# On Solutions of Variational Inequality Problems via Iterative Methods

Authors: Mohammed Ali Alghamdi; Naseer Shahzad; Habtu Zegeye
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We investigate an algorithm for a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family ofγ-inverse strongly accretive mappings. Our theorems improve and unify most of the results that have been proved in this direction for this important class of nonlinear mappings.


---

## Body

## 1. Introduction

LetC be a subset of a real Hilbert space H. Let A
:
C
→
H be a nonlinear mapping. Thevariational inequality problem for A and C is to
(1)
find
x
*
∈
C
such
that
〈
A
x
*
,
v
-
x
*
〉
≥
0
,
∀
v
∈
C
.The set of solutions of variational inequality problem is denoted byVI
⁡
(
C
,
A
); that is,
(2)
VI
⁡
(
C
,
A
)
=
{
x
*
∈
C
:
〈
A
x
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
C
}
.It is well known that variational inequality theory has emerged as an important tool in studying a wide class of numerous problems in variational inequalities, minimax problems, optimization, physics, and the Nash equilibrium problems in noncooperative games. Several numerical methods have been developed for solving variational inequalities and related optimization problems; see, for instance, [1–5] and the references therein.A mappingA
:
C
⊆
H
→
H is said to be γ
-inverse strongly accretive (or γ-inverse strongly monotone) if there exists a positive real number γ such that
(3)
〈
x
-
y
,
A
x
-
A
y
〉
≥
γ
∥
A
x
-
A
y
∥
2
,
∀
x
,
y
∈
C
.IfA is γ-inverse strongly accretive, then inequality (3) implies that A is Lipschitzian with constant L
:
=
1
/
γ; that is, ∥
A
x
-
A
y
∥
≤
(
1
/
γ
)
∥
x
-
y
∥, for all x
,
y
∈
C. If in (3) we have that γ
=
0, then A is calledaccretive (or monotone).LetC be a closed and convex subset of a real Hilbert space H. A mapping T
:
C
→
H is calleda contraction mapping if there exists L
∈
[
0,1
) such that ∥
T
x
-
T
y
∥
≤
L
∥
x
-
y
∥ for all x
,
y
∈
C. If L
=
1, then T is callednonexpansive. A mapping T
:
C
→
E is called λ
-strictly pseudocontractive of Browder-Petryshyn type [6] if and only if there exists λ
∈
(
0,1
) such that
(4)
∥
T
x
-
T
y
∥
2
≤
∥
x
-
y
∥
2
+
λ
∥
(
I
-
T
)
x
-
(
I
-
T
)
y
∥
2
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
x
,
y
∈
C
.
Tis calledpseudocontractive if
(5)
∥
T
x
-
T
y
∥
2
≤
∥
x
-
y
∥
2
+
∥
(
I
-
T
)
x
-
(
I
-
T
)
y
∥
2
,
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
x
,
y
∈
C
.We note that inequalities (4) and (5) can be equivalently written as
(6)
〈
T
x
-
T
y
,
x
-
y
〉
≤
∥
x
-
y
∥
2
-
k
∥
(
x
-
T
x
)
-
(
y
-
T
y
)
∥
2
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
h
∀
x
,
y
∈
C
,
for some k
>
0 and
(7)
〈
T
x
-
T
y
,
x
-
y
〉
≤
∥
x
-
y
∥
2
∀
x
,
y
∈
C
,
respectively. We remark that T is pseudocontractive if and only if A
:
=
(
I
-
T
) is accretive. A point x
∈
C is afixed point of
T if T
x
=
x and we denote by F
(
T
) the set of fixed points of T; that is, F
(
T
)
=
{
x
∈
C
:
T
x
=
x
}.We observe that in a real Hilbert spaceH a class of pseudocontractive mappings includes the class of λ-strictly pseudocontractive mappings and hence the classes of nonexpansive and contraction mappings.Closely related to the variational inequality problems is the problem of finding fixed points of nonexpansive mappings,λ-strict pseudocontraction mappings or pseudocontractive mappings which is the current interest in functional analysis. Several researchers considered a unified approach that approximates a common point of fixed point of nonlinear problems and solutions of variational inequality problems and solutions of variational inequality problems; see, for example, [7–18] and the references therein.In [19], Takahashi and Toyoda studied the problem of finding a common point of fixed points of a nonexpansive mapping and solutions of a variational inequality problem (1) by considering the following iterative algorithm:
(8)
x
0
∈
C
,
x
n
+
1
=
α
n
x
n
+
(
1
-
α
n
)
T
P
C
(
x
n
-
λ
n
A
x
n
)
,
n
=
0,1
,
…
,
where {
α
n
} is a sequence in (
0,1
), {
λ
n
} is a positive sequence, T
:
C
→
C is a nonexpansive mapping, and A
:
C
→
H is an γ-inverse strongly accretive mapping. They showed that the sequence {
x
n
} generated by (8) convergesweakly to some z
∈
VI
⁡
(
C
,
A
)
∩
F
(
S
) provided that the control sequences satisfy some restrictions.Iiduka and Takahashi [20] reconsidered the common element problem via the following iterative algorithm:
(9)
x
1
=
x
∈
C
,
x
n
+
1
=
α
n
x
+
(
1
-
α
n
)
T
P
C
(
x
n
-
λ
n
A
x
n
)
,
n
=
0,1
,
…
,
where T
:
C
→
C is a nonexpansive mapping, A
:
C
→
H is a γ-inverse-strongly accretive mapping, {
α
n
} is a sequence in (
0,1
), and {
λ
n
} is a sequence in (
0,2
α
). They proved that the sequence {
x
n
} strongly converges to some point z
∈
F
(
T
)
∩
VI
⁡
(
C
,
A
).Recently, Zegeye and Shahzad [21] investigated the problem of finding a common point of fixed points of a Lipschitz pseudocontractive mapping T and solutions of a variational inequality problem for γ-inverse strongly accretive mapping A by considering the following iterative algorithm:
(10)
y
n
=
(
1
-
β
n
)
x
n
+
β
n
T
x
n
,
x
n
+
1
=
P
C
[
(
1
-
α
n
)
(
δ
n
T
y
n
+
θ
n
x
n
+
γ
n
P
C
[
I
-
γ
A
]
x
n
)
]
,
where P
C is a metric projection from H onto C and {
δ
n
}
,
{
θ
n
}
,
{
γ
n
}
,
{
α
n
}
,
{
β
n
} are in (
0,1
) satisfying certain conditions. Then, they proved that the sequence {
x
n
} converges strongly to the minimum-norm point of F
(
T
)
∩
VI
⁡
(
C
,
A
).A natural question arises whether we can obtain an iterative scheme which converges strongly to a common point of fixed points of a finite family of pseudocontractive mappings and solutions of a finite family of variational inequality problems for
γ
-inverse strongly accretive mappings or not.It is our purpose in this paper to introduce an algorithm and prove that the algorithm converges strongly to a common point of fixed points of a finite family of Lipschitz pseudocontractive mappings and solutions of a finite family of variational inequality problems forγ-inverse strongly accretive mappings. The results obtained in this paper improve and extend the results of Takahashi and Toyoda [19], Iiduka and Takahashi [20], and Zegeye and Shahzad [21], Theorem 3.2 of Yao et al. [22], and some other results in this direction.


