# A New Iterative Method for the Set of Solutions of Equilibrium Problems and of Operator Equations with Inverse-Strongly Monotone Mappings

Authors: Jong Kyu Kim; Nguyen Buong; Jae Yull Sim
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

The purpose of  the paper is to present a new  iteration method  for finding a common element for the set of solutions of equilibrium problems and of operator equations with a finite family ofλ
i-inverse-strongly monotone mappings in Hilbert spaces.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with the inner product 〈
·
,
·
〉 and the norm ∥
·
∥, respectively. Let C be a nonempty closed convex subset of H, and let G be a bifunction from C
×
C into (
-
∞
,
+
∞
). The equilibrium problem for G is to find u
*
∈
C such that
(1)
G
(
u
*
,
v
)
≥
0
,
∀
v
∈
C
.
The set of solutions of (1) is denoted by EP
⁡
(
G
).Equilibrium problem (1) includes the numerous problems in physics, optimization, economics, transportation, and engineering, as special cases.Assume that the bifunctionG satisfies the following standard properties.Assumption A.
LetG
:
C
×
C
→
(
-
∞
,
+
∞
) be a bifunction satisfying the conditions (A1)–(A4):(A1)
G
(
u
,
u
)
=
0,  ∀
u
∈
C;
(A2)
G
(
u
,
v
)
+
G
(
v
,
u
)
≤
0,  ∀
(
u
,
v
)
∈
C
×
C;
(A3)
for eachu
∈
C, G
(
u
,
·
)
:
C
→
(
-
∞
,
+
∞
) is lower semicontinuous and convex;
(A4)
lim
⁡
¯
t
→
+
0
G
(
(
1
-
t
)
u
+
t
z
,
v
)
≤
G
(
u
,
v
), ∀
(
u
,
z
,
v
)
∈
C
×
C
×
C.Let{
T
i
}, i
=
1
,
…
,
N, be a finite family of k
i-strictly pseudocontractive mappings from C into C with the set of fixed points F
(
T
i
); that is,
(2)
F
(
T
i
)
=
{
x
∈
C
:
T
i
x
=
x
}
.
Assume that
(3)
S
:
=
⋂
i
=
1
N
‍
F
(
T
i
)
∩
EP
⁡
(
G
)
≠
∅
.The problem of finding an element(4)
u
*
∈
S
is studied intensively in [1–27].Recall that a mappingT in H is said to be a k-strictly pseudocontractive mapping in the terminology of Browder and Petryshyn [28] if there exists a constant 0
≤
k
<
1 such that
(5)
∥
T
x
-
T
y
∥
2
≤
∥
x
-
y
∥
2
+
k
∥
(
I
-
T
)
x
-
(
I
-
T
)
y
∥
2
,
for all x
,
y
∈
D
(
T
), the domain of T, where I is the identity operator in H. Clearly, if k
=
0, then T is nonexpansive; that is,
(6)
∥
T
(
x
)
-
T
(
y
)
∥
≤
∥
x
-
y
∥
.We know that the class ofk-strictly pseudocontractive mappings strictly includes the class of nonexpansive mappings.In the case thatT
i
≡
I, (4) is reduced to the equilibrium problem (1) and shown in [5, 23] to cover monotone inclusion problems, saddle point problems, variational inequality problems, minimization problems, Nash equilibria in noncooperative games, vector equilibrium problems, and certain fixed point problems (see also [29]). For finding approximative solutions of (1) there exist several methods: the regularization approach in [7, 9, 15, 24, 30, 31], the gap-function approach in [8, 15, 16, 18, 19], and the iterative procedure approach in [1–4, 6, 8, 11–14, 19–22, 32, 33].In the case thatG
≡
0 and N
=
1, (4) is a problem of finding a fixed point for a k-strictly pseudocontractive mapping in C and is given by Marino and Xu [17].Theorem 1 (see [17]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T
:
C
→
C be a k-strictly pseudocontractive mapping for some 0
≤
k
<
1, and assume that
(7)
F
(
T
)
≠
∅
.
