# Relaxed Iterative Algorithms for Generalized Mixed Equilibrium Problems with Constraints of Variational Inequalities and Variational Inclusions

Authors: Lu-Chuan Ceng; Chi-Ming Chen; Ching-Feng Wen; Chin-Tzong Pang
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We introduce and analyze a relaxed extragradient-like viscosity iterative algorithm for finding a solution of a generalized mixed equilibrium problem with constraints of several problems: a finite family of variational inequalities for inverse strongly monotone mappings, a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings, and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we derive the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems which also solves a variational inequality problem.


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## Body

## 1. Introduction

LetH be a real Hilbert space with inner product 〈
·
,
·
〉 and norm ∥
·
∥, C a nonempty closed convex subset of H, and P
C the metric projection of H onto C. Let S
:
C
→
H be a nonlinear mapping on C. We denote by Fix
⁡
(
S
) the set of fixed points of S and by R the set of all real numbers. A mapping V is called strongly positive on H if there exists a constant γ
¯
>
0 such that
(1)
〈
V
x
,
x
〉
≥
γ
¯
∥
x
∥
2
,
∀
x
∈
H
.
A mapping S
:
C
→
H is called L-Lipschitz continuous if there exists a constant L
≥
0 such that
(2)
∥
S
x
-
S
y
∥
≤
L
∥
x
-
y
∥
,
∀
x
,
y
∈
C
.
In particular, if L
=
1, then S is called a nonexpansive mapping; if L
∈
[
0,1
), then A is called a contraction.LetA
:
C
→
H be a nonlinear mapping on C. We consider the following variational inequality problem (VIP) [1]: find a point x
∈
C such that
(3)
〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
The solution set of VIP (3) is denoted by VI
⁡
(
C
,
A
).In 1976, Korpelevič [2] proposed an iterative algorithm for solving the VIP (3) in Euclidean space R
n:
(4)
y
n
=
P
C
(
x
n
-
τ
A
x
n
)
,
x
n
+
1
=
P
C
(
x
n
-
τ
A
y
n
)
,
∀
n
≥
0
,
with τ
>
0 a given number, which is known as the extragradient method. The literature on the VIP is vast and Korpelevich’s extragradient method has received great attention given by many authors. See, for example, [3–10] and references therein.Letφ
:
C
→
R be a real-valued function, A
:
H
→
H a nonlinear mapping, and Θ
:
C
×
C
→
R a bifunction. In 2008, Peng and Yao [11] introduced the following generalized mixed equilibrium problem (GMEP): finding  x
∈
C such that
(5)
Θ
(
x
,
y
)
+
φ
(
y
)
-
φ
(
x
)
+
〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C
.
We denote the set of solutions of GMEP
⁡ (5) by GMEP
⁡
(
Θ
,
φ
,
A
).Throughout this paper, it is assumed as in [11] that Θ
:
C
×
C
→
R is a bifunction satisfying conditions (H1)–(H4) and φ
:
C
→
R is a lower semicontinuous and convex function with restriction (H5), where(H1)
Θ
(
x
,
x
)
=
0 for all x
∈
C,
(H2)
Θ is monotone, that is, Θ
(
x
,
y
)
+
Θ
(
y
,
x
)
≤
0 for any x
,
y
∈
C,
(H3)
Θ is upper hemicontinuous, that is, for each x
,
y
,
z
∈
C,
(6)
limsup
⁡
t
→
0
+
⁡
Θ
(
t
z
+
(
1
-
t
)
x
,
y
)
≤
Θ
(
x
,
y
)
;
(H4)
Θ
(
x
,
·
) is convex and lower semicontinuous for each x
∈
C,
(H5)
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,
(7)
Θ
(
z
,
y
x
)
+
φ
(
y
x
)
-
φ
(
z
)
+
1
r
〈
y
x
-
z
,
z
-
x
〉
<
0
,given a positive number r
>
0. Let S
r
(
Θ
,
φ
)
:
H
→
C be the solution set of the auxiliary mixed equilibrium problem; that is, for each x
∈
H,
(8)
S
r
(
Θ
,
φ
)
(
x
)
:
=
{
1
r
y
∈
C
:
Θ
(
y
,
z
)
+
φ
(
z
)
-
φ
(
y
)
f
f
f
f
f
f
f
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
-
y
〉
≥
0
,
∀
z
∈
C
}
.
In particular, whenever K
(
x
)
=
(
1
/
2
)
∥
x
∥
2
,
for  all
x
∈
H, S
r
(
Θ
,
φ
) is rewritten as T
r
(
Θ
,
φ
).Let{
T
n
}
n
=
1
∞ be an infinite family of nonexpansive mappings on H and {
λ
n
}
n
=
1
∞ a sequence of nonnegative numbers in [
0,1
]. For any n
≥
1, define a mapping W
n on H as follows:
(9)
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.In 2010, for the case whereC
=
H, Yao et al. [12] proposed the following hybrid iterative algorithm:
(10)
Θ
(
y
n
,
z
)
+
φ
(
z
)
-
φ
(
y
n
)
+
1
r
〈
K
′
(
y
n
)
-
K
′
(
x
n
)
,
z
-
y
n
〉
≥
0
,
z
∈
H
,
x
n
+
1
=
α
n
(
u
+
γ
f
(
x
n
)
)
+
β
n
x
n
+
(
(
1
-
β
n
)
I
-
α
n
(
I
+
μ
V
)
)
W
n
y
n
,
∀
n
≥
1
,
where f
:
H
→
H is a contraction, K
:
H
→
R is differentiable and strongly convex, {
α
n
}, {
β
n
}
⊂
(
0,1
), and x
1, u
∈
H are given, for finding a common element of the set MEP
⁡
(
Θ
,
φ
) and the fixed point set ∩
n
=
1
∞
Fix
⁡
(
T
n
) of an infinite family of nonexpansive mappings {
T
n
}
n
=
1
∞ on H. They proved the strong convergence of the sequence generated by the hybrid iterative algorithm (10) to a point x
*
∈
Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
MEP
⁡
(
Θ
,
φ
) under some appropriate conditions. This point x
* also solves the following optimization problem:
(
OP
)
min
⁡
x
∈
Ω
μ
2
〈
V
x
,
x
〉
+
1
2
∥
x
-
u
∥
2
-
h
(
x
)
,
where h
:
H
→
R is the potential function of γ
f.On the other hand, letB be a single-valued mapping of C into H and R a set-valued mapping with D
(
R
)
=
C. Consider the following variational inclusion: find a point x
∈
C such that
(11)
0
∈
B
x
+
R
x
.
We denote by I
(
B
,
R
) the solution set of the variational inclusion (11). In particular, if B
=
R
=
0, then I
(
B
,
R
)
=
C. If B
=
0, then problem (11) becomes the inclusion problem introduced by Rockafellar [13]. Let a set-valued mapping R
:
D
(
R
)
⊂
H
→
2
H be maximal monotone. We define the resolvent operator J
R
,
λ
:
H
→
D
(
R
)
¯ associated with R and λ as follows:
(12)
J
R
,
λ
=
(
I
+
λ
R
)
-
1
,
∀
x
∈
H
,
where λ is a positive number.In 1998, Huang [14] studied problem (11) 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. [15] further studied problem (11) in the case which is more general than Huang’s one [14].Inspired by the above facts, we introduce and analyze an iterative algorithm by relaxed extragradient-like viscosity method for finding a solution of a generalized mixed equilibrium problem with constraints of several problems: a finite family of variational inequalities for inverse strongly monotone mappings, a finite family of variational inclusions for maximal monotone and inverse strongly monotone mappings, and a fixed point problem of infinitely many nonexpansive mappings in a real Hilbert space. Under some suitable conditions, we derive the strong convergence of the sequence generated by the proposed algorithm to a common solution of these problems. Such solution also solves a variational inequality problem. Several special cases are also discussed. The results presented in this paper are the supplement, extension, improvement, and generalization of the previously known results in this area.


## 2. Preliminaries

Throughout this paper, we assume thatH is a real Hilbert space whose inner product and norm are denoted by 〈
·
,
·
〉 and ∥
·
∥, respectively. Let C be a nonempty closed convex subset of H. We write 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,
(13)
ω
w
(
x
n
)


:
=
{
x
∈
H
:
x
n
i
⇀
x
for
some
subsequence
{
x
n
i
}
of
{
x
n
}
}


.Definition 1.
A mappingA
:
C
→
H is called(i)
monotone if(14)
〈
A
x
-
A
y
,
x
-
y
〉
≥
0
,
∀
x
,
y
∈
C
,
(ii)
η-strongly monotone if there exists a constant η
>
0 such that
(15)
〈
A
x
-
A
y
,
x
-
y
〉
≥
η
∥
x
-
y
∥
2
,
∀
x
,
y
∈
C
,
(iii)
ζ-inverse strongly monotone if there exists a constant ζ
>
0 such that
(16)
〈
A
x
-
A
y
,
x
-
y
〉
≥
ζ
∥
A
x
-
A
y
∥
2
,
∀
x
,
y
∈
C
.
It is easy to see that the projectionP
C is 1-ism. Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields.Definition 2.
A differentiable functionK
:
H
→
R is called(i)
convex, if(17)
K
(
y
)
-
K
(
x
)
≥
〈
K
′
(
x
)
,
y
-
x
〉
,
∀
x
,
y
∈
H
,
where K
′
(
x
) is the Frechet derivative of K at x;
(ii)
strongly convex, if there exists a constantσ
>
0 such that
(18)
K
(
y
)
-
K
(
x
)
-
〈
K
′
(
x
)
,
y
-
x
〉
≥
σ
2
∥
x
-
y
∥
2
,
∀
x
,
y
∈
H
.
It is easy to see that ifK
:
H
→
R is a differentiable strongly convex function with constant σ
>
0 then K
′
:
H
→
H is strongly monotone with constant σ
>
0.
The metric (or nearest point) 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 gathered in the following proposition.Proposition 3.
For givenx
∈
H and z
∈
C,(i)
z
=
P
C
x
⇔
〈
x
-
z
,
y
-
z
〉
≤
0
,
for
all
y
∈
C;
(ii)
z
=
P
C
x
⇔
∥
x
-
z
∥
2
≤
∥
x
-
y
∥
2
-
∥
y
-
z
∥
2
,
for
all
y
∈
C;
(iii)
〈
P
C
x
-
P
C
y
,
x
-
y
〉
≥
∥
P
C
x
-
P
C
y
∥
2,  for  all y
∈
H. (This implies that P
C is nonexpansive and monotone.)By using the technique of [16], we can readily obtain the following elementary result.Proposition 4 (see [17, Lemma 1 and Proposition 1]).
LetC be a nonempty closed convex subset of a real Hilbert space H and let φ
:
C
→
R be a lower semicontinuous and convex function. Let Θ
:
C
×
C
→
R be a bifunction satisfying the conditions (H1)–(H4). Assume that(i)
K
:
H
→
R is strongly convex with constant σ
>
0 and the function x
↦
〈
y
-
x
,
K
′
(
x
)
〉 is weakly upper semicontinuous for each y
∈
H;
(ii)
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,
(20)
Θ
(
z
,
y
x
)
+
φ
(
y
x
)
-
φ
(
z
)
+
1
r
〈
K
′
(
z
)
-
K
′
(
x
)
,
y
x
-
z
〉
<
0
.Then the following hold:(a)
for eachx
∈
H
,
S
r
(
Θ
,
φ
)
(
x
)
≠
∅,
(b)
S
r
(
Θ
,
φ
) is single valued,
(c)
S
r
(
Θ
,
φ
) is nonexpansive if K
′ is Lipschitz continuous with constant ν
>
0 and
(21)
〈
K
′
(
x
1
)
-
K
′
(
x
2
)
,
u
1
-
u
2
〉
≤
〈
K
′
(
u
1
)
-
K
′
(
u
2
)
,
u
1
-
u
2
〉
,
555555555555
555
∀
(
x
1
,
x
2
)
∈
H
×
H
,
where u
i
=
S
r
(
Θ
,
φ
)
(
x
i
) for i
=
1,2,
(d)
for alls
,
t
>
0 and x
∈
H
(22)
〈
K
′
(
S
s
(
Θ
,
φ
)
x
)
-
K
′
(
S
t
(
Θ
,
φ
)
x
)
,
S
s
(
Θ
,
φ
)
x
-
S
t
(
Θ
,
φ
)
x
〉
≤
s
-
t
s
〈
K
′
(
S
s
(
Θ
,
φ
)
x
)
-
K
′
(
x
)
,
S
s
(
Θ
,
φ
)
x
-
S
t
(
Θ
,
φ
)
x
〉
;
(e)
Fix
⁡
(
S
r
(
Θ
,
φ
)
)
=
MEP
⁡
(
Θ
,
φ
),
(f)
MEP
⁡
(
Θ
,
φ
) is closed and convex.In particular, whenever Θ
:
C
×
C
→
R is a bifunction satisfying the conditions (H1)–(H4) and K
(
x
)
=
(
1
/
2
)
∥
x
∥
2,  for  all x
∈
H, then that is, for any x
,
y
∈
H,
(23)
∥
S
r
(
Θ
,
φ
)
x
-
S
r
(
Θ
,
φ
)
y
∥
2
≤
〈
S
r
(
Θ
,
φ
)
x
-
S
r
(
Θ
,
φ
)
y
,
x
-
y
〉
(S
r
(
Θ
,
φ
) is firmly nonexpansive) and
(24)
∥
S
s
(
Θ
,
φ
)
x
-
S
t
(
Θ
,
φ
)
x
∥
≤
|
s
-
t
|
s
∥
S
s
(
Θ
,
φ
)
x
-
x
∥
,
∀
s
,
t
>
0
,
x
∈
H
.