## 2. Preliminaries

In what follows we will make use of the following lemmas.Lemma 1.
LettingH be a real Hilbert space, the following identity holds:
(11)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
x
+
y
〉
,
∀
x
,
y
∈
H
.Lemma 2 (see [23]).
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let A
:
C
→
E be a γ-inverse strongly accretive mapping. Then, for 0
<
μ
<
2
γ, the mapping A
μ
x
:
=
(
x
-
μ
A
x
) is nonexpansive.Lemma 3 (see [24]).
LetC be a nonempty, closed, and convex subset of a smooth Banach space E. Let Q
C be a sunny nonexpansive retraction from E onto C and let A be an accretive operator of C into E. Then for all λ
>
0,
(12)
V
I
(
C
,
A
)
=
F
(
Q
C
(
I
-
λ
A
)
)
.Lemma 4 (see [25]).
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T
i
:
C
→
E, i
=
1
,
…
,
N, be nonexpansive mappings such that ∩
i
=
1
N
F
(
T
i
)
≠
∅. Let T
:
=
θ
1
T
1
+
θ
2
T
2
+
⋯
+
θ
N
T
N with θ
1
+
θ
2
+
⋯
+
θ
N
=
1. Then T is nonexpansive and F
(
T
)
=
∩
i
=
1
N
F
(
T
i
).Lemma 5 (see [26]).
LetC be a convex subset of a real Hilbert space H. Let x
∈
H. Then x
0
=
P
C
x if and only if
(13)
〈
z
-
x
0
,
x
-
x
0
〉
≤
0
,
∀
z
∈
C
.Lemma 6 (see [27]).
LetC be a closed convex subset of a real Hilbert space H and A
:
C
→
C be a continuous pseudo-contractive mapping. Then, for 0
<
μ
<
2
γ, the mapping A
μ
x
:
=
(
x
-
μ
A
x
) is nonexpansive(i)
F
(
T
) is a closed convex subset of C;
(ii)
(
I
-
T
) is demiclosed at zero; that is, if {
x
n
} is a sequence in C such that x
n
⇀
x and T
x
n
-
x
n
→
0, as n
→
∞, then x
=
T
(
x
).Lemma 7 (see [28]).
LetH be a real Hilbert space. Then for all x
i
∈
H and α
i
∈
[
0,1
] for i
=
1,2
,
3 such that α
1
+
α
2
+
α
3
=
1 the following equality holds:
(14)
∥
α
1
x
1
+
α
2
x
2
+
α
3
x
3
∥
2
=
∑
i
=
1
3
‍
α
i
∥
x
i
∥
2
-
∑
1
≤
i
,
j
≤
3
‍
α
i
α
j
∥
x
i
-
x
j
∥
2
.Lemma 8 (see [29]).
Let{
a
n
} be sequences of real numbers such that there exists a subsequence {
n
i
} of {
n
} such that a
n
i
<
a
n
i
+
1 for all i
∈
N. Then there exists an increasing sequence {
m
k
}
⊂
N such that m
k
→
∞ and the following properties are satisfied by all (sufficiently large) numbers k
∈
N:
(15)
a
m
k
≤
a
m
k
+
1
,
a
k
≤
a
m
k
+
1
.
In fact,m
k is the largest number n in the set {
1,2
,
…
,
k
} such that the condition a
n
≤
a
n
+
1 holds.Lemma 9 (see [30]).
Let{
a
n
} be a sequence of nonnegative real numbers satisfying the following relation:
(16)
a
n
+
1
≤
(
1
-
α
n
)
a
n
+
α
n
δ
n
,
n
≥
n
0
,
where {
α
n
}
⊂
(
0,1
) and {
δ
n
}
⊂
R satisfying the following conditions: lim
⁡
n
→
∞
α
n
=
0, ∑
n
=
1
∞
α
n
=
∞ and limsup
⁡
n
→
∞
δ
n
≤
0. Then, lim
⁡
n
→
∞
a
n
=
0.