Let {
x
n
} be the sequence generated by the following algorithm:
(8)
x
0
∈
C
,
y
n
=
α
n
x
n
+
(
1
-
α
n
)
T
x
n
,
C
n
=
{
∥
x
n
-
T
x
n
∥
2
z
∈
C
:
∥
y
n
-
z
∥
2
≤
∥
x
n
-
z
∥
2
i
i
i
i
i
+
(
1
-
α
n
)
(
k
-
α
n
)
∥
x
n
-
T
x
n
∥
2
}
,
Q
n
=
{
z
∈
C
:
〈
x
n
-
z
,
x
0
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
0
.
Assume that the control sequence {
α
n
} is chosen so that α
n
<
1 for all n. Then {
x
n
} converges strongly to P
F
(
T
)
x
0, the projection of x
0 onto F
(
T
).For the case thatG
≡
0 and N
>
1, (4) is a problem of finding a common fixed point for a finite family of k
i-strictly pseudocontractive mappings T
i in C and is studied in [27].Letx
0
∈
C and {
α
n
}, {
β
n
}, and {
γ
n
} three sequences in [
0,1
] satisfying α
n
+
β
n
+
γ
n
=
1 for all n
≥
1, and let {
u
n
} be a sequence in C. Then the sequence {
x
n
} generated by
(9)
x
1
=
α
1
x
0
+
β
1
T
1
x
1
+
γ
1
u
1
,
x
2
=
α
2
x
1
+
β
2
T
2
x
2
+
γ
2
u
2
,
⋮
x
N
=
α
N
x
N
-
1
+
β
N
T
N
x
N
+
γ
N
u
N
,
x
N
+
1
=
α
N
+
1
x
N
+
β
N
+
1
T
1
x
N
+
1
+
γ
N
+
1
u
N
+
1
,
⋮
is called the implicit iteration process with mean errors for a finite family of strictly pseudocontractive mappings {
T
i
}
i
=
1
N.The scheme (9) can be expressed in the compact form as
(10)
x
n
=
α
n
x
n
-
1
+
β
n
T
n
x
n
+
γ
n
u
n
,
where T
n
=
T
n
mod
⁡
N.Theorem 2 (see [27]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {
T
i
}
i
=
1
N be a finite family of strictly pseudocontractive mappings of C into itself such that
(11)
⋂
i
=
1
N
‍
F
(
T
i
)
≠
∅
.
Let x
0
∈
C and let {
u
n
} be a bounded sequence in C; let {
α
n
}, {
β
n
}, and {
γ
n
} be three sequences in [
0,1
] satisfying the following conditions:(i)
α
n
+
β
n
+
γ
n
=
1, ∀
n
≥
1;
(ii)
there exist constantsσ
1
,
σ
2 such that 0
<
σ
1
≤
β
n
≤
σ
2
<
1,  ∀
n
≥
1;
(iii)
∑
n
=
1
∞
‍
γ
n
<
∞.
Then the implicit iterative sequence{
x
n
} defined by (9) converges weakly to a common fixed point of the mappings {
T
i
}
i
=
1
N. Moreover, if there exists i
0
∈
{
1,2
,
…
,
N
} such that T
i
0 is demicompact, then {
x
n
} converges strongly.IfG is an arbitrary bifunction satisfying Assumption A and N
=
1, then (4) is a problem of finding a common element of the fixed point set for a k-strictly pseudocontractive mapping in C and of the solution set of equilibrium problem for G (see [26]).Theorem 3 (see [26]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let G be a bifunction from C
×
C to (
-
∞
,
+
∞
) satisfying Assumption A, and let T be a nonexpansive mapping of C into H such that
(12)
F
(
T
)
∩
E
P
(
G
)
≠
∅
.