In this case, S
r
(
Θ
,
φ
) is rewritten as T
r
(
Θ
,
φ
). If, in addition, φ
≡
0, then T
r
(
Θ
,
φ
) is rewritten as T
r
Θ.Remark 5.
SupposeK
:
H
→
R is strongly convex with constant σ
>
0 and K
′
:
H
→
H is Lipschitz continuous with constant ν
>
0. Then K
′
:
H
→
H is σ-strongly monotone and ν-Lipschitz continuous with positive constants σ
,
ν
>
0. Utilizing Proposition 4 (d) we obtain that, for all s
,
t
>
0 and x
∈
H,
(25)
∥
S
s
(
Θ
,
φ
)
x
-
S
t
(
Θ
,
φ
)
x
∥
≤
|
s
-
t
|
s
·
ν
σ
∥
S
s
(
Θ
,
φ
)
x
-
x
∥
.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:
(26)
∥
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
-
λ
ν
∥
x
-
z
∥
2
-
μ
ν
∥
y
-
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
(27)
limsup
⁡
n
→
∞
⁡
∥
x
n
-
y
∥
2
=
limsup
⁡
n
→
∞
∥
x
n
-
x
∥
2
+
∥
x
-
y
∥
2
,
∀
y
∈
H
.
We have the following crucial lemmas concerning theW-mappings defined by (9).Lemma 8 (see [18, Lemma 3.2]).
Let{
T
n
}
n
=
1
∞ be a sequence of nonexpansive self-mappings on H 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
∈
H and k
≥
1 the limit lim
⁡
n
→
∞
U
n
,
k
x exists, where U
n
,
k is defined by (9).Lemma 9 (see [18, Lemma 3.3]).
Let{
T
n
}
n
=
1
∞ be a sequence of nonexpansive self-mappings on H 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 10 (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. Here I is the identity operator of H.Lemma 11.
LetA
:
C
→
H be a monotone mapping. In the context of the variational inequality problem the characterization of the projection (see Proposition 3 (i)) implies
(28)
u
∈
VI
⁡
(
C
,
A
)
⟺
u
=
P
C
(
u
-
λ
A
u
)
,
λ
>
0
.Lemma 12 (see [20]).
Let{
x
n
} and {
z
n
} be bounded sequences in a real Banach space X and {
β
n
} a sequence in [
0,1
] with 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
β
n
<
1. Suppose
(29)
x
n
+
1
=
β
n
x
n
+
(
1
-
β
n
)
z
n
,
∀
n
≥
0
,
limsup
⁡
n
→
∞
(
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
0
.
Then, lim
⁡
n
→
∞
∥
z
n
-
x
n
∥
=
0.Lemma 13 (see [21]).
Assume that{
a
n
} is a sequence of nonnegative real numbers such that
(30)
a
n
+
1
≤
(
1
-
δ
n
)
a
n
+
σ
n
δ
n
,
∀
n
≥
1
,
where {
δ
n
} is a sequence in [
0,1
] and {
σ
n
} is a real sequence such that(i)
∑
n
=
1
∞
‍
δ
n
=
∞;
(ii)
limsup
⁡
n
→
∞
σ
n
≤
0 or ∑
n
=
1
∞
‍
|
σ
n
δ
n
|
<
∞.Then lim
⁡
n
→
∞
a
n
=
0.Recall that a set-valued mappingT
:
D
(
T
)
⊂
H
→
2
H is called monotone if, for all x
,
y
∈
D
(
T
), f
∈
T
x and g
∈
T
y imply
(31)
〈
f
-
g
,
x
-
y
〉
≥
0
.
A set-valued mapping T is called maximal monotone if  T is monotone and (
I
+
λ
T
)
D
(
T
)
=
H for each λ
>
0, where I is the identity mapping of H. We denote by G
(
T
) the graph of T. It is known that a monotone mapping T is maximal if and only if, for (
x
,
f
)
∈
H
×
H, 〈
f
-
g
,
x
-
y
〉
≥
0 for every (
y
,
g
)
∈
G
(
T
) implies f
∈
T
x. Next we provide an example to illustrate the concept of maximal monotone mapping.LetA
:
C
→
H be a monotone, k-Lipschitz-continuous mapping and let N
C
v be the normal cone to C at v
∈
C; that is,
(32)
N
C
v
=
{
w
∈
H
:
〈
v
-
u
,
w
〉
≥
0
,
∀
u
∈
C
}
.
Define
(33)
T
v
=
{
A
v
+
N
C
v
,
if
v
∈
C
,
∅
,
if
v
∉
C
.
Then, T is maximal monotone and 0
∈
T
v if and only if v
∈
VI
⁡
(
C
,
A
); see [13].Assume thatR
:
D
(
R
)
⊂
H
→
2
H is a maximal monotone mapping. Let λ
>
0. In terms of Huang [14], there holds the following property for the resolvent operator J
R
,
λ
:
H
→
D
(
R
)
¯.Lemma 14.
J
R
,
λ is single valued and firmly nonexpansive; that is,
(34)
〈
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 15 (see [9]).
LetR be a maximal monotone mapping with D
(
R
)
=
C. Then, for any given λ
>
0,  u
∈
C is a solution of problem (11) if and only if u
∈
C satisfies
(35)
u
=
J
R
,
λ
(
u
-
λ
B
u
)
.Lemma 16 (see [15]).
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 17 (see [9]).
LetR be a maximal monotone mapping with D
(
R
)
=
C and B
:
C
→
H 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.Lemma 18 (see [22, Lemma 2.8]).
Let{
a
n
}
n
=
1
∞ be a bounded sequence of nonnegative real numbers and {
b
n
}
n
=
1
∞ a sequence of real numbers such that limsup
⁡
n
→
∞
b
n
≤
0. Then, limsup
⁡
n
→
∞
a
n
b
n
≤
0.


## 3. Main Results

We will introduce and analyze an iterative algorithm by relaxed extragradient-like viscosity method for finding a solution of a generalized equilibrium problem with constraints of several problems: a finite family of variational inclusions, a finite family of variational inequalities, and a fixed point problem in a real Hilbert space. Under appropriate conditions imposed on the parameter sequences we will prove strong convergence of the proposed algorithm.Theorem 19.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M and N be two integers. Let Θ be a bifunction from C
×
C to R satisfying (H1)–(H4) and φ
:
C
→
R a lower semicontinuous and convex functional. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
:
H
→
H, A
k
:
C
→
H, and B
i
:
C
→
H be ζ-inverse strongly monotone, ζ
k-inverse strongly monotone, and η
i-inverse strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive mappings on H and {
λ
n
} a sequence in (
0
,
b
] for some b
∈
(
0,1
). Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
) and f
:
H
→
H a ρ-contraction with ρ
∈
(
0,1
). Let W
n be the W-mapping defined by (9). Assume that Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
∩
k
=
1
M
VI
⁡
(
C
,
A
k
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
≠
∅. Define Δ
k
=
P
C
(
I − ν
k
A
k
)
⋯
P
C
(
I
-
ν
1
A
1
) and Λ
i
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
⋯
J
R
1
,
μ
1
(
I
-
μ
1
B
1
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let {
α
n
}
,
{
β
n
}, and {
γ
n
} be three sequences in [
0,1
]. Assume that(i)
K
:
H
→
R is strongly convex with constant σ
>
0 and its derivative K
′ is Lipschitz continuous with constant ν
>
0 such that the function x
↦
〈
y
-
x
,
K
′
(
x
)
〉 is weakly upper semicontinuous for each y
∈
H,
(ii)
for eachx
∈
H, there exist a bounded subset D
x
⊂
C and z
x
∈
C such that, for any y
∉
D
x,
(36)
Θ
(
y
,
z
x
)
+
φ
(
z
x
)
-
φ
(
y
)
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
x
-
y
〉
<
0
;
(iii)
{
α
n
}
⊂
[
0,1
-
ρ
] and ∑
n
=
1
∞
‍
α
n
=
∞,
(iv)
li
inf
⁡
n
→
∞
γ
n
>
0 and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1,
(v)
lim
⁡
n
→
∞
(
|
α
n
+
1
/
(
1
-
(
1
-
α
n
+
1
)
β
n
+
1
)
-
α
n
/
(
1
-
(
1
-
α
n
)
β
n
)
| + |
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
|
)
=
0,
(vi)
{
r
n
}
⊂
[
0,2
ζ
]
,
lim
⁡
n
→
∞
|
r
n
+
1
-
r
n
|
=
0 and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ. Given x
1
∈
H arbitrarily, the sequence {
x
n
}  is generated iteratively by
(37)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
y
n
=
β
n
x
n
+
γ
n
W
n
Δ
M
u
n
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
W
n
Λ
N
u
n
,
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
,
∀
n
≥
1
,
where ν
k
∈
(
0,2
ζ
k
) and μ
i
∈
(
0,2
η
i
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. If α
n
(
f
(
x
n
)
-
x
n
)
→
0 and S
r
(
Θ
,
φ
) is firmly nonexpansive, then
(38)
x
n
⟶
x
*
⟺
limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
,
where x
* is a unique solution in Ω to the  VIP (39)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.Proof.
Asliminf
⁡
n
→
∞
γ
n
>
0, 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1, and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ, we may assume, without loss of generality, that {
β
n
}
,
{
γ
n
}
⊂
[
a
,
a
^
]
⊂
(
0,1
), {
r
n
}
⊂
[
c
,
c
^
]
⊂
(
0,2
ζ
) and β
n
+
γ
n
∥
V
∥
≤
1 for all n
≥
1. Put
(40)
Δ
k
=
P
C
(
I
-
ν
k
A
k
)
⋯
P
C
(
I
-
ν
1
A
1
)
,
Λ
i
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
⋯
J
R
1
,
μ
1
(
I
-
μ
1
B
1
)
,
for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}, and Δ
0
=
Λ
0
=
I, where I is the identity mapping on H. Moreover, set v
n
=
Δ
M
u
n and w
n
=
Λ
N
u
n.
SinceV is a γ
¯-strongly positive bounded linear operator on H, we know that
(41)
∥
V
∥
=
sup
⁡
{
〈
V
u
,
u
〉
:
u
∈
H
,
∥
u
∥
=
1
}
≥
γ
¯
>
1
.
Taking into account that β
n
+
γ
n
∥
V
∥
≤
1 for all n
≥
1, we have
(42)
〈
(
(
1
-
β
n
)
I
-
γ
n
V
)
u
,
u
〉
=
1
-
β
n
-
γ
n
〈
V
u
,
u
〉
≥
1
-
β
n
-
γ
n
∥
V
∥
≥
0
.
That is, (
1
-
β
n
)
I
-
γ
n
V is positive. It follows that
(43)
∥
(
1
-
β
n
)
I
-
γ
n
V
∥
=
sup
⁡
{
〈
(
(
1
-
β
n
)
I
-
γ
n
V
)
u
,
u
〉
:
u
∈
H
,
∥
u
∥
=
1
}
=
sup
⁡
{
1
-
β
n
-
γ
n
〈
V
u
,
u
〉
:
u
∈
H
,
∥
u
∥
=
1
}
≤
1
-
β
n
-
γ
n
γ
¯
.
In the meantime, it is not hard to find thatΔ
M and Λ
N are nonexpansive. As a matter of fact, observe that, for all x
,
y
∈
C,
(44)
∥
Δ
M
x
-
Δ
M
y
∥
2
=
∥
P
C
(
I
-
ν
M
A
M
)
Δ
M
-
1
x
-
P
C
(
I
-
ν
M
A
M
)
Δ
M
-
1
y
∥
2
≤
∥
(
I
-
ν
M
A
M
)
Δ
M
-
1
x
-
(
I
-
ν
M
A
M
)
Δ
M
-
1
y
∥
2
=
∥
(
Δ
M
-
1
x
-
Δ
M
-
1
y
)
-
ν
M
(
A
M
Δ
M
-
1
x
-
A
M
Δ
M
-
1
y
)
∥
2
≤
∥
Δ
M
-
1
x
-
Δ
M
-
1
y
∥
2
+
ν
M
(
ν
M
-
2
ζ
M
)
∥
A
M
Δ
M
-
1
x
-
A
M
Δ
M
-
1
y
∥
2
≤
∥
Δ
M
-
1
x
-
Δ
M
-
1
y
∥
2
⋮
≤
∥
Δ
0
x
-
Δ
0
y
∥
2
=
∥
x
-
y
∥
2
,
∥
Λ
N
x
-
Λ
N
y
∥
2
=
∥
J
R
N
,
μ
N
(
I
-
μ
N
B
N
)
Λ
N
-
1
x
-
J
R
N
,
μ
N
(
I
-
μ
N
B
N
)
Λ
N
-
1
y
∥
2
≤
∥
(
I
-
μ
N
B
N
)
Λ
N
-
1
x
-
(
I
-
μ
N
B
N
)
Λ
N
-
1
y
∥
2
=
∥
(
Λ
N
-
1
x
-
Λ
N
-
1
y
)
-
μ
N
(
B
N
Λ
N
-
1
x
-
B
N
Λ
N
-
1
y
)
∥
2
≤
∥
Λ
N
-
1
x
-
Λ
N
-
1
y
∥
2
+
μ
N
(
μ
N
-
2
η
N
)
∥
B
N
Λ
N
-
1
x
-
B
N
Λ
N
-
1
y
∥
2
≤
∥
Λ
N
-
1
x
-
Λ
N
-
1
y
∥
2
⋮
≤
∥
Λ
0
x
-
Λ
0
y
∥
2
=
∥
x
-
y
∥
2
.
