# Relaxed Extragradient Methods with Regularization for General System of Variational Inequalities with Constraints of Split Feasibility and Fixed Point Problems

Authors: L. C. Ceng; A. Petruşel; J. C. Yao
Journal: Abstract and Applied Analysis
 (2013)

---

## Abstract

We suggest and analyze relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of a general system of variational inequalities, the solution set of a split feasibility problem, and the fixed point set of a strictly pseudocontractive mapping defined on a real Hilbert space. Here the relaxed extragradient methods with regularization are based on the well-known successive approximation method, extragradient method, viscosity approximation method, regularization method, and so on. Strong convergence of the proposed algorithms under some mild conditions is established. Our results represent the supplementation, improvement, extension, and development of the corresponding results in the very recent literature.


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## Body

## 1. Introduction

Letℋ be a real Hilbert space, whose inner product and norm are denoted by 〈
·
,
·
〉 and ∥
·
∥, respectively. Let K be a nonempty closed convex subset of ℋ. The (nearest point or metric) projection from ℋ onto Kis denoted by P
K. 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.LetC and Q be nonempty closed convex subsets of infinite-dimensional real Hilbert spaces ℋ
1 and ℋ
2, respectively. The split feasibility problem (SFP) is to find a point x
* with the property:
(1)
x
*
∈
C
,
A
x
*
∈
Q
,
where A
∈
B
(
ℋ
1
,
ℋ
2
) and B
(
ℋ
1
,
ℋ
2
) denotes the family of all bounded linear operators from ℋ
1 to ℋ
2.In 1994, the SFP was first introduced by Censor and Elfving [1], in finite-dimensional Hilbert spaces, for modeling inverse problems which arise from phase retrievals and in medical image reconstruction. A number of image reconstruction problems can be formulated as the SFP; see, for example, [2] and the references therein. Recently, it was found that the SFP can also be applied to study intensity-modulated radiation therapy; see, for example, [3–5] and the references therein. In the recent past, a wide variety of iterative methods have been used in signal processing and image reconstruction and for solving the SFP; see, for example, [2–12] and the references therein. A special case of the SFP is the following convex constrained linear inverse problem [13] of finding an element x such that
(2)
x
∈
C
,
A
x
=
b
.
It has been extensively investigated in the literature using the projected Landweber iterative method [14]. Comparatively, the SFP has received much less attention so far, due to the complexity resulting from the set Q. Therefore, whether various versions of the projected Landweber iterative method [14] can be extended to solve the SFP remains an interesting open topics. For example, it is yet not clear whether the dual approach to (1.2) of [15] can be extended to the SFP. The original algorithm given in [1] involves the computation of the inverse A
-
1 (assuming the existence of the inverse of A) and thus has not become popular. A seemingly more popular algorithm that solves the SFP is the C
Q algorithm of Byrne [2, 7] which is found to be a gradient-projection method (GPM) in convex minimization. It is also a special case of the proximal forward-backward splitting method [16]. The C
Q algorithm only involves the computation of the projections P
C and P
Q onto the sets C and Q, respectively, and is therefore implementable in the case where P
C and P
Q have closed-form expressions, for example, C and Q are closed balls or half-spaces. However, it remains a challenge how to implement the C
Q algorithm in the case where the projections P
C and/or P
Q fail to have closed-form expressions, though theoretically we can prove the (weak) convergence of the algorithm.Very recently, Xu [6] gave a continuation of the study on the C
Q algorithm and its convergence. He applied Mann’s algorithm to the SFP and purposed an averaged C
Q algorithm which was proved to be weakly convergent to a solution of the SFP. He also established the strong convergence result, which shows that the minimum-norm solution can be obtained.Furthermore, Korpelevič [17] introduced the so-called extragradient method for finding a solution of a saddle point problem. He proved that the sequences generated by the proposed iterative algorithm converge to a solution of the saddle point problem.Throughout this paper, assume that the SFP is consistent, that is, the solution setΓ of the SFP is nonempty. Let f
:
ℋ
1
→
R be a continuous differentiable function. The minimization problem
(3)
min
x
∈
C
f
(
x
)
:
=
∥
1
2
A
x
-
P
Q
A
x
∥
2
is ill posed. Therefore, Xu [6] considered the following Tikhonov regularization problem:
(4)
min
x
∈
C
f
α
(
x
)
:
=
1
2
∥
A
x
-
P
Q
A
x
∥
2
+
1
2
α
∥
x
∥
2
,
where α
>
0 is the regularization parameter. The regularized minimization (4) has a unique solution which is denoted by x
α. The following results are easy to prove.Proposition 1 (see [18, Proposition  3.1]).
Givenx
*
∈
ℋ
1, the following statements are equivalent:(i)
x
* solves the SFP;
(ii)
x
* solves the fixed point equation
(5)
P
C
(
I
-
λ
∇
f
)
x
*
=
x
*
,
where λ
>
0, ∇
f
=
A
*
(
I
-
P
Q
)
A and A
* is the adjoint of A;
(iii)
x
* solves the variational inequality problem (VIP) of finding x
*
∈
C such that
(6)
〈
∇
f
(
x
*
)
,
x
-
x
*
〉
≥
0
,
∀
x
∈
C
.It is clear from Proposition1 that
(7)
Γ
=
Fix
(
P
C
(
I
-
λ
∇
f
)
)
=
VI
(
C
,
∇
f
)
,
for all λ
>
0, where Fix
(
P
C
(
I
-
λ
∇
f
)
) and VI
(
C
,
∇
f
) denote the set of fixed points of P
C
(
I
-
λ
∇
f
) and the solution set of VIP (6), respectively.Proposition 2 (see [18]).
There hold the following statements:(i)
the gradient
(8)
∇
f
α
=
∇
f
+
α
I
=
A
*
(
I
-
P
Q
)
A
+
α
I
is (
α
+
∥
A
∥
2
)-Lipschitz continuous and α-strongly monotone;
(ii)
the mapping P
C
(
I
-
λ
∇
f
α
) is a contraction with coefficient
(9)
1
-
λ
(
2
α
-
λ
(
∥
A
∥
2
+
α
)
2
)
(
≤
1
-
α
λ
≤
1
-
1
2
α
λ
)
,
where 0
<
λ
≤
α
/
(
∥
A
∥
2
+
α
)
2;
(iii)
if the SFP is consistent, then the strong lim
α
→
0
x
α exists and is the minimum-norm solution of the SFP.Very recently, by combining the regularization method and extragradient method due to Nadezhkina and Takahashi [19], Ceng et al. [18] proposed an extragradient algorithm with regularization and proved that the sequences generated by the proposed algorithm converge weakly to an element of Fix
(
S
)
∩
Γ, where S
:
C
→
C is a nonexpansive mapping.Theorem 3 (see [18, Theorem  3.1]).
LetS
:
C
→
C be a nonexpansive mapping such that Fix
(
S
)
∩
Γ
≠
∅. Let {
x
n
} and {
y
n
} the sequences in C generated by the following extragradient algorithm:
(10)
x
0
=
x
∈
C
c
h
o
s
e
n
a
r
b
i
t
r
a
r
i
l
y
,
y
n
=
P
C
(
x
n
-
λ
n
∇
f
α
n
(
x
n
)
)
,
x
n
+
1
=
β
n
x
n
+
(
1
-
β
n
)
S
P
C
(
x
n
-
λ
n
∇
f
α
n
(
y
n
)
)
,
∀
n
≥
0
,
where ∑
n
=
0
∞
‍
α
n
<
∞
,
{
λ
n
}
⊂
[
a
,
b
] for some a
,
b
∈
(
0
,
1
/
∥
A
∥
2
) and {
β
n
}
⊂
[
c
,
d
] for some c
,
d
∈
(
0,1
). Then, both the sequences {
x
n
} and {
y
n
} converge weakly to an element x
^
∈
Fix
(
S
)
∩
Γ.On the other hand, assume thatC is a nonempty closed convex subset of ℋ and A
:
C
→
ℋ is a mapping. The classical variational inequality problem (VIP) is to find x
*
∈
C such that
(11)
〈
A
x
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
C
.
It is now well known that the variational inequalities are equivalent to the fixed point problems, the origin of which can be traced back to Lions and Stampacchia [20]. This alternative formulation has been used to suggest and analyze projection iterative method for solving variational inequalities under the conditions that the involved operator must be strongly monotone and Lipschitz continuous. Related to the variational inequalities, we have the problem of finding the fixed points of nonexpansive mappings or strict pseudocontraction mappings, which is the current interest in functional analysis. Several people considered a unified approach to solve variational inequality problems and fixed point problems; see, for example, [21–28] and the references therein.A mappingA
:
C
→
ℋ is said to be an α-inverse strongly monotone if there exists α
>
0 such that
(12)
〈
A
x
-
A
y
,
x
-
y
〉
≥
α
∥
A
x
-
A
y
∥
2
,
∀
x
,
y
∈
C
.A mappingS
:
C
→
C is said to be k-strictly pseudocontractive if there exists 0
≤
k
<
1 such that
(13)
∥
S
