# Generalized Mixed Equilibria, Variational Inclusions, and Fixed Point Problems

Authors: A. E. Al-Mazrooei; A. S. M. Alofi; A. Latif; J.-C. Yao
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We propose two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under suitable conditions.


---

## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈
·
,
·
〉 and norm ∥
·
∥, let C be a nonempty closed convex subset of H, and let P
C be the metric projection of H onto C. Let S
:
C
→
C be a self-mapping on C. Fix
⁡
(
S
) is the set of fixed points of S and R is the set of all real numbers. We recall that a mapping A
:
C
→
H is said to be L-Lipschitz continuous, if there exists a constant L
≥
0 such that
(1)
∥
A
x
-
A
y
∥
≤
L
∥
x
-
y
∥
,
∀
x
,
y
∈
C
.
If L
=
1, then A is called a nonexpansive mapping and if L
∈
[
0,1
), then A is called a contraction. We also let I denote the identity operator on the Hilbert space H.Letφ
:
C
→
R be a real-valued function, let A
:
C
→
H be a nonlinear mapping, and let F
:
C
×
C
→
R be a bifunction. The generalized mixed equilibrium problem (GMEP) introduced in [1] is to find x
∈
C such that
(2)
F
(
x
,
y
)
+
φ
(
y
)
-
φ
(
x
)
+
〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
We denote the set of solutions of GMEP (2) by GMEP
⁡
(
F
,
φ
,
A
). The GMEP covers many problems [2–6] as special cases and has been extensively studied recently.Throughout this paper, we assume as in [1] that F
:
C
×
C
→
R is a bifunction satisfying conditions (A1)–(A4) and φ
:
C
→
R is a lower semicontinuous and convex function with restriction (B1) or (B2), where(A1)
F
(
x
,
x
)
=
0, for all x
∈
C;
(A2)
F is monotone; that is, F
(
x
,
y
)
+
F
(
y
,
x
)
≤
0, for any x
,
y
∈
C;
(A3)
F is upper-hemicontinuous; that is, for each x
,
y
,
z
∈
C,
(3)
limsup
⁡
t
→
0
+
F
(
t
z
+
(
1
-
t
)
x
,
y
)
≤
F
(
x
,
y
)
;
(A4)
F
(
x
,
·
) is convex and lower semicontinuous, for each x
∈
C;
(B1)
for eachx
∈
H and r
>
0, there exists a bounded subset  D
x
⊂
C and y
x
∈
C such that, for any z
∈
C
∖
D
x,
(4)
F
(
z
,
y
x
)
+
φ
(
y
x
)
-
φ
(
z
)
+
1
r
〈
y
x
-
z
,
z
-
x
〉
<
0
;
(B2)
C is a bounded set.Next we list some elementary consequences for the mixed equilibrium problem studied in [2] where MEP
⁡
(
F
,
φ
) is the solution set.Proposition 1 (see [2]).
Assume thatF
:
C
×
C
→
R satisfies (A1)–(A4) and let φ
:
C
→
R be a proper lower semicontinuous and convex function. Assume that either (B1) or (B2) holds. For r
>
0 and x
∈
H, define a mapping T
r
(
F
,
φ
)
:
H
→
C as follows:
(5)
T
r
(
F
,
φ
)
(
x
)
=
{
1
r
z
∈
C
:
F
(
z
,
y
)
+
φ
(
y
)
-
φ
(
z
)
+
1
r
〈
y
-
z
,
z
-
x
〉
≥
0
,
∀
y
∈
C
1
r
}
,
for all
x
∈
H
. Then the following hold:
(i)
for each
x
∈
H
,
T
r
(
F
,
φ
)
(
x
)
≠
∅
;
(ii)
T
r
(
F
,
φ
)
is single-valued;
(iii)
T
r
(
F
,
φ
)
is firmly nonexpansive; that is, for any
x
,
y
∈
H
,
(6)
∥
T
r
(
F
,
φ
)
x
-
T
r
(
F
,
φ
)
y
∥
2
≤
〈
T
r
(
F
,
φ
)
x
-
T
r
(
F
,
φ
)
y
,
x
-
y
〉
;
(iv)
Fix
⁡
(
T
r
(
F
,
φ
)
)
=
MEP
⁡
(
F
,
φ
)
;
(v)
MEP
⁡
(
F
,
φ
)
is closed and convex.Next, recall some concepts.Definition 2.
LetC be a nonempty subset of a normed space X and let S
:
C
→
C be a self-mapping on C.(i)
S is asymptotically nonexpansive (see [7]), if there exists a sequence {
k
n
} of positive numbers satisfying the property lim
⁡
n
→
∞
k
n
=
1 and
(7)
∥
S
n
x
-
S
n
y
∥
≤
k
n
∥
x
-
y
∥
,
∀
n
≥
1
,
∀
x
,
y
∈
C
.
(ii)
S is asymptotically nonexpansive in the intermediate sense (see [8]) provided that S is uniformly continuous and
(8)
limsup
⁡
n
→
∞
sup
⁡
x
,
y
∈
C
(
∥
S
n
x
-
S
n
y
∥
-
∥
x
-
y
∥
)
≤
0
.
(iii)
S is uniformly Lipschitzian, if there exists a constant L
>
0 such that
(9)
∥
S
n
x
-
S
n
y
∥
≤
L
∥
x
-
y
∥
,
∀
n
≥
1
,
∀
x
,
y
∈
C
.It is clear that every nonexpansive mapping is asymptotically nonexpansive and every asymptotically nonexpansive mapping is uniformly Lipschitzian. Recently, Kim and Xu [9] introduced the concept of asymptotically κ-strict pseudocontractive mappings in a Hilbert space as follows.Definition 3.
LetC be a nonempty subset of a Hilbert space H. A mapping S
:
C
→
C is said to be an asymptotically κ-strict pseudocontractive mapping with sequence {
γ
n
}, if there exist a constant κ
∈
[
0,1
) and a sequence {
γ
n
} in [
0
,
∞
) with lim
⁡
n
→
∞
γ
n
=
0 such that
(10)
∥
S
n
x
-
S
n
y
∥
2
≤
(
1
+
γ
n
)
∥
x
-
y
∥
2
+
κ
∥
x
-
S
n
x
-
(
y
-
S
n
y
)
∥
2
,
∀
n
≥
1
,
∀
x
,
y
∈
C
.
They studied weak and strong convergence theorems for this class of mappings. It is important to note that every asymptoticallyκ-strict pseudocontractive mapping with sequence {
γ
n
} is a uniformly L-Lipschitzian mapping with L
=
sup
⁡
{
(
κ
+
1
+
(
1
-
κ
)
γ
n
)
/
(
1
+
κ
)
:
n
≥
1
}.Recently, Sahu et al. [10] considered the concept of asymptotically κ-strict pseudocontractive mappings in the intermediate sense, which are not necessarily Lipschitzian.Definition 4.
LetC be a nonempty subset of a Hilbert space H. A mapping S
:
C
→
C is said to be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
}, if there exist a constant κ
∈
[
0,1
) and a sequence {
γ
n
} in [
0
,
∞
) with lim
⁡
n
→
∞
γ
n
=
0 such that
(11)
limsup
⁡
n
→
∞
sup
⁡
x
,
y
∈
C
(
∥
S
n
x
-
S
n
y
∥
2
-
(
1
+
γ
n
)
∥
x
-
y
∥
2
-
κ
∥
x
-
S
n
x
-
(
y
-
S
n
y
)
∥
2
)
≤
0
.
Putc
n
:
=
max
⁡
{
0
,
sup
⁡
x
,
y
∈
C
(
∥
S
n
x
-
S
n
y
∥
2
-
(
1
+
γ
n
)
∥
x
-
y
∥
2
-
κ
∥
x
-
S
n
x
-
(
y
-
S
n
y
)
∥
2
)
}. Then, c
n
≥
0
(
∀
n
≥
1
), c
n
→
0
(
n
→
∞
), and (11) reduces to the relation
(12)
∥
S
n
x
-
S
n
y
∥
2
≤
(
1
+
γ
n
)
∥
x
-
y
∥
2
+
κ
∥
x
-
S
n
x
-
(
y
-
S
n
y
)
∥
2
+
c
n
,
∀
n
≥
1
,
∀
x
,
y
∈
C
.
Wheneverc
n
=
0, for all n
≥
1 in (12), then S is an asymptotically κ-strict pseudocontractive mapping with sequence {
γ
n
}.LetB be a single-valued mapping of C into H and R be a multivalued mapping with domain D
(
R
)
=
C. Consider the following variational inclusion: find a point x
∈
C such that
(13)
0
∈
B
x
+
R
x
.
We denote by I
(
B
,
R
) the solution set of the variational inclusion (13). In 1998, Huang [11] studied problem (13) in the case where R is maximal monotone and B is strongly monotone and Lipschitz continuous with D
(
R
)
=
C
=
H. Subsequently, Zeng et al. [12] further studied problem (13) in the case which is more general than that in [11]. Moreover, the authors [12] obtained the same strong convergence conclusion as in [11]. In addition, the authors also gave the geometric convergence rate estimate for approximate solutions.In this paper, inspired by the research work mentioned above, we introduce two iterative algorithms for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and an asymptoticallyκ-strict pseudocontractive mapping in the intermediate sense in a real Hilbert space. We prove some strong and weak convergence theorems for the proposed iterative algorithms under mild conditions.


## 2. Preliminaries

LetH be a real Hilbert space and C be a nonempty closed convex subset of H. We use the notation x
n
⇀
x to indicate that the sequence {
x
n
} converges weakly to x and x
n
→
x to indicate that the sequence {
x
n
} converges strongly to x. Moreover, we use ω
w
(
x
n
) to denote the weak ω-limit set of the sequence {
x
n
}; that is,
(14)
ω
w
(
x
n
)
:
=
{
x
∈
H
:
x
n
i
⇀
x
for
some
subsequence
{
x
n
i
}
of
{
x
n
}
}
.Recall that a mappingA
:
C
→
H is called(i)
Monotone, if(15)
〈
A
x
-
A
y
,
x
-
y
〉
≥
0
,
∀
x
,
y
∈
C
;
(ii)
η-strongly monotone, if there exists a constant η
>
0 such that
(16)
〈
A
x
-
A
y
,
x
-
y
〉
≥
η
∥
x
-
y
∥
2
,
∀
x
,
y
∈
C
;
(iii)
α-inverse strongly monotone, if there exists a constant α
>
0 such that
(17)
〈
A
x
-
A
y
,
x
-
y
〉
≥
α
∥
A
x
-
A
y
∥
2
,
∀
x
,
y
∈
C
.WhenA is an α-inverse strongly monotone mapping of C into H, it is easy to see that A is (
1
/
α
)-Lipschitz continuous. We also have that, for all u
,
v
∈
C and λ
>
0,
(18)
∥
(
I
-
λ
A
)
u
-
(
I
-
λ
A
)
v
∥
2
=
∥
(
u
-
v
)
-
λ
(
A
u
-
A
v
)
∥
2
=
∥
u
-
v
∥
2
-
2
λ
〈
A
u
-
A
v
,
u
-
v
〉
+
λ
2
∥
A
u
-
A
v
∥
2
≤
∥
u
-
v
∥
2
+
λ
(
λ
-
2
α
)
∥
A
u
-
A
v
∥
2
.