In addition, note that(45)
〈
(
I
-
f
)
x
-
(
I
-
f
)
y
,
x
-
y
〉
=
∥
x
-
y
∥
2
-
〈
f
(
x
)
-
f
(
y
)
,
x
-
y
〉
≥
(
1
-
ρ
)
∥
x
-
y
∥
2
,
∀
x
,
y
∈
H
,
∥
(
I
-
f
)
x
-
(
I
-
f
)
y
∥
=
∥
x
-
y
∥
+
∥
f
(
x
)
-
f
(
y
)
∥
≤
(
1
+
ρ
)
∥
x
-
y
∥
,
∀
x
,
y
∈
H
.
That is, I
-
f is strongly monotone and Lipschitz continuous. So, there exists a unique solution x
* in Ω to the VIP
(46)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.
That is, x
*
∈
VI
⁡
(
Ω
,
I
-
f
).
We divide the rest of the proof into several steps.
Step  1. We show that {
x
n
} is bounded. Indeed, take p
∈
Ω arbitrarily. Since p
=
S
r
n
(
Θ
,
φ
)
(
p
-
r
n
A
p
), A is ζ-inverse strongly monotone, and 0
≤
r
n
≤
2
ζ, we have, for any n
≥
1,
(47)
∥
u
n
-
p
∥
2
=
∥
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
p
∥
2
≤
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
∥
2
=
∥
(
x
n
-
p
)
-
r
n
(
A
x
n
-
A
p
)
∥
2
=
∥
x
n
-
p
∥
2
-
2
r
n
〈
x
n
-
p
,
A
x
n
-
A
p
〉
+
r
n
2
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
-
2
r
n
ζ
∥
A
x
n
-
A
p
∥
2
+
r
n
2
∥
A
x
n
-
A
p
∥
2
=
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
.
Since p
=
P
C
(
I
-
ν
k
A
k
)
p, Δ
k
p
=
p, and A
k is ζ
k-inverse strongly monotone, where ν
k
∈
(
0,2
ζ
k
), k
∈
{
1,2
,
…
,
M
}, by Proposition 3 we obtain that for each n
≥
1
(48)
∥
v
n
-
p
∥
2
≤
∥
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
-
(
I
-
ν
M
A
M
)
Δ
M
-
1
p
∥
2
≤
∥
Δ
M
-
1
u
n
-
Δ
M
-
1
p
∥
2
+
ν
M
(
ν
M
-
2
ζ
M
)
∥
A
M
Δ
M
-
1
u
n
-
A
M
Δ
M
-
1
p
∥
2
≤
∥
Δ
M
-
1
u
n
-
Δ
M
-
1
p
∥
2
⋮
≤
∥
Δ
0
u
n
-
Δ
0
p
∥
2
=
∥
u
n
-
p
∥
2
.
Since p
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
p, Λ
i
p
=
p, and B
i is η
i-inverse strongly monotone, where μ
i
∈
(
0,2
η
i
), i
∈
{
1,2
,
…
,
N
}, by Lemma 14 we deduce that for each n
≥
1
(49)
∥
w
n
-
p
∥
2
≤
∥
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
-
(
I
-
μ
N
B
N
)
Λ
N
-
1
p
∥
2
≤
∥
Λ
N
-
1
u
n
-
Λ
N
-
1
p
∥
2
+
μ
N
(
μ
N
-
2
η
N
)
∥
B
N
Λ
N
-
1
u
n
-
B
N
Λ
N
-
1
p
∥
2
≤
∥
Λ
N
-
1
u
n
-
Λ
N
-
1
p
∥
2
⋮
≤
∥
Λ
0
u
n
-
Λ
0
p
∥
2
=
∥
u
n
-
p
∥
2
.
Hence from (37)–(49), we have
(50)
∥
y
n
-
p
∥
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
Δ
M
u
n
-
p
)
dddddd
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
Λ
N
u
n
-
p
)
+
γ
n
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
W
n
Δ
M
u
n
-
p
∥
+
∥
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
Λ
N
u
n
-
p
)
∥
+
γ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
W
n
v
n
-
p
∥
+
(
1
-
β
n
-
γ
n
γ
-
)
∥
W
n
w
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
v
n
-
p
∥
+
(
1
-
β
n
-
γ
n
γ
-
)
∥
w
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
u
n
-
p
∥
+
(
1
-
β
n
-
γ
n
γ
-
)
∥
u
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
≤
β
n
∥
x
n
-
p
∥
+
γ
n
∥
x
n
-
p
∥
+
(
1
-
β
n
-
γ
n
γ
-
)
∥
x
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
=
(
1
-
γ
n
(
γ
-
-
1
)
)
∥
x
n
-
p
∥
+
γ
n
∥
(
I
-
V
)
p
∥
=
(
1
-
γ
n
(
γ
-
-
1
)
)
∥
x
n
-
p
∥
+
γ
n
(
γ
-
-
1
)
∥
(
I
-
V
)
p
∥
γ
-
-
1
≤
max
⁡
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
-
-
1
}
.
Since f
:
H
→
H is a ρ-contraction with ρ
∈
(
0,1
), from (37) and (50) we get
(51)
∥
x
n
+
1
-
p
∥
≤
α
n
∥
f
(
x
n
)
-
p
∥
+
(
1
-
α
n
)
∥
y
n
-
p
∥
≤
α
n
(
∥
f
(
x
n
)
-
f
(
p
)
∥
+
∥
f
(
p
)
-
p
∥
)
+
(
1
-
α
n
)
max
⁡
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
≤
α
n
(
ρ
∥
x
n
-
p
∥
+
∥
f
(
p
)
-
p
∥
)
+
(
1
-
α
n
)
max
⁡
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
≤
α
n
ρ
max
⁡
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
+
(
1
-
α
n
)
max
⁡
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
+
α
n
∥
f
(
p
)
-
p
∥
=
(
1
-
α
n
(
1
-
ρ
)
)
max
⁡
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
+
α
n
∥
(
I
-
f
)
p
∥
=
(
1
-
α
n
(
1
-
ρ
)
)
max
⁡
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
}
+
α
n
(
1
-
ρ
)
∥
(
I
-
f
)
p
∥
1
-
ρ
≤
max
⁡
{
∥
x
n
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
,
∥
(
I
-
f
)
p
∥
1
-
ρ
}
.
By induction, we get
(52)
∥
x
n
-
p
∥
≤
max
⁡
{
∥
x
1
-
p
∥
,
∥
(
I
-
V
)
p
∥
γ
¯
-
1
,
∥
(
I
-
f
)
p
∥
1
-
ρ
}
.
Therefore, {
x
n
} is bounded and so are the sequences {
u
n
}, {
v
n
}, {
w
n
}, {
f
(
x
n
)
}, and {
W
n
w
n
}.
Step
2. We show that lim
⁡
n
→
∞
∥
x
n
-
x
n
+
1
∥
=
0 and lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0. Indeed, put σ
n
=
(
1
-
α
n
)
β
n
,
for
all
n
≥
1. Then it follows from conditions (iii) and (iv) that
(53)
β
n
≥
σ
n
=
(
1
-
α
n
)
β
n
≥
(
1
-
(
1
-
ρ
)
)
β
n
=
ρ
β
n
,
∀
n
≥
1
,
and hence
(54)
0
<
liminf
⁡
n
→
∞
σ
n
≤
limsup
⁡
n
→
∞
σ
n
<
1
.
Define
(55)
x
n
+
1
=
σ
n
x
n
+
(
1
-
σ
n
)
z
n
,
∀
n
≥
1
.
Observe that
(56)
z
n
+
1
-
z
n
=
x
n
+
2
-
σ
n
+
1
x
n
+
1
1
-
σ
n
+
1
-
x
n
+
1
-
σ
n
x
n
1
-
σ
n
=
α
n
+
1
f
(
x
n
+
1
)
+
(
1
-
α
n
+
1
)
y
n
+
1
-
σ
n
+
1
x
n
+
1
1
-
σ
n
+
1
-
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
-
σ
n
x
n
1
-
σ
n
=
(
α
n
+
1
f
(
x
n
+
1
)
1
-
σ
n
+
1
-
α
n
f
(
x
n
)
1
-
σ
n
)
-
(
(
1
-
α
n
)
[
β
n
x
n
+
γ
n
W
n
v
n
h
h
h
h
h
h
h
h
h
h
h
h
g
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
W
n
w
n
]
-
σ
n
x
n
)
×
(
1
-
σ
n
)
-
1
+
(
(
1
-
α
n
+
1
)
[
β
n
+
1
x
n
+
1
+
γ
n
+
1
W
n
+
1
v
n
+
1
hhhhhhhhhhhhhh
+
[
(
1
-
β
n
+
1
)
I
-
γ
n
+
1
V
]
W
n
+
1
w
n
+
1
]
-
σ
n
+
1
x
n
+
1
β
n
+
1
)
×
(
1
-
σ
n
+
1
)
-
1
=
(
α
n
+
1
f
(
x
n
+
1
)
1
-
σ
n
+
1
-
α
n
f
(
x
n
)
1
-
σ
n
)
+
1
-
α
n
+
1
1
-
σ
n
+
1
×
(
γ
n
+
1
W
n
+
1
v
n
+
1
+
(
(
1
-
β
n
+
1
)
I
-
γ
n
+
1
V
)
W
n
+
1
w
n
+
1
)
-
1
-
α
n
1
-
σ
n
(
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
)
=
(
α
n
+
1
f
(
x
n
+
1
)
1
-
σ
n
+
1
-
α
n
f
(
x
n
)
1
-
σ
n
)
+
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
×
[
γ
n
+
1
W
n
+
1
v
n
+
1
+
(
(
1
-
β
n
+
1
)
I
-
γ
n
+
1
V
)
W
n
+
1
w
n
+
1
1
-
β
n
+
1
222222222
-
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
1
-
β
n
]
+
[
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
-
(
1
-
α
n
)
(
1
-
β
n
)
1
-
σ
n
]
×
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
1
-
β
n
=
α
n
+
1
1
-
σ
n
+
1
(
f
(
x
n
+
1
)
-
f
(
x
n
)
)
+
(
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
)
f
(
x
n
)
+
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
×
[
γ
n
+
1
1
-
β
n
+
1
W
n
+
1
v
n
+
1
-
γ
n
1
-
β
n
W
n
v
n
+
W
n
+
1
w
n
+
1
fffffffffffff
-
W
n
w
n
+
γ
n
1
-
β
n
V
W
n
w
n
-
γ
n
+
1
1
-
β
n
+
1
V
W
n
+
1
w
n
+
1
]
-
(
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
)
×
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
1
-
β
n
=
α
n
+
1
1
-
σ
n
+
1
(
f
(
x
n
+
1
)
-
f
(
x
n
)
)
+
(
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
)
f
(
x
n
)
+
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
×
[
(
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
)
(
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
)
ggggggg
+
(
W
n
+
1
v
n
+
1
-
W
n
v
n
)
γ
n
1
-
β
n
ggggggg
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
(
W
n
+
1
w
n
+
1
-
W
n
w
n
)
1
-
β
n
]
-
(
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
)
×
γ
n
W
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
1
-
β
n
,
and hence
(57)
∥
z
n
+
1
-
z
n
∥
≤
α
n
+
1
1
-
σ
n
+
1
∥
f
(
x
n
+
1
)
-
f
(
x
n
)
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
(
1
-
α
n
+
1
)
(
1
-
β
n
+
1
)
1
-
σ
n
+
1
×
[
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
dddddddd
+
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
γ
n
1
-
β
n
dddddddd
+
∥
(
(
1
-
β
n
)
I
-
γ
n
V
)
(
W
n
+
1
w
n
+
1
-
W
n
w
n
)
∥
1
-
β
n
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
γ
n
v
n
+
(
(
1
-
β
n
)
I
-
γ
n
V
)
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
ρ
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
gggggggg
+
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
γ
n
1
-
β
n
gggggggg
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
W
n
+
1
w
n
+
1
-
W
n
w
n
∥
1
-
β
n
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
×
γ
n
∥
W
n
v
n
∥
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
+
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
γ
n
1
-
β
n
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
W
n
+
1
w
n
+
1
-
W
n
w
n
∥
1
-
β
n
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
.
