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,
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
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
.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
,
{
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,
/
∅
}
∈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
≤ 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
≥ 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
x ∈ X
1
: y ∈ A
x
, ∀y ∈ X
1
,
B
−1
y
x ∈ X
2
: y ∈ B
x
, ∀y ∈ X
2
.
2.6
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
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
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
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
− u
2
t
X
1
≥ ζ
A
t
u
1
t
− u
2
t
2
X
1
,
2.11
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
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
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
f
1
p
t, y
1
t
,g
1
p
t, y
1
t
,x
1
t
− y
1
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
t
X
1
≤ a
A
t
x
1
t
− y
1
t
X
1
,
1
f
1
y
1
,u
1
,x
1
− y
1
X
1
≥ α
A
x
1
− y
1
2
X
x
1
− H
1
u
1
,g
1
y
1
,x
1
− y
1
X
1
≥−β
A
x
1
− y
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
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
,
1
−→ CB
X
1
,
t, x
1
−→
E
t,x
1
αx
1
, ∀
t, x
1
∈ Ω × X
1
,
F
∗
It follows that
E
∗
t, x
1
E
t,x
1
αx
1
{
z
1
∈ X
1
:
E
t,x
1
z
1
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
,
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
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
,y
t
A
p
t, u
t
,
0 ∈ N
s
N
t, v
t
,x
t
,y
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
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
− 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
·,·
A,λ
H
1
f
1
p
t, u
t
,g
1
p
t, u
t
− λM
s
M
f
2
q
t, v
t
,g
2
q
t, v
t
− ρN
s
N
t, v
t
,x
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
1
f
1
p
t, u
t
,g
1
p
t, u
t
λA
p
t, u
t
t, v
t
,x
t
,y
t
∈ H
2
f
2
q
t, v
t
,g
2
q
,y
t
A
p
t, u
t
, ∀t ∈ Ω,
0 ∈ N
s
N
t, v
t
,x
t
,y
, 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:
t, u
n
t
,g
1
p
t, u
n
t
− λM
s
M
t, u
n
t
,x
n
·,·
B,ρ
H
2
f
2
q
t, v
n
t
,g
2
q
t, v
n
t
− ρN
∗
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
,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
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
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
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
respect to g
2
,whereα
B
>β
B
,
If
A
t
λ
α
A
− β
A
β
M
t
ξ
E
t
2
a
A
t
2
1
α
A
− β
A
2
1 − 2λζ
A
t
λ
2
ξ
M
t
2
,
B
β
N
t
ξ
E
t
,
D
t
ρ
α
B
− β
B
η
N
t
ξ
F
t
t
2
1
α
B
− β
B
2
1 − 2ρζ
B
t
ρ
2
ξ
N
t
2
,
0 <A
t
t
, lim
n →∞
y
n
t
y
t
, lim
n →∞
u
n
t
u
t
, lim
n →∞
v
n
p
t, u
n
t
,g
1
p
t, u
n
t
− λM
s
M
t, u
n
t
,x
,g
2
q
t, v
n
t
− ρN
s
N
t, v
n
t
,x
n
t
,y
n
t
t
−q
t, v
n
t
R
H
2
·,·
B,ρ
t
n
t
.
4.4
We use ·
1
to replace ·
X
1
and ·
2
to replace ·
T
n
t
. 4.6
By the definition of S
n
t, T
n
t, s
n
,andt
n
,wehavethefollowing
S
n
t
1
S
n−1
tS
n
t − S
n−1
t
R
H
1
·,·
A,λ
s
n
t
− R
H
1
·,·
A,λ
s
n−1
t
1
2
≤
v
n
t − v
n−1
t −
q
t, v
n
t
− q
t, v
n−1
t
2
n
t} and {B
n
t} in 0, 1, such that
u
n1
t − u
n
t
1
≤ A
n
t
u
n
t − u
n−1
t
1
B
n
t
u
n
t − u
n−1
t
2
1
− 2
p
t, u
n
t
− p
t, u
n−1
t
,u
n
t
p
t
2
u
n
t
− u
n−1
t
2
1
.
4.10
12 Journal of Inequalities and Applications
For the second term in 4.7, it follows from Lemma 3.2 that
R
H
1
≤
1
α
A
− β
A
H
1
f
1
p
t, u
n
t
,g
1
p
t, u
n
f
1
p
t, u
n−1
t
,g
1
p
t, u
n−1
t
−λM
s
M
t, u
n−1
− u
n−1
t
−
H
1
f
1
p
t, u
n
t
,g
1
p
t, u
n
1
−u
n
t−u
n−1
t−λ
M
s
M
t, u
n
t
,x
n
t
,y
n
t
t, u
n−1
t,x
n
t,y
n
t − M
s
M
t, u
n−1
t
,x
n−1
t
,y
n
t
1
n−1
t
,x
n−1
t
,y
n−1
t
1
.