So, if λ
≤
2
α, then I
-
λ
A is a nonexpansive mapping from C to H.The metric projection fromH onto C is the mapping P
C
:
H
→
C which assigns to each point x
∈
H the unique point P
C
x
∈
C satisfying the property
(19)
∥
x
-
P
C
x
∥
=
inf
⁡
y
∈
C
∥
x
-
y
∥
=
:
d
(
x
,
C
)
.Some important properties of projections are listed in the following proposition.Proposition 5.
For givenx
∈
H and z
∈
C,(i)
z
=
P
C
x
⇔
〈
x
-
z
,
y
-
z
〉
≤
0, ∀
y
∈
C;
(ii)
z
=
P
C
x
⇔
∥
x
-
z
∥
2
≤
∥
x
-
y
∥
2
-
∥
y
-
z
∥
2, ∀
y
∈
C;
(iii)
〈
P
C
x
-
P
C
y
,
x
-
y
〉
≥
∥
P
C
x
-
P
C
y
∥
2, ∀
y
∈
H.
Consequently,P
C is nonexpansive and monotone. It is easy to see that the projection P
C is 1-ism.We need some facts and tools in a real Hilbert spaceH which are listed as lemmas below.Lemma 6.
LetX be a real inner product space. Then there holds the following inequality:
(20)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
x
+
y
〉
,
∀
x
,
y
∈
X
.Lemma 7.
LetH be a real Hilbert space. Then the following hold:(a)
∥
x
-
y
∥
2
=
∥
x
∥
2
-
∥
y
∥
2
-
2
〈
x
-
y
,
y
〉, for all x
,
y
∈
H;
(b)
∥
λ
x
+
μ
y
+
ν
z
∥
2
=
λ
∥
x
∥
2
+
μ
∥
y
∥
2
+
ν
∥
z
∥
2
-
λ
μ
∥
x
-
y
∥
2
-
μ
ν
∥
y
-
z
∥
2
-
λ
ν
∥
x
-
z
∥
2, for all x
,
y
,
z
∈
H and λ
,
μ
,
ν
∈
[
0,1
] with λ
+
μ
+
ν
=
1;
(c)
if{
x
n
} is a sequence in H such that x
n
⇀
x, it follows that
(21)
limsup
⁡
n
→
∞
∥
x
n
-
y
∥
2
=
limsup
⁡
n
→
∞
∥
x
n
-
x
∥
2
+
∥
x
-
y
∥
2
,
∀
y
∈
H
.Lemma 8 (see [10, Lemma 2.5]).
LetH be a real Hilbert space. Given a nonempty closed convex subset C of H and points x
,
y
,
z
∈
H and given also a real number a
∈
R, the set
(22)
{
v
∈
C
:
∥
y
-
v
∥
2
≤
∥
x
-
v
∥
2
+
〈
z
,
v
〉
+
a
}
is convex and closed.Lemma 9 (see [10, Lemma 2.6]).
LetC be a nonempty subset of a Hilbert space H and let S
:
C
→
C be an asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
}. Then
(23)
∥
S
n
x
-
S
n
y
∥
≤
1
1
-
κ
×
(
κ
∥
x
-
y
∥
+
(
1
+
(
1
-
κ
)
γ
n
)
∥
x
-
y
∥
2
+
(
1
-
κ
)
c
n
)
,
for all x
,
y
∈
C and n
≥
1.Lemma 10 (see [10, Lemma 2.7]).
LetC be a nonempty subset of a Hilbert space H and let S
:
C
→
C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
}. Let {
x
n
} be a sequence in C such that ∥
x
n
-
x
n
+
1
∥
→
0 and ∥
x
n
-
S
n
x
n
∥
→
0 as n
→
∞. Then ∥
x
n
-
S
x
n
∥
→
0 as n
→
∞.Lemma 11 (demiclosedness principle [10, Proposition 3.1]).
LetC be a nonempty closed convex subset of a Hilbert space H and let S
:
C
→
C be a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
}. Then I
-
S is demiclosed at zero in the sense that if {
x
n
} is a sequence in C such that x
n
⇀
x
∈
C and limsup
⁡
m
→
∞
limsup
⁡
n
→
∞
∥
x
n
-
S
m
x
n
∥
=
0, then (
I
-
S
)
x
=
0.Lemma 12 (see [10, Proposition 3.2]).
LetC be a nonempty closed convex subset of a Hilbert space H and let S
:
C
→
C be a continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense with sequence {
γ
n
} such that Fix
⁡
(
S
)
≠
∅. Then, Fix
⁡
(
S
) is closed and convex.Lemma 13 (see [13, page 80]).
Let{
a
n
}
n
=
1
∞
,
{
b
n
}
n
=
1
∞, and {
δ
n
}
n
=
1
∞ be sequences of nonnegative real numbers satisfying the inequality
(24)
a
n
+
1
≤
(
1
+
δ
n
)
a
n
+
b
n
,
∀
n
≥
1
.
If ∑
n
=
1
∞
‍
δ
n
<
∞ and ∑
n
=
1
∞
‍
b
n
<
∞, then lim
⁡
n
→
∞
a
n exists. If, in addition, {
a
n
}
n
=
1
∞ has a subsequence which converges to zero, then lim
⁡
n
→
∞
a
n
=
0.Corollary 14 (see [14, page 303]).
Let{
a
n
}
n
=
0
∞ and {
b
n
}
n
=
0
∞ be two sequences of nonnegative real numbers satisfying the inequality
(25)
a
n
+
1
≤
a
n
+
b
n
,
∀
n
≥
0
.
If ∑
n
=
0
∞
‍
b
n converges, then lim
⁡
n
→
∞
a
n exists.Recall that a Banach spaceX is said to satisfy the Opial condition [15], if for any given sequence {
x
n
}
⊂
X which converges weakly to an element x
∈
X, there holds the inequality
(26)
limsup
⁡
n
→
∞
∥
x
n
-
x
∥
<
limsup
⁡
n
→
∞
∥
x
n
-
y
∥
,
∀
y
∈
X
,
y
≠
x
.
It is well known in [15] that every Hilbert space H satisfies the Opial condition.Lemma 15 (see [16, Proposition 3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let {
x
n
} be a sequence in H. Suppose that
(27)
∥
x
n
+
1
-
p
∥
2
≤
(
1
+
λ
n
)
∥
x
n
-
p
∥
2
+
δ
n
,
∀
p
∈
C
,
n
≥
1
,
where {
λ
n
} and {
δ
n
} are sequences of nonnegative real numbers such that ∑
n
=
1
∞
‍
λ
n
<
∞ and ∑
n
=
1
∞
‍
δ
n
<
∞. Then {
P
C
x
n
} converges strongly in C.Lemma 16 (see [17]).
LetC be a closed convex subset of a real Hilbert space H. Let {
x
n
} be a sequence in H and u
∈
H. Let q
=
P
C
u. If {
x
n
} is such that ω
w
(
x
n
)
⊂
C and satisfies the condition
(28)
∥
x
n
-
u
∥
≤
∥
u
-
q
∥
,
∀
n
,
then x
n
→
q as n
→
∞.
Let{
T
n
}
n
=
1
∞ be an infinite family of nonexpansive self-mappings on C and let {
λ
n
}
n
=
1
∞ be a sequence of nonnegative numbers in [
0,1
]. For any n
≥
1, define a self-mapping W
n on C as follows:
(29)
U
n
,
n
+
1
=
I
,

U
n
,
n
=
λ
n
T
n
U
n
,
n
+
1
+
(
1
-
λ
n
)
I
,
U
n
,
n
-
1
=
λ
n
-
1
T
n
-
1
U
n
,
n
+
(
1
-
λ
n
-
1
)
I
,
⋮
U
n
,
k
=
λ
k
T
k
U
n
,
k
+
1
+
(
1
-
λ
k
)
I
,
U
n
,
k
-
1
=
λ
k
-
1
T
k
-
1
U
n
,
k
+
(
1
-
λ
k
-
1
)
I
,
⋮
U
n
,
2
=
λ
2
T
2
U
n
,
3
+
(
1
-
λ
2
)
I
,

W
n
=
U
n
,
1
=
λ
1
T
1
U
n
,
2
+
(
1
-
λ
1
)
I
.
Such a mapping W
n is called the W-mapping generated by T
n
,
T
n
-
1
,
…
,
T
1 and λ
n
,
λ
n
-
1
,
…
,
λ
1.Lemma 17 (see [18, Lemma 3.2]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive self-mappings on C such that ⋂
n
=
1
∞
Fix
⁡
(
T
n
)
≠
∅ and let {
λ
n
} be a sequence in (
0
,
b
] for some b
∈
(
0,1
). Then, for every x
∈
C and k
≥
1, the limit lim
⁡
n
→
∞
U
n
,
k
x exists.Utilizing Lemma17, we can define a mapping W
:
H
→
H as follows:
(30)
W
x
=
lim
⁡
n
→
∞
W
n
x
=
lim
⁡
n
→
∞
U
n
,
1
x
,
∀
x
∈
H
.
Such a W is called the W-mapping generated by T
1
,
T
2
,
… and λ
1
,
λ
2
,
…. We remark that since W
n is nonexpansive, W
:
H
→
H is also nonexpansive.Lemma 18 (see [18, Lemma 3.3]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive self-mappings on C such that ⋂
n
=
1
∞
Fix
⁡
(
T
n
)
≠
∅, and let {
λ
n
} be a sequence in (
0
,
b
], for some b
∈
(
0,1
). Then, Fix
⁡
(
W
)
=
⋂
n
=
1
∞
Fix
⁡
(
T
n
).Lemma 19 (see [19, demiclosedness principle]).
