Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2010, Article ID 728452, 15 pages
doi:10.1155/2010/728452
Research Article
A Generalized Nonlinear Random Equations with
Random Fuzzy Mappings in Uniformly Smooth
Banach Spaces
Nawitcha Onjai-Uea
1, 2
and Poom Kumam
1, 2
1
Department of Mathematics, Faculty of Science, King Mongkut’s University of
Technology Thonburi (KMUTT), Bangmod, Bangkok 10140, Thailand
2
Centre of Excellence in Mathematics, CHE, Sriayudthaya Road, Bangkok 10400, Thailand
Correspondence should be addressed to Poom Kumam, [email protected]
Received 26 July 2010; Accepted 31 October 2010
Academic Editor: Yeol J. E. Cho
Copyright q 2010 N. Onjai-Uea and P. Kumam. 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 the general nonlinear random H, η-accretive equations with random
fuzzy mappings. By using the resolvent technique for the H, η-accretive operators, we prove
the existence theorems and convergence theorems of the generalized random iterative algorithm
for this nonlinear random equations with random fuzzy mappings in q-uniformly smooth Banach
spaces. Our result in this paper improves and generalizes some known corresponding results in
the literature.
1. Introduction
Fuzzy Set Theory was formalised by Professor Lofti Zadeh at the University of California
product and the norm on E, respectively. In the sequel, we denote 2
E
,CBE and
H by 2
E
{A : A ∈ E},CBE{A ⊂ E : A is nonempty, bounded and closed} and the Hausdorff
metric on CBE, respectively.
Next, we will use the following definitions and lemmas.
Definition 2.1. An operator x : Ω → E is said to be measurable if, for any B ∈BE, {t ∈ Ω :
xt ∈ B}∈A.
Definition 2.2. A operator F : Ω × E → E is called a random operator if for any x ∈ E, Ft, x
yt is measurable. A random operator F is said to be continuous resp. linear, bounded if for
any t ∈ Ω, the operator Ft, · :
E → E is continuous resp. linear, bounded.
Similarly, we can define a random operator a : Ω × E × E → E. We will write F
t
x
Ft, xt and a
t
x, yat, xt,yt for all t ∈ Ω and xt,yt ∈ E.
It is well known that a measurable operator i s necessarily a random operator.
Definition 2.3. A multivalued operator G : Ω → 2
E
is said to be measurable if, for any B ∈
BE, G
−1
B{t ∈ Ω : Gt ∩ B
/
M
t
,V
t
.
2.1
Journal of Inequalities and Applications 3
Definition 2.7. A multivalued operator F : Ω × E → 2
E
is called a random multivalued operator
if, for any x ∈ E, F·,x is measurable. A random multivalued operator F : Ω × E → CBE
is said to be H-continuous if, for any t ∈ Ω, Ft, · is continuous in
H·, ·, where
H·, · is the
Hausdorff metric on CBE defined as follows: for any given A, B ∈ CBE,
H
A, B
max
sup
α
{
x ∈ E : A
x
≥ α
}
2.3
is called a α-cut set of fuzzy set A.
i A fuzzy mapping F : Ω →FE is called measurable if, for any given α ∈ 0, 1,
F·
α
: Ω → 2
E
is a measurable multivalued mapping.
ii A fuzzy mapping F : Ω × E →FE is called a random fuzzy mapping if, for any
x ∈ E, F·,x : Ω →FE is a measurable fuzzy mapping.
Let K, T, G : Ω × E →FE be three random fuzzy mappings satisfying the following
condition C: there exists three mappings a, b, c : E → 0, 1, such that
K
t,x
ax
∈ CB
E
T and
G as follows, respectively.
K : Ω × E −→ CB
E
,
t, x
−→
K
t,x
ax
, ∀
t, x
∈ Ω × E,
T : Ω × E −→ CB
E
,
t, x
∈ Ω × E.
2.5
It means that
K
t, x
K
t,x
ax
{
z ∈ E,
K
t,x
z
≥ a
x
}
∈ CB
E
,
G
t, x
G
t,x
cx
{
z ∈ E,
G
t,x
z
≥ c
x
Find measurable mappings x, u, v, w : Ω → E such that for all t ∈ Ω, xt ∈ E,
K
t,xt
ut ≥ axt, T
t,xt
vt ≥ bxt, G
t,xt
wt ≥ cxt and
0 ∈ N
t, S
t, x
t
,u
t
,v
t
M
t, p
t, x
t
M
t, x
t
,G
t, x
t
, 2.8
for all t ∈ Ω and u ∈ Mt, xt. The problem 2.8 was considered and studied by Agarwal
et al. 1, when G ≡ I.