x
-
S
y
∥
2
≤
∥
x
-
y
∥
2
+
k
∥
(
I
-
S
)
x
-
(
I
-
S
)
y
∥
2
,
∀
x
,
y
∈
C
.
In this case, we also say that S is a k-strict pseudo-contraction. In particular, whenever k
=
0, S becomes a nonexpansive mapping from C into itself. It is clear that every inverse strongly monotone mapping is a monotone and Lipschitz continuous mapping. We denote by VI
(
C
,
A
) and Fix
(
S
) the solution set of problem (11) and the set of all fixed points of S, respectively.For finding an element ofFix
(
S
)
∩
VI
(
C
,
A
) under the assumption that a set C
⊂
ℋ is nonempty, closed, and convex, a mapping S
:
C
→
C is nonexpansive, and a mapping A
:
C
→
ℋ is α-inverse strongly monotone, Takahashi and Toyoda [29] introduced an iterative scheme and studied the weak convergence of the sequence generated by the proposed scheme to a point of Fix
(
S
)
∩
VI
(
C
,
A
). Recently, Iiduka and Takahashi [30] presented another iterative scheme for finding an element of Fix
(
S
)
∩
VI
(
C
,
A
) and showed that the sequence generated by the scheme converges strongly to P
Fix
(
S
)
∩
VI
(
C
,
A
)
u, where u is the initially chosen point in the iterative scheme and P
K denotes the metric projection of ℋ onto K.Based on Korpelevič’s extragradient method [17], Nadezhkina and Takahashi [19] introduced an iterative process for finding an element of Fix
(
S
)
∩
VI
(
C
,
A
) and proved the weak convergence of the sequence to a point of Fix
(
S
)
∩
VI
(
C
,
A
). Zeng and Yao [27] presented an iterative scheme for finding an element of Fix
(
S
)
∩
VI
(
C
,
A
) and proved that two sequences generated by the method converges strongly to an element of Fix
(
S
)
∩
VI
(
C
,
A
). Recently, Bnouhachem et al. [31] suggested and analyzed an iterative scheme for finding a common element of the fixed point set Fix
(
S
) of a nonexpansive mapping S and the solution set VI
(
C
,
A
) of the variational inequality (11) for an inverse strongly monotone mapping A
:
C
→
ℋ.Furthermore, as a much more general generalization of the classical variational inequality problem (11), Ceng et al. [23] introduced and considered the following problem of finding (
x
*
,
y
*
)
∈
C
×
C such that
(14)
〈
μ
1
B
1
y
*
+
x
*
-
y
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
C
,
〈
μ
2
B
2
x
*
+
y
*
-
x
*
,
x
-
y
*
〉
≥
0
,
∀
x
∈
C
,
which is called a general system of variational inequalities (GSVI), which μ
1
>
0 and μ
2
>
0 are two constants. The set of solutions of problem (14) is denoted by GSVI
(
C
,
B
1
,
B
2
). In particular, if B
1
=
B
2, then problem (14) reduces to the new system of variational inequalities, introduced and studied by Verma [32]. Recently, Ceng et al. [23] transformed problem (14) into a fixed point problem in the following way.Lemma 4 (see [23]).
For givenx
-
,
y
-
∈
C
,
(
x
-
,
y
-
) is a solution of problem (14) if and only if x
- is a fixed point of the mapping G
:
C
→
C defined by
(15)
G
(
x
)
=
P
C
[
P
C
(
x
-
μ
2
B
2
x
)
-
μ
1
B
1
P
C
(
x
-
μ
2
B
2
x
)
]
,
∀
x
∈
C
,
where y
-
=
P
C
(
x
-
-
μ
2
B
2
x
-
).
In particular, if the mappingB
i
:
C
→
ℋ is β
i-inverse strongly monotone for i
=
1,2, then the mapping G is nonexpansive provided μ
i
∈
(
0,2
β
i
) for i
=
1,2.Utilizing Lemma4, they introduced and studied a relaxed extragradient method for solving problem (14). Throughout this paper, the set of fixed points of the mapping G is denoted by Ξ. Based on the relaxed extragradient method and viscosity approximation method, Yao et al. [26] proposed and analyzed an iterative algorithm for finding a common solution of the GSVI (14) and the fixed point problem of a strictly pseudo-contractive mapping S
:
C
→
C. Subsequently, Ceng et al. [33] further presented and analyzed an iterative scheme for finding a common element of the solution set of the VIP (11), the solution set of the GSVI (14), and the fixed point set of a strictly pseudo-contractive mapping S
:
C
→
C.Theorem 5 (see [33, Theorem  3.1]).
LetC be a nonempty closed convex subset of a real Hilbert space ℋ. Let A
:
C
→
ℋ be α-inverse strongly monotone and B
i
:
C
→
ℋ be β
i-inverse strongly monotone for i
=
1,2. Let S
:
C
→
C be a k-strictly pseudo-contractive mapping such that Fix
(
S
)
∩
Ξ
∩
V
I
(
C
,
A
)
≠
∅. Let Q
:
C
→
C be a ρ-contraction with ρ
∈
[
0
,
1
/
2
). For given x
0
∈
C arbitrarily, let the sequences {
x
n
}
,
{
u
n
}
,
{
u
~
n
} be generated by the relaxed extragradient iterative scheme:
(16)
u
n
=
P
C
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
,
u
~
n
=
P
C
(
u
n
-
λ
n
A
u
n
)
,
y
n
=
α
n
Q
x
n
+
(
1
-
α
n
)
P
C
(
u
n
-
λ
n
A
u
~
n
)
,
x
n
+
1
=
β
n
x
n
+
γ
n
y
n
+
δ
n
S
y
n
,
∀
n
≥
0
,
where μ
i
∈
(
0,2
β
i
) for i
=
1,2, and the following conditions hold for five sequences {
λ
n
}
⊂
(
0
,
α
] and {
α
n
}
,
{
β
n
}
,
{
γ
n
}
,
{
δ
n
}
⊂
[
0,1
]:(i)
β
n
+
γ
n
+
δ
n
=
1 and (
γ
n
+
δ
n
)
k
≤
γ
n for all n
≥
0;
(ii)
lim
n
→
∞
α
n
=
0 and ∑
n
=
0
∞
‍
α
n
=
∞;
(iii)
0
<
lim
inf
n
→
∞
β
n
≤
limsup
n
→
∞
β
n
<
1 and liminf
n
→
∞
δ
n
>
0;
(iv)
lim
n
→
∞
(
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
)
=
0;
(v)
0
<
lim
inf
n
→
∞
λ
n
≤
limsup
n
→
∞
λ
n
<
α and lim
n
→
∞
|
λ
n
+
1
-
λ
n
|
=
0.
Then the sequences{
x
n
}
,
{
u
n
}
,
{
u
~
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Ξ
∩
V
I
(
C
,
A
)
Q
x
- if and only if lim
n
→
∞
∥
u
n
+
1
-
u
n
∥
=
0. Furthermore, (
x
-
,
y
-
) is a solution of the GSVI (14), where y
-
=
P
C
(
x
-
-
μ
2
B
2
x
-
).Motivated and inspired by the research going on this area, we propose and analyze the following relaxed extragradient iterative algorithms with regularization for finding a common element of the solution set of the GSVI (14), the solution set of the SFP (1), and the fixed point set of a strictly pseudo-contractive mapping S
:
C
→
C.Algorithm 6.
Letμ
i
∈
(
0,2
β
i
) for i
=
1,2
,
{
α
n
}
⊂
(
0
,
∞
)
,
{
λ
n
}
⊂
(
0
,
1
/
∥
A
∥
2
) and {
σ
n
}
,
{
β
n
}
,
{
γ
n
}
,
{
δ
n
}
⊂
[
0,1
] such that β
n
+
γ
n
+
δ
n
=
1 for all n
≥
0. For given x
0
∈
C arbitrarily, let {
x
n
}
,
{
u
n
}
,
{
u
~
n
} be the sequences generated by the following relaxed extragradient iterative scheme with regularization:
(17)
u
n
=
P
C
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
,
u
~
n
=
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
n
)
)
,
y
n
=
σ
n
Q
x
n
+
(
1
-
σ
n
)
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
,
x
n
+
1
=
β
n
x
n
+
γ
n
y
n
+
δ
n
S
y
n
,
∀
n
≥
0
.
Under mild assumptions, it is proven that the sequences{
x
n
}
,
{
u
n
}
,
{
u
~
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Ξ
∩
Γ
Q
x
- if and only if lim
n
→
∞
∥
u
n
+
1
-
u
n
∥
=
0. Furthermore, (
x
-
,
y
-
) is a solution of the GSVI (14), where y
-
=
P
C
(
x
-
-
μ
2
B
2
x
-
).Algorithm 7.
Letμ
i
∈
(
0,2
β
i
) for i
=
1,2
,
{
α
n
}
⊂
(
0
,
∞
)
,
{
λ
n
}
⊂
(
0
,
1
/
∥
A
∥
2
) and {
σ
n
}
,
{
τ
n
}
,
{
β
n
}
,
{
γ
n
}
,
{
δ
n
}
⊂
[
0,1
] such that σ
n
+
τ
n
≤
1 and β
n
+
γ
n
+
δ
n
=
1 for all n
≥
0. For given x
0
∈
C arbitrarily, let {
x
n
}
,
{
y
n
}
,
{
z
n
} be the sequences generated by the following relaxed extragradient iterative scheme with regularization:
(18)
z
n
=
P
C
(
x
n
-
λ
n
∇
f
α
n
(
x
n
)
)
,
y
n
=
σ
n
Q
x
n
+
τ
n
P
C
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
+
(
1
-
σ
n
-
τ
n
)
P
C
[
P
C
(
z
n
-
μ
2
B
2
z
n
)
-
μ
1
B
1
P
C
(
z
n
-
μ
2
B
2
z
n
)
]
,
x
n
+
1
=
β
n
x
n
+
γ
n
y
n
+
δ
n
S
y
n
,
∀
n
≥
0
,
Also, under appropriate conditions, it is shown that the sequences{
x
n
}
,
{
y
n
}
,
{
z
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Ξ
∩
Γ
Q
x
- if and only if lim
n
→
∞
∥
z
n
+
1
-
z
n
∥
=
0. Furthermore, (
x
-
,
y
-
) is a solution of the GSVI (14), where y
-
=
P
C
(
x
-
-
μ
2
B
2
x
-
).
Note that both [6, Theorem  5.7] and [18, Theorem  3.1] are weak convergence results for solving the SFP (1). Beyond question our strong convergence results are very interesting and quite valuable. Because our relaxed extragradient iterative schemes (17) and (18) with regularization involve a contractive self-mapping Q, a k-strictly pseudo-contractive self-mapping S and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [6, Theorem  5.7] and [18, Theorem  3.1], respectively. Furthermore, the relaxed extragradient iterative scheme (16) is extended to develop our relaxed extragradient iterative schemes (17) and (18) with regularization. All in all, our results represent the modification, supplementation, extension, and improvement of [6, Theorem  5.7], [18, Theorem  3.1], and [33, Theorem  3.1].