LetC be a nonempty closed convex subset of a real Hilbert space H. Let T be a nonexpansive self-mapping on C. Then I
-
T is demiclosed. That is, whenever {
x
n
} is a sequence in C weakly converging to some x
∈
C and the sequence {
(
I
-
T
)
x
n
} strongly converges to some y, it follows that (
I
-
T
)
x
=
y.Recall that a set-valued mappingR
:
D
(
R
)
⊂
H
→
2
H is called monotone, if, for all x
,
y
∈
D
(
R
)
,
f
∈
R
(
x
), and g
∈
R
(
y
), we have
(31)
〈
f
-
g
,
x
-
y
〉
≥
0
.
A set-valued mapping R is called maximal monotone, if R is monotone and (
I
+
λ
R
)
D
(
R
)
=
H, for each λ
>
0. We denote by G
(
R
) the graph of R. It is known that a monotone mapping R is maximal if and only if, for (
x
,
f
)
∈
H
×
H
,
〈
f
-
g
,
x
-
y
〉
≥
0, for every (
y
,
g
)
∈
G
(
R
), we have f
∈
R
(
x
).Assume thatR
:
D
(
R
)
⊂
H
→
2
H is a maximal monotone mapping. Then, for λ
>
0, associated with R, the resolvent operator J
R
,
λ can be defined as
(32)
J
R
,
λ
x
=
(
I
+
λ
R
)
-
1
x
,
∀
x
∈
H
.
We have the following property for the resolvent operator J
R
,
λ
:
H
→
D
(
R
)
¯.Lemma 20 (see [11]).
J
R
,
λ is single-valued and firmly nonexpansive; that is,
(33)
〈
J
R
,
λ
x
-
J
R
,
λ
y
,
x
-
y
〉
≥
∥
J
R
,
λ
x
-
J
R
,
λ
y
∥
2
,
∀
x
,
y
∈
H
.Consequently,J
R
,
λ is nonexpansive and monotone.Lemma 21 (see [20]).
LetR be a maximal monotone mapping with D
(
R
)
=
C. Then, for any given λ
>
0, u
∈
C is a solution of the variational inclusion: find a point x
∈
C such that
(34)
0
∈
B
x
+
R
x
,
whereB is a single-valued mapping of C into H if and only if u
∈
C satisfies
(35)
u
=
J
R
,
λ
(
u
-
λ
B
u
)
.Lemma 22 (see [12]).
LetR be a maximal monotone mapping with D
(
R
)
=
C and let B
:
C
→
H be a strongly monotone, continuous, and single-valued mapping. Then, for each z
∈
H, the equation z
∈
(
B
+
λ
R
)
x has a unique solution x
λ , for λ
>
0.Lemma 23 (see [20]).
LetR be a maximal monotone mapping with D
(
R
)
=
C and let B
:
C
→
H be a monotone, continuous, and single-valued mapping. Then (
I
+
λ
(
R
+
B
)
)
C
=
H, for each λ
>
0. In this case, R
+
B is maximal monotone.


## 3. Strong Convergence Theorem

In this section, we prove a strong convergence theorem for an iterative algorithm for finding a common element of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and asymptoticallyκ-strict pseudocontractive mapping S
:
C
→
C in the intermediate sense in a real Hilbert space. This iterative algorithm is based on the extragradient method [21], Mann-type iterative method, and shrinking projection method. For more recent related results, see [22] and the references therein.Theorem 24.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M
,
N be two integers. Let F
k be a bifunction from C
×
C to R satisfying (A1)–(A4) and let φ
k
:
C
→
R
∪
{
+
∞
} be a proper lower semicontinuous and convex function, where k
∈
{
1,2
,
…
,
M
}. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
k
:
H
→
H and B
i
:
C
→
H be μ
k-inverse strongly monotone and η
i-inverse strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let S
:
C
→
C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense, for some 0
≤
κ
<
1, with sequence {
γ
n
}
⊂
[
0
,
∞
) such that lim
⁡
n
→
∞
γ
n
=
0 and {
c
n
}
⊂
[
0
,
∞
) such that lim
⁡
n
→
∞
c
n
=
0. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive self-mappings on C and let {
λ
n
} be a sequence in (
0
,
b
] for some b
∈
(
0,1
). Assume that F
:
=
⋂
n
=
1
∞
Fix
⁡
(
T
n
)
∩
⋂
k
=
1
M
GMEP
⁡
(
F
k
,
φ
k
,
A
k
)
∩
⋂
i
=
1
N
I
(
B
i
,
R
i
)
∩
Fix
⁡
(
S
) is nonempty and bounded and that either (B1) or (B2) holds. Let {
α
n
}
,
{
β
n
}, and {
δ
n
} be sequences in [
0,1
] such that α
n
+
β
n
≤
1, 0
<
a
≤
α
n
≤
1, and κ
≤
δ
n
≤
d
<
1. Pick any x
0
∈
H and set C
1
=
C
,
x
1
=
P
C
1
x
0. Let {
x
n
} be a sequence generated by the following algorithm:
(36)
u
n
=
T
r
M
,
n
(
F
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
T
r
M
-
1
,
n
(
F
M
-
1
,
φ
M
-
1
)
×
(
I
-
r
M
-
1
,
n
A
M
-
1
)
⋯
T
r
1
,
n
(
F
1
,
φ
1
)
(
I
-
r
1
,
n
A
1
)
x
n
,
z
n
=
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
J
R
N
-
1
,
λ
N
-
1
,
n
×
(
I
-
λ
N
-
1
,
n
B
N
-
1
)
⋯
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
u
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
,
y
n
=
(
1
-
α
n
-
β
n
)
x
n
+
α
n
k
n
+
β
n
W
n
z
n
,
C
n
+
1
=
{
z
∈
C
n
:
∥
y
n
-
z
∥
2
≤
∥
x
n
-
z
∥
2
+
θ
n
}
,
x
n
+
1
=
P
C
n
+
1
x
0
,
∀
n
≥
0
,
where W
n is the W-mapping generated by (2.2), θ
n
=
γ
n
Δ
n
2
+
c
n, and Δ
n
=
sup
⁡
{
∥
x
n
-
p
∥
:
p
∈
F
}
<
∞. Assume that the following conditions hold:(i)
0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
α
n
+
β
n
)
<
1;
(ii)
{
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
)
,
∀
i
∈
{
1,2
,
…
,
N
};
(iii)
{
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
)
,
∀
k
∈
{
1,2
,
…
,
M
}.Then {
x
n
} converges strongly to P
F
x
0.Proof.
We divide the proof into several steps.
Step  1. First note that the defining inequality in C
n is equivalent to the inequality
(37)
〈
2
(
x
n
-
z
n
)
,
z
〉
≤
∥
x
n
∥
2
-
∥
z
n
∥
2
+
θ
n
.
So, by Lemma 8, C
n is closed and convex, for every n
≥
1.
We next show thatF
⊂
C
n, for all n
≥
1. Put
(38)
Θ
n
k
=
T
r
k
,
n
(
F
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
T
r
k
-
1
,
n
(
F
k
-
1
,
φ
k
-
1
)
×
(
I
-
r
k
-
1
,
n
A
k
-
1
)
⋯
T
r
1
,
n
(
F
1
,
φ
1
)
(
I
-
r
1
,
n
A
1
)
x
n
,
for all k
∈
{
1,2
,
…
,
M
} and n
≥
1;
(39)
Ω
n
i
=
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
J
R
i
-
1
,
λ
i
-
1
,
n
×
(
I
-
λ
i
-
1
,
n
B
i
-
1
)
⋯
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
,
for all i
∈
{
1,2
,
…
,
N
} and n
≥
1, and Θ
n
0
=
Ω
n
0
=
I, where I is the identity mapping on H. Then we have that u
n
=
Θ
n
M
x
n and z
n
=
Ω
n
N
u
n. Suppose that F
⊂
C
n for some n
≥
1. Take p
∈
F arbitrarily. Then, from (18) and Proposition 1(iii), we have
(40)
∥
u
n
-
p
∥
=
∥
T
r
M
,
n
(
F
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
Θ
n
M
-
1
x
n
∥
u
n
-
p
∥
-
T
r
M
,
n
(
F
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
Θ
n
M
-
1
p
∥
∥
u
n
-
p
∥
≤
∥
(
I
-
r
M
,
n
A
M
)
Θ
n
M
-
1
x
n
-
(
I
-
r
M
,
n
A
M
)
Θ
n
M
-
1
p
∥
∥
u
n
-
p
∥
≤
∥
Θ
n
M
-
1
x
n
-
Θ
n
M
-
1
p
∥
∥
u
n
-
p
∥
⋮
∥
u
n
-
p
∥
≤
∥
Θ
n
0
x
n
-
Θ
n
0
p
∥
∥
u
n
-
p
∥
=
∥
x
n
-
p
∥
.
Similarly, we have
(41)
∥
z
n
-
p
∥
=
∥
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
Ω
n
N
-
1
u
n
∥
z
n
-
p
∥
-
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
Ω
n
N
-
1
p
∥
∥
z
n
-
p
∥
≤
∥
(
I
-
λ
N
,
n
B
N
)
Ω
n
N
-
1
u
n
-
(
I
-
λ
N
,
n
B
N
)
Ω
n
N
-
1
p
∥
∥
z
n
-
p
∥
≤
∥
Ω
n
N
-
1
u
n
-
Ω
n
N
-
1
p
∥
∥
z
n
-
p
∥
⋮
∥
z
n
-
p
∥
≤
∥
Ω
n
0
x
n
-
Ω
n
0
p
∥
∥
z
n
-
p
∥
=
∥
u
n
-
p
∥
.
Combining (40) and (41), we have
(42)
∥
z
n
-
p
∥
≤
∥
x
n
-
p
∥
.