4.11
There are four terms in 4.11. Since H
1
f
1
,g
1
is μ
A
t-strongly monotone and a
n
t
H
1
f
1
p
t, u
n−1
t
,g
1
p
t, u
n−1
t
n
t
,g
1
p
t, u
n
t
−H
1
f
1
p
t, u
n−1
t
n
t
,g
1
p
t, u
n
t
− H
1
f
1
p
t, u
n−1
t
u
n
t − u
n−1
t
2
1
.
4.12
For the second term, Since M is ζ
A
t-monotone with respect to the mapping s
M
in the first
argument and ξ
M
t-Lipschitz continuous with respect to mapping s
M
in the first argument,
Journal of Inequalities and Applications 13
so
u
n
t − u
n−1
t
,x
n
t
,y
n
t
2
1
u
n
t − u
n−1
t
2
1
− 2λ
,x
n
t
,y
n
t
,u
n
t
− u
n−1
t
1
λ
2
M
n
t
,y
n
t
2
1
≤
u
n
t − u
n−1
t
2
1
− 2λζ
A
t
2
1
≤
1 − 2λζ
A
t
λ
2
ξ
M
t
2
u
n
t
− u
n−1
t
t,y
n
t
1
≤ β
M
t
x
n
t − x
n−1
t
1
≤ β
M
t
1 ε
n
D
E
n
t − u
n−1
t
1
.
4.14
Similarly, because η
M
t-Lipschitz continuous with respect to the third argument and F
∗
is
ξ
F
t-D-Lipschitz continuous, so we can derive
M
s
M
t, u
n−1
t
,x
n−1
1
≤ η
M
t
y
n
t − y
n−1
t
2
≤ η
M
t
1 ε
n
D
F
∗
t − v
n−1
t
2
.
4.15
If we let
A
n
t
λ
α
A
− β
A
β
M
t
1 ε
n
1 − 2μ
A
t
a
A
t
2
1
α
A
− β
A
2
1 − 2λζ
A
t
λ
2
ξ
,
4.17
14 Journal of Inequalities and Applications
then, from 4.5 to 4.17, we can have
u
n1
t − u
n
t
1
≤ A
n
t
u
n
t − u
n−1
t
1
B
n
t
n
t
u
n
t − u
n−1
t
1
D
n
t
v
n
t − v
n−1
t
2
. 4.19
We now can claim easily that u
n
t, x
n
A
n
t
C
n
t
u
n
t − u
n−1
t
1
B
n
t
D
n
t
u
n
t
− u
n−1
t
1
v
n
t
− v
n−1
t
2
.
4.20
t
max
A
t
C
t
,B
t
D
t
,
4.21
then we have the following:
lim
n →∞
A
n
t
D
n
t
D
t
, lim
n →∞
θ
n
t
θ
t
.
4.22
It follows from the assumptions of Theorem 4.1 that 0 <θt < 1, for all t ∈ Ω,andso
{u
n
t} and {v
n
t} are both Cauchy sequences. For sequences {x
n
} and {y
t
≤ 2ξ
E
t
u
n1
t − u
n
t
1
,
y
n1
t − y
n
t
2
≤
1 ε
t
2
,
4.23
thus, {x
n
t} and {y
n
t} are also Cauchy sequences in Hilbert spaces X
1
and X
2
, respectively.
We now show that 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 and
lim
n →∞
x
n
t
x
t
v
t
,
4.24
Journal of Inequalities and Applications 15
where {x
n
t}, {y
n
t}, {u
n
t},and{v
n
t} are four iterative sequences generated by
Algorithm 3.7.
Because X
1
,X
2
are two Hilbert spaces and {x
n
t},{y
n
t},{u
n
t},and{v
n
t} are f our
u
t
, lim
n →∞
v
n
t
v
t
.
4.25
Furthermore,
d
x
t
,E
∗
t, u
t
t
,E
∗
t, u
t
≤
xt − x
n
t
1
D
E
∗
t, u
n
t
,E
∗
t, u
n
tut,andE
∗
t,ut ∈ CBX
1
, we have the
following:
x
t
∈ E
∗
t, u
t
. 4.27
Similar argument leads to the fact that
y
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
R
H
2
·,·
B,ρ
H
2
f
2
q
t, v
t
,g
2
q
t, v
t
− ρN
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