## 3. Main Result

For the rest of this paper, let{
a
n
}
,
{
b
n
}
,
{
c
n
}
,
⊂
(
c
,
1
)
⊂
(
0,1
), for some c
∈
(
0,1
), and {
α
n
}
⊂
(
0
,
b
)
⊂
(
0,1
), for some b
∈
(
0,1
), satisfy (i) a
n
+
b
n
+
c
n
=
1; (ii) lim
⁡
n
→
∞
α
n
=
0; and (iii) ∑
α
n
=
∞.Theorem 10.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T
j
:
C
→
C, j
=
1,2
,
…
,
M, be Lipschitz pseudocontractive mappings with Lipschitz constants L
i, respectively. Let A
j
:
C
→
H, for j
=
1,2
,
…
,
N, be γ
j-inverse strongly accretive mappings. Let f
:
C
→
C be a contraction with constant α. Assume that F
=
[
∩
j
=
1
M
F
(
T
j
)
]
⋂
[
∩
j
=
1
N
V
I
(
C
,
A
j
)
] is nonempty. Let a sequence {
x
n
} be generated from an arbitrary x
0
∈
C by
(17)
y
n
=
(
1
-
λ
n
)
x
n
+
λ
n
T
n
x
n
;
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
(
a
n
x
n
+
b
n
T
n
y
n
+
c
n
G
x
n
)
,
where T
n
=
T
n
(
m
o
d
M
) and G
:
=
e
0
I
+
e
1
P
C
[
I
-
γ
A
1
]
+
e
2
P
C
[
I
-
γ
A
2
]
+
⋯
+
e
N
P
C
[
I
-
γ
A
r
], for γ
∈
(
0,2
γ
0
), for γ
0
:
=
min
⁡
1
≤
j
≤
N
{
γ
j
} with e
0
+
e
1
+
⋯
+
e
r
=
1 and b
n
+
c
n
≤
λ
n
≤
λ
<
1
/
(
1
+
L
2
+
1
)
,
∀
n
≥
0, for L
=
max
⁡
{
L
j
:
1
≤
j
≤
M
}. Then, {
x
n
} converges strongly to a point x
*
∈
F which is the unique solution of the variational inequality 〈
(
I
-
f
)
(
x
*
)
,
x
-
x
*
〉
≥
0 for all x
∈
F.Proof.
From Lemmas2, 4, and 3 we get that G is nonexpansive mapping with F
(
G
)
=
∩
j
=
1
N
VI
⁡
(
C
,
A
j
). Let p
∈
F. Then from (17), (5), and Lemma 7 we have that
(18)
∥
y
n
-
p
∥
2
=
∥
(
1
-
λ
n
)
(
x
n
-
p
)
+
λ
n
(
T
n
x
n
-
p
)
∥
2
=
(
1
-
λ
n
)
∥
x
n
-
p
∥
2
+
λ
n
∥
T
n
x
n
-
p
∥
2
-
λ
n
(
1
-
λ
n
)
∥
x
n
-
T
n
x
n
∥
2
≤
(
1
-
λ
n
)
∥
x
n
-
p
∥
2
+
λ
n
[
∥
x
n
-
p
∥
2
+
∥
x
n
-
T
n
x
n
∥
2
]
-
λ
n
(
1
-
λ
n
)
∥
x
n
-
T
n
x
n
∥
2
=
∥
x
n
-
p
∥
2
+
λ
n
2
∥
x
n
-
T
n
x
n
∥
2
,
(19)
∥
x
n
+
1
-
p
∥
2
=
∥
α
n
f
(
x
n
)
+
(
1
-
α
n
)
(
a
n
x
n
+
b
n
T
n
y
n
+
c
n
G
x
n
)
-
p
∥
2
=
∥
α
n
(
f
(
x
n
)
-
p
)
+
(
1
-
α
n
)
×
(
a
n
(
x
n
-
p
)
+
b
n
(
T
n
y
n
-
p
)
+
c
n
(
G
x
n
-
p
)
)
∥
2
≤
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
×
∥
a
n
(
x
n
-
p
)
+
b
n
(
T
n
y
n
-
p
)
+
c
n
(
G
x
n
-
p
)
∥
2
≤
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
×
[
a
n
∥
x
n
-
p
∥
2
+
b
n
∥
T
n
y
n
-
p
∥
2
+
c
n
∥
G
x
n
-
p
∥
2
]
-
(
1
-
α
n
)
b
n
a
n
∥
T
n
y
n
-
x
n
∥
2
≤
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
[
(
a
n
+
c
n
)
∥
x
n
-
p
∥
2
hhhhhhhhhhhhhhhhhhhhhhhhhh
+
b
n
∥
T
n
y
n
-
p
∥
2
]
-
(
1
-
α
n
)
b
n
a
n
∥
T
n
y
n
-
x
n
∥
2
≤
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
[
(
a
n
+
c
n
)
∥
x
n
-
p
∥
2
hhhhhhhhhhhhh
+
b
n
(
∥
y
n
-
p
∥
2
+
∥
y
n
-
T
n
y
n
∥
2
)
]
-
(
1
-
α
n
)
b
n
a
n
∥
T
n
y
n
-
x
n
∥
2
.
Now, substituting (18) in (19) we get that
(20)
∥
x
n
+
1
-
p
∥
2
≤
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
[
(
a
n
+
c
n
)
∥
x
n
-
p
∥
2
hhhhhhhhhhhh
+
b
n
(
∥
x
n
-
p
∥
2
+
λ
n
2
∥
x
n
-
T
n
x
n
∥
2
)
hhhhhhhhhhhh
+
b
n
∥
y
n
-
T
n
y
n
∥
2
]
-
(
1
-
α
n
)
b
n
a
n
∥
T
n
y
n
-
x
n
∥
2
=
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
(
1
-
α
n
)
λ
n
2
b
n
∥
x
n
-
T
n
x
n
∥
2
+
(
1
-
α
n
)
b
n
∥
y
n
-
T
n
y
n
∥
2
-
(
1
-
α
n
)
b
n
a
n
∥
T
n
y
n
-
x
n
∥
2
.
Moreover, from (17), Lemma 7, and Lipschitz property of T
n we get that
(21)
∥
y
n
-
T
n
y
n
∥
2
=
∥
(
1
-
λ
n
)
(
x
n
-
T
n
y
n
)
+
λ
n
(
T
n
x
n
-
T
n
y
n
)
∥
2
=
(
1
-
λ
n