From (9), since W
n, T
n, and U
n
,
i are all nonexpansive, we have
(58)
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
≤
∥
W
n
+
1
v
n
+
1
-
W
n
+
1
v
n
∥
+
∥
W
n
+
1
w
n
-
W
n
v
n
∥
=
∥
W
n
+
1
v
n
+
1
-
W
n
+
1
v
n
∥
+
∥
λ
1
T
1
U
n
+
1,2
v
n
-
λ
1
T
1
U
n
,
2
v
n
∥
≤
∥
v
n
+
1
-
v
n
∥
+
λ
1
∥
U
n
+
1,2
v
n
-
U
n
,
2
v
n
∥
=
∥
v
n
+
1
-
v
n
∥
+
λ
1
∥
λ
2
T
2
U
n
+
1,3
v
n
-
λ
2
T
2
U
n
,
3
v
n
∥
≤
∥
v
n
+
1
-
v
n
∥
+
λ
1
λ
2
∥
U
n
+
1,3
v
n
-
U
n
,
3
v
n
∥
⋮
≤
∥
v
n
+
1
-
v
n
∥
+
λ
1
λ
2
⋯
λ
n
∥
U
n
+
1
,
n
+
1
v
n
-
U
n
,
n
+
1
v
n
∥
≤
∥
v
n
+
1
-
v
n
∥
+
M
~
∏
i
=
1
n
‍
λ
i
,
∥
W
n
+
1
w
n
+
1
-
W
n
w
n
∥
≤
∥
W
n
+
1
w
n
+
1
-
W
n
+
1
w
n
∥
+
∥
W
n
+
1
w
n
-
W
n
w
n
∥
=
∥
W
n
+
1
w
n
+
1
-
W
n
+
1
w
n
∥
+
∥
λ
1
T
1
U
n
+
1,2
w
n
-
λ
1
T
1
U
n
,
2
w
n
∥
≤
∥
w
n
+
1
-
w
n
∥
+
λ
1
∥
U
n
+
1,2
w
n
-
U
n
,
2
w
n
∥
=
∥
w
n
+
1
-
w
n
∥
+
λ
1
∥
λ
2
T
2
U
n
+
1,3
w
n
-
λ
2
T
2
U
n
,
3
w
n
∥
≤
∥
w
n
+
1
-
w
n
∥
+
λ
1
λ
2
∥
U
n
+
1,3
w
n
-
U
n
,
3
w
n
∥
⋮
≤
∥
w
n
+
1
-
w
n
∥
+
λ
1
λ
2
⋯
λ
n
∥
U
n
+
1
,
n
+
1
w
n
-
U
n
,
n
+
1
w
n
∥
≤
∥
w
n
+
1
-
w
n
∥
+
M
~
∏
i
=
1
n
‍
λ
i
,
where sup
⁡
n
≥
1
{
∥
U
n
+
1
,
n
+
1
v
n
∥
+
∥
U
n
,
n
+
1
v
n
∥
}
≤
M
~ and sup
⁡
n
≥
1
{
∥
U
n
+
1
,
n
+
1
w
n
∥
+
∥
U
n
,
n
+
1
w
n
∥
}
≤
M
~ for some M
~
>
0.
On the other hand, we estimate∥
v
n
+
1
-
v
n
∥ and ∥
w
n
+
1
-
w
n
∥. First observe that
(59)
∥
v
n
+
1
-
v
n
∥
2
=
∥
P
C
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
+
1
-
P
C
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
∥
2
≤
∥
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
+
1
-
(
I
-
ν
M
A
M
)
Δ
M
-
1
u
n
∥
2
=
∥
(
Δ
M
-
1
u
n
+
1
-
Δ
M
-
1
u
n
)
d
d
d
-
ν
M
(
A
M
Δ
M
-
1
u
n
+
1
-
A
M
Δ
M
-
1
u
n
)
∥
2
≤
∥
Δ
M
-
1
u
n
+
1
-
Δ
M
-
1
u
n
∥
2
+
ν
M
(
ν
M
-
2
ζ
M
)
∥
A
M
Δ
M
-
1
u
n
+
1
-
A
M
Δ
M
-
1
u
n
∥
2
≤
∥
Δ
M
-
1
u
n
+
1
-
Δ
M
-
1
u
n
∥
2
⋮
≤
∥
Δ
0
u
n
+
1
-
Δ
0
u
n
∥
2
=
∥
u
n
+
1
-
u
n
∥
2
.
Utilizing Remark 5 and Lemma 14, we have
(60)
∥
w
n
+
1
-
w
n
∥
2
=
∥
J
R
N
,
μ
N
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
+
1
d
d
d
-
J
R
N
,
μ
N
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
∥
2
≤
∥
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
+
1
-
(
I
-
μ
N
B
N
)
Λ
N
-
1
u
n
∥
2
=
∥
(
Λ
N
-
1
u
n
+
1
-
Λ
N
-
1
u
n
)
d
d
d
-
μ
N
(
B
N
Λ
N
-
1
u
n
+
1
-
B
N
Λ
N
-
1
u
n
)
∥
2
≤
∥
Λ
N
-
1
u
n
+
1
-
Λ
N
-
1
u
n
∥
2
+
μ
N
(
μ
N
-
2
η
N
)
∥
B
N
Λ
N
-
1
u
n
+
1
-
B
N
Λ
N
-
1
u
n
∥
2
≤
∥
Λ
N
-
1
u
n
+
1
-
Λ
N
-
1
u
n
∥
2
⋮
≤
∥
Λ
0
u
n
+
1
-
Λ
0
u
n
∥
2
=
∥
u
n
+
1
-
u
n
∥
2
,
∥
(
I
-
r
n
+
1
A
)
x
n
+
1
-
(
I
-
r
n
A
)
x
n
∥
=
∥
x
n
+
1
-
x
n
-
r
n
+
1
(
A
x
n
+
1
-
A
x
n
)
+
(
r
n
-
r
n
+
1
)
A
x
n
∥
≤
∥
x
n
+
1
-
x
n
-
r
n
+
1
(
A
x
n
+
1
-
A
x
n
)
∥
+
|
r
n
+
1
-
r
n
|
∥
A
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
∥
A
x
n
∥
,
∥
u
n
+
1
-
u
n
∥
=
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
+
1
A
)
x
n
+
1
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
=
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
+
1
A
)
x
n
+
1
-
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
222222
+
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
≤
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
+
1
A
)
x
n
+
1
-
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
+
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
≤
∥
(
I
-
r
n
+
1
A
)
x
n
+
1
-
(
I
-
r
n
A
)
x
n
∥
+
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
∥
A
x
n
∥
+
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
∥
A
x
n
∥
+
|
r
n
+
1
-
r
n
|
r
n
+
1
·
ν
σ
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
x
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
×
(
∥
A
x
n
∥
+
ν
c
σ
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
x
n
∥
)
≤
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
M
~
1
,
where sup
⁡
n
≥
1
{
∥
A
x
n
∥
+
(
ν
/
c
σ
)
∥
S
r
n
+
1
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
x
n
∥
}
≤
M
~
1 for some M
~
1
>
0. So, combining (57)–(60) we get
(61)
∥
z
n
+
1
-
z
n
∥
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
ddddddd
+
∥
W
n
+
1
v
n
+
1
-
W
n
v
n
∥
γ
n
1
-
β
n
ddddddd
+
1
-
β
n
-
γ
n
γ
¯
1
-
β
n
∥
W
n
+
1
w
n
+
1
-
W
n
w
n
∥
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
(
∥
w
n
+
1
-
w
n
∥
+
M
~
∏
i
=
1
n
‍
λ
i
)
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
ddddddd
+
(
∥
v
n
+
1
-
v
n
∥
+
M
~
∏
i
=
1
n
‍
λ
i
)
γ
n
1
-
β
n
ddddddd
+
1
-
β
n
-
γ
n
γ
¯
1
-
β
n
(
∥
w
n
+
1
-
w
n
∥
+
M
~
∏
i
=
1
n
‍
λ
i
)
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
(
∥
u
n
+
1
-
u
n
∥
+
M
~
∏
i
=
1
n
‍
λ
i
)
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
ddddddd
+
(
∥
u
n
+
1
-
u
n
∥
+
M
~
∏
i
=
1
n
‍
λ
i
)
γ
n
1
-
β
n
ddddddd
+
1
-
β
n
-
γ
n
γ
¯
1
-
β
n
(
∥
u
n
+
1
-
u
n
∥
+
M
~
∏
i
=
1
n
‍
λ
i
)
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
=
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
×
[
∏
i
=
1
n
‍
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
ddddddd
+
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
1
-
β
n
∥
u
n
+
1
-
u
n
∥
ddddddd
+
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
1
-
β
n
M
~
∏
i
=
1
n
‍
λ
i
]
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
∥
u
n
+
1
-
u
n
∥
+
M
~
∏
i
=
1
n
‍
λ
i
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
α
n
+
1
1
-
σ
n
+
1
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
+
1
-
α
n
+
1
-
σ
n
+
1
1
-
σ
n
+
1
(
∥
x
n
+
1
-
x
n
∥
+
|
r
n
+
1
-
r
n
|
M
~
1
)
+
M
~
∏
i
=
1
n
‍
λ
i
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
β
n
≤
∥
x
n
+
1
-
x
n
∥
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
f
(
x
n
)
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
W
n
+
1
v
n
+
1
-
V
W
n
+
1
w
n
+
1
∥
+
|
r
n
+
1
-
r
n
|
M
~
1
+
M
~
∏
i
=
1
n
‍
λ
i
+
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
1
-
a
^
≤
∥
x
n
+
1
-
x
n
∥
+
M
~
2
(
|
α
n
+
1
1
-
σ
n
+
1
-
α
n
1
-
σ
n
|
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
+
|
r
n
+
1
-
r
n
|
+
b
n
)
,
where sup
⁡
n
≥
1
{
∥
f
(
x
n
)
∥
+
(
(
∥
W
n
v
n
∥
+
∥
W
n
w
n
∥
)
/
(
1
-
a
^
)
)
+
∥
W
n
v
n
-
V
W
n
w
n
∥
+
M
~
1
+
M
~
}
≤
M
~
2 for some M
~
2
>
0. Thus, from (61), b
∈
(
0,1
), and conditions (v)-(vi) it follows that
(62)
limsup
⁡
n
→
∞
(
∥
z
n
+
1
-
z
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
0
.
Since x
n
+
1
=
σ
n
x
n
+
(
1
-
σ
n
)
z
n for all n
≥
1, by Lemma 12 we obtain from 0
<
liminf
⁡
n
→
∞
σ
n
≤
limsup
⁡
n
→
∞
σ
n
<
1 that
(63)
lim
⁡
n
→
∞
∥
z
n
-
x
n
∥
=
0
,
which immediately yields
(64)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
lim
⁡
n
→
∞
(
1
-
σ
n
)
∥
z
n
-
x
n
∥
=
0
.
Note that
(65)
ρ
∥
y
n
-
x
n
∥
=
(
1
-
(
1
-
ρ
)
)
∥
y
n
-
x
n
∥
≤
(
1
-
α
n
)
∥
y
n
-
x
n
∥
=
∥
x
n
+
1
-
x
n
-
α
n
(
f
(
x
n
)
-
x
n
)
∥
≤
∥
x
n
+
1
-
x
n
∥
+
∥
α
n
(
f
(
x
n
)
-
x
n
)
∥
.
Consequently, it follows from (64) and α
n
(
f
(
x
n
)
-
x
n
)
→
0 that
(66)
lim
⁡
n
→
∞
∥
y
n
-
x
n
∥
=
0
.
Step
3. We prove lim
⁡
n
→
∞
∥
x
n
-
u
n
∥
=
0.
Indeed, for anyp
∈
Ω, we find that
(67)
∥
u
n
-
p
∥
2
=
∥
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
p
∥
2
≤
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
∥
2
=
∥
x
n
-
p
-
r
n
(
A
x
n
-
A
p
)
∥
2
≤
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
.
From (37), (48), (49), and (67), we obtain
(68)
∥
y
n
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
v
n
-
p
)
22222
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
+
γ
n
(
I
-
V
)
p
∥
2
≤
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
v
n
-
p
)
22222
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
W
n
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
×
∥
1
1
-
β
n
-
γ
n
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
(
1
-
β
n
-
γ
n
γ
¯
)
2
(
1
-
β
n
-
γ
n
)
2
∥
W
n
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
2
1
-
β
n
-
γ
n
∥
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
u
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
×
∥
u
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
×
[
∥
x
n
-
p
∥
2
+
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
]
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
r
n
×
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which immediately yields
(69)
(
1
-
β
n
-
γ
n
∥
V
∥
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting p
=
x
* we have
(70)
(
1
-
β
n
-
γ
n
∥
V
∥
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
x
*
∥
2
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
∥
(
I
-
V
)
x
*
∥
∥
y
n
-
x
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
=
∥
x
n
-
y
n
∥
×
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
.
Since limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1, 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ, and limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0, we obtain from (66) and the boundedness of {
x
n
} and {
y
n
} that
(71)
lim
⁡
n
→
∞
∥
A
x
n
-
A
x
*
∥
=
0
.