Let f be a contraction of H into itself and let {
x
n
} and {
u
n
} be sequences generated by x
1
∈
H and
(13)
G
(
u
n
,
y
)
+
1
r
n
〈
y
-
u
n
,
u
n
-
x
n
〉
≥
0
,
∀
y
∈
C
,
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
T
u
n
,
for all n
∈  N, where {
α
n
}
⊂
[
0,1
] and {
r
n
}
⊂
(
0
,
∞
) satisfy
(14)
lim
⁡
n
→
∞
α
n
=
0
,
∑
n
=
1
∞
‍
α
n
=
∞
,
∑
n
=
1
∞
|
α
n
+
1
-
α
n
|
<
∞
,
liminf
⁡
n
→
∞
r
n
>
0
,
∑
n
=
1
∞
|
r
n
+
1
-
r
n
|
<
∞
.
Then, {
x
n
} and {
u
n
} converge strongly to z
∈
F
(
T
)
∩
E
P
(
G
), where
(15)
z
=
P
F
(
T
)
∩
E
P
(
G
)
f
(
z
)
.SetA
i
=
I
-
T
i. Obviously, A
i are λ
i-inverse-strongly monotone; that is,
(16)
〈
A
i
(
x
)
-
A
i
(
y
)
,
x
-
y
〉
≥
λ
i
∥
A
i
(
x
)
-
A
i
(
y
)
∥
2
,
iiiiiiiiiiiiiiiiiiiiiiiii
∀
x
,
y
∈
D
(
A
i
)
,
λ
i
=
1
-
k
i
2
.From now on, let{
A
i
}
i
=
1
N be a finite family of λ
i-inverse-strongly monotone mappings in H with C
⊂
⋂
i
=
1
N
‍
D
(
A
i
) and λ
i
>
0, i
=
1
,
…
,
N. On the other hand, if there exists i
0
∈
{
1,2
,
…
,
N
} such that λ
i
0
>
1, then A
i
0 is a contraction; that is, ∥
A
i
0
(
x
)
-
A
i
0
(
y
)
∥
≤
(
1
/
λ
i
0
)
∥
x
-
y
∥ with 1
/
λ
i
0
<
1. And hence, A
i
0 has only one solution and, consequently, the stated problem does not have sense. So, without loss of generality, assume that 0
<
λ
i
≤
1, i
=
1
,
…
,
N.Set(17)
S
=
⋂
i
=
1
N
‍
S
i
,
where S
i
=
{
x
∈
C
:
A
i
(
x
)
=
0
} is the solution set of A
i in C.Assume thatEP
⁡
(
G
)
∩
S
≠
∅.Our problem is to find an element(18)
u
*
∈
EP
⁡
(
G
)
∩
S
.Since the mappingA
=
I
-
T is (
1
/
2
)-inverse-strongly monotone for each nonexpansive mapping T, the problem of finding an element u
*
∈
C, which is not only a solution of a variational inequality involving an inverse-strongly monotone mapping but also a fixed point of a nonexpansive mapping, is a particular case of (18).For instance, the case thatG
(
u
,
v
)
≡
〈
A
(
u
)
,
v
-
u
〉, where A is some inverse-strongly monotone mapping and N
=
1, is studied in [25].Theorem 4 (see [25]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let λ
>
0. Let A be a λ-inverse-strongly monotone mapping of C into H, and let T be a nonexpansive mapping of C into itself such that
(19)
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅
,
where V
I
(
C
,
A
) denotes the solution set of the following variational inequality: find x
*
∈
C such that
(20)
〈
A
(
x
*
)
,
x
-
x
*
〉
≥
0
,
∀
x
∈
C
.