By Lemma 7(b), we deduce from (36) and (42) that
(43)
∥
k
n
-
p
∥
2
=
∥
δ
n
(
z
n
-
p
)
+
(
1
-
δ
n
)
(
S
n
z
n
-
p
)
∥
2
=
δ
n
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
∥
S
n
z
n
-
p
∥
2
-
δ
n
(
1
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
≤
δ
n
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
×
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
κ
∥
z
n
-
S
n
z
n
∥
2
+
c
n
]
-
δ
n
(
1
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
=
[
1
+
γ
n
(
1
-
δ
n
)
]
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
(
κ
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
(
1
-
δ
n
)
c
n
≤
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
(
κ
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
c
n
≤
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
.
It follows from (42), (43), and the convexity of ∥
·
∥
2 that
(44)
∥
y
n
-
p
∥
2
=
∥
(
1
-
α
n
-
β
n
)
(
x
n
-
p
)
+
α
n
(
k
n
-
p
)
+
β
n
(
W
n
z
n
-
p
)
∥
2
≤
∥
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
+
β
n
∥
W
n
z
n
-
p
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
(
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
)
+
β
n
∥
z
n
-
p
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
(
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
)
+
β
n
∥
x
n
-
p
∥
2
=
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
(
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
)
=
∥
x
n
-
p
∥
2
+
α
n
γ
n
∥
x
n
-
p
∥
2
+
α
n
c
n
≤
∥
x
n
-
p
∥
2
+
γ
n
∥
x
n
-
p
∥
2
+
c
n
≤
∥
x
n
-
p
∥
2
+
θ
n
.
Hence, p
∈
C
n
+
1. This implies that F
⊂
C
n, for all n
≥
1.
Step  2. We prove that ∥
x
n
-
k
n
∥
→
0 as n
→
∞.
Indeed, letv
=
P
F
x
0. From x
n
=
P
C
n
x
0 and v
∈
F
⊂
C
n, we obtain
(45)
∥
x
n
-
x
0
∥
≤
∥
v
-
x
0
∥
.
This implies that {
x
n
} is bounded and, hence, {
u
n
}, {
z
n
}, {
k
n
}, and {
y
n
} are also bounded. Since x
n
+
1
∈
C
n
+
1
⊂
C
n and x
n
=
P
C
n
x
0, we have
(46)
∥
x
n
-
x
0
∥
≤
∥
x
n
+
1
-
x
0
∥
,
∀
n
≥
1
.
Therefore, lim
⁡
n
→
∞
∥
x
n
-
x
0
∥ exists. From x
n
=
P
C
n
x
0
,
x
n
+
1
∈
C
n
+
1
⊂
C
n, by Proposition 5(ii) we obtain
(47)
∥
x
n
+
1
-
x
n
∥
2
≤
∥
x
0
-
x
n
+
1
∥
2
-
∥
x
0
-
x
n
∥
2
,
which implies
(48)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
It follows from x
n
+
1
∈
C
n
+
1 that ∥
y
n
-
x
n
+
1
∥
2
≤
∥
x
n
-
x
n
+
1
∥
2
+
θ
n and hence
(49)
∥
x
n
-
y
n
∥
2
≤
2
(
∥
x
n
-
x
n
+
1
∥
2
+
∥
x
n
+
1
-
y
n
∥
2
)
≤
2
(
∥
x
n
-
x
n
+
1
∥
2
+
∥
x
n
-
x
n
+
1
∥
2
+
θ
n
)
=
2
(
2
∥
x
n
-
x
n
+
1
∥
2
+
θ
n
)
.
From (48) and lim
⁡
n
→
∞
θ
n
=
0, we have
(50)
lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0
.
Also, utilizing Lemma 7(b), we obtain from (36), (42), and (43) that
(51)
∥
y
n
-
p
∥
2
=
∥
(
1
-
α
n
-
β
n
)
(
x
n
-
p
)
+
α
n
(
k
n
-
p
)
+
β
n
(
W
n
z
n
-
p
)
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
+
β
n
∥
W
n
z
n
-
p
∥
2
-
β
n
(
1
-
α
n
-
β
n
)
∥
x
n
-
W
n
z
n
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
]
+
β
n
∥
z
n
-
p
∥
2
-
β
n
(
1
-
α
n
-
β
n
)
∥
x
n
-
W
n
z
n
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
]
+
β
n
∥
x
n
-
p
∥
2
-
β
n
(
1
-
α
n
-
β
n
)
∥
x
n
-
W
n
z
n
∥
2
=
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
]
-
β
n
(
1
-
α
n
-
β
n
)
∥
x
n
-
W
n
z
n
∥
2
=
∥
x
n
-
p
∥
2
+
α
n
γ
n
∥
x
n
-
p
∥
2
+
α
n
c
n
-
β
n
(
1
-
α
n
-
β
n
)
∥
x
n
-
W
n
z
n
∥
2
≤
∥
x
n
-
p
∥
2
+
θ
n
-
β
n
(
1
-
α
n
-
β
n
)
∥
x
n
-
W
n
z
n
∥
2
,
which, hence, yields
(52)
β
n
(
1
-
α
n
-
β
n
)
∥
x
n
-
W
n
z
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
θ
n
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
θ
n
.
Since θ
n
→
0 and {
x
n
}
,
{
y
n
} are bounded, it follows from (50) and condition (i) that
(53)
lim
⁡
n
→
∞
∥
x
n
-
W
n
z
n
∥
=
0
.
Note that
(54)
y
n
-
x
n
=
α
n
(
k
n
-
x
n
)
+
β
n
(
W
n
z
n
-
x
n
)
,
which leads to
(55)
α
n
∥
k
n
-
x
n
∥
=
∥
y
n
-
x
n
-
β
n
(
W
n
z
n
-
x
n
)
∥
≤
∥
y
n
-
x
n
∥
+
β
n
∥
W
n
z
n
-
x
n
∥
≤
∥
y
n
-
x
n
∥
+
∥
W
n
z
n
-
x
n
∥
.
So, from (50), (53), and 0
<
a
≤
α
n
≤
1, we get
(56)
lim
⁡
n
→
∞
∥
x
n
-
k
n
∥
=
0
.
Step  3. We prove that ∥
x
n
-
u
n
∥
→
0, ∥
u
n
-
z
n
∥
→
0, ∥
z
n
-
W
n
z
n
∥
→
0, and ∥
z
n
-
S
n
z
n
∥
→
0 as n
→
∞.
Indeed, from (42) and (43), it follows that
(57)
∥
k
n
-
p
∥
2
≤
[
1
+
γ
n
(
1
-
δ
n
)
]
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
(
κ
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
(
1
-
δ
n
)
c
n
≤
∥
z
n
-
p
∥
2
+
γ
n
∥
z
n
-
p
∥
2
+
c
n
≤
∥
z
n
-
p
∥
2
+
γ
n
∥
x
n
-
p
∥
2
+
c
n
≤
∥
z
n
-
p
∥
2
+
θ
n
.
Next, we prove that
(58)
lim
⁡
n
→
∞
∥
Θ
n
k
x
n
-
Θ
n
k
-
1
x
n
∥
=
0
,
k
=
1,2
,
…
,
M
.
For p
∈
F, it follows from (18) that
(59)
∥
Θ
n
k
x
n
-
p
∥
2
=
∥
T
r
k
,
n
(
F
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
Θ
n
k
-
1
x
n
-
T
r
k
,
n
(
F
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
p
∥
2
≤
∥
(
I
-
r
k
,
n
A
k
)
Θ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
∥
2
≤
∥
Θ
n
k
-
1
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
2
≤
∥
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
2
.
By (40), (41), (57), and (59), we obtain
(60)
∥
k
n
-
p
∥
2
≤
∥
z
n
-
p
∥
2
+
θ
n
≤
∥
u
n
-
p
∥
2
+
θ
n
≤
∥
Θ
n
k
x
n
-
p
∥
2
+
θ
n
≤
∥
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
2
+
θ
n
,
which implies that
(61)
r
k
,
n
(
2
μ
k
-
r
k
,
n
)
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
k
n
-
p
∥
2
+
θ
n
≤
∥
x
n
-
k
n
∥
(
∥
x
n
-
p
∥
+
∥
k
n
-
p
∥
)
+
θ
n
.
Since {
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
), k
∈
{
1,2
,
…
,
M
}, lim
⁡
n
→
∞
θ
n
=
0, and (56), we have
(62)
lim
⁡
n
→
∞
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
=
0
,
k
=
1,2
,
…
,
M
.
By Proposition 1(iii) and Lemma 7(a), we have
(63)
∥
Θ
n
k
x
n
-
p
∥
2
=
∥
T
r
k
,
n
(
F
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
Θ
n
k
-
1
x
n
-
T
r
k
,
n
(
F
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
p
∥
2
≤
〈
(
I
-
r
k
,
n
A
k
)
Θ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
,
Θ
n
k
x
n
-
p
〉
=
1
2
(
∥
(
I
-
r
k
,
n
A
k
)
Θ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
∥
2
+
∥
Θ
n
k
x
n
-
p
∥
2
-
∥
(
I
-
r
k
,
n
A
k
)
Θ
n
k
-
1
x
n
-
(
I
-
r
k
,
n
A
k
)
p
-
(
Θ
n
k
x
n
-
p
)
∥
2
)
≤
1
2
(
∥
Θ
n
k
-
1
x
n
-
p
∥
2
+
∥
Θ
n
k
x
n
-
p
∥
2
-
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
-
r
k
,
n
(
A
k
Θ
n
k
-
1
x
n
-
A
k
p
)
∥
2
)
,
which implies that
(64)
∥
Θ
n
k
x
n
-
p
∥
2
≤
∥
Θ
n
k
-
1
x
n
-
p
∥
2
-
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
-
r
k
,
n
(
A
k
Θ
n
k
-
1
x
n
-
A
k
p
)
∥
2
=
∥
Θ
n
k
-
1
x
n
-
p
∥
2
-
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
2
-
r
k
,
n
2
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
2
+
2
r
k
,
n
〈
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
,
A
k
Θ
n
k
-
1
x
n
-
A
k
p
〉
≤
∥
Θ
n
k
-
1
x
n
-
p
∥
2
-
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
≤
∥
x
n
-
p
∥
2
-
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
.
Combining (57) and (64), we have
(65)
∥
k
n
-
p
∥
2
≤
∥
z
n
-
p
∥
2
+
θ
n
≤
∥
u
n
-
p
∥
2
+
θ
n
≤
∥
Θ
n
k
x
n
-
p
∥
2
+
θ
n
≤
∥
x
n
-
p
∥
2
-
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
+
θ
n
,
which implies
(66)
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
k
n
-
p
∥
2
+
2
r
k
,
n
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
+
θ
n
≤
∥
x
n
-
k
n
∥
(
∥
x
n
-
p
∥
+
∥
k
n
-
p
∥
)
+
2
r
k
,
n
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
+
θ
n
.