## 2. Preliminaries

LetK be a nonempty, closed, and convex subset of a real Hilbert space ℋ. Now we present some known results and definitions which will be used in the sequel.The metric (or nearest point) projection fromℋ onto K is the mapping P
K
:
ℋ
→
K which assigns to each point x
∈
ℋ the unique point P
K
x
∈
K satisfying the property
(19)
∥
x
-
P
K
x
∥
=
inf
y
∈
K
∥
x
-
y
∥
=
:
d
(
x
,
K
)
.The following properties of projections are useful and pertinent to our purpose.Proposition 8 (see [34]).
For givenx
∈
ℋ and z
∈
K:(i)
z
=
P
K
x
⇔
〈
x
-
z
,
y
-
z
〉
≤
0
,
∀
y
∈
K;
(ii)
z
=
P
K
x
⇔
∥
x
-
z
∥
2
≤
∥
x
-
y
∥
2
-
∥
y
-
z
∥
2
,
∀
y
∈
K;
(iii)
〈
P
K
x
-
P
K
y
,
x
-
y
〉
≥
∥
P
K
x
-
P
K
y
∥
2
,
∀
y
∈
ℋ, which hence implies that P
K is nonexpansive and monotone.Definition 9.
A mappingT
:
ℋ
→
ℋ is said to be(a)
nonexpansive if(20)
∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥
,
∀
x
,
y
∈
ℋ
;
(b)
firmly nonexpansive if2
T
-
I is nonexpansive, or equivalently,
(21)
〈
x
-
y
,
T
x
-
T
y
〉
≥
∥
T
x
-
T
y
∥
2
,
∀
x
,
y
∈
ℋ
;alternatively, T is firmly nonexpansive if and only if T can be expressed as
(22)
T
=
1
2
(
I
+
S
)
,
where S
:
ℋ
→
ℋ is nonexpansive; projections are firmly nonexpansive.Definition 10.
LetT be a nonlinear operator with domain D
(
T
)
⊆
ℋ and range R
(
T
)
⊆
ℋ.(a)
T is said to be monotone if
(23)
〈
x
-
y
,
T
x
-
T
y
〉
≥
0
,
∀
x
,
y
∈
D
(
T
)
.
(b)
Given a numberβ
>
0, T is said to be β-strongly monotone if
(24)
〈
x
-
y
,
T
x
-
T
y
〉
≥
β
∥
x
-
y
∥
2
,
∀
x
,
y
∈
D
(
T
)
.
(c)
Given a numberν
>
0, T is said to be ν-inverse strongly monotone (ν-ism) if
(25)
〈
x
-
y
,
T
x
-
T
y
〉
≥
ν
∥
T
x
-
T
y
∥
2
,
∀
x
,
y
∈
D
(
T
)
.
It can be easily seen that ifS is nonexpansive, then I
-
S is monotone. It is also easy to see that a projection P
K is 1-ism.
Inverse strongly monotone (also referred to as cocoercive) operators have been applied widely in solving practical problems in various fields, for instance, in traffic assignment problems; see, for example, [35, 36].Definition 11.
A mappingT
:
ℋ
→
ℋ is said to be an averaged mapping if it can be written as the average of the identity I and a nonexpansive mapping, that is,
(26)
T
≡
(
1
-
α
)
I
+
α
S
,
where α
∈
(
0,1
) and S
:
ℋ
→
ℋis nonexpansive. More precisely, when the last equality holds, we say that T is α-averaged. Thus firmly nonexpansive mappings (in particular, projections) are 1
/
2-averaged maps.Proposition 12 (see [7]).
LetT
:
ℋ
→
ℋ be a given mapping.(i)
T is nonexpansive if and only if the complement I
-
T is 1
/
2-ism.
(ii)
IfT is ν-ism, then for γ
>
0,  γ
T is ν
/
γ-ism.
(iii)
T is averaged if and only if the complement I
-
T is ν-ism for some ν
>
1
/
2. Indeed, for α
∈
(
0,1
)
,
T is α-averaged if and only if I
-
T is 1
/
2
α-ism.Proposition 13 (see [7, 37]).
LetS
,
T
,
V
:
ℋ
→
ℋ be given operators.(i)
If T
=
(
1
-
α
)
S
+
α
V for some α
∈
(
0,1
) and if S is averaged and V is nonexpansive, then T is averaged.
(ii)
T is firmly nonexpansive if and only if the complement I
-
T is firmly nonexpansive.
(iii)
If T
=
(
1
-
α
)
S
+
α
V for some α
∈
(
0,1
) and if S is firmly nonexpansive and V is nonexpansive, then T is averaged.
(iv)
The composite of finitely many averaged mappings is averaged. That is, if each of the mappings {
T
i
}
i
=
1
N is averaged, then so is the composite T
1
∘
T
2
∘
⋯
∘
T
N. In particular, if T
1 is α
1-averaged and T
2 is α
2-averaged, where α
1
,
α
2
∈
(
0,1
), then the composite T
1
∘
T
2 is α-averaged, where α
=
α
1
+
α
2
-
α
1
α
2.
(v)
If the mappings {
T
i
}
i
=
1
N are averaged and have a common fixed point, then
(27)
⋂
i
=
1
N
‍
Fix
(
T
i
)
=
Fix
(
T
1
⋯
T
N
)
.The notation Fix
(
T
) denotes the set of all fixed points of the mapping T, that is, Fix
(
T
)
=
{
x
∈
ℋ
:
T
x
=
x
}.It is clear that, in a real Hilbert spaceℋ, S
:
C
→
C is k-strictly pseudo-contractive if and only if there holds the following inequality:
(28)
〈
S
x
-
S
y
,
x
-
y
〉
≤
∥
x
-
y
∥
2
-
1
-
k
2
∥
(
I
-
S
)
x
-
(
I
-
S
)
y
∥
2
,
∀
x
,
y
∈
C
.
This immediately implies that if S is a k-strictly pseudo-contractive mapping, then I
-
S is (
1
-
k
)
/
2-inverse strongly monotone; for further detail, we refer to [38] and the references therein. It is well known that the class of strict pseudo-contractions strictly includes the class of nonexpansive mappings.In order to prove the main result of this paper, the following lemmas will be required.Lemma 14 (see [39]).
Let{
x
n
} and {
y
n
} be bounded sequences in a Banach space X and let {
β
n
} be a sequence in [
0,1
] with 0
<
liminf
n
→
∞
β
n
≤
limsup
n
→
∞
β
n
<
1. Suppose x
n
+
1
=
(
1
-
β
n
)
y
n
+
β
n
x
n for all integers n
≥
0 and limsup
n
→
∞
(
∥
y
n
+
1
-
y
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
0. Then, lim
n
→
∞
∥
y
n
-
x
n
∥
=
0.Lemma 15 (see [38, Proposition  2.1]).
LetC be a nonempty closed convex subset of a real Hilbert space ℋ and S
:
C
→
C a mapping.(i)
IfS is a k-strict pseudo-contractive mapping, then S satisfies the Lipschitz condition
(29)
∥
S
x
-
S
y
∥
≤
1
+
k
1
-
k
∥
x
-
y
∥
,
∀
x
,
y
∈
C
.
(ii)
IfS is a k-strict pseudo-contractive mapping, then the mapping I
-
S is semiclosed at 0, that is, if {
x
n
} is a sequence in C such that x
n
→
x
~ weakly and (
I
-
S
)
x
n
→
0 strongly, then (
I
-
S
)
x
~
=
0.
(iii)
IfS is k-(quasi-)strict pseudo-contraction, then the fixed point set Fix
(
S
) of S is closed and convex so that the projection P
Fix
(
S
) is well defined.The following lemma plays a key role in proving strong convergence of the sequences generated by our algorithms.Lemma 16 (see [34]).
Let{
a
n
} be a sequence of nonnegative real numbers satisfying the property
(30)
a
n
+
1
≤
(
1
-
s
n
)
a
n
+
s
n
t
n
+
r
n
,
∀
n
≥
0
,
where {
s
n
}
⊂
(
0,1
] and {
t
n
} are such that(i)
∑
n
=
0
∞
‍
s
n
=
∞;
(ii)
eitherlimsup
n
→
∞
t
n
≤
0 or ∑
n
=
0
∞
‍
|
s
n
t
n
|
<
∞;
(iii)
∑
n
=
0
∞
‍
r
n
<
∞ where r
n
≥
0
,
∀
n
≥
0. Then, lim
n
→
∞
a
n
=
0.Lemma 17 (see [26]).
LetC be a nonempty closed convex subset of a real Hilbert space ℋ. Let S
:
C
→
C be a k-strictly pseudo-contractive mapping. Let γ and δ be two nonnegative real numbers such that (
γ
+
δ
)
k
≤
γ. Then
(31)
∥
γ
(
x
-
y
)
+
δ
(
S
x
-
S
y
)
∥
≤
(
γ
+
δ
)
∥
x
-
y
∥
,
∀
x
,
y
∈
C
.The following lemma is an immediate consequence of an inner product.Lemma 18.
In a real Hilbert spaceℋ, there holds the inequality
(32)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
x
+
y
〉
,
∀
x
,
y
∈
ℋ
.LetK be a nonempty closed convex subset of a real Hilbert space ℋ and let F
:
K
→
ℋ be a monotone mapping. The variational inequality problem (VIP) is to find x
∈
K such that
(33)
〈
F
x
,
y
-
x
〉
≥
0
,
∀
y
∈
K
.
The solution set of the VIP is denoted by VI
(
K
,
F
). It is well known that
(34)
x
∈
VI
(
K
,
F
)
⟺
x
=
P
K
(
x
-
λ
F
x
)
,
∀
λ
>
0
.A set-valued mappingT
:
ℋ
→
2
ℋ is called monotone if for all x
,
y
∈
ℋ
,
f
∈
T
x and g
∈
T
y imply that 〈
x
-
y
,
f
-
g
〉
≥
0. A monotone set-valued mapping T
:
ℋ
→
2
ℋ is called maximal if its graph Gph
(
T
) is not properly contained in the graph of any other monotone set-valued mapping. It is known that a monotone set-valued mapping T
:
ℋ
→
2
ℋ is maximal if and only if for (
x
,
f
)
∈
ℋ
×
ℋ
,
〈
x
-
y
,
f
-
g
〉
≥
0 for every (
y
,
g
)
∈
Gph
(
T
) implies that f
∈
T
x. Let F
:
K
→
ℋ be a monotone and Lipschitz continuous mapping and let N
K
v be the normal cone to K at v
∈
K, that is,
(35)
N
K
v
=
{
w
∈
ℋ
:
〈
v
-
u
,
w
〉
≥
0
,
∀
u
∈
K
}
.
Define
(36)
T
v
=
{
F
v
+
N
K
v
,
if
v
∈
K
,
∅
,
if
v
∉
K
.
It is known that in this case the mapping T is maximal monotone, and 0
∈
T
v if and only if v
∈
VI
(
K
,
F
); see further details, one refers to [40] and the references therein.