Let {
x
n
} be a sequence defined by
(21)
x
0
∈
C
,
x
n
+
1
=
α
n
x
n
+
(
1
-
α
n
)
T
P
C
(
x
n
-
λ
n
A
(
x
n
)
)
,
for every n
=
0,1
,
…, where {
λ
n
}
⊂
[
a
,
b
] for some a
,
b
∈
(
0,2
λ
) and {
α
n
}
⊂
(
c
,
d
) for some c
,
d
∈
(
0,1
). Then, {
x
n
} converges weakly to z
∈
F
(
T
)
∩
V
I
(
C
,
A
), where
(22)
z
=
lim
⁡
n
→
∞
P
F
(
T
)
∩
V
I
(
C
,
A
)
x
n
.The following theorem is an improvement of Theorem4 for the case of nonself-mapping.Theorem 5 (see [34]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let A be a λ-inverse-strongly monotone mapping of C into H, and let T be a nonexpansive nonself-mapping of C into H such that
(23)
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅
.
Suppose that x
1
=
x
∈
C and {
x
n
} is given by
(24)
x
n
+
1
=
P
C
(
α
n
x
+
(
1
-
α
n
)
T
P
C
(
x
n
-
λ
n
A
(
x
n
)
)
)
for every n
=
1,2
,
…, where {
α
n
} is a sequence in [
0,1
) and {
λ
n
} is a sequence in [
0,2
α
]. If {
α
n
} and {
λ
n
} are chosen so that λ
n
∈
[
a
,
b
] for some a, b with 0
<
a
<
b
<
2
α,
(25)
lim
⁡
n
→
∞
α
n
=
0
,
∑
n
=
1
∞
‍
α
n
=
∞
,
∑
n
=
1
∞
‍
|
α
n
+
1
-
α
n
|
<
∞
,
∑
n
=
1
∞
‍
|
λ
n
+
1
-
λ
n
|
<
∞
,
then {
x
n
} converges strongly to P
F
(
T
)
∩
V
I
(
C
,
A
)
x.We know thatλ-inverse-strongly monotone mapping is (
1
/
λ
)-Lipschitz continuous and monotone. Therefore, for the case that G
(
u
,
v
)
≡
〈
A
(
u
)
,
v
-
u
〉, where A is not inverse-strongly monotone, but Lipschitz continuous and monotone, Nadezhkina and Takahashi [35] prove the following theorem.Theorem 6 (see [35]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let A be a monotone and k-Lipschitz continuous mapping of C into H, and let T be a nonexpansive mapping of C into itself such that
(26)
F
(
T
)
∩
V
I
(
C
,
A
)
≠
∅
.
Let {
x
n
}, {
y
n
}, and {
z
n
} be sequences generated by
(27)
x
0
=
x
∈
C
,
y
n
=
P
C
(
x
n
-
λ
n
A
(
x
n
)
)
,
z
n
=
P
C
(
x
n
-
λ
n
A
(
y
n
)
)
,
C
n
=
{
z
∈
C
:
∥
z
n
-
z
∥
≤
∥
x
n
-
z
∥
}
,
Q
n
=
{
z
∈
C
:
〈
x
n
-
z
,
x
-
x
n
〉
≥
0
}
,
x
n
+
1
=
P
C
n
∩
Q
n
x
for every n
=
0,1
,
…, where {
λ
n
}
⊂
[
a
,
b
] for some a
,
b
∈
(
0,1
/
k
) and α
n
⊂
[
0
,
c
] for some c
∈
[
0,1
). Then the sequences {
x
n
}, {
y
n
}, and {
z
n
} converge strongly to P
F
(
T
)
∩
V
I
(
C
,
A
)
x.Some similar results are also considered in [36, 37].Buong [38] introduced two new implicit iteration methods for solving problem (18).We construct a regularization solutionu
n of the following single equilibrium problem: find u
n
∈
C such that
(28)
F
(
u
n
,
v
)
≥
0
,
∀
v
∈
C
,
where
(29)
F
(
u
,
v
)
≔
G
(
u
,
v
)
+
∑
i
=
1
N
‍
α
n
μ
i
G
i
(
u
,
v
)
+
α
n
〈
u
,
v
-
u
〉
,
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
α
n
>
0
,
G
i
(
u
,
v
)
=
〈
A
i
(
u
)
,
v
-
u
〉
,
i
=
1
,
…
,
N
,
0
<
μ
i
<
μ
i
+
1
<
1
,
i
=
2
,
…
,
N
-
1
,
and {
α
n
} is the positive sequence of regularization parameters that converges to 0, as n
→
+
∞.The first one is the following theorem.Theorem 7 (see [38]).
