2t ≤ ξ E tx1t − x2t X1, 2.4

for all t ∈ Ω and x1t, x2t ∈ X1

Definition 2.8 A set-valued mapping A : X1 → 2X1is said to be monotone if for all x1, y1∈ X1

and u1∈ Ax1, v1∈ Ay1,



u1− v1, x1− y1



Definition 2.9 Let f1, g1: X1 → X1and H1 : X1× X1 → X1be three single-valued mappings

and A : X1 → 2X1be a set-valued mapping A is said to be H1·, ·-monotone with respect to operators f1and g1if A is monotone andH1f1, g1  λAX1 1, for every λ > 0.

Definition 2.10 The inverses of A : X1 → 2X1 and B : X2 → 2X2 are defined as follows, respectively,

A−1

y

x ∈ X1 : y ∈ Ax, ∀y ∈ X1,

B−1

y

x ∈ X2 : y ∈ Bx, ∀y ∈ X2.

2.6

Definition 2.11 p : Ω × X1 → X1is said to be

1 monotone if



p t, x1t − pt, x2t, x1t − x2tX

1≥ 0, ∀t ∈ Ω, ∀x1t, x2t ∈ X1, 2.7

2 strictly monotone if p is monotone and



p t, x1t − pt, x2t, x1t − x2tX1 1 2t, ∀t ∈ Ω, ∀x1t, x2t ∈ X1,

2.8

3 δ p t-strongly monotone if there exists some measurable function δ p : Ω →

0, ∞, such that



p t, x1t − pt, x2t, x1t − x2tX

1≥ δ p tx1t − x2t2

x1, ∀t ∈ Ω, ∀x1t, x2t ∈ X1,

2.9

4 σ p t-Lipschitz continuous if there exists some measurable function σ p : Ω →

0, ∞, such that

p t, x1t − pt, x2t

X1≤ σ p tx1t − x2t X1, ∀t ∈ Ω, ∀x1t, x2t ∈ X1. 2.10

1 ζ A t-strongly monotone with respect to the random single-valued mapping s M :

Ω × X1 → X1 in the first argument if there exists some measurable function ζ A :

Ω → 0, ∞, such that

Ms M t, u1t, ·, · − Ms M t, u2t, ·, ·, u1t − u2 X1 ≥ ζ A tu1t − u2t2

X1, 2.11

for all t ∈ Ω and u1t, u2t ∈ X1,

2 ξ M t-Lipschitz continuous with respect to the random single-valued mapping s M:

Ω × X1 → X1 in its first argument if there exists some measurable function ξ M :

Ω → 0, ∞, such that

Ms M t, u1t, ·, · − Ms M t, u2t, ·, · X1≤ ξ M tu1t − u2t X1, 2.12

for all t ∈ Ω and u1t, u2t ∈ X1,

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·, x1t, · − M·, x2t, · X1 ≤ β M tx1t − x2t X1, 2.13

for all t ∈ Ω and x1t, x2t ∈ X1,

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 ·, ·, y1t − M

·, ·, y2t

X1 ≤ η M t y1t − y2t

for all t ∈ Ω and y1t, y2t ∈ X2;

X1, g1 : X1 → X1, and H1f1, g1 : X1 → X1are three single-valued mappings, H1f1, g1 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



H1

f1

p t, x1t , g1

p t, x1t − H1



f1

p

t, y1t , g1

p

t, y1t , x1t − y1tX

1

≥ μ A t x1t − y1t 2

X1,

2.15

for all t ∈ Ω and x1t, y1t ∈ X1,

2 a A t-Lipschitz continuous with respect to the mapping p if there exists some measurable function a A:Ω → 0, ∞ such that

H1

f1



p t, x1t , g1



p t, x1t − H1



f1



p

t, y1t , g1



p

t, y1t

X1

≤ a A t x1t − y1t

for all t ∈ Ω and x1t, y1t ∈ X1.

3 α A -strongly monotone with respect to f1 in the first argument if there exists a

positive constant α A, such that



H1

f1x1, u1

− H1



f1

y1

, u1

, x1− y1



X1≥ α A x1− y1 2

for all x1, y1, u1∈ X1,

4 β A -relaxed monotone with respect to g1 in the second argument if there exists a

positive constant β A, such that



H1

u1, g1x1 − H1



u1, g1

y1

, x1− y1



X1 ≥ −β A x1− y1 2

for all x1, y1, u1∈ X1.