Furthermore, from the firm nonexpansivity of S
r
n
(
Θ
,
φ
), we have
(72)
∥
u
n
-
p
∥
2
=
∥
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
-
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
p
∥
2
≤
〈
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
,
u
n
-
p
〉
=
1
2
[
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
∥
2
+
∥
u
n
-
p
∥
2
-
∥
(
I
-
r
n
A
)
x
n
-
(
I
-
r
n
A
)
p
-
(
u
n
-
p
)
∥
2
]
≤
1
2
[
∥
x
n
-
p
∥
2
+
∥
u
n
-
p
∥
2
-
∥
x
n
-
u
n
-
r
n
(
A
x
n
-
A
p
)
∥
2
]
=
1
2
[
∥
x
n
-
p
∥
2
+
∥
u
n
-
p
∥
2
-
∥
x
n
-
u
n
∥
2
222222
+
2
r
n
〈
A
x
n
-
A
p
,
x
n
-
u
n
〉
-
r
n
2
∥
A
x
n
-
A
p
∥
2
]
,
which implies that
(73)
∥
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
-
u
n
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
.
From (68) and (73), we have
(74)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
u
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
×
[
∥
x
n
-
p
∥
2
-
∥
x
n
-
u
n
∥
2
55555
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
]
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
-
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
u
n
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
-
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
u
n
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which immediately yields
(75)
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
u
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
r
n
∥
A
x
n
-
A
p
∥
∥
x
n
-
u
n
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting p
=
x
* we have
(76)
(
1
-
β
n
-
γ
n
∥
V
∥
)
∥
x
n
-
u
n
∥
2
≤
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
u
n
∥
2
≤
∥
x
n
-
x
*
∥
2
-
∥
y
n
-
x
*
∥
2
+
2
r
n
∥
A
x
n
-
A
x
*
∥
∥
x
n
-
u
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
∥
(
I
-
V
)
x
*
∥
∥
y
n
-
x
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
+
2
r
n
∥
A
x
n
-
A
x
*
∥
∥
x
n
-
u
n
∥
=
∥
x
n
-
y
n
∥
×
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
+
2
r
n
∥
A
x
n
-
A
x
*
∥
∥
x
n
-
u
n
∥
.
Since limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1 and limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0, we deduce from (66) and (71) and the boundedness of {
u
n
}
,
{
x
n
}, and {
y
n
} that
(77)
lim
⁡
n
→
∞
∥
x
n
-
u
n
∥
=
0
.
Step  4. We prove that lim
⁡
n
→
∞
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
=
0 and lim
⁡
n
→
∞
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
=
0 for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}.
Indeed, let us show thatlim
⁡
n
→
∞
∥
A
k
Δ
k
u
n
-
A
k
x
*
∥
=
0 and lim
⁡
n
→
∞
∥
B
i
Λ
i
u
n
-
B
i
x
*
∥
=
0 for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Observe that, for any p
∈
Ω,
(78)
∥
Δ
k
u
n
-
p
∥
2
=
∥
P
C
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
P
C
(
I
-
ν
k
A
k
)
p
∥
2
≤
∥
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
(
I
-
ν
k
A
k
)
p
∥
2
≤
∥
Δ
k
-
1
u
n
-
p
∥
2
+
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
≤
∥
u
n
-
p
∥
2
+
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
≤
∥
x
n
-
p
∥
2
+
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
,
∥
Λ
i
u
n
-
p
∥
2
=
∥
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
p
∥
2
≤
∥
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
(
I
-
μ
i
B
i
)
p
∥
2
≤
∥
Λ
i
-
1
u
n
-
p
∥
2
+
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
u
n
-
p
∥
2
+
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
+
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
.
From (68) and (78) we have
(79)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
(
1
-
β
n
-
γ
n
γ
¯
)
2
(
1
-
β
n
-
γ
n
)
2
∥
W
n
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
Δ
k
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
2
1
-
β
n
-
γ
n
∥
Λ
i
u
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
[
∥
x
n
-
p
∥
2
+
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
]
+
(
1
-
β
n
-
γ
n
γ
¯
)
×
[
∥
x
n
-
p
∥
2
+
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
]
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
+
γ
n
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
+
γ
n
ν
k
(
ν
k
-
2
ζ
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
μ
i
(
μ
i
-
2
η
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which leads to
(80)
γ
n
ν
k
(
2
ζ
k
-
ν
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
μ
i
(
2
η
i
-
μ
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting p
=
x
* we have
(81)
γ
n
ν
k
(
2
ζ
k
-
ν
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
2
+
(
1
-
β
n
-
γ
n
∥
V
∥
)
μ
i
(
2
η
i
-
μ
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
2
≤
γ
n
ν
k
(
2
ζ
k
-
ν
k
)
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
μ
i
(
2
η
i
-
μ
i
)
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
2
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
∥
(
I
-
V
)
x
*
∥
∥
y
n
-
x
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
=
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
.
Since 0
<
liminf
⁡
n
→
∞
γ
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1 and limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0, we obtain from (66), ν
k
∈
(
0,2
ζ
k
)
,
μ
i
∈
(
0,2
η
i
), k
∈
{
1,2
,
…
,
M
}, i
∈
{
1,2
,
…
,
N
}, and the boundedness of {
x
n
} and {
y
n
} that
(82)
lim
⁡
n
→
∞
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
=
0
,
lim
⁡
n
→
∞
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
=
0
,
for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}.
Furthermore, by Proposition3(iii) and Lemma 7(a), we obtain
(83)
∥
Δ
k
u
n
-
p
∥
2
=
∥
P
C
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
P
C
(
I
-
ν
k
A
k
)
p
∥
2
≤
〈
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
(
I
-
ν
k
A
k
)
p
,
Δ
k
u
n
-
p
〉
=
1
2
(
∥
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
-
(
I
-
ν
k
A
k
)
p
∥
2
5555
5
+
∥
Δ
k
u
n
-
p
∥
2
5555
5
-
∥
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n
5555
5
5
5
5
5
-
(
I
-
ν
k
A
k
)
p
-
(
Δ
k
u
n
-
p
)
∥
2
)
≤
1
2
(
∥
Δ
k
-
1
u
n
-
p
∥
2
+
∥
Δ
k
u
n
-
p
∥
2
5555
5
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
-
ν
k
(
A
k
Δ
k
-
1
u
n
-
A
k
p
)
∥
2
)
≤
1
2
(
∥
u
n
-
p
∥
2
+
∥
Δ
k
u
n
-
p
∥
2
5555
5
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
-
ν
k
(
A
k
Δ
k
-
1
u
n
-
A
k
p
)
∥
2
)
≤
1
2
(
∥
x
n
-
p
∥
2
+
∥
Δ
k
u
n
-
p
∥
2
5555
5
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
-
ν
k
(
A
k
Δ
k
-
1
u
n
-
A
k
p
)
∥
2
)
,
which implies that
(84)
∥
Δ
k
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
-
ν
k
(
A
k
Δ
k
-
1
u
n
-
A
k
p
)
∥
2
=
∥
x
n
-
p
∥
2
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
-
ν
k
2
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
2
+
2
ν
k
〈
Δ
k
-
1
u
n
-
Δ
k
u
n
,
A
k
Δ
k
-
1
u
n
-
A
k
p
〉
≤
∥
x
n
-
p
∥
2
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
.
By Lemmas 7(a) and 14, we get
(85)
∥
Λ
i
u
n
-
p
∥
2
=
∥
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
p
∥
2
≤
〈
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
(
I
-
μ
i
B
i
)
p
,
Λ
i
u
n
-
p
〉
=
1
2
(
∥
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
(
I
-
μ
i
B
i
)
p
∥
2
+
∥
Λ
i
u
n
-
p
∥
2
5
-
∥
(
I
-
μ
i
B
i
)
Λ
i
-
1
u
n
-
(
I
-
μ
i
B
i
)
p
-
(
Λ
i
u
n
-
p
)
∥
2
)
≤
1
2
(
∥
Λ
i
-
1
u
n
-
p
∥
2
+
∥
Λ
i
u
n
-
p
∥
2
5
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
-
μ
i
(
B
i
Λ
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
u
n
-
p
∥
2
+
∥
Λ
i
u
n
-
p
∥
2
5
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
-
μ
i
(
B
i
Λ
i
-
1
u
n
-
B
i
p
)
∥
2
)
≤
1
2
(
∥
x
n
-
p
∥
2
+
∥
Λ
i
u
n
-
p
∥
2
5
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
-
μ
i
(
B
i
Λ
i
-
1
u
n
-
B
i
p
)
∥
2
)
,
which implies that
(86)
∥
Λ
i
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
-
μ
i
(
B
i
Λ
i
-
1
u
n
-
B
i
p
)
∥
2
=
∥
x
n
-
p
∥
2
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
-
μ
i
2
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
2
+
2
μ
i
〈
Λ
i
-
1
u
n
-
Λ
i
u
n
,
B
i
Λ
i
-
1
u
n
-
B
i
p
〉
≤
∥
x
n
-
p
∥
2
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
.
From (68), (84), and (86) we have
(87)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
(
1
-
β
n
-
γ
n
γ
¯
)
2
(
1
-
β
n
-
γ
n
)
2
∥
W
n
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
[
∥
x
n
-
p
∥
2
-
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
2222222
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
]
+
(
1
-
β
n
-
γ
n
γ
¯
)
×
[
∥
x
n
-
p
∥
2
-
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
22222
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
]
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
-
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
-
(
1
-
β
n
-
γ
n
γ
¯
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
-
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
-
(
1
-
β
n
-
γ
n
γ
¯
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which hence implies that
(88)
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
+
(
1
-
β
n
-
γ
n
∥
V
∥
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
≤
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
p
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
p
∥
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting p
=
x
* we have
(89)
γ
n
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
2
+
(
1
-
β
n
-
γ
n
∥
V
∥
)
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
2
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
×
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
ν
k
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
∥
A
k
Δ
k
-
1
u
n
-
A
k
x
*
∥
+
2
μ
i
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
∥
B
i
Λ
i
-
1
u
n
-
B
i
x
*
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
.
Since 0
<
liminf
⁡
n
→
∞
γ
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1 and limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0, we obtain from (66) and (82) and the boundedness of {
u
n
}, {
x
n
}, and {
y
n
} that
(90)
lim
⁡
n
→
∞
∥
Δ
k
-
1
u
n
-
Δ
k
u
n
∥
=
0
,
lim
⁡
n
→
∞
∥
Λ
i
-
1
u
n
-
Λ
i
u
n
∥
=
0
,
for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Consequently, from (90) it follows that
(91)
∥
u
n
-
v
n
∥
=
∥
Δ
0
u
n
-
Δ
M
u
n
∥
≤
∥
Δ
0
u
n
-
Δ
1
u
n
∥
+
∥
Δ
1
u
n
-
Δ
2
u
n
∥
+
⋯
+
∥
Δ
M
-
1
u
n
-
Δ
M
u
n
∥
⟶
0
as
n
⟶
∞
,
(92)
∥
u
n
-
w
n
∥
=
∥
Λ
0
u
n
-
Λ
N
u
n
∥
≤
∥
Λ
0
u
n
-
Λ
1
u
n
∥
+
∥
Λ
1
u
n
-
Λ
2
u
n
∥
+
⋯
+
∥
Λ
N
-
1
u
n
-
Λ
N
u
n
∥
⟶
0
as
n
⟶
∞
.
By (77), (91), and (92), we have
(93)
lim
⁡
n
→
∞
∥
x
n
-
v
n
∥
=
0
,
lim
⁡
n
→
∞
∥
x
n
-
w
n
∥
=
0
.
Step  5. We show that lim
⁡
n
→
∞
∥
v
n
-
W
v
n
∥
=
0. Indeed, utilizing Lemma 7(b), from (37), (48), (49), and (67) we obtain that, for any p
∈
Ω,
(94)
∥
y
n
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
v
n
-
p
)
5555
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
+
γ
n
(
I
-
V
)
p
∥
2
≤
∥
β
n
(
x
n
-
p
)
+
γ
n
(
W
n
v
n
-
p
)
5555
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
W
n
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
×
∥
1
1
-
β
n
-
γ
n
[
(
1
-
β
n
)
I
-
γ
n
V
]
(
W
n
w
n
-
p
)
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
v
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
)
(
1
-
β
n
-
γ
n
γ
¯
)
2
(
1
-
β
n
-
γ
n
)
2
∥
W
n
w
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
2
1
-
β
n
-
γ
n
∥
w
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
γ
n
∥
u
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
γ
¯
)
∥
u
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
u
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
=
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
p
∥
2
-
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
,
which implies that
(95)
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
y
n
-
p
∥
2
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
p
∥
+
∥
y
n
-
p
∥
)
+
2
γ
n
〈
(
I
-
V
)
p
,
y
n
-
p
〉
.
In particular, putting p
=
x
* we have
(96)
β
n
γ
n
∥
x
n
-
W
n
v
n
∥
2
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
)
+
2
γ
n
∥
(
I
-
V
)
x
*
∥
∥
y
n
-
x
n
∥
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
=
∥
x
n
-
y
n
∥
(
∥
x
n
-
x
*
∥
+
∥
y
n
-
x
*
∥
+
2
∥
(
I
-
V
)
x
*
∥
)
+
2
γ
n
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
.
Since liminf
⁡
n
→
∞
β
n
>
0, liminf
⁡
n
→
∞
γ
n
>
0, and limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0, we deduce from (66) and the boundedness of {
x
n
} and {
y
n
} that
(97)
lim
⁡
n
→
∞
∥
x
n
-
W
n
v
n
∥
=
0
.