For eachα
n
>
0, problem (28) has a unique solution u
n such that(i)
lim
⁡
n
→
+
∞
u
n
=
u
*, u
*
∈
E
P
(
G
)
∩
S, ∥
u
*
∥
≤
∥
y
∥,  ∀
y
∈
E
P
(
G
)
∩
S;
(ii)
(30)
∥
u
n
-
u
m
∥
≤
(
∥
u
*
∥
+
d
N
)
|
α
n
-
α
m
|
α
n
,
where d is a positive constant.Next, we introduce the second result. Let{
c
~
n
} and {
γ
n
} be some sequences of positive numbers, and let z
0 and z
1 be two arbitrary elements in C. Then, the sequence {
z
n
} of iterations is defined by the following equilibrium problem: find z
n
+
1
∈
C such that
(31)
c
~
n
(
G
(
z
n
+
1
,
v
)
+
∑
i
=
1
N
‍
α
n
μ
i
G
i
(
z
n
+
1
,
v
)
+
α
n
〈
z
n
+
1
,
v
-
z
n
+
1
〉
)
+
〈
z
n
+
1
-
z
n
,
v
-
z
n
+
1
〉
-
γ
n
〈
z
n
-
z
n
-
1
,
v
-
z
n
+
1
〉
≥
0
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
∀
v
∈
C
.Theorem 8 (see [38]).
Assume that the parametersc
~
n
,
γ
n, and α
n are chosen such that(i)
0
<
c
0
<
c
~
n
,
0
≤
γ
n
<
γ
0,
(ii)
∑
n
=
1
∞
‍
b
n
=
+
∞, b
n
=
c
~
n
α
n
/
(
1
+
c
~
n
α
n
),
(iii)
∑
n
=
1
∞
‍
γ
n
b
n
-
1
∥
z
n
-
z
n
-
1
∥
<
+
∞,
(iv)
lim
⁡
n
→
∞
α
n
=
0
,
lim
⁡
n
→
∞
(
|
α
n
-
α
n
+
1
|
/
α
n
b
n
)
=
0.
Then, the sequence{
z
n
} defined by (31) converges strongly to the element u
*, as n
→
+
∞.In this paper, we consider the new another iteration method: for an arbitrary elementx
0 in H, the sequence {
x
n
} of iterations is defined by finding u
n
∈
C such that
(32)
G
(
u
n
,
y
)
+
〈
u
n
-
x
n
,
y
-
u
n
〉
≥
0
,
∀
y
∈
C
,
x
n
+
1
=
P
C
(
x
n
-
β
n
[
x
n
-
u
n
+
∑
i
=
1
N
‍
α
n
μ
i
A
i
(
x
n
)
+
α
n
x
n
]
)
=
P
C
(
x
n
-
β
n
[
∑
i
=
1
N
‍
α
n
μ
i
A
i
(
x
n
)
+
(
1
+
α
n
)
x
n
-
u
n
]
)
,
where P
C is the metric projection of H onto C and {
α
n
} and {
β
n
} are sequences of positive numbers.The strong convergence of the sequence{
x
n
} defined by (32) is proved under some suitable conditions on {
α
n
} and {
β
n
} in the next section.