## 3. Main Results

In this section, we first prove the strong convergence of the sequences generated by the relaxed extragradient iterative algorithm (17) with regularization.Theorem 19.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ
1. Let A
∈
B
(
ℋ
1
,
ℋ
2
), and let B
i
:
C
→
ℋ
1 be β
i-inverse strongly monotone for i
=
1,2. Let S
:
C
→
C be a k-strictly pseudo-contractive mapping such that Fix
(
S
)
∩
Ξ
∩
Γ
≠
∅. Let Q
:
C
→
C be a ρ-contraction with ρ
∈
[
0
,
1
/
2
). For given x
0
∈
C arbitrarily, let the sequences {
x
n
}
,
{
u
n
}
,
{
u
~
n
} be generated by the relaxed extragradient iterative algorithm (17) with regularization, where μ
i
∈
(
0,2
β
i
) for i
=
1,2, {
α
n
}
⊂
(
0
,
∞
)
,
{
λ
n
}
⊂
(
0
,
1
/
∥
A
∥
2
) and {
σ
n
}
,
{
β
n
}
,
{
γ
n
}
,
{
δ
n
}
⊂
[
0,1
] such that(i)
∑
n
=
0
∞
‍
α
n
<
∞;
(ii)
β
n
+
γ
n
+
δ
n
=
1 and (
γ
n
+
δ
n
)
k
≤
γ
n for all n
≥
0;
(iii)
lim
n
→
∞
σ
n
=
0 and ∑
n
=
0
∞
‍
σ
n
=
∞;
(iv)
0
<
lim
inf
n
→
∞
β
n
≤
lim
sup
n
→
∞
β
n
<
1 and liminf
n
→
∞
δ
n
>
0;
(v)
lim
n
→
∞
(
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
)
=
0;
(vi)
0
<
lim
inf
n
→
∞
λ
n
≤
lim
sup
n
→
∞
λ
n
<
1
/
∥
A
∥
2 and lim
n
→
∞
|
λ
n
+
1
-
λ
n
|
=
0.
Then the sequences{
x
n
}
,
{
u
n
}
,
{
u
~
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Ξ
∩
Γ
Q
x
- if and only if lim
n
→
∞
∥
u
n
+
1
-
u
n
∥
=
0. Furthermore, (
x
-
,
y
-
) is a solution of the GSVI (14), where y
-
=
P
C
(
x
-
-
μ
2
B
2
x
-
).Proof.
First, taking into account0
<
liminf
n
→
∞
λ
n
≤
limsup
n
→
∞
λ
n
<
1
/
∥
A
∥
2, without loss of generality we may assume that {
λ
n
}
⊂
[
a
,
b
] for some a
,
b
∈
(
0
,
1
/
∥
A
∥
2
).
Now, let us show thatP
C
(
I
-
λ
∇
f
α
) is ζ-averaged for each λ
∈
(
0
,
2
/
(
α
+
∥
A
∥
2
)
), where
(37)
ζ
=
2
+
λ
(
α
+
∥
A
∥
2
)
4
.
Indeed, it is easy to see that∇
f
=
A
*
(
I
-
P
Q
)
A is 1
/
∥
A
∥
2-ism, that is,
(38)
〈
∇
f
(
x
)
-
∇
f
(
y
)
,
x
-
y
〉
≥
1
∥
A
∥
2
∥
∇
f
(
x
)
-
∇
f
(
y
)
∥
2
.
Observe that
(39)
(
α
+
∥
A
∥
2
)
〈
∇
f
α
(
x
)
-
∇
f
α
(
y
)
,
x
-
y
〉
=
(
α
+
∥
A
∥
2
)
[
α
∥
x
-
y
∥
2
=
(
α
+
∥
A
∥
2
)
+
〈
∇
f
(
x
)
-
∇
f
(
y
)
,
x
-
y
〉
{
∥
x
-
y
∥
2
}
]
=
α
2
∥
x
-
y
∥
2
+
α
〈
∇
f
(
x
)
-
∇
f
(
y
)
,
x
-
y
〉
+
α
∥
A
∥
2
∥
x
-
y
∥
2
+
∥
A
∥
2
〈
∇
f
(
x
)
-
∇
f
(
y
)
,
x
-
y
〉
≥
α
2
∥
x
-
y
∥
2
+
2
α
〈
∇
f
(
x
)
-
∇
f
(
y
)
,
x
-
y
〉
+
∥
∇
f
(
x
)
-
∇
f
(
y
)
∥
2
=
∥
α
(
x
-
y
)
+
∇
f
(
x
)
-
∇
f
(
y
)
∥
2
=
∥
∇
f
α
(
x
)
-
∇
f
α
(
y
)
∥
2
.
Hence, it follows that ∇
f
α
=
α
I
+
A
*
(
I
-
P
Q
)
A is 1
/
(
α
+
∥
A
∥
2
)-ism. Thus, λ
∇
f
α is 1
/
λ
(
α
+
∥
A
∥
2
)-ism according to Proposition 12(ii). By Proposition 12(iii) the complement I
-
λ
∇
f
α is λ
(
α
+
∥
A
∥
2
)
/
2-averaged. Therefore, noting that P
C is 1
/
2-averaged, and utilizing Proposition 13(iv), we know that for each λ
∈
(
0
,
2
/
(
α
+
∥
A
∥
2
)
)
,
P
C
(
I
-
λ
∇
f
α
) is ζ-averaged with
(40)
ζ
=
1
2
+
λ
(
α
+
∥
A
∥
2
)
2
-
1
2
·
λ
(
α
+
∥
A
∥
2
)
2
=
2
+
λ
(
α
+
∥
A
∥
2
)
4
∈
(
0,1
)
.
This shows that P
C
(
I
-
λ
∇
f
α
) is nonexpansive. Furthermore, for {
λ
n
}
⊂
[
a
,
b
] with a
,
b
∈
(
0
,
1
/
∥
A
∥
2
), we have
(41)
a
≤
inf
n
≥
0
λ
n
≤
sup
n
≥
0
λ
n
≤
b
<
1
∥
A
∥
2
=
lim
n
→
∞
1
α
n
+
∥
A
∥
2
.
Without loss of generality we may assume that
(42)
a
≤
inf
n
≥
0
λ
n
≤
sup
n
≥
0
λ
n
≤
b
<
1
α
n
+
∥
A
∥
2
,
∀
n
≥
0
.
Consequently, it follows that for each integer n
≥
0
,
P
C
(
I
-
λ
n
∇
f
α
n
) is ζ
n-averaged with
(43)
ζ
n
=
1
2
+
λ
n
(
α
n
+
∥
A
∥
2
)
2
-
1
2
·
λ
n
(
α
n
+
∥
A
∥
2
)
2
=
2
+
λ
n
(
α
n
+
∥
A
∥
2
)
4
∈
(
0,1
)
.
This immediately implies that P
C
(
I
-
λ
n
∇
f
α
n
) is nonexpansive for all n
≥
0.
Next we divide the remainder of the proof into several steps.
Step  1. {
x
n
} is bounded.
Indeed, take an arbitraryp
∈
Fix
(
S
)
∩
Ξ
∩
Γ. Then, we get S
p
=
p,  P
C
(
I
-
λ
∇
f
)
p
=
p for λ
∈
(
0
,
2
/
∥
A
∥
2
), and
(44)
p
=
P
C
[
P
C
(
p
-
μ
2
B
2
p
)
-
μ
1
B
1
P
C
(
p
-
μ
2
B
2
p
)
]
.
From (17) it follows that
(45)
∥
u
~
n
-
p
∥
=
∥
P
C
(
I
-
λ
n
∇
f
α
n
)
u
n
-
P
C
(
I
-
λ
n
∇
f
)
p
∥
≤
∥
P
C
(
I
-
λ
n
∇
f
α
n
)
u
n
-
P
C
(
I
-
λ
n
∇
f
α
n
)
p
∥
+
∥
P
C
(
I
-
λ
n
∇
f
α
n
)
p
-
P
C
(
I
-
λ
n
∇
f
)
p
∥
≤
∥
u
n
-
p
∥
+
∥
(
I
-
λ
n
∇
f
α
n
)
p
-
(
I
-
λ
n
∇
f
)
p
∥
≤
∥
u
n
-
p
∥
+
λ
n
α
n
∥
p
∥
.
Utilizing Lemma 18 we also have
(46)
∥
u
~
n
-
p
∥
2
=
∥
P
C
(
I
-
λ
n
∇
f
α
n
)
u
n
-
P
C
(
I
-
λ
n
∇
f
)
p
∥
2
=
∥
P
C
(
I
-
λ
n
∇
f
α
n
)
u
n
-
P
C
(
I
-
λ
n
∇
f
α
n
)
p
+
P
C