Also, observe that
(98)
∥
v
n
-
W
n
v
n
∥
≤
∥
v
n
-
u
n
∥
+
∥
u
n
-
x
n
∥
+
∥
x
n
-
W
n
v
n
∥
.
Thus, from (77), (91), and (97) it follows that
(99)
lim
⁡
n
→
∞
∥
v
n
-
W
n
v
n
∥
=
0
.
Moreover, note that
(100)
∥
v
n
-
W
v
n
∥
≤
∥
v
n
-
W
n
v
n
∥
+
∥
W
n
v
n
-
W
v
n
∥
.
From (99), [23, Remark 3.2], and the boundedness of {
v
n
} we immediately obtain
(101)
lim
⁡
n
→
∞
∥
v
n
-
W
v
n
∥
=
0
.
Step  6. We show that limsup
⁡
n
→
∞
〈
(
f
-
I
)
x
*
,
x
n
-
x
*
〉
≤
0.
Indeed, we observe that there exists a subsequence{
x
n
i
} of {
x
n
} such that
(102)
limsup
⁡
n
→
∞
〈
(
f
-
I
)
x
*
,
x
n
-
x
*
〉
=
lim
⁡
i
→
∞
〈
(
f
-
I
)
x
*
,
x
n
i
-
x
*
〉
.
Since {
x
n
i
} is bounded, there exists a subsequence {
x
n
i
j
} of {
x
n
i
} which converges weakly to some w. Without loss of generality, we may assume that x
n
i
⇀
w. From (77), (90), and (93), we have that u
n
i
⇀
w, v
n
i
⇀
w, Δ
k
u
n
i
⇀
w, and Λ
m
u
n
i
⇀
w, where k
∈
{
1,2
,
…
,
M
} and m
∈
{
1,2
,
…
,
N
}. By (101) we have that ∥
v
n
-
W
v
n
∥
→
0 as n
→
∞. Then, by Lemma 10 we obtain w
∈
Fix
⁡
(
W
)
=
∩
n
=
1
∞
Fix
⁡
(
T
n
) (due to Lemma 9). Next we prove that w
∈
∩
k
=
1
M
VI
⁡
(
C
,
A
k
). Let
(103)
T
~
k
v
=
{
A
k
v
+
N
C
v
,
v
∈
C
,
∅
,
v
∉
C
,
where k
∈
{
1,2
,
…
,
M
}. Let (
v
,
u
)
∈
G
(
T
~
k
). Since u
-
A
k
v
∈
N
C
v and Δ
k
u
n
∈
C, we have
(104)
〈
v
-
Δ
k
u
n
,
u
-
A
k
v
〉
≥
0
.
Also, from Δ
k
u
n
=
P
C
(
I
-
ν
k
A
k
)
Δ
k
-
1
u
n and v
∈
C, we have
(105)
〈
v
-
Δ
k
u
n
,
Δ
k
u
n
-
(
Δ
k
-
1
u
n
-
ν
k
A
k
Δ
k
-
1
u
n
)
〉
≥
0
,
and hence
(106)
〈
v
-
Δ
k
u
n
,
Δ
k
u
n
-
Δ
k
-
1
u
n
ν
k
+
A
k
Δ
k
-
1
u
n
〉
≥
0
.
Therefore we have
(107)
〈
v
-
Δ
k
u
n
i
,
u
〉
≥
〈
v
-
Δ
k
u
n
i
,
A
k
v
〉
≥
〈
v
-
Δ
k
u
n
i
,
A
k
v
〉
-
〈
v
-
Δ
k
u
n
i
,
Δ
k
u
n
i
-
Δ
k
-
1
u
n
i
ν
k
+
A
k
Δ
k
-
1
u
n
i
〉
=
〈
v
-
Δ
k
u
n
i
,
A
k
v
-
A
k
Δ
k
u
n
i
〉
+
〈
v
-
Δ
k
u
n
i
,
A
k
Δ
k
u
n
i
-
A
k
Δ
k
-
1
u
n
i
〉
-
〈
v
-
Δ
k
u
n
i
,
Δ
k
u
n
i
-
Δ
k
-
1
u
n
i
ν
k
〉
≥
〈
v
-
Δ
k
u
n
i
,
A
k
Δ
k
u
n
i
-
A
k
Δ
k
-
1
u
n
i
〉
-
〈
v
-
Δ
k
u
n
i
,
Δ
k
u
n
i
-
Δ
k
-
1
u
n
i
ν
k
〉
.
From (90) and since A
k is uniformly continuous, we obtain that lim
⁡
n
→
∞
∥
A
k
Δ
k
u
n
-
A
k
Δ
k
-
1
u
n
∥
=
0. From Δ
k
u
n
i
⇀
w, ν
k
∈
(
0,2
ζ
k
), for
all
k
∈
{
1,2
,
…
,
M
} and (90), we have
(108)
〈
v
-
w
,
u
〉
≥
0
.
Since T
~
k is maximal monotone, we have w
∈
T
~
k
-
1
0 and hence w
∈
VI
⁡
(
C
,
A
k
), k
=
1,2
,
…
,
M, which implies w
∈
∩
k
=
1
M
VI
⁡
(
C
,
A
k
). 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 17 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 Λ
m
u
n
=
J
R
m
,
μ
m
(
I
-
μ
m
B
m
)
Λ
m
-
1
u
n, n
≥
1, m
∈
{
1,2
,
…
,
N
}, we have
(109)
Λ
m
-
1
u
n
-
μ
m
B
m
Λ
m
-
1
u
n
∈
(
I
+
μ
m
R
m
)
Λ
m
u
n
.
That is,
(110)
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
-
μ
m
B
m
Λ
m
-
1
u
n
)
∈
R
m
Λ
m
u
n
.
In terms of the monotonicity of R
m, we get
(111)
〈
1
μ
m
v
-
Λ
m
u
n
,
g
-
B
m
v
-
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
-
μ
m
B
m
Λ
m
-
1
u
n
)
〉
≥
0
and hence
(112)
〈
v
-
Λ
m
u
n
,
g
〉
≥
〈
1
μ
m
v
-
Λ
m
u
n
,
22222
B
m
v
+
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
-
μ
m
B
m
Λ
m
-
1
u
n
)
〉
=
〈
1
μ
m
v
-
Λ
m
u
n
,
B
m
v
-
B
m
Λ
m
u
n
+
B
m
Λ
m
u
n
22222
-
B
m
Λ
m
-
1
u
n
+
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
)
〉
≥
〈
v
-
Λ
m
u
n
,
B
m
Λ
m
u
n
-
B
m
Λ
m
-
1
u
n
〉
+
〈
v
-
Λ
m
u
n
,
1
μ
m
(
Λ
m
-
1
u
n
-
Λ
m
u
n
)
〉
.
In particular,
(113)
〈
v
-
Λ
m
u
n
i
,
g
〉
≥
〈
v
-
Λ
m
u
n
i
,
B
m
Λ
m
u
n
i
-
B
m
Λ
m
-
1
u
n
i
〉
+
〈
v
-
Λ
m
u
n
i
,
1
μ
m
(
Λ
m
-
1
u
n
i
-
Λ
m
u
n
i
)
〉
.
Since ∥
Λ
m
u
n
-
Λ
m
-
1
u
n
∥
→
0 (due to (90)) and ∥
B
m
Λ
m
u
n
-
B
m
Λ
m
-
1
u
n
∥
→
0 (due to the Lipschitz continuity of B
m), we conclude from Λ
m
u
n
i
⇀
w and μ
m
∈
(
0,2
η
m
), m
∈
{
1,2
,
…
,
N
}, that
(114)
lim
⁡
i
→
∞
〈
v
-
Λ
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 show thatw
∈
GMEP
⁡
(
Θ
,
φ
,
A
). In fact, from u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n, we know that
(115)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
K
′
(
u
n
)
-
K
′
(
x
n
)
,
y
-
u
n
〉
≥
0
,
∀
y
∈
C
.
From (H2) it follows that
(116)
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
K
′
(
u
n
)
-
K
′
(
x
n
)
,
y
-
u
n
〉
≥
Θ
(
y
,
u
n
)
,
∀
y
∈
C
.
Replacing n by n
i, we have
(117)
φ
(
y
)
-
φ
(
u
n
i
)
+
〈
A
x
n
i
,
y
-
u
n
i
〉
+
〈
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
r
n
i
,
y
-
u
n
i
〉
≥
Θ
(
y
,
u
n
i
)
,
22222222222222222222222222222222
22222
∀
y
∈
C
.
Put u
t
=
t
y
+
(
1
-
t
)
w for all t
∈
(
0,1
] and y
∈
C. Then, from (117) we have
(118)
〈
u
t
-
u
n
i
,
A
u
t
〉
≥
〈
u
t
-
u
n
i
,
A
u
t
〉
-
φ
(
u
t
)
+
φ
(
u
n
i
)
-
〈
u
t
-
u
n
i
,
A
x
n
i
〉
-
〈
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
r
n
i
,
u
t
-
u
n
i
〉
+
Θ
(
u
t
,
u
n
i
)
≥
〈
u
t
-
u
n
i
,
A
u
t
-
A
u
n
i
〉
+
〈
u
t
-
u
n
i
,
A
u
n
i
-
A
x
n
i
〉
-
φ
(
u
t
)
+
φ
(
u
n
i
)
-
〈
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
r
n
i
,
u
t
-
u
n
i
〉
+
Θ
(
u
t
,
u
n
i
)
.
Since ∥
u
n
i
-
x
n
i
∥
→
0 as i
→
∞, we deduce from the Lipschitz continuity of A and K
′ that ∥
A
u
n
i
-
A
x
n
i
∥
→
0 and ∥
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
∥
→
0 as i
→
∞. Further, from the monotonicity of A, we have 〈
u
t
-
u
n
i
,
A
u
t
-
A
u
n
i
〉
≥
0. So, from (H4), the weakly lower semicontinuity of φ
,
(
K
′
(
u
n
i
)
-
K
′
(
x
n
i
)
)
/
r
n
i
→
0, and u
n
i
⇀
w, we have
(119)
〈
u
t
-
w
,
A
u
t
〉
≥
-
φ
(
u
t
)
+
φ
(
w
)
+
Θ
(
u
t
,
w
)
,
as
i
⟶
∞
.
From (H1), (H4), and (119) we also have
(120)
0
=
Θ
(
u
t
,
u
t
)
+
φ
(
u
t
)
-
φ
(
u
t
)
≤
t
Θ
(
u
t
,
y
)
+
(
1
-
t
)
Θ
(
u
t
,
w
)
+
t
φ
(
y
)
+
(
1
-
t
)
φ
(
w
)
-
φ
(
u
t
)
=
t
[
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
]
+
(
1
-
t
)
[
Θ
(
u
t
,
w
)
+
φ
(
w
)
-
φ
(
w
)
-
φ
(
u
t
)
]
≤
t
[
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
]
+
(
1
-
t
)
〈
u
t
-
w
,
A
u
t
〉
=
t
[
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
]
+
(
1
-
t
)
t
〈
y
-
w
,
A
u
t
〉
,
and hence
(121)
0
≤
Θ
(
u
t
,
y
)
+
φ
(
y
)
-
φ
(
u
t
)
+
(
1
-
t
)
〈
y
-
w
,
A
u
t
〉
.
Letting t
→
0, we have, for each y
∈
C,
(122)
0
≤
Θ
(
w
,
y
)
+
φ
(
y
)
-
φ
(
w
)
+
〈
A
w
,
y
-
w
〉
.
This implies that w
∈
GMEP
⁡
(
Θ
,
φ
,
A
). Therefore, w
∈
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ,φ,A)∩
∩
k
=
1
M
VI
⁡
(
A
k,C
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
=
Ω. This shows that ω
w
(
x
n
)
⊂
Ω. Consequently, from (102) and x
*
∈
VI
⁡
(
Ω
,
I
-
f
), we have
(123)
limsup
⁡
n
→
∞
〈
(
f
-
I
)
x
*
,
x
n
-
x
*
〉
=
〈
(
f
-
I
)
x
*
,
w
-
x
*
〉
≤
0
.
Step  7. Finally, we show that x
n
→
x
*
∈
Ω as n
→
∞.
Indeed, in terms of (68) we get
(124)
∥
y
n
-
x
*
∥
2
≤
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
(
1
-
β
n
-
γ
n
(
γ
¯
-
1
)
)
r
n
(
r
n
-
2
ζ
)
∥
A
x
n
-
A
x
*
∥
2
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
≤
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
,
which, together with (37), implies that
(125)
∥
x
n
+
1
-
x
*
∥
2
=
∥
α
n
(
f
(
x
n
)
-
f
(
x
*
)
)
+
(
1
-
α
n
)
(
y
n
-
x
*
)
+
α
n
(
f
(
x
*
)
-
x
*
)
∥
2
≤
∥
α
n
(
f
(
x
n
)
-
f
(
x
*
)
)
+
(
1
-
α
n
)
(
y
n
-
x
*
)
∥
2
+
2
α
n
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
≤
α
n
∥
f
(
x
n
)
-
f
(
x
*
)
∥
2
+
(
1
-
α
n
)
∥
y
n
-
x
*
∥
2
+
2
α
n
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
≤
α
n
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
×
[
(
1
-
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
2
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
]
+
2
α
n
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
=
(
1
-
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
2
(
1
-
α
n
)
γ
n
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
+
2
α
n
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
=
(
1
-
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
)
∥
x
n
-
x
*
∥
2
+
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
×
[
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
(
γ
¯
-
1
)
+
2
α
n
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
]
=
(
1
-
δ
n
)
∥
x
n
-
x
*
∥
2
+
σ
n
δ
n
,
where δ
n
=
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
) and
(126)
σ
n
=
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
γ
¯
-
1
+
2
α
n
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
.