## 2. Main Results

We formulate the following lemmas for the proof of our main theorems.Lemma 9 (see [9]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let G be a bifunction of C
×
C into (
-
∞
,
+
∞
) satisfying Assumption A. Let r
>
0 and x
∈
H. Then, there exists z
∈
C such that
(33)
G
(
z
,
y
)
+
1
r
〈
z
-
x
,
y
-
z
〉
≥
0
,
∀
y
∈
C
.Lemma 10 (see [9]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Assume that G
:
C
×
C
→
(
-
∞
,
+
∞
) satisfies Assumption A. For r
>
0 and x
∈
H, define a mapping T
r
:
H
→
C as follows:
(34)
T
r
(
x
)
=
{
z
∈
C
:
G
(
z
,
y
)
+
1
r
〈
z
-
x
,
y
-
z
〉
≥
0
}

,


∀
y
∈
C
.
Then, the following statements hold: (i)
T
r is single valued;
(ii)
T
r is firmly nonexpansive; that is, for any x
,
y
∈
H,
(35)
∥
T
r
(
x
)
-
T
r
(
y
)
∥
2
≤
〈
T
r
(
x
)
-
T
r
(
y
)
,
x
-
y
〉
;
(iii)
F
(
T
r
)
=
E
P
(
G
);
(iv)
E
P
(
G
) is closed and convex.Lemma 11 (see [36]).
Let{
a
n
}
,
{
b
n
}, and {
c
n
} be the sequences of positive numbers satisfying the following conditions: (i)
a
n
+
1
≤
(
1
-
b
n
)
a
n
+
c
n,
(ii)
∑
n
=
0
∞
‍
b
n
=
+
∞, b
n
<
1, lim
⁡
n
→
+
∞
(
c
n
/
b
n
)
=
0.
Then,lim
⁡
n
→
+
∞
a
n
=
0.Lemma 12 (see [38]).
LetA be any inverse-strongly monotone mapping from C into H with the solution set S
A
:
=
{
x
∈
C
:
A
(
x
)
=
0
}, and let C
0 be a closed convex subset of C such that
(36)
S
A
∩
C
0
≠
∅
.
Then, the solution set of the following variational inequality
(37)
〈
A
(
y
~
)
,
x
-
y
~
〉
≥
0
,
∀
x
∈
C
0
,
y
~
∈
C
0
,
is coincided with S
A
∩
C
0.From Lemma9, we can consider the firmly nonexpansive mapping T
0 defined by
(38)
T
0
(
x
)
=
{
z
∈
C
:
G
(
z
,
y
)
+
〈
z
-
x
,
y
-
z
〉
≥
0
,
∀
y
∈
C
}
,
iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii
∀
x
∈
H
.
From Lemma 10, we know that T
0 is nonexpansive. Consequently, A
0
:
=
I
-
T
0 is (
1
/
2
)-inverse-strongly monotone. Let
(39)
S
0
:
=
{
x
∈
C
:
A
0
(
x
)
=
0
}
.
Then, S
0
=
EP
⁡
(
G
) and problem (18) are equivalent to finding
(40)
u
*
∈
S
0
∩
S
.Now, we construct a regularization solutiony
n for (40) by solving the following variational inequality problem: find y
n
∈
C such that
(41)
〈
∑
i
=
0
N
‍
α
n
μ
i
A
i
(
y
n
)
+
α
n
y
n
,
v
-
y
n
〉
≥
0
,
∀
v
∈
C
,
μ
0
=
0
<
μ
1
<
⋯
<
μ
N
<
1
,
where the positive regularization parameter α
n
→
0, as n
→
+
∞.Now we are in a position to introduce and prove the main results.Theorem 13.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let G be a bifunction from C
×
C to (
-
∞
,
+
∞
) satisfying Assumption A and let {
A
i
}
i
=
1
N be a finite family of λ
i-inverse-strongly monotone mappings in H with C
⊂
⋂
i
=
1
N
‍
D
(
A
i
) and λ
i
>
0, i
=
1
,
…
,
N, such that
(42)
E
P
(
G
)
∩
S
≠
∅
,
where E
P
(
G
) denotes the set of solutions for (1) and
(43)
S
=
⋂
i
=
1
N
‍
S
i
,
S
i
=
{
x
∈
C
:
A
i
(
x
)
=
0
}
.