)
∥
x
n
-
T
n
y
n
∥
2
+
λ
n
∥
T
n
x
n
-
T
n
y
n
∥
2
-
λ
n
(
1
-
λ
n
)
∥
x
n
-
T
n
x
n
∥
2
≤
(
1
-
λ
n
)
∥
x
n
-
T
n
y
n
∥
2
+
λ
n
L
2
∥
x
n
-
y
n
∥
2
-
λ
n
(
1
-
λ
n
)
∥
x
n
-
T
n
x
n
∥
2
=
(
1
-
λ
n
)
∥
x
n
-
T
n
y
n
∥
2
+
λ
n
3
L
2
∥
x
n
-
T
n
x
n
∥
2
-
λ
n
(
1
-
λ
n
)
∥
x
n
-
T
n
x
n
∥
2
=
(
1
-
λ
n
)
∥
x
n
-
T
n
y
n
∥
2
-
λ
n
(
1
-
L
2
λ
n
2
-
λ
n
)
∥
x
n
-
T
n
x
n
∥
2
.
Substituting (21) into (20) we obtain that
(22)
∥
x
n
+
1
-
p
∥
2
≤
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
(
1
-
α
n
)
b
n
λ
n
2
∥
x
n
-
T
n
x
n
∥
2
+
(
1
-
α
n
)
b
n
[
(
1
-
λ
n
)
∥
x
n
-
T
n
y
n
∥
2
-
λ
n
(
1
-
L
2
λ
n
2
-
λ
n
)
∥
x
n
-
T
n
x
n
∥
2
]
-
(
1
-
α
n
)
b
n
a
n
∥
T
n
y
n
-
x
n
∥
2
,
=
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
∥
x
n
-
p
∥
2
-
(
1
-
α
n
)
λ
n
b
n
[
1
-
L
2
λ
n
2
-
2
λ
n
]
∥
x
n
-
T
n
x
n
∥
2
+
(
1
-
α
n
)
b
n
[
(
1
-
a
n
)
-
λ
n
]
∥
T
n
y
n
-
x
n
∥
2
=
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
∥
x
n
-
p
∥
2
-
(
1
-
α
n
)
λ
n
b
n
[
1
-
L
2
λ
n
2
-
2
λ
n
]
∥
x
n
-
T
n
x
n
∥
2
+
(
1
-
α
n
)
b
n
[
b
n
+
c
n
-
λ
n
]
∥
T
n
y
n
-
x
n
∥
2
.
But, from the hypothesis we have that(23)
1
-
2
λ
n
-
L
2
λ
n
2
≥
1
-
2
λ
-
L
2
λ
2
>
0
,
b
n
+
c
n
≤
λ
n
,
∀
n
≥
0
,
and hence inequality (22) gives that
(24)
∥
x
n
+
1
-
p
∥
2
≤
α
n
∥
f
(
x
n
)
-
p
∥
2
+
(
1
-
α
n
)
∥
x
n
-
p
∥
2
.
But we have that(25)
∥
f
(
x
n
)
-
p
∥
2
=
[
∥
f
(
x
n
)
-
f
(
p
)
∥
+
∥
f
(
p
)
-
p
∥
]
2
≤
[
α
∥
x
n
-
p
∥
+
∥
f
(
p
)
-
p
∥
]
2
≤
α
2
∥
x
n
-
p
∥
2
+
∥
f
(
p
)
-
p
∥
2
+
2
α
∥
x
n
-
p
∥
∥
f
(
p
)
-
p
∥
≤
α
(
1
+
α
)
∥
x
n
-
p
∥
2
+
(
1
+
α
)
∥
f
(
p
)
-
p
∥
2
.
Substituting (25) into (24) we get that
(26)
∥
x
n
+
1
-
p
∥
2
≤
(
1
-
α
n
(
1
-
α
(
1
+
α
)
)
)
∥
x
n
-
p
∥
2
+
α
n
(
1
+
α
)
∥
f
(
p
)
-
p
∥
2
.
Therefore, by induction we get that(27)
∥
x
n
+
1
-
p
∥
2
≤
max
⁡
{
∥
x
0
-
p
∥
2
,
1
+
α
1
-
α
(
1
+
α
)
∥
f
(
p
)
-
p
∥
2
}
,
hhhhhhhhhhhhhhhhhhhhhhhhhhhhhh
∀
n
≥
0
,
which implies that {
x
n
} and hence {
y
n
} are bounded.
Letx
*
=
P
F
f
(
x
*
). Then, from (17), Lemmas 1 and 7, and the methods used to get (22) we obtain that
(28)
∥
x
n
+
1
-
x
*
∥
2
=
∥
α
n
(
f
(
x
n
)
-
x
*
)
+
(
1
-
α
n
)
[
a
n
x
n
+
b
n
T
n
y
n
+
c
n
G
x
n
-
x
*
]
∥
2
≤
(
1
-
α
n
)
∥
a
n
(
x
n
-
x
*
)
+
b
n
(
T
n
y
n
-
x
*
)
h
h
h
h
h
h
h
h
h
h
h
+
c
n
(
G
x
n
-
x
*
)
∥
2
+
2
α
n
〈
f
(
x
n
)
-
x
*
,
x
n
+
1
-
x
*
〉
≤
(
1
-
α
n
)
b
n
∥
T
n
y
n
-
x
*
∥
2
+
(
1
-
α
n
)
a
n
∥
x
n
-
x
*
∥
2
×
(
1
-
α
n
)
c
n
∥
G
x
n
-
p
∥
2
-
(
1
-
α
n
)
a
n
b
n
∥
T
n
y
n
-
x
n
∥
2
-
(
1
-
α
n
)
a
n
c
n
∥
G
x
n
-
x
n
∥
2
+
2
α
n
〈
f
(
x
n
)
-
x
*
,
x
n
+
1
-
x
*
〉
,
∥
x
n
+
1
-
x
*
∥
2
≤
(
1
-
α
n
)
b
n
[
∥
y
n
-
x
*
∥
2
+
∥
y
n
-
T
n
y
n
∥
2
]
+
(
1
-
α
n
)
(
a
n
+
c
n
)
∥
x
n
-
x
*
∥
2
-
(
1
-
α
n
)
a
n
b
n
∥
T
n
y
n
-
x
n
∥
2
-
(
1
-
α
n
)
a
n
c
n
∥
G
x
n
-
x
n
∥
2
+
2
α
n
〈
f
(
x
n
)
-
x
*
,
x
n
+
1
-
x
*
〉
,
≤
(
1
-
α
n
)
b
n
[
∥
x
n
-
x
*
∥
2
+
λ
n
2
∥
x
n
-
T
n
x
n
∥
2
]
+
(
1
-
α
n
)
b
n
[
(
1
-
λ
n
)
∥
x
n
-
T
n
y
n
∥
2
hhhhhhhhhhhhhhh
-
λ
n
(
1
-
L
2
λ
n
2
-
λ
n
)
hhhhhhhhhhhhhhh
×
∥
x
n
-
T
n
x
n
∥
2
]
+
(
1
-
α
n
)
(
a
n
+
c
n
)
∥
x
n
-
x
*
∥
2
-
(
1
-
α
n
)
a
n
b
n
∥
T
n
y
n
-
x
n
∥
2
-
(
1
-
α
n
)
a
n
c
n
∥
G
x
n
-
x
n
∥
2
+
2
α
n
〈
f
(
x
n
)
-
x
*
,
x
n
+
1
-
x
*
〉
which implies that
(29)
∥
x
n
+
1
-
x
*
∥
2
≤
(
1
-
α
n
)
∥
x
n
-
x
*
∥
2
-
(
1
-
α
n
)
b
n
λ
n
[
1
-
L
2
λ
n
2
-
2
λ
n
]
×
∥
x
n
-
T
n
x
n
∥
2