From lim
⁡
n
→
∞
θ
n
=
0, (56), and (62), we know that (58) holds. Hence, we obtain
(67)
∥
x
n
-
u
n
∥
=
∥
Θ
n
0
x
n
-
Θ
n
M
x
n
∥
≤
∥
Θ
n
0
x
n
-
Θ
n
1
x
n
∥
+
∥
Θ
n
1
x
n
-
Θ
n
2
x
n
∥
+
⋯
+
∥
Θ
n
M
-
1
x
n
-
Θ
n
M
x
n
∥
⟶
0
as
n
⟶
∞
.
Next, we show thatlim
⁡
n
→
∞
∥
B
i
Ω
n
i
u
n
-
B
i
p
∥
=
0, i
=
1,2
,
…
,
N. It follows from Lemma 20 and (18) that
(68)
∥
Ω
n
i
u
n
-
p
∥
2
=
∥
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
Ω
n
i
-
1
u
n
-
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
∥
(
I
-
λ
i
,
n
B
i
)
Ω
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
∥
Ω
n
i
-
1
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
u
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
2
.
Combining (57) and (68), we have
(69)
∥
k
n
-
p
∥
2
≤
∥
z
n
-
p
∥
2
+
θ
n
≤
∥
Ω
n
i
u
n
-
p
∥
2
+
θ
n
≤
∥
x
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
2
+
θ
n
,
which implies
(70)
λ
i
,
n
(
2
η
i
-
λ
i
,
n
)
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
k
n
-
p
∥
2
+
θ
n
≤
∥
x
n
-
k
n
∥
(
∥
x
n
-
p
∥
+
∥
k
n
-
p
∥
)
+
θ
n
.
From {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
), i
∈
{
1,2
,
…
,
N
}, lim
⁡
n
→
∞
θ
n
=
0, and (56), we obtain
(71)
lim
⁡
n
→
∞
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
=
0
,
i
=
1,2
,
…
,
N
.
By Lemmas 20 and 7(a), we obtain
(72)
∥
Ω
n
i
u
n
-
p
∥
2
=
∥
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
Ω
n
i
-
1
u
n
-
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
p
∥
2
≤
〈
(
I
-
λ
i
,
n
B
i
)
Ω
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
,
Ω
n
i
u
n
-
p
〉
=
1
2
(
∥
(
I
-
λ
i
,
n
B
i
)
Ω
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
∥
2
+
∥
Ω
n
i
u
n
-
p
∥
2
-
∥
(
I
-
λ
i
,
n
B
i
)
Ω
n
i
-
1
u
n
-
(
I
-
λ
i
,
n
B
i
)
p
-
(
Ω
n
i
u
n
-
p
)
∥
2
)
≤
1
2
(
∥
Ω
n
i
-
1
u
n
-
p
∥
2
+
∥
Ω
n
i
u
n
-
p
∥
2
-
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
-
λ
i
,
n
(
B
i
Ω
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
u
n
-
p
∥
2
+
∥
Ω
n
i
u
n
-
p
∥
2
-
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
-
λ
i
,
n
(
B
i
Ω
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
x
n
-
p
∥
2
+
∥
Ω
n
i
u
n
-
p
∥
2
-
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
-
λ
i
,
n
(
B
i
Ω
n
i
-
1
u
n
-
B
i
p
)
∥
2
)
,
which implies
(73)
∥
Ω
n
i
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
-
λ
i
,
n
(
B
i
Ω
n
i
-
1
u
n
-
B
i
p
)
∥
2
=
∥
x
n
-
p
∥
2
-
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
2
-
λ
i
,
n
2
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
2
+
2
λ
i
,
n
〈
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
,
B
i
Ω
n
i
-
1
u
n
-
B
i
p
〉
≤
∥
x
n
-
p
∥
2
-
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
.
Combining (57) and (73), we get
(74)
∥
k
n
-
p
∥
2
≤
∥
z
n
-
p
∥
2
+
θ
n
≤
∥
Ω
n
i
u
n
-
p
∥
2
+
θ
n
≤
∥
x
n
-
p
∥
2
-
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
+
θ
n
,
which implies
(75)
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
k
n
-
p
∥
2
+
2
λ
i
,
n
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
+
θ
n
≤
∥
x
n
-
k
n
∥
(
∥
x
n
-
p
∥
+
∥
k
n
-
p
∥
)
+
2
λ
i
,
n
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
+
θ
n
.
From (56), (71), and lim
⁡
n
→
∞
θ
n
=
0, we have
(76)
lim
⁡
n
→
∞
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
=
0
,
i
=
1,2
,
…
,
N
.
From (76), we get
(77)
∥
u
n
-
z
n
∥
=
∥
Ω
n
0
u
n
-
Ω
n
N
u
n
∥
∥
u
n
-
z
n
∥
≤
∥
Ω
n
0
u
n
-
Ω
n
1
u
n
∥
+
∥
Ω
n
1
u
n
-
Ω
n
2
u
n
∥
∥
u
n
-
z
n
∥
+
⋯
+
∥
Ω
n
N
-
1
u
n
-
Ω
n
N
u
n
∥
∥
u
n
-
z
n
∥
⟶
0
as
n
⟶
∞
.
By (67) and (77), we have
(78)
∥
x
n
-
z
n
∥
≤
∥
x
n
-
u
n
∥
+
∥
u
n
-
z
n
∥
∥
x
n
-
z
n
∥
⟶
0
as
n
⟶
∞
.
From (48) and (78), we have
(79)
∥
z
n
+
1
-
z
n
∥
≤
∥
z
n
+
1
-
x
n
+
1
∥
+
∥
x
n
+
1
-
x
n
∥
+
∥
x
n
-
z
n
∥
⟶
0
as
n
⟶
∞
.
By (56), (67), and (77), we get
(80)
∥
k
n
-
z
n
∥
≤
∥
k
n
-
x
n
∥
+
∥
x
n
-
u
n
∥
+
∥
u
n
-
z
n
∥
∥
k
n
-
z
n
∥
⟶
0
as
n
⟶
∞
.
We observe that
(81)
k
n
-
z
n
=
(
1
-
δ
n
)
(
S
n
z
n
-
z
n
)
.
From δ
n
≤
d
<
1 and (80), we have
(82)
lim
⁡
n
→
∞
∥
S
n
z
n
-
z
n
∥
=
0
.
We note that
(83)
∥
S
n
z
n
-
S
n
+
1
z
n
∥
≤
∥
S
n
z
n
-
z
n
∥
+
∥
z
n
-
z
n
+
1
∥
+
∥
z
n
+
1
-
S
n
+
1
z
n
+
1
∥
+
∥
S
n
+
1
z
n
+
1
-
S
n
+
1
z
n
∥
.
From (79), (82), and Lemma 9, we obtain
(84)
lim
⁡
n
→
∞
∥
S
n
z
n
-
S
n
+
1
z
n
∥
=
0
.
On the other hand, we note that
(85)
∥
z
n
-
S
z
n
∥
≤


∥
z
n
-
S
n
z
n
∥
+
∥
S
n
z
n
-
S
n
+
1
z
n
∥
+
∥
S
n
+
1
z
n
-
S
z
n
∥
.
From (82), (84), and the uniform continuity of S, we have
(86)
lim
⁡
n
→
∞
∥
z
n
-
S
z
n
∥
=
0
.
In addition, note that
(87)
∥
z
n
-
W
z
n
∥
≤
∥
z
n
-
W
n
z
n
∥
+
∥
W
n
z
n
-
W
z
n
∥
≤
∥
z
n
-
x
n
∥
+
∥
x
n
-
W
n
z
n
∥
+
∥
W
n
z
n
-
W
z
n
∥
.
So, from (53), (78), and [4, Remark 3.2], it follows that
(88)
lim
⁡
n
→
∞
∥
z
n
-
W
z
n
∥
=
0
.
Step  4. Finally we prove that x
n
→
v
=
P
F
x
0 as n
→
∞.
Indeed, since{
x
n
} is bounded, there exists a subsequence {
x
n
i
} which converges weakly to some w. From (58) and (76)–(78), we have that Θ
n
i
k
x
n
i
⇀
w
,
Ω
n
i
m
u
n
i
⇀
w and z
n
i
⇀
w, where k
∈
{
1,2
,
…
,
M
} and m
∈
{
1,2
,
…
,
N
}. Since S is uniformly continuous, by (86), we get lim
⁡
n
→
∞
∥
z
n
-
S
m
z
n
∥
=
0, for any m
≥
1. Hence, from Lemma 11, we obtain w
∈
Fix
⁡
(
S
). In the meantime, utilizing Lemma 19, we deduce from (88) and z
n
i
⇀
w that w
∈
Fix
⁡
(
W
)
=
⋂
n
=
1
∞
Fix
⁡
(
T
n
). Next, we prove that w
∈
⋂
m
=
1
N
I
(
B
m
,
R
m
). As a matter of fact, since B
m is η
m-inverse strongly monotone, B
m is a monotone and Lipschitz continuous mapping. It follows from Lemma 23 that R
m
+
B
m is maximal monotone. Let (
v
,
g
)
∈
G
(
R
m
+
B
m
); that is, g
-
B
m
v
∈
R
m
v. Again, since Ω
n
m
u
n
=
J
R
m
,
λ
m
,
n
(
I
-
λ
m
,
n
B
m
)
Λ
n
m
-
1
u
n
,
n
≥
1, m
∈
{
1,2
,
…
,
N
}, we have
(89)
Ω
n
m
-
1
u
n
-
λ
m
,
n
B
m
Ω
n
m
-
1
u
n
∈
(
I
+
λ
m
,
n
R
m
)
Ω
n
m
u
n
;
that is,
(90)
1
λ
m
,
n
(
Ω
n
m
-
1
u
n
-
Ω
n
m
u
n
-
λ
m
,
n
B
m
Ω
n
m
-
1
u
n
)
∈
R
m
Ω
n
m
u
n
.