(
I
-
λ
n
∇
f
α
n
)
p
-
P
C
(
I
-
λ
n
∇
f
)
p
∥
2
≤
∥
P
C
(
I
-
λ
n
∇
f
α
n
)
u
n
-
P
C
(
I
-
λ
n
∇
f
α
n
)
p
∥
2
+
2
〈
P
C
(
I
-
λ
n
∇
f
α
n
)
p
-
P
C
(
I
-
λ
n
∇
f
)
p
,
u
~
n
-
p
{
(
I
-
λ
n
∇
f
α
n
)
}
〉
≤
∥
u
n
-
p
∥
2
+
2
∥
P
C
(
I
-
λ
n
∇
f
α
n
)
p
≤
∥
u
n
-
p
∥
2
+
2
-
P
C
(
I
-
λ
n
∇
f
)
p
∥
∥
u
~
n
-
p
∥
≤
∥
u
n
-
p
∥
2
+
2
∥
(
I
-
λ
n
∇
f
α
n
)
p
≤
∥
u
n
-
p
∥
2
+
2
-
(
I
-
λ
n
∇
f
)
p
{
(
I
-
λ
n
∇
f
α
n
)
p
}
∥
∥
u
~
n
-
p
∥
=
∥
u
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
.
For simplicity, we write
(47)
q
=
P
C
(
p
-
μ
2
B
2
p
)
,
x
~
n
=
P
C
(
x
n
-
μ
2
B
2
x
n
)
,
u
-
n
=
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
,
for each n
≥
0. Then y
n
=
σ
n
Q
x
n
+
(
1
-
σ
n
)
u
-
n for each n
≥
0. Since B
i
:
C
→
ℋ
1 is β
i-inverse strongly monotone for i
=
1,2 and 0
<
μ
i
<
2
β
i for i
=
1,  2, we know that for all n
≥
0,
(48)
∥
u
n
-
p
∥
2
=
∥
P
C
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
-
p
∥
2
=
∥
P
C
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
-
P
C
[
P
C
(
p
-
μ
2
B
2
p
)
-
μ
1
B
1
P
C
(
p
-
μ
2
B
2
p
)
]
∥
2
≤
∥
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
-
[
P
C
(
p
-
μ
2
B
2
p
)
-
μ
1
B
1
P
C
(
p
-
μ
2
B
2
p
)
]
∥
2
=
∥
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
P
C
(
p
-
μ
2
B
2
p
)
]
-
μ
1
[
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
B
1
P
C
(
p
-
μ
2
B
2
p
)
]
∥
2
≤
∥
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
P
C
(
p
-
μ
2
B
2
p
)
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
(
2
β
1
-
μ
1
)
-
B
1
P
C
(
p
-
μ
2
B
2
p
)
∥
2
≤
∥
(
x
n
-
μ
2
B
2
x
n
)
-
(
p
-
μ
2
B
2
p
)
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
x
~
n
-
B
1
q
∥
2
=
∥
(
x
n
-
p
)
-
μ
2
(
B
2
x
n
-
B
2
p
)
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
x
~
n
-
B
1
q
∥
2
≤
∥
x
n
-
p
∥
2
-
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
x
n
-
B
2
p
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
x
~
n
-
B
1
q
∥
2
≤
∥
x
n
-
p
∥
2
.
Furthermore, by Proposition 8(ii), we have
(49)
∥
u
-
n
-
p
∥
2
≤
∥
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
-
p
∥
2
-
∥
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
-
u
-
n
∥
2
=
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
-
n
∥
2
+
2
λ
n
〈
∇
f
α
n
(
u
~
n
)
,
p
-
u
-
n
〉
=
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
-
n
∥
2
+
2
λ
n
(
{
1
2
}
〈
∇
f
α
n
(
u
~
n
)
-
∇
f
α
n
(
p
)
,
p
-
u
~
n
〉
+
〈
∇
f
α
n
(
p
)
,
p
-
u
~
n
〉
+
〈
∇
f
α
n
(
u
~
n
)
,
u
~
n
-
u
-
n
〉
{
1
2
}
)
≤
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
-
n
∥
2
+
2
λ
n
(
〈
∇
f
α
n
(
p
)
,
p
-
u
~
n
〉
+
〈
∇
f
α
n
(
u
~
n
)
,
u
~
n
-
u
-
n
〉
)
=
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
-
n
∥
2
+
2
λ
n
[
{
〈
∇
f
α
n
(
u
~
n
)
,
u
~
n
-
u
-
n
〉
}
〈
(
α
n
I
+
∇
f
)
p
,
p
-
u
~
n
〉
+
〈
∇
f
α
n
(
u
~
n
)
,
u
~
n
-
u
-
n
〉
]
≤
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
-
n
∥
2
+
2
λ
n
[
α
n
〈
p
,
p
-
u
~
n
〉
+
〈
∇
f
α
n
(
u
~
n
)
,
u
~
n
-
u
-
n
〉
]
=
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
~
n
∥
2
-
2
〈
u
n
-
u
~
n
,
u
~
n
-
u
-
n
〉
-
∥
u
~
n
-
u
-
n
∥
2
+
2
λ
n
[
α
n
〈
p
,
p
-
u
~
n
〉
+
〈
∇
f
α
n
(
u
~
n
)
,
u
~
n
-
u
-
n
〉
]
=
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
~
n
∥
2
-
∥
u
~
n
-
u
-
n
∥
2
+
2
〈
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
-
u
~
n
,
u
-
n
-
u
~
n
〉
+
2
λ
n
α
n
〈
p
,
p
-
u
~
n
〉
.
Further, by Proposition 8(i), we have
(50)
〈
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
-
u
~
n
,
u
-
n
-
u
~
n
〉
=
〈
u
n
-
λ
n
∇
f
α
n
(
u
n
)
-
u
~
n
,
u
-
n
-
u
~
n
〉
+
〈
λ
n
∇
f
α
n
(
u
n
)
-
λ
n
∇
f
α
n
(
u
~
n
)
,
u
-
n
-
u
~
n
〉
≤
〈
λ
n
∇
f
α
n
(
u
n
)
-
λ
n
∇
f
α
n
(
u
~
n
)
,
u
-
n
-
u
~
n
〉
≤
λ
n
∥
∇
f
α
n
(
u
n
)
-
∇
f
α
n
(
u
~
n
)
∥
∥
u
-
n
-
u
~
n
∥
≤
λ
n
(
α
n
+
∥
A
∥
2
)
∥
u
n
-
u
~
n
∥
∥
u
-
n
-
u
~
n
∥
.
So, from (45) we obtain
(51)
∥
u
-
n
-
p
∥
2
≤
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
~
n
∥
2
-
∥
u
~
n
-
u
-
n
∥
2
+
2
〈
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
-
u
~
n
,
u
-
n
-
u
~
n
〉
+
2
λ
n
α
n
〈
p
,
p
-
u
~
n
〉
≤
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
~
n
∥
2
-
∥
u
~
n
-
u
-
n
∥
2
+
2
λ
n
(
α
n
+
∥
A
∥
2
)
×
∥
u
n
-
u
~
n
∥
∥
u
-
n
-
u
~
n
∥
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
≤
∥
u
n
-
p
∥
2
-
∥
u
n
-
u
~
n
∥
2
-
∥
u
~
n
-
u
-
n
∥
2
+
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