If Mt, x, sMt, x for all t
∈ Ω, x, s ∈ E and, for all t ∈ Ω, Mt, · : E → 2
E
is a
H
t
,η-accretive mapping, then 2.7 reduces to the following generalized nonlinear random
multivalued operator equation involving H
t
,η-accretive mapping in Banach spaces:
Find x, v : Ω → E such that vt ∈ Tt, xt and
0 ∈ N
∗
is defined by
J
q
x
f
∗
∈ E
∗
:
x, f
∗
x
q
,
f
∗
The modules of smoothness of E is the function ρ
E
: 0, ∞ → 0, ∞ defined by
ρ
E
t
sup
1
2
x y
x − y
− 1:
x
≤ 1,
> 0 such that, for all x, y ∈ E and
j
q
x ∈ J
q
x, the following inequality holds:
x y
q
≤
x
q
q
y, j
q
x
c
q
y
t
≥ α
t
x
t
− y
t
2
,
2.13
for all xt,yt ∈ E and t ∈ Ω, where αt > 0 is a real-valued random variable;
b β-Lipschitz continuous if there exists a real-valued random variable βt > 0 such that
g
t
2
xt −
yt ∈ J
2
xt − yt such that
N
t
S
t
x
, ·, ·
− N
t
S
t
y
, ·, ·
,j
2
x
N
t
x, ·, ·
− N
t
y, ·, ·
≤
t
x
t
− y
t
t
y where M
t
zMt, zt,for
all t ∈ Ω;
b strictly η-accretive if
u
t
− v
t
,η
t
x, y
≥ 0, 2.18
for all xt,yt ∈ E, ut ∈ M
t
x, vt ∈ M
t
y and t ∈ Ω and the equality holds if
and only if utvt for all t ∈ Ω;
c r-strongly η-accretive if there exists a real-valued random variable rt > 0 such that,
for any t ∈ Ω,
,
2.19
for all xt,yt ∈ E, ut ∈ M
t
x, vt ∈ M
t
y and t ∈ Ω.
Definition 2.12. Let η : Ω × E × E → E be a single-valued mapping, A : Ω × E → E be a
single-valued mapping, M : Ω × E → 2
E
be a multivalued mapping.
i M
t
is said to be m-accretive if M
t
is accretive and I ρtMt, ·EE for all t ∈ Ω
and ρt > 0, where I is identity operator on E;
ii M
t
is said to be generalized m-accretive if M
t
is η-accretive and I ρtMt, ·EE
for all t ∈ Ω and ρt > 0;
iii M
t
is said to be H
t
-accretive if M
t
is accretive and H
η
t
x, y
≤ τ
t
x
t
− y
t
, 2.20
for all xt,yt ∈ E and t ∈ Ω.
Journal of Inequalities and Applications 7
Definition 2.15. A multivalued measurable operator T : Ω × E → CBE is said to be γ-
H-
Lipschitz continuous if there exists a measurable function γ : Ω → 0, ∞ such that, for any
, 2.21
for all xt,yt ∈ E.
Definition 2.16. Let M : Ω × E × E → 2
E
be a H
t
,η-accretive random operator and H :
Ω × E → E be r-strongly monotone random operator. Then the proximal operator J
ρt,H
t
M
t·,x
is
defined as follows:
J
ρt,H
t
M
t·,x
z
H
t
ρ
t
t
M
t
x
− J
ρt,H
t
M
t
y
≤
τ
q−1
r
x − y
, ∀x, y ∈ E, t ∈ Ω. 2.23
3. Random Iterative Algorithms
In this section, we suggest and analyze a new class of iterative methods and construct some
new random iterative algorithms for solving 2.7.
t
x
,u,v
. 3.1
Proof. The proof directly follows from the definition of J
ρt,H
t
M
t·,w
as follows:
p
t
x
J
ρt,H
t
M
t·,w
H
t
p
t
t
M
t
−1
H
t
p
t
x
− ρ
t
N
t
S
t
x
,u,v
x
ρ
t
M
t
p
t
x
,w
⇐⇒ 0 ∈ M
t
p
t
x
,w
N
t
and Rangep
dom Mt, ·,s
/
∅. For any given measurable mapping x
0
: Ω → E,
the multivalued mappings
K·,x
0
·,
T·,x
0
·,
G·,x
0
· : Ω → E are measurable by
Lemma 2.5. Then, there exists measurable selections u
0
· ∈
K·,x
0
·,v
0
· ∈
− J
ρt,H
t
M
t
·,w
0
H
t
p
t
x
0
− ρ
t
N
t
S
t
t
x
0
∈ CBE and w
0
t ∈
G
t
x
0
∈
CBE,byLemma 2.6, there exist measurable selections u
1
t ∈
K
t
x
1
,v
1
t ∈
T
t
x
1
and
w
K
t
x
0
,
K
t
x
1
,
v
0
t
− v
1
t
≤
− w
1
t
≤
1
1
1
H
G
t
x
0
,
G
t
x
1
x
n
− J
ρt,H
t
M
t·,w
n
H
t
p
t
x
n
− ρ
t
N
t
S
t
− u
n1
t
≤
1
1
n 1
H
K
t
x
n
,
K
t
≤
1
1
n 1
H
T
t
x
n
,
T
t
x
n1
,
w
n
H
G
t
x
n
,
G
t
x
n1
.