Then, for each α
n
>
0, problem (41) has a unique solution y
n such that (i)
lim
⁡
n
→
+
∞
y
n
=
u
*, u
*
∈
E
P
(
G
)
∩
S,
(ii)
∥
u
*
∥
≤
∥
y
∥,
∀
y
∈
E
P
(
G
)
∩
S,
(iii)
(44)
∥
y
n
-
y
m
∥
≤
|
α
n
-
α
m
|
α
n
(
∥
u
*
∥
+
d
N
)
,
where d is some positive constant.Proof.
From Lemma12, we know that S
0 is the set of solutions for the following variational inequality problem: find u
*
∈
C such that
(45)
〈
A
0
(
u
*
)
,
v
-
u
*
〉
≥
0
,
∀
v
∈
C
.
If we define the new bifunction G
0
(
u
,
v
) by
(46)
G
0
(
u
,
v
)
=
〈
A
0
(
u
*
)
,
v
-
u
*
〉
,
then problem (41) is the same as (28) with a new G
(
u
,
v
), and the proof for the theorem is a complete repetition of the proof for Theorem  2.1 in [38].
Set(47)
L
=
max
⁡
{
2
,
1
λ
i
,
i
=
1
,
…
,
N
}
.Theorem 14.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let G be a bifunction from C
×
C to (
-
∞
,
+
∞
) satisfying Assumption A and let {
A
i
}
i
=
1
N be a finite family of λ
i-inverse-strongly monotone mappings in H with C
⊂
⋂
i
=
1
N
‍
D
(
A
i
) and λ
i
>
0, i
=
1
,
…
,
N, such that
(48)
E
P
(
G
)
∩
S
≠
∅
,
where E
P
(
G
) denotes the set of solutions for (1) and
(49)
S
=
⋂
i
=
1
N
‍
S
i
,
S
i
=
{
x
∈
C
:
A
i
(
x
)
=
0
}
.
Suppose that α
n
,
β
n satisfy the following conditions:
(50)
α
n
,
β
n
>
0
(
α
n
≤
1
)
,
lim
⁡
n
→
∞
α
n
=
0
,
lim
⁡
n
→
∞
|
α
n
-
α
n
+
1
|
α
n
2
β
n
=
0
,
∑
n
=
0
∞
‍
α
n
β
n
=
∞
,
lim
⁡
¯
n
→
∞
β
n
(
L
(
N
+
1
)
+
α
n
)
2
α
n
<
1
.
Then, the sequence {
x
n
} defined by (32) converges strongly to u
*
∈
E
P
(
G
)
∩
S; that is,
(51)
lim
⁡
n
→
∞
x
n
=
u
*
∈
E
P
(
G
)
∩
S
.Proof.
Lety
n be the solution of (41). Then,
(52)
y
n
=
P
C
(
y
n
-
β
n
[
∑
i
=
0
N
‍
α
n
μ
i
A
i
(
y
n
)
+
α
n
y
n
]
)
.
Set Δ
n
=
∥
x
n
-
y
n
∥. Obviously,
(53)
Δ
n
+
1
=
∥
x
n
+
1
-
y
n
+
1
∥
≤
∥
x
n
+
1
-
y
n
∥
+
∥
y
n
+
1
-
y
n
∥
.