∥
u
n
-
u
~
n
∥
2
+
∥
u
~
n
-
u
-
n
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
=
∥
u
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
+
(
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
-
1
)
∥
u
n
-
u
~
n
∥
2
≤
∥
u
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
≤
∥
u
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
[
{
1
2
}
∥
u
n
-
p
∥
+
λ
n
α
n
∥
p
∥
]
≤
∥
u
n
-
p
∥
2
+
4
λ
n
α
n
∥
p
∥
∥
u
n
-
p
∥
+
4
λ
n
2
α
n
2
∥
p
∥
2
=
(
∥
u
n
-
p
∥
+
2
λ
n
α
n
∥
p
∥
)
2
.
Hence it follows from (48) and (51) that
(52)
∥
y
n
-
p
∥
=
∥
σ
n
(
Q
x
n
-
p
)
+
(
1
-
σ
n
)
(
u
-
n
-
p
)
∥
≤
σ
n
∥
Q
x
n
-
p
∥
+
(
1
-
σ
n
)
∥
u
-
n
-
p
∥
≤
σ
n
(
∥
Q
x
n
-
Q
p
∥
+
∥
Q
p
-
p
∥
)
+
(
1
-
σ
n
)
(
∥
u
n
-
p
∥
+
2
λ
n
α
n
∥
p
∥
)
≤
σ
n
(
ρ
∥
x
n
-
p
∥
+
∥
Q
p
-
p
∥
)
+
(
1
-
σ
n
)
(
∥
x
n
-
p
∥
+
2
λ
n
α
n
∥
p
∥
)
≤
(
1
-
(
1
-
ρ
)
σ
n
)
∥
x
n
-
p
∥
+
σ
n
∥
Q
p
-
p
∥
+
2
λ
n
α
n
∥
p
∥
=
(
1
-
(
1
-
ρ
)
σ
n
)
∥
x
n
-
p
∥
+
(
1
-
ρ
)
σ
n
∥
Q
p
-
p
∥
1
-
ρ
+
2
λ
n
α
n
∥
p
∥
≤
max
{
∥
x
n
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
λ
n
α
n
∥
p
∥
.
Since (
γ
n
+
δ
n
)
k
≤
γ
n for all n
≥
0, utilizing Lemma 17 we obtain from (52)
(53)
∥
x
n
+
1
-
p
∥
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
∥
≤
β
n
∥
x
n
-
p
∥
+
∥
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
∥
≤
β
n
∥
x
n
-
p
∥
+
(
γ
n
+
δ
n
)
∥
y
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
+
(
γ
n
+
δ
n
)
[
max
{
∥
x
n
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
λ
n
α
n
∥
p
∥
]
≤
β
n
∥
x
n
-
p
∥
+
(
γ
n
+
δ
n
)
max
{
∥
x
n
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
λ
n
α
n
∥
p
∥
≤
max
{
∥
x
n
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
α
n
.
Now, we claim that
(54)
∥
x
n
+
1
-
p
∥
≤
max
{
∥
x
0
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
∑
j
=
0
n
‍
α
j
.
As a matter of fact, if n
=
0, then it is clear that (54) is valid, that is,
(55)
∥
x
1
-
p
∥
≤
max
{
∥
x
0
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
∑
j
=
0
0
‍
α
j
.
Assume that (54) holds for n
≥
1, that is,
(56)
∥
x
n
-
p
∥
≤
max
{
∥
x
0
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
∑
j
=
0
n
-
1
‍
α
j
.
Then, we conclude from (53) and (56) that
(57)
∥
x
n
+
1
-
p
∥
≤
max
{
∥
x
n
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
α
n
≤
max
{
{
∑
j
=
0
n
-
1
α
j
,
∥
Q
p
-
p
∥
1
-
ρ
}
max
{
∥
x
0
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
∑
j
=
0
n
-
1
α
j
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
α
n
≤
max
{
∥
x
0
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
∑
j
=
0
n
-
1
‍
α
j
+
2
b
∥
p
∥
α
n
=
max
{
∥
x
0
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
∑
j
=
0
n
α
j
.
By induction, we conclude that (54) is valid. Hence, {
x
n
} is bounded. Since P
C
,
∇
f
α
n
,
B
1 and B
2 are Lipschitz continuous, it is easy to see that {
u
n
}
,
{
u
~
n
}
,
{
u
-
n
}
,
{
y
n
} and {
x
~
n
} are bounded, where x
~
n
=
P
C
(
x
n
-
μ
2
B
2
x
n
) for all n
≥
0.
Step  2.
lim
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0.
Indeed, definex
n
+
1
=
β
n
x
n
+
(
1
-
β
n
)
w
n for all n
≥
0. It follows that
(58)
w
n
+
1
-
w
n
=
x
n
+
2
-
β
n
+
1
x
n
+
1
1
-
β
n
+
1
-
x
n
+
1
-
β
n
x
n
1
-
β
n
=
γ
n
+
1
y
n
+
1
+
δ
n
+
1
S
y
n
+
1
1
-
β
n
+
1
-
γ
n
y
n
+
δ
n
S
y
n
1
-
β
n
=
γ
n
+
1
(
y
n
+
1
-
y
n
)
+
δ
n
+
1
(
S
y
n
+
1
-
S
y
n
)
1
-
β
n
+
1
+
(
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
)
y
n
+
(
δ
n
+
1
1
-
β
n
+
1
-
δ
n
1
-
β
n
)
S
y
n
.
Since (
γ
n
+
δ
n
)
k
≤
γ
n for all n
≥
0, utilizing Lemma 17 we have
(59)
∥
γ
n
+
1
(
y
n
+
1
-
y
n
)
+
δ
n
+
1
(
S
y
n
+
1
-
S
y
n
)
∥
≤
(
γ
n
+
1
+
δ
n
+
1
)
∥
y
n
+
1
-
y
n
∥
.
Next, we estimate ∥
y
n
+
1
-
y
n
∥. Observe that
(60)
∥
u
~
n
+
1
-
u
~
n
∥
=
∥
P
C
(
u
n
+
1
-
λ
n
+
1
∇
f
α
n
+
1
(
u
n
+
1
)
)
-
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
n
)
)
∥
≤
∥
P
C
(
I
-
λ
n
+
1
∇
f
α
n
+
1
)
u
n
+
1
-
P
C
(
I
-
λ
n
+
1
∇
f
α
n
+
1
)
u
n
∥
+
∥
P
C
(
I
-
λ
n
+
1
∇
f
α
n
+
1
)
u
n
-
P
C
(
I
-
λ
n
∇
f
α
n
)
u
n
∥
≤
∥
u
n
+
1
-
u
n
∥
+
∥
(
I
-
λ
n
+
1
∇
f
α
n
+
1
)
u
n
-
(
I
-
λ
n
∇
f
α
n
)
u
n
∥
=
∥
u
n
+
1
-
u
n
∥
+
∥
λ
n
+
1
(
α
n
+
1
I
+
∇
f
)
u
n
-
λ
n
(
α
n
I
+
∇
f
)
u
n
∥
≤
∥
u
n
+
1
-
u
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
u
n
)
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
u
n
∥
,
(61)
∥
u
n
+
1
-
u
n
∥
2
=
∥
{
1
2
}
P
C
[
P
C
(
x
n
+
1