3.5
From Algorithm 3.2 , we can get the following algorithms.
Journal of Inequalities and Applications 9
Algorithm 3.3. Suppose that E, M, η, S,
K,
T and λ are the same as in Algorithm 3.2.Let
G : Ω × E → E be a random single-valued operator, p ≡ I and Nt, x, y, zft, zgt, x, y
for all t ∈ Ω and x, y, z ∈ E. Then, for given measurable x
x
n
t
− ρ
t
f
t
v
n
g
t
S
t
x
n
,u
n
1
1
n 1
H
K
t
x
n
,
K
t
x
n1
,
v
n
t
∈
T
t
x
n
,
T
t
x
n1
.
3.6
Algorithm 3.4. Let M : Ω×E → 2
E
be a random multivalued operator such that for each fixed
t ∈ Ω, Mt, · : E → 2
E
is a H, η-accretive mapping and Rangep
dom Mt, ·
/
∅.IfS, p,
η, N,
K,
t·,w
n
p
t
x
n
− ρ
t
N
t
S
t
x
n
,u
n
,v
n
1
n 1
H
K
t
x
n
,
K
t
x
n1
,
v
n
t
∈
t
x
n
,
T
t
x
n1
.
3.7
4. Existence and Convergence Theorems
In this section, we prove the existence and convergence theorems of the generalized random
iterative algorithm for this nonlinear random equations with random fuzzy mappings in q-
uniformly smooth Banach spaces.
Theorem 4.1. Suppose that E is a q-uniformly smooth and separable real Banach space, p : Ω ×E →
E is α-strongly accretive and β-Lipschitz continuous, η : Ω × E × E → E be τ-Lipschitz continuous,
H : Ω × E → E is r-strongly η-accretive and μ
A
-Lipschitz continuous and M : Ω × E × E → 2
E
is a random multivalued operator such that for each fixed t ∈ Ω and s ∈ E, Mt, ·,s : E → 2
E
is a
H
J
ρt,H
t
M
t·,x
z
− J
ρt,H
t
M
t·,y
z
≤ π
t
x − y
μ
A
t
β
t
ρ
t
t
σ
t
μ
t
μ
K
t
λt,ρt > 0 then there exist x
∗
t ∈ E, u
∗
t ∈ K
t
x
∗
,v
∗
t ∈ T
t
x
∗
and w
∗
t ∈ G
t
x
∗
such
that x
∗
t,u
∗
t,v
∗
t,w
∗
t is solution of 2.7 and
,w
n
t
−→ w
∗
t
4.3
as n →∞,where{x
n
t}, {u
n
t}, {v
n
t} and {w
n
t} are iterative sequences generated by
Algorithm 3.2.
Proof. From Algorithm 3.2, Lemma 2.17 and 4.1, we compute
x
n1
t
− x
n
H
t
p
t
x
n
− ρ
t
N
t
S
t
x
n
,u
n
,v
n
t
x
n−1
− ρ
t
N
t
S
t
x
n−1
,u
n−1
,v
n−1
≤
λ
t
J
ρt,H
t
M
t·,w
n
H
t
p
t
x
n
− ρ
t
x
n−1
− ρ
t
N
t
S
t
x
n−1
,u
n−1
,v
n−1
≤
x
t
J
ρt,H
t
M
t·,w
n
H
t
p
t
x
n
− ρ
t
N
t
− ρ
t
N
t
S
t
x
n−1
,u
n−1
,v
n−1
λ
t
,u
n−1
,v
n−1
− J
ρt,H
t
M
t·,w
n−1
H
t
p
t
x
n−1
− ρ
t
N
− λ
t
p
t
x
n
− p
t
x
n−1
λ
t
τ
t
q−1
n
,v
n
−
H
t
p
t
x
n−1
− ρ
t
N
t
S
t
x
n−1
,u
n
t
− x
n−1
t
λ
t
x
n
t
− x
n−1
t
−
p
t
H
t
p
t
x
n
− H
t
p
t
x
n−1
ρ
t
N
t
N
t
S
t
x
n−1
,u
n
,v
n
− N
t
S
t
x
n−1
,u
n−1
,v
n
,u
n−1
,v
n−1
λ
t
π
t
w
n
− w
n−1
.