+

(
1

-

α
n
)
b
n
(
b
n
+
c
n
-
λ
n
)
∥
x
n
-
T
n
y
n
∥
2
-
(
1
-
α
n
)
a
n
d
n
∥
G
x
n
-
x
n
∥
2
+
2
α
n
〈
f
(
x
n
)
-
x
*
,
x
n
+
1
-
x
*
〉
(30)
≤
(
1
-
α
n
)
∥
x
n
-
x
*
∥
2
+
2
α
n
〈
f
(
x
n
)
-
x
*
,
x
n
+
1
-
x
*
〉
.
But(31)
〈
f
(
x
n
)
-
x
*
,
x
n
+
1
-
x
*
〉
=
〈
f
(
x
n
)
-
x
*
,
x
n
-
x
*
〉
+
〈
f
(
x
n
)
-
x
*
,
x
n
+
1
-
x
n
〉
≤
〈
f
(
x
n
)
-
f
(
x
*
)
,
x
n
-
x
*
〉
+
〈
f
(
x
*
)
-
x
*
,
x
n
-
x
*
〉
+
∥
x
n
+
1
-
x
n
∥
∥
f
(
x
n
)
-
x
*
∥
≤
α
∥
x
n
-
x
*
∥
2
+
〈
f
(
x
*
)
-
x
*
,
x
n
-
x
*
〉
+
∥
x
n
+
1
-
x
n
∥
∥
f
(
x
n
)
-
x
*
∥
.
Thus, substituting (31) in (30) we obtain that
(32)
∥
x
n
+
1
-
x
*
∥
2
≤
(
1
-
α
n
(
1
-
2
α
)
)
∥
x
n
-
x
*
∥
2
+
2
α
n
〈
f
(
x
*
)
-
x
*
,
x
n
-
x
*
〉
+
2
α
n
∥
x
n
+
1
-
x
n
∥
·
∥
f
(
x
n
)
-
x
*
∥
.
Next, we consider two cases.
Case
1. Suppose that there exists n
0
∈
N such that {
∥
x
n
-
x
*
∥
} is decreasing for all n
≥
n
0. Then, we get that {
∥
x
n
-
x
*
∥
} is convergent. Thus, from (29) and (23) we have that
(33)
x
n
-
T
n
x
n
⟶
0
,
G
x
n
-
x
n
⟶
0
as
n
⟶
∞
.
Furthermore, from (17) and (33) we obtain that
(34)
∥
y
n
-
x
n
∥
=
λ
n
∥
x
n
-
T
n
x
n
∥
⟶
0
as
n
⟶
∞
,
and hence Lipschitz continuity of T
n, (34), and (33) implies that
(35)
∥
T
n
y
n
-
x
n
∥
≤
∥
T
n
y
n
-
T
n
x
n
∥
+
∥
T
n
x
n
-
x
n
∥
≤
L
∥
y
n
-
x
n
∥
+
∥
T
n
x
n
-
x
n
∥
⟶
0
hhhhhhhhhhhhhhhhhhh
as
n
⟶
∞
.
Thus, from (33) and (35) we have that
(36)
∥
x
n
+
1
-
x
n
∥
=
∥
α
n
(
f
(
x
n
)
-
x
n
)
+
(
1
-
α
n
)
×
(
a
n
x
n
+
b
n
T
n
y
n
+
c
n
G
x
n
)
-
x
n
∥
≤
α
n
∥
f
(
x
n
)
-
x
n
∥
+
(
1
-
α
n
)
b
n
∥
T
n
y
n
-
x
n
∥
+
(
1
-
α
n
)
c
n
∥
G
x
n
-
x
n
∥
⟶
0
as
n
⟶
∞
.
Therefore,∥
x
n
+
j
-
x
n
∥
→
0, as n
→
∞, for all j
=
1,2
,
…
,
M, and hence
(37)
∥
x
n
-
T
n
+
j
x
n
∥
≤
∥
x
n
-
x
n
+
j
∥
+
∥
x
n
+
j
-
T
n
+
j
x
n
+
j
∥
+
L
∥
x
n
+
j
-
x
n
∥
⟶
0
,
as n
→
∞, for all j
∈
{
1,2
,
…
,
M
}.
Now, since{
x
n
} is bounded subset of H, we can choose a subsequence {
x
n
m
} of {
x
n
} such that x
n
m
⇀
x and limsup
⁡
n
→
∞
〈
f
(
x
*
)
-
x
*
,
x
n
-
x
*
〉
=
lim
⁡
m
→
∞
〈
f
(
x
*
)
-
x
*
,
x
n
m
-
x
*
〉. Then, from (37) and Lemma 6 we have that x
∈
F
(
T
j
), for each j
=
1,2
,
…
,
M. Hence, x
∈
∩
j
=
1
M
F
(
T
j
).
In addition, sinceG is nonexpansive, from Lemma 6 we get that x
∈
F
(
G
) and hence by Lemmas 4 and 3 we obtain that x
∈
VI
⁡
(
C
,
A
j
), for each j
∈
{
1,2
,
…
,
N
}.
Therefore, by Lemma5, we immediately obtain that
(38)
limsup
⁡
n
→
∞
〈
f
(
x
*
)
-
x
*
,
x
n
-
x
*
〉
=
lim
⁡
m
→
∞
〈
f
(
x
*
)
-
x
*
,
x
n
m
-
x
*
〉
=
〈
f
(
x
*
)
-
x
*
,
x
-
x
*
〉
≤
0
.Then, it follows from (32), (38), and Lemma 9 that ∥
x
n
-
x
*
∥
→
0 as n
→
∞. Consequently, x
n
→
x
*
=
P
F
(
f
(
x
*
)
).Case
2. Suppose that there exists a subsequence {
n
i
} of {
n
} such that
(39)
∥
x
n
i
-
x
*
∥
<
∥
x
n
i
+
1
-
x
*
∥
,
for all i
∈
N. Then, by Lemma 8, there exists a nondecreasing sequence {
m
k
}
⊂
N such that m
k
→
∞, and
(40)
∥
x
m
k
-
x
*
∥
≤
∥
x
m
k
+
1
-
x
*
∥
,
∥
x
k
-
x
*
∥
≤
∥
x
m
k
+
1
-
x
*
∥
,
for all k
∈
N. Now, from (29) and (23) we get that x
m
k
-
T
m
k
x
m
k
→
0 and G
x
n
k
-
x
n
k
→
0 as k
→
∞. Thus, following the method in Case 1, we obtain that x
m
k
+
1
-
x
m
k
→
0, x
m
k
-
T
j
x
m
k
→
0, and
(41)
limsup
⁡
k
→
∞
〈
f
(
x
*
)
-
x
*
,
x
m