LetFX1 be a collection of all fuzzy sets over X1 A mapping F from Ω into FX1 is

called a fuzzy mapping If F is a fuzzy mapping on X1, then for any given t ∈ Ω, Ftdenote

it by F tin the sequel is a fuzzy set on X1and F t y is the membership function of y in F t

Let A ∈ FX1, α ∈ 0, 1, then the set

is called an α-cut set of fuzzy set A.

given α ∈ 0, 1, F· α:Ω → 2X1is a measurable set-valued mapping

for any given x1 ∈ X1, E·, x1 : Ω → FX1 is a measurable fuzzy mapping

and notations for operators in X2

Let E : Ω × X1 → FX1 and F : Ω × X2 → FX2 be two random fuzzy mappings satisfying the following condition∗∗:

∗∗ there exist two mappings α : X1 → 0, 1 and β : X2 → 0, 1, such that

E t,x1αx1∈ CBX1, ∀t, x1 ∈ Ω × X1,

F t,x2βx2∈ CBX2, ∀t, x2 ∈ Ω × X2. 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 × X1−→ CBX1, t, x1 −→ E t,x1αx1, ∀t, x1 ∈ Ω × X1,

F∗:Ω × X2−→ CBX2, t, x2 −→ E t,x2αx2, ∀t, x2 ∈ Ω × X2.

2.21

It follows that

E∗t, x1 t,x1αx1 1∈ X1:E t,x1z1 ≥ αx1},

F∗t, x2 t,x2βx2 z2∈ X2:F t,x2z2 ≥ βx2. 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.

Problem 1 Let f1, g1: X1 → X1be two single-valued mappings and s M , p : Ω × X1 → X1be

two random single-valued mappings Let f2, g2 : X2 → X2be two single-valued mappings

and s N , q : Ω × X2 → X2be two random single-valued mappings Let H1 : X1× X1 → X1,

H2 : X2× X2 → X2, M : X1× X1× X2 → X1and N : X2× X1× X2 → X2 be four

single-valued mappings Suppose that A : X1 → 2X1is an H1·, ·-monotone mapping with respect

to f1and g1and B : X2 → 2X2 is an H2·, ·-monotone mapping with respect to f2 and g2

E : Ω × X1 → FX1 and F : Ω × X2 → FX2 are two random fuzzy mappings, α, β, E∗, and

F∗are the same as the above Assume that p

for all t∈ Ω We consider the following problem

Find four measurable mappings u, x : Ω → X1and v, y : Ω → X2, such that

E t,ut xt ≥ αut,

F t,vt

y t ≥ βvt,

0∈ Ms M t, ut, xt, yt  Ap t, ut ,

0∈ Ns N t, vt, xt, yt  Bq t, vt ,

2.23

for all t∈ Ω

Problem1 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 Problem1

3 Random Iterative Algorithm

In order to prove the main results, we need the following lemmas

Lemma 3.1 see 23 Let H1, f1, g1, and A be defined as in Problem 1 Let H1f1, g1 be α A -strongly monotone with respect to f1, β A -relaxed monotone with respect to g1, where α A > β A

g1, then the resolvent operator R H1·,·

Lemma 3.2 see 23 Let H1, f1, g1, A be defined as in Problem 1 Let H1f1, g1 be α A -strongly monotone with respect to f1, β A -relaxed monotone with respect to g1, where α A > β A Suppose that

A : X1 → 2X1 is an H1·, ·-monotone set-valued mapping with respect to f1 and g1 Then, the resolvent operator R H1·,·

A,λ is 1/ α A − β A -Lipschitz continuous.

the resolvent operator R H1·,·

A,λ can be found in23

Lemma 3.4 see Chang 8 Let V : Ω × X1 → CBX1 be a D-continuous random set-valued

Ω → CBX1 is measurable.