In terms of the monotonicity of R
m, we get
(91)
〈
1
λ
m
,
n
v
-
Ω
n
m
u
n
,
g
-
B
m
v
-
1
λ
m
,
n
(
Ω
n
m
-
1
u
n
-
Ω
n
m
u
n
-
λ
m
,
n
B
m
Ω
n
m
-
1
u
n
)
〉
≥
0
and, hence,
(92)
〈
v
-
Ω
n
m
u
n
,
g
〉
≥
〈
1
λ
m
,
n
v
-
Ω
n
m
u
n
,
B
m
v
+
1
λ
m
,
n
(
Ω
n
m
-
1
u
n
-
Ω
n
m
u
n
-
λ
m
,
n
B
m
Ω
n
m
-
1
u
n
)
〉
=
〈
1
λ
m
,
n
v
-
Ω
n
m
u
n
,
B
m
v
-
B
m
Ω
n
m
u
n
+
B
m
Ω
n
m
u
n
-
B
m
Ω
n
m
-
1
u
n
+
1
λ
m
,
n
(
Ω
n
m
-
1
u
n
-
Ω
n
m
u
n
)
〉
≥
〈
v
-
Ω
n
m
u
n
,
B
m
Ω
n
m
u
n
-
B
m
Ω
n
m
-
1
u
n
〉
+
〈
v
-
Ω
n
m
u
n
,
1
λ
m
,
n
(
Ω
n
m
-
1
u
n
-
Ω
n
m
u
n
)
〉
.
In particular,
(93)
〈
v
-
Ω
n
i
m
u
n
i
,
g
〉
≥
〈
v
-
Ω
n
i
m
u
n
i
,
B
m
Ω
n
i
m
u
n
i
-
B
m
Ω
n
i
m
-
1
u
n
i
〉
+
〈
v
-
Ω
n
i
m
u
n
i
,
1
λ
m
,
n
i
(
Ω
n
i
m
-
1
u
n
i
-
Ω
n
i
m
u
n
i
)
〉
.
Since ∥
Ω
n
m
u
n
-
Ω
n
m
-
1
u
n
∥
→
0 (due to (76)) and ∥
B
m
Ω
n
m
u
n
-
B
m
Ω
n
m
-
1
u
n
∥
→
0 (due to the Lipschitz continuity of B
m), we conclude from Ω
n
i
m
u
n
i
⇀
w and condition (ii) that
(94)
lim
⁡
i
→
∞
〈
v
-
Ω
n
i
m
u
n
i
,
g
〉
=
〈
v
-
w
,
g
〉
≥
0
.
It follows from the maximal monotonicity of B
m
+
R
m that 0
∈
(
R
m
+
B
m
)
w; that is, w
∈
I
(
B
m
,
R
m
). Therefore, w
∈
⋂
m
=
1
N
I
(
B
m
,
R
m
). Next, we prove that w
∈
⋂
k
=
1
M
GMEP
⁡
(
F
k
,
φ
k
,
A
k
). Since Θ
n
k
x
n
=
T
r
k
,
n
(
F
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
Θ
n
k
-
1
x
n, n
≥
1, k
∈
{
1,2
,
…
,
M
}, we have
(95)
F
k
(
Θ
n
k
x
n
,
y
)
+
φ
k
(
y
)
-
φ
k
(
Θ
n
k
x
n
)
+
〈
A
k
Θ
n
k
-
1
x
n
,
y
-
Θ
n
k
x
n
〉
+
1
r
k
,
n
〈
y
-
Θ
n
k
x
n
,
Θ
n
k
x
n
-
Θ
n
k
-
1
x
n
〉
≥
0
.
By (A2), we have
(96)
φ
k
(
y
)
-
φ
k
(
Θ
n
k
x
n
)
+
〈
A
k
Θ
n
k
-
1
x
n
,
y
-
Θ
n
k
x
n
〉
+
1
r
k
,
n
〈
y
-
Θ
n
k
x
n
,
Θ
n
k
x
n
-
Θ
n
k
-
1
x
n
〉
≥
F
k
(
y
,
Θ
n
k
x
n
)
.
Let z
t
=
t
y
+
(
1
-
t
)
w, for all t
∈
(
0,1
] and y
∈
C. This implies that z
t
∈
C. Then, we have
(97)
〈
z
t
-
Θ
n
k
x
n
,
A
k
z
t
〉
≥
φ
k
(
Θ
n
k
x
n
)
-
φ
k
(
z
t
)
+
〈
z
t
-
Θ
n
k
x
n
,
A
k
z
t
〉
-
〈
z
t
-
Θ
n
k
x
n
,
A
k
Θ
n
k
-
1
x
n
〉
-
〈
z
t
-
Θ
n
k
x
n
,
Θ
n
k
x
n
-
Θ
n
k
-
1
x
n
r
k
,
n
〉
+
F
k
(
z
t
,
Θ
n
k
x
n
)
=
φ
k
(
Θ
n
k
x
n
)
-
φ
k
(
z
t
)
+
〈
z
t
-
Θ
n
k
x
n
,
A
k
z
t
-
A
k
Θ
n
k
x
n
〉
+
〈
z
t
-
Θ
n
k
x
n
,
A
k
Θ
n
k
x
n
-
A
k
Θ
n
k
-
1
x
n
〉
-
〈
z
t
-
Θ
n
k
x
n
,
Θ
n
k
x
n
-
Θ
n
k
-
1
x
n
r
k
,
n
〉
+
F
k
(
z
t
,
Θ
n
k
x
n
)
.
By (58), we have ∥
A
k
Θ
n
k
x
n
-
A
k
Θ
n
k
-
1
x
n
∥
→
0 as n
→
∞. Furthermore, by the monotonicity of A
k, we obtain 〈
z
t
-
Θ
n
k
x
n
,
A
k
z
t
-
A
k
Θ
n
k
x
n
〉
≥
0. Then, by (A4), we obtain
(98)
〈
z
t
-
w
,
A
k
z
t
〉
≥
φ
k
(
w
)
-
φ
k
(
z
t
)
+
F
k
(
z
t
,
w
)
.
Utilizing (A1), (A4), and (98), we obtain
(99)
0
=
F
k
(
z
t
,
z
t
)
+
φ
k
(
z
t
)
-
φ
k
(
z
t
)
≤
t
F
k
(
z
t
,
y
)
+
(
1
-
t
)
F
k
(
z
t
,
w
)
+
t
φ
k
(
y
)
+
(
1
-
t
)
φ
k
(
w
)
-
φ
k
(
z
t
)
≤
t
[
F
k
(
z
t
,
y
)
+
φ
k
(
y
)
-
φ
k
(
z
t
)
]
+
(
1
-
t
)
〈
z
t
-
w
,
A
k
z
t
〉
=
t
[
F
k
(
z
t
,
y
)
+
φ
k
(
y
)
-
φ
k
(
z
t
)
]
+
(
1
-
t
)
t
〈
y
-
w
,
A
k
z
t
〉
,
and, hence,
(100)
0
≤
F
k
(
z
t
,
y
)
+
φ
k
(
y
)
-
φ
k
(
z
t
)
+
(
1
-
t
)
〈
y
-
w
,
A
k
z
t
〉
.
Letting t
→
0, we have, for each y
∈
C,
(101)
0
≤
F
k
(
w
,
y
)
+
φ
k
(
y
)
-
φ
k
(
w
)
+
〈
y
-
w
,
A
k
w
〉
.
This implies that w
∈
GMEP
⁡
(
F
k
,
φ
k
,
A
k
) and, hence, w
∈
⋂
k
=
1
M
GMEP
⁡
(
F
k
,
φ
k
,
A
k
). Consequently, w
∈
F
=
⋂
n
=
1
∞
Fix
⁡
(
T
n
)
∩
⋂
k
=
1
M
GMEP
⁡
(
F
k
,
φ
k
,
A
k
)
∩
⋂
m
=
1
N
I
(
B
m
,
R
m
)
∩
Fix
⁡
(
S
). This shows that ω
w
(
{
x
n
}
)
⊂
F. From (45) and Lemma 16, we infer that x
n
→
v
=
P
F
x
0 as n
→
∞. This completes the proof.


## 4. Weak Convergence Theorem

In this section, we prove a weak convergence theorem for an iterative algorithm for finding a common element of the set of solutions of the set of solutions of finite generalized mixed equilibrium problems, the set of solutions of finite variational inclusions for maximal monotone and inverse strong monotone mappings, and the set of common fixed points of infinite nonexpansive mappings and asymptoticallyκ-strict pseudocontractive mapping S
:
C
→
C in the intermediate sense in a real Hilbert space. This iterative algorithm is based on the extragradient method and Mann-type iterative method.Theorem 25.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M
,
N be two integers. Let F
k be a bifunction from C
×
C to R satisfying (A1)–(A4) and let φ
k
:
C
→
R
∪
{
+
∞
} be a proper lower semicontinuous and convex function, where k
∈
{
1,2
,
…
,
M
}. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
k
:
H
→
H and B
i
:
C
→
H be μ
k-inverse strongly monotone and η
i-inverse strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
}, i
∈
{
1,2
,
…
,
N
}. Let S
:
C
→
C be a uniformly continuous asymptotically κ-strict pseudocontractive mapping in the intermediate sense, for some 0
≤
κ
<
1, with sequences {
γ
n
}
⊂
[
0
,
∞
) and {
c
n
}
⊂
[
0
,
∞
). Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive self-mappings on C and {
λ
n
} be a sequence in (
0
,
b
], for some b
∈
(
0,1
). Assume that F
:
=
⋂
n
=
1
∞
Fix
⁡
(
T
n
)
∩
⋂
k
=
1
M
GMEP
⁡
(
F
k
,
φ
k
,
A
k
)
∩
⋂
i
=
1
N
I
(
B
i
,
R
i
)
∩
Fix
⁡
(
S
) is nonempty and that either (B1) or (B2) holds. Let {
α
n
}
,
{
β
n
}, and {
δ
n
} be sequences in [
0,1
] such that α
n
+
β
n
≤
1,0
<
a
≤
α
n
≤
1 and κ
≤
δ
n
≤
d
<
1. Pick any x
1
∈
H and let {
x
n
} be a sequence generated by the following algorithm:
(102)
u
n
=
T
r
M
,
n
(
F
M
,
φ
M
)
(
I
-
r
M
,
n
A
M
)
T
r
M
-
1
,
n
(
F
M
-
1
,
φ
M
-
1
)
×
(
I
-
r
M
-
1
,
n
A
M
-
1
)
⋯
T
r
1
,
n
(
F
1
,
φ
1
)
(
I
-
r
1
,
n
A
1
)
x
n
,
z
n
=
J
R
N
,
λ
N
,
n
(
I
-
λ
N
,
n
B
N
)
J
R
N
-
1
,
λ
N
-
1
,
n
×
(
I
-
λ
N
-
1
,
n
B
N
-
1
)
⋯
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
u
n
,
k
n
=
δ
n
z
n
+
(
1
-
δ
n
)
S
n
z
n
,
y
n
=
(
1
-
α
n
-
β
n
)
x
n
+
α
n
k
n
+
β
n
W
n
z
n
,
where W
n is the W-mapping generated by (2.2). Assume that the following conditions hold:(i)
∑
n
=
1
∞
‍
γ
n
<
∞ and ∑
n
=
1
∞
‍
c
n
<
∞;
(ii)
0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
α
n
+
β
n
)
<
1;
(iii)
{
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
), ∀
i
∈
{
1,2
,
…
,
N
};
(iv)
{
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
), ∀
k
∈
{
1,2
,
…
,
M
}.Then, {
x
n
} converges weakly to w
=
lim
⁡
n
→
∞
P
F
x
n.Proof.