k
-
x
*
〉
≤
0
.Furthermore, from (32) and (40) we obtain that
(42)
α
m
k
(
1
-
2
α
)
∥
x
m
k
-
x
*
∥
2
≤
∥
x
m
k
-
x
*
∥
2
-
∥
x
m
k
+
1
-
x
*
∥
2
+
2
α
m
k
〈
f
(
x
*
)
-
x
*
,
x
m
k
-
x
*
〉
+
2
α
m
k
∥
x
m
k
+
1
-
x
m
k
∥
∥
f
(
x
m
k
)
-
x
*
∥
≤
2
α
m
k
〈
f
(
x
*
)
-
x
*
,
x
m
k
-
x
*
〉
+
2
α
m
k
∥
x
m
k
+
1
-
x
m
k
∥
∥
f
(
x
m
k
)
-
x
*
∥
.Now, using the fact thatα
m
k
>
0 and (41) we get that
(43)
∥
x
m
k
-
x
*
∥
2
⟶
0
as
k
⟶
∞
,
and this together with (32) implies that ∥
x
m
k
+
1
-
x
*
∥
→
0 as k
→
∞. Since ∥
x
k
-
x
*
∥
≤
∥
x
m
k
+
1
-
x
*
∥ for all k
∈
N, we obtain that x
k
→
x
*. Hence, from the above two cases, we can conclude that {
x
n
} converges strongly to a point x
*
=
P
F
f
(
x
*
), which satisfies the variational inequality 〈
(
I
-
f
)
(
x
*
)
,
x
-
x
*
〉
≥
0, for all x
∈
F. The proof is complete.If, in Theorem10, we assume that f
(
x
)
=
u
∈
C, a constant mapping, then we get the following corollary.Corollary 11.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T
j
:
C
→
C, j
=
1,2
,
…
,
M, be Lipschitz pseudocontractive mappings with Lipschitz constants L
j, respectively. Let A
j
:
C
→
H, for j
=
1,2
,
…
,
N, be γ
j-inverse strongly accretive mappings. Assume that F
=
[
∩
j
=
1
M
F
(
T
j
)
]
⋂
[
∩
j
=
1
N
V
I
(
C
,
A
j
)
] is nonempty. Let a sequence {
x
n
} be generated from an arbitrary x
0
,
u
∈
C by
(44)
y
n
=
(
1
-
λ
n
)
x
n
+
λ
n
T
n
x
n
;
x
n
+
1
=
α
n
u
+
(
1
-
α
n
)
(
a
n
x
n
+
b
n
T
n
y
n
+
c
n
G
x
n
)
,
where T
n
=
T
n
(
m
o
d
M
), G
:
=
e
0
I
+
e
1
P
C
[
I
-
γ
A
1
]
+
e
2
P
C
[
I
-
γ
A
2
]
+
⋯
+
e
N
P
C
[
I
-
γ
A
r
], for γ
∈
(
0,2
γ
0
), for γ
0
:
=
min
⁡
1
≤
j
≤
N
{
γ
j
} with e
0
+
e
1
+
⋯
+
e
r
=
1, and b
n
+
c
n
≤
λ
n
≤
λ
<
1
/
(
1
+
L
2
+
1
)
,
∀
n
≥
0, for L
=
max
⁡
{
L
j
:
1
≤
j
≤
M
}. Then, {
x
n
} converges strongly to a unique point x
*
∈
C satisfying x
*
=
P
F
(
u
), which is the unique solution of the variational inequality 〈
x
*
-
u
,
x
-
x
*
〉
≥
0 for all x
∈
F.If, in Theorem10, we assume that N
=
1 and M
=
1, then we get the following corollary which is Theorem 3.1 of [21].Corollary 12.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T
:
C
→
C be Lipschitz pseudocontractive mappings with Lipschitz constant L and A
:
C
→
H an γ-inverse strongly accretive mapping. Let f
:
C
→
C be a contraction with constant α. Assume that F
=
F
(
T
)
⋂
V
I
(
C
,
A
) is nonempty. Let a sequence {
x
n
} be generated from an arbitrary x
0
∈
C by
(45)
y
n
=
(
1
-
λ
n
)
x
n
+
λ
n
T
x
n
;
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
(
a
n
x
n
+
b
n
T
y
n
+
c
n
P
C
[
I
-
r
A
]
x
n
)
,
where r
∈
(
0,2
γ
) and b
n
+
c
n
≤
λ
n
≤
λ
<
1
/
(
1
+
L
2
+
1
)
,
∀
n
≥
0. Then, {
x
n
} converges strongly to a point x
*
∈
F, which is the unique solution of the variational inequality 〈
(
I
-
f
)
(
x
*
)
,
x
-
x
*
〉
≥
0 for all x
∈
F.If, in Theorem10, we assume that T
i
′
s are strictly pseudocontractive mappins, then we get the following corollary.Corollary 13.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T
i
:
C
→
C, i
=
1,2
,
…
M, be λ
i-strictly pseudocontractive mappings and let A
i
:
C
→
H, for i
=
1,2
,
…
,
N, be an γ
i-inverse strongly accretive mappings. Let f
:
C
→
C be a contraction with constant α. Assume that F