-

μ
2
B
2
x
n
+
1
)

-

μ
1
B
1
P
C
(
x
n
+
1

-

μ
2
B
2
x
n
+
1
)
]
-
P
C
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
{
1
2
}
∥
2
≤
∥
{
1
2
}
[
P
C
(
x
n
+
1
-
μ
2
B
2
x
n
+
1
)
-
μ
1
B
1
P
C
(
x
n
+
1
-
μ
2
B
2
x
n
+
1
)
]
-
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
{
1
2
}
∥
2
=
∥
{
1
2
}
[
P
C
(
x
n
+
1
-
μ
2
B
2
x
n
+
1
)
-
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
-
μ
1
[
B
1
P
C
(
x
n
+
1

-

μ
2
B
2
x
n
+
1
)

-

B
1
P
C
(
x
n

-

μ
2
B
2
x
n
)
]
{
1
2
}
∥
2
≤
∥
P
C
(
x
n
+
1
-
μ
2
B
2
x
n
+
1
)
-
P
C
(
x
n
-
μ
2
B
2
x
n
)
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
×
∥
B
1
P
C
(
x
n
+
1
-
μ
2
B
2
x
n
+
1
)
-
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
∥
2
≤
∥
P
C
(
x
n
+
1
-
μ
2
B
2
x
n
+
1
)
-
P
C
(
x
n
-
μ
2
B
2
x
n
)
∥
2
≤
∥
(
x
n
+
1
-
μ
2
B
2
x
n
+
1
)
-
(
x
n
-
μ
2
B
2
x
n
)
∥
2
=
∥
(
x
n
+
1
-
x
n
)
-
μ
2
(
B
2
x
n
+
1
-
B
2
x
n
)
∥
2
≤
∥
x
n
+
1
-
x
n
∥
2
-
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
x
n
+
1
-
B
2
x
n
∥
2
≤
∥
x
n
+
1
-
x
n
∥
2
,
and hence
(62)
∥
u
-
n
+
1
-
u
-
n
∥
=
∥
P
C
(
u
n
+
1
-
λ
n
+
1
∇
f
α
n
+
1
(
u
~
n
+
1
)
)
-
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
∥
≤
∥
(
u
n
+
1
-
λ
n
+
1
∇
f
α
n
+
1
(
u
~
n
+
1
)
)
-
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
∥
=
∥
(
I
-
λ
n
+
1
∇
f
α
n
+
1
)
u
n
+
1
-
(
I
-
λ
n
+
1
∇
f
α
n
+
1
)
u
n
+
λ
n
+
1
(
∇
f
α
n
+
1
(
u
n
+
1
)
-
∇
f
α
n
+
1
(
u
n
)
)
-
λ
n
+
1
∇
f
α
n
+
1
(
u
~
n
+
1
)
+
λ
n
∇
f
α
n
(
u
~
n
)
∥
≤
∥
(
I
-
λ
n
+
1
∇
f
α
n
+
1
)
u
n
+
1
-
(
I
-
λ
n
+
1
∇
f
α
n
+
1
)
u
n
∥
+
λ
n
+
1
∥
∇
f
α
n
+
1
(
u
n
+
1
)
-
∇
f
α
n
+
1
(
u
n
)
∥
+
∥
λ
n
+
1
∇
f
α
n
+
1
(
u
~
n
+
1
)
-
λ
n
∇
f
α
n
(
u
~
n
)
∥
≤
∥
u
n
+
1
-
u
n
∥
+
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
∥
u
n
+
1
-
u
n
∥
+
∥
λ
n
+
1
(
α
n
+
1
I
+
∇
f
)
(
u
~
n
+
1
)
-
λ
n
(
α
n
I
+
∇
f
)
(
u
~
n
)
∥
≤
∥
u
n
+
1
-
u
n
∥
+
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
∥
u
n
+
1
-
u
n
∥
+
∥
λ
n
+
1
α
n
+
1
u
~
n
+
1
-
λ
n
α
n
u
~
n
∥
+
∥
λ
n
+
1
∇
f
(
u
~
n
+
1
)
-
λ
n
∇
f
(
u
~
n
)
∥
≤
∥
u
n
+
1
-
u
n
∥
+
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
∥
u
n
+
1
-
u
n
∥
+
∥
λ
n
+
1
α
n
+
1
u
~
n
+
1
-
λ
n
α
n
u
~
n
∥
+
λ
n
+
1
∥
∇
f
(
u
~
n
+
1
)
-
∇
f
(
u
~
n
)
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
u
~
n
)
∥
≤
∥
x
n
+
1
-
x
n
∥
+
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
∥
u
n
+
1
-
u
n
∥
+
∥
λ
n
+
1
α
n
+
1
u
~
n
+
1
-
λ
n
α
n
u
~
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
u
~
n
)
∥
+
λ
n
+
1
∥
A
∥
2
∥
u
~
n
+
1
-
u
~
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
×
(
∥
u
n
+
1
-
u
n
∥
+
∥
u
~
n
+
1
-
u
~
n
∥
)
+
∥
λ
n
+
1
α
n
+
1
u
~
n
+
1
-
λ
n
α
n
u
~
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
u
~
n
)
∥
.
Combining (60) with (61), we get
(63)
∥
u
~
n
+
1
-
u
~
n
∥
≤
∥
u
n
+
1
-
u
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
u
n
)
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
u
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
u
n
)
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
u
n
∥
.
This together with (62) implies that
(64)
∥
y
n
+
1
-
y
n
∥
=
∥
u
-
n
+
1
+
σ
n
+
1
(
Q
x
n
+
1
-
u
-
n
+
1
)
-
u
-
n
-
σ
n
(
Q
x
n
-
u
-
n
)
∥
≤
∥
u
-
n
+
1
-
u
-
n
∥
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
-
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
-
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
(
∥
u
n
+
1
-
u
n
∥
+
∥
u
~
n
+
1
-
u
~
n
∥
)
+
∥
λ
n
+
1
α
n
+
1
u
~
n
+
1
-
λ
n
α
n
u
~
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
u
~
n
)
∥
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
-
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
-
n
∥
.
Hence it follows from (58), (59), and (64) that
(65)
∥
w
n
+
1
-
w
n
∥
≤
∥
γ
n
+
1
(
y
n
+
1
-
y
n
)
+
δ
n
+
1
(
S
y
n
+
1
-
S
y
n
)
∥
1
-
β
n
+
1
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
y
n
∥
+
|
δ
n
+
1
1
-
β
n
+
1
-
δ
n
1
-
β
n
|
∥
S
y
n
∥
≤
γ
n
+
1
+
δ
n
+
1
1
-
β
n
+
1
∥
y
n
+
1
-
y
n
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
(
∥
y
n
∥
+
∥
S
y
n
∥
)
=
∥
y
n
+
1
-
y
n
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
(
∥
y
n
∥
+
∥
S
y
n
∥
)
≤
∥
x
n
+
1
-
x
n
∥
+
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
(
∥
u
n
+
1
-
u
n
∥
+
∥
u
~
n
+
1
-
u
~
n
∥
)
+
∥
λ
n
+
1
α
n
+
1
u
~
n
+
1
-
λ
n
α
n
u
~
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
u
~
n
)
∥
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
-
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
-
n
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
(
∥
y
n
∥
+
∥
S
y
n
∥
)
.
From (60), we deduce from condition (vi) that lim
n
→
∞
∥
u
~
n
+
1
-
u
~
n
∥
=
0. Since {
x
n
}
,
{
u
n
}, {
u
~
n
}
,
{
y
n
} and {
u
-
n
} are bounded, it follows from conditions (i), (iii), (v), and (vi) that
(66)
limsup
n
→
∞
(
∥
w
n
+
1
-
w
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
lim
sup
n
→
∞
{
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
≤
lim
sup
n
→
∞
×
(
∥
u
n
+
1
-
u
n
∥
+
∥
u
~
n
+
1
-
u
~
n
∥
)
≤
lim
sup
n
→
∞
+
∥
λ
n
+
1
α
n
+
1
u
~
n
+
1
-
λ
n
α
n
u
~
n
∥
≤
lim
sup
n
→
∞
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
u
~
n
)
∥
≤
lim
sup
n
→
∞
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
-
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
-
n
∥
≤
lim
sup
n
→
∞
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
(
∥
y
n
∥
+
∥
S
y
n
∥
)
}
=
0
.
Hence by Lemma 14 we get lim
n
→
∞
∥
w
n
-
x
n
∥
=
0. Thus,
(67)
lim
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
lim
n
→
∞
(
1
-
β
n
)
∥
w
n
-
x
n
∥
=
0
.
Step  3.
lim
n
→
∞
∥
B
2
x
n
-
B
2
p
∥
=
0,  lim
n
→
∞
∥
B
1
x
~
n
-
B
1
q
∥
=
0, and lim
n
→
∞
∥
u
n
-
u
~
n
∥
=
0, where q
=
P
C
(
p
-
μ
2
B
2
p
).
Indeed, utilizing Lemma17 and the convexity of ∥
·
∥
2, we obtain from (17), (48), and (51) that
(68)
∥
x
n
+
1
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
∥
1
γ
n
+
δ
n
[
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
]
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
[
σ
n
∥
Q
x
n
-
p
∥
2
+
(
1
-
σ
n
)
∥
u
-
n
-
p
∥
2
]
≤
β
n
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
∥
u
-
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
[
{
∥
u
n
-
u
~
n
∥
2
}
∥
u
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
+
(
γ
n
+
δ
n
)
+
(
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
-
1
)
∥
u
n
-
u
~
n
∥
2
]
≤
β
n
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
×
[
{
1
2
}
∥
x
n
-
p
∥
2
-
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
x
n
-
B
2
p
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
x
~
n
-
B
1
q
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
+
(
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
-
1
)
∥
u
n
-
u
~
n
∥
2
{
1
2
}
]
≤
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
-
(
γ
n
+
δ
n
)
×
[
{
1
2
}
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
x
n
-
B
2
p
∥
2
+
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
x
~
n
-
B
1
q
∥
2
+
(
1
-
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
)
∥
u
n
-
u
~
n
∥
2
{
1
2
}
]
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
.
Therefore,
(69)
(
γ
n
+
δ
n
)
[
{
(
1
-
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
)
}
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
x
n
-
B
2
p
∥
2
(
γ
n
+
δ
n
)
+
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
x
~
n
-
B
1
q
∥
2
(
γ
n
+
δ
n
)
+
(
1
-
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
)
∥
u
n
-
u
~
n
∥
2
]
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
≤
(
∥
x
n
-
p
∥
+
∥
x
n
+
1
-
p
∥
)
∥
x
n
-
x
n
+
1
∥
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
.
Since α
n
→
0
,
σ
n
→
0,  ∥
x
n
-
x
n
+
1
∥
→
0,  liminf
n
→
∞
(
γ
n
+
δ
n
)
>
0 and {
λ
n
}
⊂
[
a
,
b
] for some a
,
b
∈
(
0
,
1
/
∥
A
∥
2
), it follows that
(70)
lim
n
→
∞
∥
u
n
-
u
~
n
∥
=
0
,
lim
n
→
∞
∥
B
1
x
~
n
-
B
1
q
∥
=
0
,
lim
n
→
∞
∥
B
2
x
n
-
B
2
p
∥
=
0
.
Step  4.
lim
n
→
∞
∥
S
y
n
-
y
n
∥
=
0.
Indeed, observe that(71)
∥
u
-
n
-
u
~
n
∥
=
∥
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
-
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
n
)
)
∥
≤
∥
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
-
(
u
n
-
λ
n
∇
f
α
n
(
u
n
)
)
∥
=
λ
n
∥
∇
f
α
n
(
u
~
n
)
-
∇
f
α
n
(
u
n
)
∥
≤
λ
n
(
α
n
+
∥
A
∥
2
)
∥
u
~
n
-
u
n
∥
.
This together with ∥
u
~
n
-
u
n
∥
→
0 implies that lim
n
→
∞
∥
u
-
n
-
u
~
n
∥
=
0 and hence lim
n
→
∞
∥
u
-
n
-
u
n
∥
=
0. By firm nonexpansiveness of P
C, we have
(72)
∥
x
~
n
-
q
∥
2
=
∥
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
P
C
(
p
-
μ
2
B
2
p
)
∥
2
≤
〈
(
x
n
-
μ
2
B
2
x
n
)
-
(
p
-
μ
2
B
2
p
)
,
x
~
n
-
q
〉
=
1
2
[
∥
x
n
-
p
-
μ
2
(
B
2
x
n
-
B
2
p
)
∥
2
+
∥
x
~
n
-
q
∥
2
-
∥
(
x
n
-
p
)
-
μ
2
(
B
2
x
n
-
B
2
p
)
-
(
x
~
n
-
q
)
∥
2
]
≤
1
2
[
∥
x
n
-
p
∥
2
+
∥
x
~
n
-
q
∥
2
-
∥
(
x
n
-
x
~
n
)
-
μ
2
(
B
2
x
n
-
B
2
p
)
-
(
p
-
q
)
∥
2
]
=
1
2
[
∥
x
n