Since {
ρ
n
}
⊂
[
0,1
-
ρ
],  {
γ
n
}
⊂
[
a
,
a
^
]
⊂
(
0,1
), and ∑
n
=
1
∞
‍
α
n
=
∞, we deduce that ∑
n
=
1
∞
‍
δ
n = ∑
n
=
1
∞
‍
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
=
∞ and 2
α
n
/
(
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
)
≤
2
/
ρ
a
(
γ
¯
-
1
). Note that
(127)
limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
=
limsup
⁡
n
→
∞
[
〈
(
I
-
V
)
x
*
,
y
n
-
x
n
〉
+
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
]
≤
0
.
Hence from (123) and Lemma 18 it follows that
(128)
limsup
⁡
n
→
∞
σ
n
=
limsup
⁡
n
→
∞
[
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
(
γ
¯
-
1
)
+
2
α
n
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
5555555555
×
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
(
γ
¯
-
1
)
]
≤
limsup
⁡
n
→
∞
2
〈
(
I
-
V
)
x
*
,
y
n
-
x
*
〉
γ
¯
-
1
+
limsup
⁡
n
→
∞
2
α
n
(
1
-
α
n
)
γ
n
(
γ
¯
-
1
)
〈
(
f
-
I
)
x
*
,
x
n
+
1
-
x
*
〉
≤
0
.
Applying Lemma 13 to (125), we infer that the sequence {
x
n
} converges strongly to x
*. This completes the proof.Corollary 20.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ be a bifunction from C
×
C to R satisfying (H1)–(H4) and φ
:
C
→
R a lower semicontinuous and convex functional. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
:
H
→
H, A
1
:
C
→
H, and B
i
:
C
→
H be ζ-inverse strongly monotone, ζ
1-inverse strongly monotone, and η
i-inverse strongly monotone, respectively, where i
∈
{
1,2
,
…
,
N
}. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive mappings on H and {
λ
n
} a sequence in (
0
,
b
] for some b
∈
(
0,1
). Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
) and f
:
H
→
H a ρ-contraction with ρ
∈
(
0,1
). Let W
n be the W-mapping defined by (9). Assume that Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ,φ,A
)
∩
VI
⁡
(
C
,
A
1
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
≠
∅. Define Λ
i
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
⋯
J
R
1
,
μ
1
(
I
-
μ
1
B
1
) for each i
∈
{
1,2
,
…
,
N
}. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three sequences in [
0,1
]. Assume that(i)
K
:
H
→
R is strongly convex with constant σ
>
0 and its derivative K
′ is Lipschitz continuous with constant ν
>
0 such that the function x
↦
〈
y
-
x
,
K
′
(
x
)
〉 is weakly upper semicontinuous for each y
∈
H,
(ii)
for eachx
∈
H, there exist a bounded subset D
x
⊂
C and z
x
∈
C such that, for any y
∉
D
x,
(129)
Θ
(
y
,
z
x
)
+
φ
(
z
x
)
-
φ
(
y
)
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
x
-
y
〉
<
0
;
(iii)
{
α
n
}
⊂
[
0,1
-
ρ
] and ∑
n
=
1
∞
‍
α
n
=
∞,
(iv)
liminf
⁡
n
→
∞
γ
n
>
0 and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1,
(v)
lim
⁡
n
→
∞
(
|
α
n
+
1
/
(
1
-
(
1
-
α
n
+
1
)
β
n
+
1
)
-
α
n
/
(
1
-
(
1
-
α
n
)
β
n
)
|
+
|
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
|
)
=
0,
(vi)
{
r
n
}
⊂
[
0,2
ζ
], lim
⁡
n
→
∞
|
r
n
+
1
-
r
n
|
=
0, and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ.Given x
1
∈
H arbitrarily, the sequence {
x
n
}  is generated iteratively by
(130)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
y
n
=
β
n
x
n
+
γ
n
P
C
(
I
-
ν
1
A
1
)
u
n
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
W
n
Λ
N
u
n
,
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
,
∀
n
≥
1
,
where ν
1
∈
(
0,2
ζ
1
) and μ
i
∈
(
0,2
η
i
) for each i
∈
{
1,2
,
…
,
N
}. If α
n
(
f
(
x
n
)
-
x
n
)
→
0 and S
r
(
Θ
,
φ
) is firmly nonexpansive, then
(131)
x
n
⟶
x
*
⟺
limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
,
where x
* is a unique solution in Ω to the  VIP(132)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.Corollary 21.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let N be an integer. Let Θ be a bifunction from C
×
C to R satisfying (H1)–(H4) and φ
:
C
→
R a lower semicontinuous and convex functional. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
:
H
→
H and B
i
:
C
→
H be ζ-inverse strongly monotone and η
i-inverse strongly monotone, respectively, where i
∈
{
1,2
,
…
,
N
}. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive mappings on H and {
λ
n
} a sequence in (
0
,
b
] for some b
∈
(
0,1
). Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
) and f
:
H
→
H a ρ-contraction with ρ
∈
(
0,1
). Let W
n be the W-mapping defined by (9). Assume that Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ,φ,A
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
≠
∅. Define Λ
i
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
⋯
J
R
1
,
μ
1
(
I
-
μ
1
B
1
) for each i
∈
{
1,2
,
…
,
N
}. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three sequences in [
0,1
]. Assume that(i)
K
:
H
→
R is strongly convex with constant σ
>
0 and its derivative K
′ is Lipschitz continuous with constant ν
>
0 such that the function x
↦
〈
y
-
x
,
K
′
(
x
)
〉 is weakly upper semicontinuous for each y
∈
H,
(ii)
for eachx
∈
H, there exist a bounded subset D
x
⊂
C and z
x
∈
C such that, for any y
∉
D
x,
(133)
Θ
(
y
,
z
x
)
+
φ
(
z
x
)
-
φ
(
y
)
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
x
-
y
〉
<
0
;
(iii)
{
α
n
}
⊂
[
0,1
-
ρ
] and ∑
n
=
1
∞
‍
α
n
=
∞,
(iv)
liminf
⁡
n
→
∞
γ
n
>
0 and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1,
(v)
lim
⁡
n
→
∞
(
|
α
n
+
1
/
(
1
-
(
1
-
α
n
+
1
)
β
n
+
1
)
-
α
n
/
(
1
-
(
1
-
α
n
)
β
n
)
|
+
|
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
|
)
=
0,
(vi)
{
r
n
}
⊂
[
0,2
ζ
], lim
⁡
n
→
∞
|
r
n
+
1
-
r
n
|
=
0, and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ.Given x
1
∈
H arbitrarily, the sequence {
x
n
}  is generated iteratively by
(134)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
y
n
=
β
n
x
n
+
γ
n
W
n
u
n
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
W
n
Λ
N
u
n
,
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
,
∀
n
≥
1
,
where μ
i
∈
(
0,2
η
i
) for each i
∈
{
1,2
,
…
,
N
}. If α
n
(
f
(
x
n
)
-
x
n
)
→
0 and S
r
(
Θ
,
φ
) is firmly nonexpansive, then
(135)
x
n
⟶
x
*
⟺
limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
,
where x
* is a unique solution in Ω to the VIP(136)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.Proof.
In Theorem19, putting A
k
≡
0 for each k
=
1,2
,
…
,
M, we know that the iterative scheme (37) reduces to (134). In this case, we get ∩
k
=
1
M
VI
⁡
(
C
,
A
k
)
=
C. Utilizing Theorem 19 we derive the desired result.Corollary 22.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M be an integer. Let Θ be a bifunction from C
×
C to R satisfying (H1)–(H4) and φ
:
C
→
R a lower semicontinuous and convex functional. Let R
1
:
C
→
2
H be a maximal monotone mapping and let A
:
H
→
H, A
k
:
C
→
H, and B
1
:
C
→
H be ζ-inverse strongly monotone, ζ
k-inverse strongly monotone, and η
1-inverse strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
}. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive mappings on H and {
λ
n
} a sequence in (
0
,
b
] for some b
∈
(
0,1
). Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
) and f
:
H
→
H a ρ-contraction with ρ
∈
(
0,1
). Let W
n be the W-mapping defined by (9). Assume that Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
∩
k
=
1
M
VI
⁡
(
C
,
A
k
)
∩
I
(
B
1
,
R
1
)
≠
∅. Define Δ
k
=
P
C
(
I
-
ν
k
A
k
)
⋯
P
C
(
I
-
ν
1
A
1
) for each k
∈
{
1,2
,
…
,
M
}. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three sequences in [
0,1
]. Assume that(i)
K
:
H
→
R is strongly convex with constant σ
>
0 and its derivative K
′ is Lipschitz continuous with constant ν
>
0 such that the function x
↦
〈
y
-
x
,
K
′
(
x
)
〉 is weakly upper semicontinuous for each y
∈
H,
(ii)
for eachx
∈
H, there exist a bounded subset D
x
⊂
C and z
x
∈
C such that, for any y
∉
D
x,
(137)
Θ
(
y
,
z
x
)
+
φ
(
z
x
)
-
φ
(
y
)
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
x
-
y
〉
<
0
;
(iii)
{
α
n
}
⊂
[
0,1
-
ρ
] and ∑
n
=
1
∞
‍
α
n
=
∞,
(iv)
liminf
⁡
n
→
∞
γ
n
>
0 and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1,
(v)
lim
⁡
n
→
∞
(
|
α
n
+
1
/
(
1
-
(
1
-
α
n
+
1
)
β
n
+
1
)
-
α
n
/
(
1
-
(
1
-
α
n
)
β
n
)
|
+
|
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
|
)
=
0,
(vi)
{
r
n
}
⊂
[
0,2
ζ
], lim
⁡
n
→
∞
|
r
n
+
1
-
r
n
|
=
0, and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ.Given x
1
∈
H arbitrarily, the sequence {
x
n
}is generated iteratively by
(138)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
y
n
=
β
n
x
n
+
γ
n
W
n
Δ
M
u
n
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
W
n
J
R
1
,
μ
1
(
I
-
μ
1
B
1
)
u
n
,
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
,
∀
n
≥
1
,
where ν
k
∈
(
0,2
ζ
k
) and μ
1
∈
(
0,2
η
1
) for each k
∈
{
1,2
,
…
,
M
}. If α
n
(
f
(
x
n
)
-
x
n
)
→
0 and S
r
(
Θ
,
φ
) is firmly nonexpansive, then
(139)
x
n
⟶
x
*
⟺
limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
,
where x
* is a unique solution in Ω to the VIP(140)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.Corollary 23.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M and N be two integers. Let Θ be a bifunction from C
×
C to R satisfying (H1)–(H4) and φ
:
C
→
R a lower semicontinuous and convex functional. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
k
:
C
→
H and B
i
:
C
→
H be ζ
k-inverse strongly monotone and η
i-inverse strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive mappings on H and {
λ
n
} a sequence in (
0
,
b
] for some b
∈
(
0,1
). Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
) and f
:
H
→
H a ρ-contraction with ρ
∈
(
0,1
). Let W
n be the W-mapping defined by (9). Assume that Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
MEP
⁡
(
Θ
,
φ
)
∩
∩
k
=
1
M
VI
⁡
(
C
,
A
k
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
≠
∅. Define Δ
k
=
P
C
(
I
-
ν
k
A
k
)
⋯
P
C
(
I
-
ν
1
A
1
) and Λ
i
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
⋯
J
R
1
,
μ
1
(
I
-
μ
1
B
1
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three sequences in [
0,1
]. Assume that(i)
K
:
H
→
R is strongly convex with constant σ
>
0 and its derivative K
′ is Lipschitz continuous with constant ν
>
0 such that the function x
↦
〈
y
-
x
,
K
′
(
x
)
〉 is weakly upper semicontinuous for each y
∈
H,
(ii)
for eachx
∈
H, there exist a bounded subset D
x
⊂
C and z
x
∈
C such that, for any y
∉
D
x,
(141)
Θ
(
y
,
z
x
)
+
φ
(
z
x
)
-
φ
(
y
)
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
x
-
y
〉
<
0
;
(iii)
{
α
n
}
⊂
[
0,1
-
ρ
] and ∑
n
=
1
∞
‍
α
n
=
∞,
(iv)
liminf
⁡
n
→
∞
γ
n
>
0 and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1,
(v)
lim
⁡
n
→
∞
(
|
α
n
+
1
/
(
1
-
(
1
-
α
n
+
1
)
β
n
+
1
)
-
α
n
/
(
1
-
(
1
-
α
n
)
β
n
)
|
+
|
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
|
)
=
0,
(vi)
{
r
n
} is a bounded sequence in (
0
,
∞
) such that
(142)
liminf
⁡
n
→
∞
r
n
>
0
,
lim
⁡
n
→
∞
|
r
n
+
1
-
r
n
|
=
0
.Given x
1
∈
H arbitrarily, the sequence {
x
n
}is generated iteratively by
(143)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
1
r
n
〈
K
′
(
u
n
)
-
K
′
(
x
n
)
,
y
-
u
n
〉
≥
0
,
555555555555555555555555555
∀
y
∈
C
,
y
n
=
β
n
x
n
+
γ
n
W
n
Δ
M
u
n
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
W
n
Λ
N
u
n
,
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
,
∀
n
≥
1
,
where ν
k
∈
(
0,2
ζ
k
) and μ
i
∈
(
0,2
η
i
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. If α
n
(
f
(
x
n
)
-
x
n
)
→
0 and S
r
(
Θ
,
φ
) is firmly nonexpansive, then
(144)
x
n
⟶
x
*
⟺
limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
,
where x
* is a unique solution in Ω to the VIP(145)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.Proof.