4.4
By using p
t
is a strongly accretive and Lipschitz continuous, we have
x
n
p
t
x
n
− p
t
x
n−1
,j
q
x
n
t
− x
n−1
t
c
q
n
t − x
n−1
t
q
,
4.5
that is
x
n
t
− x
n−1
t
−
p
t
x
n
− p
n−1
t
,
4.6
where c
q
is the same as in Lemma 2.8.
By Lipschitz continuity of N in the first, second and third argument, S, p, H is σ-
Lipschitz continuous, β-Lipschitz continuous, μ
A
-Lipschitz continuous, respectively, and
K,
T,and
G are H-Lipschitz continuous, we have
N
t
S
t
x
n
− S
t
x
n−1
≤
t
σ
t
x
n
t
− x
n−1
t
,
t
u
n
− u
n−1
≤ μ
t
1
1
n
H
K
t
x
n−1
,
K
t
,
12 Journal of Inequalities and Applications
N
t
S
t
x
n−1
,u
n−1
,v
n
− N
t
S
t
x
n−1
x
n−1
,
T
t
x
n
≤
1
1
n
ν
t
μ
T
t
x
n−1
≤ μ
A
t
p
t
x
n
− p
t
x
n−1
≤ μ
A
n
H
G
t
x
n−1
,
G
t
x
n
≤
1
1
n
μ
G
n
t
x
n
t
− x
n−1
t
, 4.8
where
θ
n
t
1 − λ
t
λ
t
1
n
π
t
μ
G
t
τ
t
q−1
r
t
μ
A
t
β
K
t
ν
t
μ
T
t
.
4.9
Letting
κ
t
1 − qα
t
μ
A
t
β
t
ρ
t
t
σ
t
μ
t
μ
K
t
4.10
Thus κ
n
t → κt, θ
n
t → θt as n →∞.From4.2, we know that 0 <θt < 1, for
all t ∈ Ω. Using the same arguments as those used in the proof of Lan et al. 11, Theorem
Journal of Inequalities and Applications 13
3.1, page 14 it follows that {x
n
t}, {u
n
t}, {v
n
t} and {w
n
t} are Cauchy sequences. T hus
by the completeness of E, there exist u
∗
t,v
∗
t,w
∗
t ∈ E such that u
n
t → u
∗
t,v
n
t →
u
∗
t
− y
: y ∈ K
t
x
∗
≤
u
∗
t
− u
n
t
d
x
n
,K
t
x
∗
≤
u
∗
t
− u
n
t
μ
K
t
x
n
t ∈ Ω. Therefore, from Algorithm 3.2 and the continuity of J
ρt,H
t
M
t·,w
, p, N and S,weobtain
p
t
x
J
ρt,H
t
M
t·,w
H
t
p
t
x
− ρ
t
t
,
K,
T and λ are the same as in Theorem 4.1. Assume that M : Ω × E ×
E → 2
E
is a random multivalued operator such that, for each fixed t ∈ Ω and s ∈ E, Mt, ·,s : E →
2
E
is a H
t
,η-accretive mapping. Let f : Ω × E → E be ν-Lipschitz continuous, S : Ω × E → E
be a σ-Lipschitz continuous random operator,
G : Ω × E → E be μ
G
-Lipschitz continuous and
g : Ω × E × E → E be -Lipschitz continuous in the first argument and μ-Lipschitz continuous in
the second argument, respectively. If there exist real-valued random variables ρt > 0 and πt > 0
such that 4.13 holds:
1 ρ
t
t
1 − π
t
μ
G
t
τ
t
q−1
4.13
for all t ∈ Ω,wherec
q
isthesameasinLemma 2.8, for any t ∈ Ω, the iterative sequences {x
n
t},
{u
n
t} and {v
n
t} defined by Algorithm 3.3 converge strongly to the solution x
∗
t,u
q
β
t
q
1/q
< 1,
μ
A
t
β
t
ρ
t
t
σ
t
t
q−1
,
4.14
where c
q
isthesameasinLemma 2.8, for any t ∈ Ω, the iterative sequences {x
n
t}, {u
n
t} and
{v
n
t} defined by Algorithm 3.4 converge strongly to the solution x
∗
t,u
∗
t,v
∗
t of 2.9.
Remark 4.5. We note that for suitable choices of the mappings S, p, H, M, K, T, G, η and space
E. Theorems 4.1–4.4 reduces to many known results of generalized variational inclusions as
special cases see 1, 3, 4, 20–25, 32 and the references therein.
Acknowledgments
This research is supported by the “Centre of Excellence in Mathematics”, the Commission on
High Education, Thailand. Moreover, N. Onjai-Uea is supported by the “Centre of Excellence
in Mathematics”, the Commission on High Education, Thailand for Ph.D. Program at King
Mongkuts University of Technology Thonburi KMUTT.
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