From the nonexpansivity of P
C, the monotone and Lipschitz continuous properties of A
i, i
=
0
,
…
,
N, (41), (52), and y
n
=
T
0
(
x
n
), we have
(54)
∥
x
n
+
1
-
y
n
∥
≤
∥
x
n
-
y
n
-
β
n
∑
i
=
0
N
‍
[
∑
i
=
0
N
‍
α
n
μ
i
(
A
i
(
x
n
)
-
A
i
(
y
n
)
)
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
i
∑
i
=
0
N
‍
+
α
n
(
x
n
-
y
n
)
]
∥
,
∥
x
n
-
y
n
-
β
n
[
∑
i
=
0
N
‍
α
n
μ
i
(
A
i
(
x
n
)
-
A
i
(
y
n
)
)
+
α
n
(
x
n
-
y
n
)
]
∥
2
=
∥
x
n
-
y
n
∥
2
+
β
n
2
∥
[
∑
i
=
0
N
‍
α
n
μ
i
(
A
i
(
x
n
)
-
A
i
(
y
n
)
)
+
α
n
(
x
n
-
y
n
)
]
∥
2
-
2
β
n
〈
∑
i
=
0
N
‍
α
n
μ
i
(
A
i
(
x
n
)
-
A
i
(
y
n
)
)
∑
i
i
i
i
i
i
i
i
i
=
0
N
‍
iiiiiiiiiiiiiiiiiiiiiiiii
+
α
n
(
x
n
-
y
n
)
,
x
n
-
y
n
〉
≤
∥
x
n
-
y
n
∥
2
[
1
-
2
β
n
α
n
+
β
n
2
(
2
+
∑
i
=
1
N
‍
α
n
μ
i
1
λ
i
+
α
n
)
2
]
.
Thus,
(55)
∥
x
n
+
1
-
y
n
∥
≤
Δ
n
(
1
-
2
β
n
α
n
+
β
n
2
(
L
(
N
+
1
)
+
α
n
)
2
)
1
/
2
.
Therefore,
(56)
Δ
n
+
1
≤
Δ
n
(
1
-
2
β
n
α
n
+
β
n
2
(
L
(
N
+
1
)
+
α
n
)
2
)
1
/
2
+
|
α
n
-
α
n
+
1
|
α
n
(
∥
u
*
∥
+
d
N
)
≤
Δ
n
(
1
-
α
n
β
n
)
1
/
2
+
|
α
n
-
α
n
+
1
|
α
n
(
∥
u
*
∥
+
d
N
)
.
We note that, forε
>
0, a
>
0, b
>
0, the inequality
(57)
(
a
+
b
)
2
≤
(
1
+
ε
)
(
a
2
+
b
2
ε
)
holds. Thus, applying inequality (57) for ε
=
α
n
β
n
/
2, we obtain
(58)
0
≤
Δ
n
+
1
2
≤
Δ
n
2
(
1
-
α
n
β
n
)
(
1
+
1
2
α
n
β
n
)
+
(
α
n
-
α
n
+
1
α
n
(
∥
u
*
∥
+
d
N
)
)
2
2
α
n
β
n
(
1
+
1
2
α
n
β
n
)
=
Δ
n
2
(
1
-
1
2
α
n
β
n
-
1
2
(
α
n
β
n
)
2
)
+
(
α
n
-
α
n
+
1
α
n
2
β
n
(
∥
u
*
∥
+
d
N
)
)
2
2
α
n
β
n
(
1
+
1
2
α
n
β
n
)
.
Set(59)
b
n
=
α
n
β
n
(
1
2
+
1
2
α
n
β
n
)
c
n
=
(
α
n
-
α
n
+
1
α
n
2
β
n
(
∥
u
*
∥
+
d
N
)
)
2
2
α
n
β
n
(
1
+
1
2
α
n
β
n
)
.
Then, it is not difficult to check that b
n and c
n satisfy the conditions in Lemma 11 for sufficiently large n. Hence, lim
⁡
n
→
+
∞
Δ
n
2
=
0. Since lim
⁡
n
→
∞
y
n
=
u
*, we have
(60)
lim
⁡
n
→
∞
x
n
=
u
*
∈
EP
⁡
(
G
)
∩
S
.
This completes the proof.Remark 15.
The sequencesα
n
=
(
1
+
n
)
-
p
,
0
<
p
<
1
/
2, and β
n
=
γ
0
α
n with
(61)
0
<
γ
0
<
1
(
L
(
N
+
1
)
+
α
0
)
2
satisfy all the necessary conditions in Theorem 14.



---

*Source: 595673-2014-06-16.xml*