In Theorem19, for all n
≥
1
,
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n is equivalent to
(146)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
〈
A
x
n
,
y
-
u
n
〉
+
1
r
n
〈
K
′
(
u
n
)
-
K
′
(
x
n
)
,
y
-
u
n
〉
≥
0
,
∀
y
∈
C
.
Put A
≡
0. Then it follows that
(147)
Θ
(
u
n
,
y
)
+
φ
(
y
)
-
φ
(
u
n
)
+
1
r
n
〈
K
′
(
u
n
)
-
K
′
(
x
n
)
,
y
-
u
n
〉
≥
0
,
∀
y
∈
C
.
Observe that for all ζ
∈
(
0
,
∞
)
(148)
〈
A
x
-
A
y
,
x
-
y
〉
≥
ζ
∥
A
x
-
A
y
∥
2
,
∀
x
,
y
∈
H
.
So, whenever 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ for some ζ
∈
(
0
,
∞
), we obtain the desired result by using Theorem 19.LetT
:
H
→
H be a κ-strictly pseudocontractive mapping. For recent convergence result for strictly pseudocontractive mappings, we refer to [24]. Putting A
=
I
-
T, we know that for all x
,
y
∈
H
(149)
∥
(
I
-
A
)
x
-
(
I
-
A
)
y
∥
2
≤
∥
x
-
y
∥
2
+
κ
∥
A
x
-
A
y
∥
2
.
Note that
(150)
∥
(
I
-
A
)
x
-
(
I
-
A
)
y
∥
2
=
∥
x
-
y
∥
2
+
∥
A
x
-
A
y
∥
2
-
2
〈
A
x
-
A
y
,
x
-
y
〉
.
Hence we have for all x
,
y
∈
H
(151)
〈
A
x
-
A
y
,
x
-
y
〉
≥
1
-
κ
2
∥
A
x
-
A
y
∥
2
.
Consequently, if T
:
H
→
H is a κ-strictly pseudocontractive mapping, then the mapping A
=
I
-
T is (
1
-
κ
)
/
2-inverse strongly monotone.Corollary 24.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M and N be two integers. Let Θ be a bifunction from C
×
C to R satisfying (H1)–(H4) and φ
:
C
→
R a lower semicontinuous and convex functional. Let R
i
:
C
→
2
H be a maximal monotone mapping and let T
:
H
→
H
,
A
k
:
C
→
H, and B
i
:
C
→
H be κ-strictly pseudocontractive, ζ
k-inverse strongly monotone, and η
i-inverse strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive mappings on H and {
λ
n
} a sequence in (
0
,
b
] for some b
∈
(
0,1
). Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
) and f
:
H
→
H a ρ-contraction with ρ
∈
(
0,1
). Let W
n be the W-mapping defined by (9). Assume that Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
∩
k
=
1
M
VI
⁡
(
C
,
A
k
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
≠
∅ where A
=
I
-
T. Define Δ
k
=
P
C
(
I
-
ν
k
A
k
)
⋯
P
C
(
I
-
ν
1
A
1
) and Λ
i
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
⋯
J
R
1
,
μ
1
(
I
-
μ
1
B
1
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three sequences in [
0,1
]. Assume that(i)
K
:
H
→
R is strongly convex with constant σ
>
0 and its derivative K
′ is Lipschitz continuous with constant ν
>
0 such that the function x
↦
〈
y
-
x
,
K
′
(
x
)
〉 is weakly upper semicontinuous for each y
∈
H,
(ii)
for eachx
∈
H, there exist a bounded subset D
x
⊂
C and z
x
∈
C such that, for any y
∉
D
x,
(152)
Θ
(
y
,
z
x
)
+
φ
(
z
x
)
-
φ
(
y
)
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
x
-
y
〉
<
0
;
(iii)
{
α
n
}
⊂
[
0,1
-
ρ
] and ∑
n
=
1
∞
‍
α
n
=
∞,
(iv)
liminf
⁡
n
→
∞
γ
n
>
0 and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1,
(v)
lim
⁡
n
→
∞
(
|
α
n
+
1
/
(
1
-
(
1
-
α
n
+
1
)
β
n
+
1
)
-
α
n
/
(
1
-
(
1
-
α
n
)
β
n
)
|
+
|
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
|
)
=
0,
(vi)
{
r
n
}
⊂
[
0,1
-
κ
], lim
⁡
n
→
∞
|
r
n
+
1
-
r
n
|
=
0, and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
1
-
κ. Given x
1
∈
H arbitrarily, the sequence {
x
n
}is generated iteratively by
(153)
u
n
=
S
r
n
(
Θ
,
φ
)
(
(
1
-
r
n
)
x
n
+
r
n
T
x
n
)
,
y
n
=
β
n
x
n
+
γ
n
W
n
Δ
M
u
n
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
W
n
Λ
N
u
n
,
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
,
∀
n
≥
1
,where ν
k
∈
(
0,2
ζ
k
) and μ
i
∈
(
0,2
η
i
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. If α
n
(
f
(
x
n
)
-
x
n
)
→
0 and S
r
(
Θ
,
φ
) is firmly nonexpansive, then
(154)
x
n
⟶
x
*
⟺
limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
,
where x
* is a unique solution in Ω to the VIP(155)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.Proof.
SinceT is a κ-strictly pseudocontractive mapping, the mapping A
=
I
-
T is (
1
-
κ
)
/
2-inverse strongly monotone. In this case, put ζ
=
(
1
-
κ
)
/
2. Moreover, we obtain that
(156)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
=
S
r
n
(
Θ
,
φ
)
(
x
n
-
r
n
(
I
-
T
)
x
n
)
=
S
r
n
(
Θ
,
φ
)
(
(
1
-
r
n
)
x
n
+
r
n
T
x
n
)
.
So, from Theorem 19, we obtain the desired result.Corollary 25.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M and N be two integers. Let Θ be a bifunction from C
×
C to R satisfying (H1)–(H4) and φ
:
C
→
R a lower semicontinuous and convex functional. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
:
H
→
H, A
k
:
C
→
H, and B
i
:
C
→
H be ζ-inverse strongly monotone, ζ
k-inverse strongly monotone, and η
i-inverse strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
) and f
:
H
→
H a ρ-contraction with ρ
∈
(
0,1
). Assume that Ω
:
=
GMEP
⁡
(
Θ
,
φ
,
A
)
∩
∩
k
=
1
M
VI
⁡
(
C
,
A
k
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
≠
∅. Define Δ
k
=
P
C
(
I
-
ν
k
A
k
)
⋯
P
C
(
I
-
ν
1
A
1
) and Λ
i
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
⋯
J
R
1
,
μ
1
(
I
-
μ
1
B
1
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three sequences in [
0,1
]. Assume that(i)
K
:
H
→
R is strongly convex with constant σ
>
0 and its derivative K
′ is Lipschitz continuous with constant ν
>
0 such that the function x
↦
〈
y
-
x
,
K
′
(
x
)
〉 is weakly upper semicontinuous for each y
∈
H,
(ii)
for eachx
∈
H, there exist a bounded subset D
x
⊂
C and z
x
∈
C such that, for any y
∉
D
x,
(157)
Θ
(
y
,
z
x
)
+
φ
(
z
x
)
-
φ
(
y
)
+
1
r
〈
K
′
(
y
)
-
K
′
(
x
)
,
z
x
-
y
〉
<
0
;
(iii)
{
α
n
}
⊂
[
0,1
-
ρ
] and ∑
n
=
1
∞
‍
α
n
=
∞,
(iv)
liminf
⁡
n
→
∞
γ
n
>
0 and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1;
(v)
lim
⁡
n
→
∞
(
|
α
n
+
1
/
(
1
-
(
1
-
α
n
+
1
)
β
n
+
1
)
-
α
n
/
(
1
-
(
1
-
α
n
)
β
n
)
|
+
|
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
|
)
=
0,
(vi)
{
r
n
}
⊂
[
0,2
ζ
], lim
⁡
n
→
∞
|
r
n
+
1
-
r
n
|
=
0, and 0
<
liminf
⁡
n
→
∞
r
n
≤
limsup
⁡
n
→
∞
r
n
<
2
ζ.Given x
1
∈
H arbitrarily, the sequence {
x
n
}is generated iteratively by
(158)
u
n
=
S
r
n
(
Θ
,
φ
)
(
I
-
r
n
A
)
x
n
,
y
n
=
β
n
x
n
+
γ
n
Δ
M
u
n
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
Λ
N
u
n
,
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
,
∀
n
≥
1
,
where ν
k
∈
(
0,2
ζ
k
) and μ
i
∈
(
0,2
η
i
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. If α
n
(
f
(
x
n
)
-
x
n
)
→
0 and S
r
(
Θ
,
φ
) is firmly nonexpansive, then
(159)
x
n
⟶
x
*
⟺
limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
,
where x
* is a unique solution in Ω to the VIP(160)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.Proof.
PutT
n
x
=
x for all integers n
≥
1 and all x
∈
H. Then, the desired result follows from Theorem 19.Corollary 26.
LetC be a nonempty closed convex subset of a real Hilbert space H. Let M and N be two integers. Let R
i
:
C
→
2
H be a maximal monotone mapping and let A
k
:
C
→
H and B
i
:
C
→
H be ζ
k-inverse strongly monotone and η
i-inverse strongly monotone, respectively, where k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let {
T
n
}
n
=
1
∞ be a sequence of nonexpansive mappings on H and {
λ
n
} a sequence in (
0
,
b
] for some b
∈
(
0,1
). Let V be a γ
¯-strongly positive bounded linear operator with γ
¯
∈
(
1,2
) and f
:
H
→
H a ρ-contraction with ρ
∈
(
0,1
). Let W
n be the W-mapping defined by (9). Assume that Ω
:
=
∩
n
=
1
∞
Fix
⁡
(
T
n
)
∩
∩
k
=
1
M
VI
⁡
(
C
,
A
k
)
∩
∩
i
=
1
N
I
(
B
i
,
R
i
)
≠
∅. Define Δ
k
=
P
C
(
I
-
ν
k
A
k
)
⋯
P
C
(
I
-
ν
1
A
1
) and Λ
i
=
J
R
i
,
μ
i
(
I
-
μ
i
B
i
)
⋯
J
R
1
,
μ
1
(
I
-
μ
1
B
1
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. Let {
α
n
}, {
β
n
}, and {
γ
n
} be three sequences in [
0,1
]. Assume that(i)
{
α
n
}
⊂
[
0,1
-
ρ
] and ∑
n
=
1
∞
‍
α
n
=
∞,
(ii)
liminf
⁡
n
→
∞
γ
n
>
0 and 0
<
liminf
⁡
n
→
∞
β
n
≤
limsup
⁡
n
→
∞
(
β
n
+
γ
n
∥
V
∥
)
<
1,
(iii)
lim
⁡
n
→
∞
(
|
α
n
+
1
/
(
1
-
(
1
-
α
n
+
1
)
β
n
+
1
)
-
α
n
/
(
1
-
(
1
-
α
n
)
β
n
)
|
+
|
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
|
)
=
0.Given x
1
∈
H arbitrarily, the sequence {
x
n
}is generated iteratively by
(161)
y
n
=
β
n
x
n
+
γ
n
W
n
Δ
M
x
n
+
[
(
1
-
β
n
)
I
-
γ
n
V
]
W
n
Λ
N
x
n
,
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
y
n
,
∀
n
≥
1
,
where ν
k
∈
(
0,2
ζ
k
) and μ
i
∈
(
0,2
η
i
) for each k
∈
{
1,2
,
…
,
M
} and i
∈
{
1,2
,
…
,
N
}. If α
n
(
f
(
x
n
)
-
x
n
)
→
0 and S
r
(
Θ
,
φ
) is firmly nonexpansive, then
(162)
x
n
⟶
x
*
⟺
limsup
⁡
n
→
∞
〈
(
I
-
V
)
x
*
,
x
n
-
x
*
〉
≤
0
,
where x
* is a unique solution in Ω to the VIP(163)
〈
(
I
-
f
)
x
*
,
x
*
-
x
〉
≤
0
,
∀
x
∈
Ω
.Proof.
PutΘ
(
x
,
y
)
=
0
,
φ
(
x
)
=
0, for all x
,
y
∈
C, A
x
=
0 for all x
∈
H, and r
n
=
1. Take K
(
x
)
=
(
1
/
2
)
∥
x
∥
2 for all x
∈
H. Then we get u
n
=
x
n in Theorem 19 and the conclusion follows.



---

*Source: 345212-2014-03-16.xml*