=
[
∩
i
=
1
M
F
(
T
i
)
]
⋂
[
∩
i
=
1
N
V
I
(
C
,
A
i
)
] is nonempty. Let a sequence {
x
n
} be generated from an arbitrary x
0
∈
C by
(46)
y
n
=
(
1
-
λ
n
)
x
n
+
λ
n
T
n
x
n
;
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
(
a
n
x
n
+
b
n
T
n
y
n
+
c
n
G
x
n
)
,
where T
n
=
T
n
(
m
o
d
M
), G
:
=
c
0
I
+
e
1
P
C
[
I
-
γ
A
1
]
+
e
2
P
C
[
I
-
γ
A
2
]
+
⋯
+
e
N
P
C
[
I
-
γ
A
r
], for γ
∈
(
0,2
γ
0
), for γ
0
:
=
min
⁡
1
≤
i
≤
N
{
γ
i
} with e
0
+
e
1
+
⋯
+
e
r
=
1, and b
n
+
c
n
≤
λ
n
≤
λ
<
1
/
(
1
+
L
2
+
1
)
,
∀
n
≥
0, L
=
max
⁡
{
(
1
+
λ
i
)
/
λ
i
}. Then, {
x
n
} converges strongly to a point x
*
∈
F, which is the unique solution of the variational inequality 〈
(
I
-
f
)
(
x
*
)
,
x
-
x
*
〉
≥
0 for all x
∈
F.If, in Theorem10, we assume that T
i
′
s are nonexpansive mapping, then we get the following corollary.Corollary 14.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T
i
:
C
→
C, i
=
1,2
,
…
,
M, be nonexpansive mappings and let A
i
:
C
→
H, for i
=
1,2
,
…
,
N, be an γ
i-inverse strongly accretive mappings. Let f
:
C
→
C be a contraction with constant α. Assume that F
=
[
∩
i
=
1
M
F
(
T
i
)
]
⋂
‍
[
∩
i
=
1
N
V
I
(
C
,
A
i
)
] is nonempty. Let a sequence {
x
n
} be generated from an arbitrary x
0
∈
C by
(47)
y
n
=
(
1
-
λ
n
)
x
n
+
λ
n
T
n
x
n
;
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
(
a
n
x
n
+
b
n
T
n
y
n
+
c
n
G
x
n
)
,
where T
n
=
T
n
(
m
o
d
M
), G
=
c
0
I
+
e
1
P
C
[
I
-
γ
A
1
]
+
e
2
P
C
[
I
-
γ
A
2
]
+
⋯
+
e
N
P
C
[
I
-
γ
A
r
], for γ
∈
(
0,2
γ
0
), for γ
0
:
=
min
⁡
1
≤
i
≤
N
{
γ
i
} with e
0
+
e
1
+
⋯
+
e
r
=
1, a
n
+
b
n
+
c
n
=
1, and b
n
+
c
n
≤
λ
n
≤
λ
<
1
/
(
2
+
1
)
,
∀
n
≥
0. Then, {
x
n
} converges strongly to point x
*
∈
F, which is the unique solution of the variational inequality 〈
(
I
-
f
)
(
x
*
)
,
x
-
x
*
〉
≥
0 for all x
∈
F.We note that the method of proof of Theorem10 provides the following theorem which is a convergence theorem for a minimum norm point of common fixed points of a finite family of Lipschitz pseudocontractive mappings and common solutions of a finite family of variational inequality problems for accretive mappings.Theorem 15.
LetC be a nonempty, closed, and convex subset of a real Hilbert space H. Let T
j
:
C
→
C, j
=
1,2
,
…
,
M, be Lipschitz pseudocontractive mappings with Lipschitz constants L
j, respectively. Let A
j
:
C
→
H, for j
=
1,2
,
…
,
N, be γ
j-inverse strongly accretive mappings. Assume that F
=
[
∩
j
=
1
M
F
(
T
j
)
]
⋂
‍
[
∩
j
=
1
N
V
I
(
C
,
A
j
)
] is nonempty. Let a sequence {
x
n
} be generated from an arbitrary x
0
∈
C by
(48)
y
n
=
(
1
-
λ
n
)
x
n
+
λ
n
T
n
x
n
;
x
n
+
1
=
P
C
[
(
1
-
α
n
)
(
a
n
x
n
+
b
n
T
n
y
n
+
c
n
G
x
n
)
]
,
where T
n
=
T
n
(
m
o
d
M
), G
:
=
e
0
I
+
e
1
P
C
[
I
-
γ
A
1
]
+
e
2
P
C
[
I
-
γ
A
2
]
+
⋯
+
e
N
P
C
[
I
-
γ
A
r
], for γ
∈
(
0,2
γ
0
), for γ
0
:
=
min
⁡
1
≤
j
≤
N
{
γ
j
} with e
0
+
e
1
+
⋯
+
e
r
=
1, and b
n
+
c
n
≤
λ
n
≤
λ
<
1
/
(
1
+
L
2
+
1
)
,
∀
n
≥
0, for L
=
max
⁡
{
L
j
:
1
≤
j
≤
M
}. Then, {
x
n
} converges strongly to a unique minimum norm point x
* of F
(i.e., x
*
=
P
F
(
0
)
), which is the unique solution of the variational inequality 〈
x
*
,
x
-
x
*
〉
≥
0 for all x
∈
F.