Lemma 3.5 see Chang 8 Let V, W : Ω → CBX1 be two measurable set-valued mappings,

measurable selection v : Ω → X1of W, such that for all t ∈ Ω,

Lemma 3.6 The four measurable mappings x, u : Ω → X1 and y, v : Ω → X2 are solution of Problem 1 if and only if, for all t ∈ Ω,

x t ∈ E∗t, ut,

x t ∈ F∗t, vt,

p H1·,·

A,λ H1



f1



p t, ut , g1



p t, ut − λMs M t, ut, xt, yt ,

q H2·,·

B,ρ H2



f2



q t, vt , g2



q t, vt − ρNs N t, vt, xt, yt ,

3.2

where R H1·,·

A,λ 1f1, g1  λA−1and R H2·,·

B,ρ 2f2, g2  ρB−1are two resolvent operators Proof From the definitions of R H1·,·

A,λ and R H2·,·

B,ρ , one has

H1



f1



p t, ut , g1



p t, ut − λMs M t, ut, xt, yt

∈ H1



f1

p t, ut , g1

p t, ut  λAp t, ut , ∀t ∈ Ω,

H2

f2

q t, vt , g2

q t, vt − ρNs N t, vt, xt, yt

∈ H2



f2



q t, vt , g2



q t, vt  ρBq t, vt , ∀t ∈ Ω.

3.3

Hence,

0∈ Ms M t, ut, xt, yt  Ap t, ut , ∀t ∈ Ω,

0∈ Ns N t, vt, xt, yt  Bq t, vt , ∀t ∈ Ω. 3.4

Thus,x, y, u, v is a set of solution of Problem1 This completes the proof

Now we use Lemma3.6to construct the following algorithm

Let u0:Ω → X1and v0:Ω → X2be two measurable mappings, then by Himmelberg

35, there exist x0 : Ω → X1, a measurable selection of E∗·, u0· : Ω → CBX1 and

y0 : Ω → X2, a measurable selection of F∗·, v0· : Ω → CBX2 We now propose the following algorithm

Algorithm 3.7 For any given measurable mappings u0:Ω → X1and v0 :Ω → X2, iterative sequences that attempt to solve Problem1are defined as follows:

u n1 n t − pt, u n t

 R H1·,·

A,λ H1

f1

p t, u n t , g1

p t, u n t − λMs M t, u n t, x n t, y n t ,

v n1 n t − qt, v n t

 R H2·,·

B,ρ H2

f2

q t, v n t , g2

q t, v n t − ρNs 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 X1 ≤ 1  ε n1 DE∗t, u n1 t, E∗t, u n t,

y n1 t − y n t

X2 ≤ 1  ε n1 DF∗t, v n1 t, F∗t, v n t, 3.6 for any t

4 Existence and Convergence

Theorem 4.1 Let X1and X2be two separable real Hilbert spaces Suppose that s M , p : Ω×X1 → X1

X1× X1 → X1, f2, g2 : X2 → X2, H2 : X2× X2 → X2are six single-valued mappings Assume that

1 A : X1 → 2X1is an H1·, ·-monotone with respect to operators f1and g1,

2 B : X2 → 2X2is an H2·, ·-monotone with respect to operators f2and g2,

4 M : X1× X1× X2 → X1 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 : X2× X1× X2 → X2 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 × X1 → FX1 and F : Ω × X2 → FX2 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, H1f1, g1 is μ A t-strongly monotone

with respect to the mapping p and a A t-Lipschitz continuous with respect to the mapping

p,

8 H1f1, g1 is α A -strongly monotone with respect to f1, and β A -relaxed monotone with respect to g1, where α 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, H2f2, g2 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 H2f2, g2 is α B -strongly monotone with respect to f2 and β B -relaxed monotone with respect to g2, where α B > β B ,

If

α A − β A

β M tξ E t 2

1− 2δ p t  σ p t2

α A − β A

2



1− 2μ A t  a A t2

α A − β A

2



1− 2λζ A t  λ2ξ M t2,

α A − β A

η M tξ F t,

α B − β B

β N tξ E t,

α B − β B

η N tξ F t 2

1− 2δ q t  σ q t2

α B − β B

2



1− 2μ B t  a B t2

α B − β B

2



1− 2ρζ B t  ρ2ξ N t2,

0 < At  Ct < 1, ∀t ∈ Ω,

0 < Bt  Dt < 1, ∀t ∈ Ω,

4.1

that x, y, u, v is a set of solution of Problem 1 Moreover,

lim

n→ ∞x n lim

n→ ∞y n lim

n→ ∞u n lim

n→ ∞ v n

4.2

where x n t, y n t, u n t, and v n t are defined as in Algorithm 3.7

2.23

for all t∈ Ω

Problem1 is called a system of generalized random nonlinear variational inclusions

involving random fuzzy mappings and set-valued mappings with H·,... u1∈ X1,

4 β A< /small> -relaxed monotone with... u2t ∈ X1,

3 β M t-Lipschitz