First, let us show thatlim
⁡
n
→
∞
∥
x
n
-
p
∥ exists, for any p
∈
F. Put
(103)
Θ
n
k
=
T
r
k
,
n
(
F
k
,
φ
k
)
(
I
-
r
k
,
n
A
k
)
T
r
k
-
1
,
n
(
F
k
-
1
,
φ
k
-
1
)
×
(
I
-
r
k
-
1
,
n
A
k
-
1
)
⋯
T
r
1
,
n
(
F
1
,
φ
1
)
(
I
-
r
1
,
n
A
1
)
x
n
,
for all k
∈
{
1,2
,
…
,
M
}, n
≥
1;
(104)
Ω
n
i
=
J
R
i
,
λ
i
,
n
(
I
-
λ
i
,
n
B
i
)
J
R
i
-
1
,
λ
i
-
1
,
n
×
(
I
-
λ
i
-
1
,
n
B
i
-
1
)
⋯
J
R
1
,
λ
1
,
n
(
I
-
λ
1
,
n
B
1
)
,
for all i
∈
{
1,2
,
…
,
N
}, n
≥
1, Θ
n
0
=
Ω
n
0
=
I, where I is the identity mapping on H. Then, we have that u
n
=
Θ
n
M
x
n and z
n
=
Ω
n
N
u
n. Take p
∈
F arbitrarily. Similar to the proof of Theorem 24, we obtain that
(105)
∥
u
n
-
p
∥
≤
∥
x
n
-
p
∥
,
(106)
∥
z
n
-
p
∥
≤
∥
u
n
-
p
∥
,
(107)
∥
Θ
n
k
x
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
2
,
k
∈
{
1,2
,
…
,
M
}
,
(108)
∥
Θ
n
k
x
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
×
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
,
k
=
1,2
,
…
,
M
,
(109)
∥
Ω
n
i
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
2
,
i
∈
{
1,2
,
…
,
N
}
,
(110)
∥
Ω
n
i
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
×
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
,
i
∈
{
1,2
,
…
,
N
}
.
We observe that
(111)
∥
k
n
-
p
∥
2
=
∥
δ
n
(
z
n
-
p
)
+
(
1
-
δ
n
)
(
S
n
z
n
-
p
)
∥
2
=
δ
n
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
∥
S
n
z
n
-
p
∥
2
-
δ
n
(
1
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
≤
δ
n
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
×
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
κ
∥
z
n
-
S
n
z
n
∥
2
+
c
n
]
-
δ
n
(
1
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
=
[
1
+
γ
n
(
1
-
δ
n
)
]
∥
z
n
-
p
∥
2
+
(
1
-
δ
n
)
(
κ
-
δ
n
)
∥
z
n
-
S
n
z
n
∥
2
+
(
1
-
δ
n
)
c
n
≤
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
.
It follows from (105), (106), and (111) that
(112)
∥
x
n
+
1
-
p
∥
2
=
∥
(
1
-
α
n
-
β
n
)
(
x
n
-
p
)
+
α
n
(
k
n
-
p
)
+
β
n
(
W
n
z
n
-
p
)
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
+
β
n
∥
W
n
z
n
-
p
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
]
+
β
n
∥
z
n
-
p
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
]
+
β
n
∥
x
n
-
p
∥
2
=
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
]
=
(
1
+
α
n
γ
n
)
∥
x
n
-
p
∥
2
+
α
n
c
n
≤
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
.
By Lemma 13 and condition (i), we deduce that lim
⁡
n
→
∞
∥
x
n
-
p
∥ exists. Hence, {
x
n
} is bounded and so are {
u
n
}, {
z
n
}, and {
k
n
}.
In addition, by Lemma7(b), we obtain from (105), (106), and (111) that
(113)
∥
x
n
+
1
-
p
∥
2
=
∥
(
1
-
α
n
-
β
n
)
(
x
n
-
p
)
+
α
n
(
k
n
-
p
)
+
β
n
(
W
n
z
n
-
p
)
∥
2
=
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
(
α
n
+
β
n
)
∥
α
n
(
k
n
-
p
)
+
β
n
(
W
n
z
n
-
p
)
α
n
+
β
n
∥
2
-
(
α
n
+
β
n
)
(
1
-
α
n
-
β
n
)
∥
α
n
(
k
n
-
x
n
)
+
β
n
(
W
n
z
n
-
x
n
)
α
n
+
β
n
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
(
α
n
+
β
n
)
[
α
n
α
n
+
β
n
∥
k
n
-
p
∥
2
+
β
n
α
n
+
β
n
∥
W
n
z
n
-
p
∥
2
]
-
(
α
n
+
β
n
)
(
1
-
α
n
-
β
n
)
∥
x
n
+
1
-
x
n
α
n
+
β
n
∥
2
=
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
+
β
n
∥
W
n
z
n
-
p
∥
2
-
1
-
α
n
-
β
n
α
n
+
β
n
∥
x
n
+
1
-
x
n
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
]
+
β
n
∥
z
n
-
p
∥
2
-
1
-
α
n
-
β
n
α
n
+
β
n
∥
x
n
+
1
-
x
n
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
]
+
β
n
∥
x
n
-
p
∥
2
-
1
-
α
n
-
β
n
α
n
+
β
n
∥
x
n
+
1
-
x
n
∥
2
=
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
]
-
1
-
α
n
-
β
n
α
n
+
β
n
∥
x
n
+
1
-
x
n
∥
2
=
(
1
+
α
n
γ
n
)
∥
x
n
-
p
∥
2
+
α
n
c
n
-
1
-
α
n
-
β
n
α
n
+
β
n
∥
x
n
+
1
-
x
n
∥
2
≤
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
-
(
1
-
α
n
-
β
n
)
∥
x
n
+
1
-
x
n
∥
2
,
which immediately yields
(114)
(
1
-
α
n
-
β
n
)
∥
x
n
+
1
-
x
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
γ
n
∥
x
n
-
p
∥
2
+
c
n
.
From lim
⁡
n
→
∞
γ
n
=
0, lim
⁡
n
→
∞
c
n
=
0, limsup
⁡
n
→
∞
(
α
n
+
β
n
)
<
1, and the existence of lim
⁡
n
→
∞
∥
x
n
-
p
∥, it follows that
(115)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
Again, utilizing Lemma 7(b), we obtain from (105), (106), and (111) that
(116)
∥
x
n
+
1
-
p
∥
2
=
∥
(
1
-
α
n
-
β
n
)
(
x
n
-
p
)
+
α
n
(
k
n
-
p
)
+
β
n
(
W
n
z
n
-
p
)
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
∥
k
n
-
p
∥
2
+
β
n
∥
W
n
z
n
-
p
∥
2
-
β
n
(
1
-
α
n
-
β
n
)
∥
W
n
z
n
-
x
n
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
z
n
-
p
∥
2
+
c
n
]
+
β
n
∥
z
n
-
p
∥
2
-
β
n
(
1
-
α
n
-
β
n
)
∥
W
n
z
n
-
x
n
∥
2
≤
(
1
-
α
n
-
β
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
]
+
β
n
∥
x
n
-
p
∥
2
-
β
n
(
1
-
α
n
-
β
n
)
∥
W
n
z
n
-
x
n
∥
2
=
(
1
-
α
n
)
∥
x
n
-
p
∥
2
+
α
n
[
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
]
-
β
n
(
1
-
α
n
-
β
n
)
∥
W
n
z
n
-
x
n
∥
2
=
(
1
+
α
n
γ
n
)
∥
x
n
-
p
∥
2
+
α
n
c
n
-
β
n
(
1
-
α
n
-
β
n
)
∥
W
n
z
n
-
x
n
∥
2
≤
(
1
+
γ
n
)
∥
x
n
-
p
∥
2
+
c
n
-
β
n
(
1
-
α
n
-
β
n
)
∥
W
n
z
n
-
x
n
∥
2
,
which leads to
(117)
β
n
(
1
-
α
n
-
β
n
)
∥
W
n
z
n
-
x
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
γ
n
∥
x
n
-
p
∥
2
+
c
n
.
From lim
⁡
n
→
∞
γ
n
=
0, lim
⁡
n
→
∞
c
n
=
0, condition (ii), and the existence of lim
⁡
n
→
∞
∥
x
n
-
p
∥, it follows that
(118)
lim
⁡
n
→
∞
∥
W
n
z
n
-
x
n
∥
=
0
.
Note that
(119)
x
n
+
1
-
x
n
=
α
n
(
k
n
-
x
n
)
+
β
n
(
W
n
z
n
-
x
n
)
.
Hence, it is easy to see from 0
<
a
≤
α
n
≤
1 that
(120)
a
∥
x
n
-
k
n
∥
≤
α
n
∥
k
n
-
x
n
∥
≤
∥
x
n
+
1
-
x
n
-
β
n
(
W
n
z
n
-
x
n
)
∥
≤
∥
x
n
+
1
-
x
n
∥
+
β
n
∥
W
n
z
n
-
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
∥
W
n
z
n
-
x
n
∥
.
From (115) and (118), it follows that
(121)
lim
⁡
n
→
∞
∥
k
n
-
x
n
∥
=
0
.