## 4. Numerical Example

Now, we give an example of two Lipschitz pseudocontractive mappings and twoγ-inverse strongly accretive mappings satisfying Theorem 10 and some numerical experiment result to explain the conclusion of the theorem as follows.Example 1.
LetH
=
R with absolute value norm. Let C
=
[
-
2,2
] and let T
1
,
T
2
:
C
→
C be defined by
(49)
T
1
x
:
=
{
x
+
x
2
,
x
∈
[
-
2,0
]
,
x
,
x
∈
(
0,2
]
,
T
2
x
:
=
{
x
,
x
∈
[
-
2
,
1
2
]
,
x
-
(
16
9
)
(
x
-
1
2
)
2
,
x
∈
(
1
2
,
2
]
.
Clearly, forx
,
y
∈
C we have that
(50)
〈
(
I
-
T
1
)
x
-
(
I
-
T
1
)
y
,
x
-
y
〉
≥
0
,
〈
(
I
-
T
2
)
x
-
(
I
-
T
2
)
y
,
x
-
y
〉
≥
0
which show that both mappings are pseudocontractive. Next, we show that T
1 is Lipschitz with L
=
5. If x
,
y
∈
[
-
2,0
], then
(51)
|
T
1
x
-
T
1
y
|
=
|
x
+
x
2
-
y
-
y
2
|
=
|
(
x
+
y
)
+
1
|
|
x
-
y
|
≤
3
|
x
-
y
|
.
Ifx
,
y
∈
(
0,2
], then
(52)
|
T
1
x
-
T
1
y
|
=
|
x
-
y
|
.
Ifx
∈
[
-
2,0
] and y
∈
(
0,2
], then
(53)
|
T
1
x
-
T
1
y
|
=
|
x
+
x
2
-
y
|
=
|
x
-
y
+
x
2
|
=
|
x
-
y
+
x
2
-
y
2
+
y
2
|
=
|
x
-
y
+
x
2
-
y
2
|
+
y
2
≤
|
x
+
y
+
1
|
·
|
x
-
y
|
+
|
y
-
x
|
2
=
(
|
x
+
y
+
1
|
+
|
x
+
y
|
)
·
|
x
-
y
|
≤
5
|
x
-
y
|
.
Thus, we get thatT
1 is Lipschitz pseudocontractive with L
=
5 and F
(
T
1
)
=
[
0,2
] which is not nonexpansive, since if we take x
=
-
2 and y
=
-
1.9, we have that |
T
1
x
-
T
2
y
|
=
0.29
>
0.1
=
|
x
-
y
|. Similarly, we can show that T
2 is Lipschitz pseudocontractive with L
=
4 and F
(
T
2
)
=
[
-
2,1
/
2
] which is not nonexpansive.
Furthermore, forC
=
[
-
2,2
], let A
1
,
A
2
:
C
→
R be defined by
(54)
A
1
x
:
=
{
-
(
x
-
1
2
)
2
,
x
∈
[
-
2
,
1
2
)
,
0
,
x
∈
[
1
2
,
2
]
,
A
2
x
:
=
{
0
,
x
∈
[
-
2
,
2
3
]
,
3
(
x
-
2
3
)
2
,
x
∈
(
2
3
,
2
]
.
Then we first show thatA
1 is γ-inverse strongly accretive mapping with γ
=
1
/
5.
Ifx
,
y
∈
[
-
2
,
1
/
2
), then
(55)
〈
A
1
x
-
A
1
y
,
x
-
y
〉
=
〈
-
(
x
-
1
2
)
2
+
(
y
-
1
2
)
2
,
x
-
y
〉
=
[
(
x
-
1
2
)
2
-
(
y
-
1
2
)
2
]
(
y
-
x
)
=
[
(
x
-
1
2
)
2
-
(
y
-
1
2
)
2
]
[
(
y
-
1
2
)
-
(
x
-
1
2
)
]
=
[
(
x
-
1
2
)
2
-
(
y
-
1
2
)
2
]
[
(
y
-
1
/
2
)
2
-
(
x
-
1
/
2
)
2
]
(
y
-
1
/
2
)
+
(
x
-
1
/
2
)
=
[
(
x
-
1
2
)
2
-
(
y
-
1
2
)
2
]
[
(
x
-
1
/
2
)
2
-
(
y
-
1
/
2
)
2
]
(
1
/
2
-
x
)
+
(
1
/
2
-
y
)
≥
1
5
|
(
x
-
1
2
)
2
-
(
y
-
1
2
)
2
|
2
=
1
5
|
A
1
x
-
A
1
y
|
2
.
Ifx
∈
[
-
2
,
1
/
2
) and y
∈
[
1
/
2
,
2
], we get that
(56)
〈
A
1
x
-
A
1
y
,
x
-
y
〉
=
〈
-
(
x
-
1
2
)
2
,
x
-
y
〉
=
(
x
-
1
2
)
2
(
y
-
x
)
=
(
x
-
1
2
)
2
[
(
y
-
1
2
)
-
(
x
-
1
2
)
]
≥
(
x
-
1
2
)
2
(
1
2
-
x
)
=
(
x
-
1
2
)
2
(
1
/
2
-
x
)
2
(
1
/
2
-
x
)
≥
2
5
|
(
x
-
1
2
)
2
|
2
≥
1
5
|
A
1
x
-
A
1
y
|
2
.
Ifx
,
y
∈
[
1
/
2
,
2
], then we get that |
A
1
x
-
A
1
y
|
=
0 and hence
(57)
〈
A
1
x
-
A
1
y
,
x
-
y
〉
≥
1
5
|
A
1
x
-
A
1
y
|
2
.
Therefore,A
1 is γ-inverse strongly accretive mapping with γ
=
1
/
5 and VI
⁡
(
C
,
A
1
)
=
[
1
/
2
,
2
]. Similarly, we can show that A
2 is γ-inverse strongly accretive mapping with γ
=
1
/
2 and VI
⁡
(
C
,
A
2
)
=
[
-
2
,
2
/
3
].
Note that we haveF
(
T
1
)
∩
F
(
T
2
)
∩
VI
⁡
(
C
,
A
1
)
∩
VI
⁡
(
C
,
A
2
)
=
{
1
/
2
}.
Thus, takingα
n
=
1
/
(
10
n
+
100
), λ
n
=
2
/
(
n
+
100
)
+
0.065, b
n
=
c
n
=
1
/
(
n
+
100
)
+
0.01, a
n
=
1
-
2
/
(
n
+
100
)
-
0.02, and f
(
x
)
=
u
∈
C, we observe that conditions of Theorem 10 are satisfied and Scheme (17) provides the following Table 1 and Figures 1(a) and 1(b) for u
=
0.6 and u
=
0.8, respectively.
We observe that the data provides strong convergence of the sequence to the common point of fixed points of both pseudocontractive mappings and solutions of both variational inequality problems forγ-inverse strongly accretive mappings.Table 1
u
=
0.6
x
0
=
1
u
=
0.8
x
0
=
-
1
n
x
n
n
x
n
0
1.0000
0
−1.0000
500
0.6112
5000
0.0627
10,000
0.5137
10,000
0.4282
12,000
0.5121
15,000
0.4540
14,000
0.5110
20,000
0.4686
18,000
0.5093
25,000
0.4782
20,000
0.5087
35,000
0.4905(a)
(b)Remark 2.
Theorem10 provides an iteration scheme which converges strongly to a common point of fixed points of a finite family of Lipschitzian pseudocontractive mappings and solutions of a finite family of variational inequality problems in Hilbert spaces.Remark 3.
Theorem10 improves Theorem 3.1 of Takahashi and Toyoda [19], Iiduka and Takahashi [20], and Zegeye and Shahzad [21] and Theorem 3.2 of Yao et al. [22] in the sense that our convergence is to a common point of fixed points of a finite family of Lipschitzian pseudocontractive mappings and solutions of a finite family of variational inequality problems.



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*Source: 424875-2014-08-04.xml*