Combining (107) and (111), we have
(122)
∥
k
n
-
p
∥
2
≤
∥
z
n
-
p
∥
2
+
γ
n
∥
z
n
-
p
∥
2
+
c
n
≤
∥
u
n
-
p
∥
2
+
γ
n
∥
z
n
-
p
∥
2
+
c
n
≤
∥
Θ
n
k
x
n
-
p
∥
2
+
γ
n
M
1
+
c
n
≤
∥
x
n
-
p
∥
2
+
r
k
,
n
(
r
k
,
n
-
2
μ
k
)
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
2
+
γ
n
M
1
+
c
n
,
where M
1
=
sup
⁡
n
≥
1
∥
z
n
-
p
∥
2, which implies
(123)
r
k
,
n
(
2
μ
k
-
r
k
,
n
)
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
k
n
-
p
∥
2
+
γ
n
M
1
+
c
n
≤
∥
x
n
-
k
n
∥
(
∥
x
n
-
p
∥
+
∥
k
n
-
p
∥
)
+
γ
n
M
1
+
c
n
.
From {
r
k
,
n
}
⊂
[
e
k
,
f
k
]
⊂
(
0,2
μ
k
), k
∈
{
1,2
,
…
,
M
}, lim
⁡
n
→
∞
γ
n
=
0, lim
⁡
n
→
∞
c
n
=
0, and (121), we have
(124)
lim
⁡
n
→
∞
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
=
0
,
k
=
1,2
,
…
,
M
.
Combining (108) and (111), we have
(125)
∥
k
n
-
p
∥
2
≤
∥
Θ
n
k
x
n
-
p
∥
2
+
γ
n
M
1
+
c
n
≤
∥
x
n
-
p
∥
2
-
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
2
+
2
r
k
,
n
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
+
γ
n
M
1
+
c
n
,
which implies
(126)
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
k
n
-
p
∥
2
+
2
r
k
,
n
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
+
γ
n
M
1
+
c
n
≤
∥
x
n
-
k
n
∥
(
∥
x
n
-
p
∥
+
∥
k
n
-
p
∥
)
+
2
r
k
,
n
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
∥
A
k
Θ
n
k
-
1
x
n
-
A
k
p
∥
+
γ
n
M
1
+
c
n
.
From lim
⁡
n
→
∞
γ
n
=
0
,
lim
⁡
n
→
∞
c
n
=
0, (121), and (124), we get
(127)
lim
⁡
n
→
∞
∥
Θ
n
k
-
1
x
n
-
Θ
n
k
x
n
∥
=
0
,
k
=
1,2
,
…
,
M
.
From (127), we have
(128)
∥
x
n
-
u
n
∥
=
∥
Θ
n
0
x
n
-
Θ
n
M
x
n
∥
≤
∥
Θ
n
0
x
n
-
Θ
n
1
x
n
∥
+
∥
Θ
n
1
x
n
-
Θ
n
2
x
n
∥
+
⋯
+
∥
Θ
n
M
-
1
x
n
-
Θ
n
M
x
n
∥
⟶
as
n
⟶
∞
.
Combining (109) and (111), we obtain
(129)
∥
k
n
-
p
∥
2
≤
∥
z
n
-
p
∥
2
+
γ
n
∥
z
n
-
p
∥
2
+
c
n
≤
∥
Ω
n
i
u
n
-
p
∥
2
+
γ
n
M
1
+
c
n
≤
∥
x
n
-
p
∥
2
+
λ
i
,
n
(
λ
i
,
n
-
2
η
i
)
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
2
+
γ
n
M
1
+
c
n
,
where i
∈
{
1,2
,
…
,
N
}, which implies
(130)
λ
i
,
n
(
2
η
i
-
λ
i
,
n
)
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
k
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
p
-
t
n
∥
+
γ
n
∥
z
n
-
p
∥
2
+
c
n
≤
∥
x
n
-
k
n
∥
(
∥
x
n
-
p
∥
+
∥
k
n
-
p
∥
)
+
γ
n
M
1
+
c
n
.
From {
λ
i
,
n
}
⊂
[
a
i
,
b
i
]
⊂
(
0,2
η
i
), i
∈
{
1,2
,
…
,
N
}, lim
⁡
n
→
∞
γ
n
=
0, lim
⁡
n
→
∞
c
n
=
0, and (121), we get
(131)
lim
⁡
n
→
∞
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
=
0
,
i
∈
{
1,2
,
…
,
N
}
.
Combining (110) and (111), we have
(132)
∥
k
n
-
p
∥
2
≤
∥
z
n
-
p
∥
2
+
γ
n
∥
z
n
-
p
∥
2
+
c
n
≤
∥
Ω
n
i
u
n
-
p
∥
2
+
γ
n
M
1
+
c
n
≤
∥
x
n
-
p
∥
2
-
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
2
+
2
λ
i
,
n
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
+
γ
n
M
1
+
c
n
,
which implies
(133)
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
k
n
-
p
∥
2
+
2
λ
i
,
n
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
+
γ
n
M
1
+
c
n
≤
∥
x
n
-
k
n
∥
(
∥
x
n
-
p
∥
+
∥
k
n
-
p
∥
)
+
2
λ
i
,
n
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
∥
B
i
Ω
n
i
-
1
u
n
-
B
i
p
∥
+
γ
n
M
1
+
c
n
.
From lim
⁡
n
→
∞
γ
n
=
0, lim
⁡
n
→
∞
c
n
=
0, (121), and (131), we obtain
(134)
lim
⁡
n
→
∞
∥
Ω
n
i
-
1
u
n
-
Ω
n
i
u
n
∥
=
0
,
i
∈
{
1,2
,
…
,
N
}
.
By (134), we have
(135)
∥
u
n
-
z
n
∥
=
∥
Ω
n
0
u
n
-
Ω
n
N
u
n
∥
≤
∥
Ω
n
0
u
n
-
Ω
n
1
u
n
∥
+
∥
Ω
n
1
u
n
-
Ω
n
2
u
n
∥
+
⋯
+
∥
Ω
n
N
-
1
u
n
-
Ω
n
N
u
n
∥
⟶
0
as
n
⟶
∞
.
From (128) and (135), we have
(136)
∥
x
n
-
z
n
∥
≤
∥
x
n
-
u
n
∥
+
∥
u
n
-
z
n
∥
⟶
0
as
n
⟶
∞
.
By (121) and (136), we obtain
(137)
∥
k
n
-
z
n
∥
≤
∥
k
n
-
x
n
∥
+
∥
x
n
-
z
n
∥
⟶
0
as
n
⟶
∞
.
We note that
(138)
k
n
-
z
n
=
(
1
-
δ
n
)
(
S
n
z
n
-
z
n
)
.
From δ
n
≤
d
<
1 and (137), we have
(139)
lim
⁡
n
→
∞
∥
S
n
z
n
-
z
n
∥
=
0
.
On the other hand, we observe that
(140)
∥
z
n
+
1
-
z
n
∥
≤
∥
z
n
+
1
-
x
n
+
1
∥
+
∥
x
n
+
1
-
x
n
∥
+
∥
x
n
-
z
n
∥
.
By (115) and (136), we have
(141)
lim
⁡
n
→
∞
∥
z
n
+
1
-
z
n
∥
=
0
.
We note that
(142)
∥
z
n
-
S
z
n
∥
≤
∥
z
n
-
z
n
+
1
∥
+
∥
z
n
+
1
-
S
n
+
1
z
n
+
1
∥
+
∥
S
n
+
1
z
n
+
1
-
S
n
+
1
z
n
∥
+
∥
S
n
+
1
z
n
-
S
z
n
∥
.
From (139), (141), Lemma 9, and the uniform continuity of S, we obtain
(143)
lim
⁡
n
→
∞
∥
z
n
-
S
z
n
∥
=
0
.
In addition, note that
(144)
∥
z
n
-
W
z
n
∥
≤
∥
z
n
-
W
n
z
n
∥
+
∥
W
n
z
n
-
W
z
n
∥
≤
∥
z
n
-
x
n
∥
+
∥
x
n
-
W
n
z
n
∥
+
∥
W
n
z
n
-
W
z
n
∥
.
So, from (118), (136), and [4, Remark 3.2], it follows that
(145)
lim
⁡
n
→
∞
∥
z
n
-
W
z
n
∥
=
0
.
Since{
x
n
} is bounded, there exists a subsequence {
x
n
i
} of {
x
n
} which converges weakly to w. From (136), we have that z
n
i
⇀
w. From (143) and the uniform continuity of S, we have lim
⁡
n
→
∞
∥
z
n
-
S
m
z
n
∥
=
0, for any m
≥
1. So, from Lemma 11, we have w
∈
Fix
⁡
(
S
). Utilizing the similar arguments to those in the proof of Theorem 24, we can derive w
∈
⋂
k
=
1
M
GMEP
⁡
(
F
k
,
φ
k
,
A
k
)
∩
⋂
i
=
1
N
I
(
B
i
,
R
i
)
∩
Fix
⁡
(
W
). Consequently, w
∈
F. This shows that ω
w
(
x
n
)
⊂
F.
Next, let us show thatω
w
(
x
n
) is a single-point set. As a matter of fact, let {
x
n
j
} be another subsequence of {
x
n
} such that x
n
j
⇀
w
′. Then, we get w
′
∈
F. If w
≠
w
′, from the Opial condition, we have
(146)
lim
⁡
n
→
∞
∥
x
n
-
w
∥
=
lim
⁡
i
→
∞
∥
x
n
i
-
w
∥
<
lim
⁡
i
→
∞
∥
x
n
i
-
w
′
∥
=
lim
⁡
n
→
∞
∥
x
n
-
w
′
∥
=
lim
⁡
j
→
∞
∥
x
n
j
-
w
′
∥
<
lim
⁡
j
→
∞
∥
x
n
j
-
w
∥
=
lim
⁡
n
→
∞
∥
x
n
-
w
∥
.
This attains a contradiction. So we have w
=
w
′. Put v
n
=
P
F
(
x
n
). Since w
∈
F, we have 〈
x
n
-
v
n
,
v
n
-
w
〉
≥
0. By Lemma 15, we have that {
v
n
} converges strongly to some w
0
∈
F. Since {
x
n
} converges weakly to w, we have
(147)
〈
w
-
w
0
,
w
0
-
w
〉
≥
0
.
Therefore, we obtain w
=
w
0
=
lim
⁡
n
→
∞
P
F
x
n. This completes the proof.



---

*Source: 251065-2014-06-12.xml*