-

p
∥
2

+

∥
x
~
n

-

q
∥
2

-

∥
x
n

-

x
~
n

-

(
p

-

q
)
∥
2
+
2
μ
2
〈
x
n
-
x
~
n
-
(
p
-
q
)
,
B
2
x
n
-
B
2
p
〉
-
μ
2
2
∥
B
2
x
n
-
B
2
p
∥
2
]
≤
1
2
[
∥
x
n
-
p
∥
2
+
∥
x
~
n
-
q
∥
2
-
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
2
+
2
μ
2
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
∥
B
2
x
n
-
B
2
p
∥
]
,
that is,
(73)
∥
x
~
n
-
q
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
2
+
2
μ
2
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
∥
B
2
x
n
-
B
2
p
∥
.
Moreover, using the argument technique similar to the above one, we derive
(74)
∥
u
n
-
p
∥
2
=
∥
P
C
(
x
~
n
-
μ
1
B
1
x
~
n
)
-
P
C
(
q
-
μ
1
B
1
q
)
∥
2
≤
〈
(
x
~
n
-
μ
1
B
1
x
~
n
)
-
(
q
-
μ
1
B
1
q
)
,
u
n
-
p
〉
=
1
2
[
∥
x
~
n
-
q
-
μ
1
(
B
1
x
~
n
-
B
1
q
)
∥
2
+
∥
u
n
-
p
∥
2
-
∥
(
x
~
n
-
q
)
-
μ
1
(
B
1
x
~
n
-
B
1
q
)
-
(
u
n
-
p
)
∥
2
]
≤
1
2
[
∥
x
~
n
-
q
∥
2
+
∥
u
n
-
p
∥
2
-
∥
(
x
~
n
-
u
n
)
-
μ
1
(
B
1
x
~
n
-
B
1
q
)
+
(
p
-
q
)
∥
2
]
=
1
2
[
∥
x
~
n
-
q
∥
2
+
∥
u
n
-
p
∥
2
-
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
〈
x
~
n
-
u
n
+
(
p
-
q
)
,
B
1
x
~
n
-
B
1
q
〉
-
μ
1
2
∥
B
1
x
~
n
-
B
1
q
∥
2
]
≤
1
2
[
∥
x
~
n
-
q
∥
2
+
∥
u
n
-
p
∥
2
-
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
x
~
n
-
B
1
q
∥
{
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
2
}
]
,
that is,
(75)
∥
u
n
-
p
∥
2
≤
∥
x
~
n
-
q
∥
2
-
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
x
~
n
-
B
1
q
∥
.
Utilizing (46), (73), and (75), we have
(76)
∥
u
-
n
-
p
∥
2
=
∥
u
~
n
-
p
+
u
-
n
-
u
~
n
∥
2
≤
∥
u
~
n
-
p
∥
2
+
2
〈
u
-
n
-
u
~
n
,
u
-
n
-
p
〉
≤
∥
u
~
n
-
p
∥
2
+
2
∥
u
-
n
-
u
~
n
∥
∥
u
-
n
-
p
∥
≤
∥
u
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
+
2
∥
u
-
n
-
u
~
n
∥
∥
u
-
n
-
p
∥
≤
∥
x
~
n
-
q
∥
2
-
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
x
~
n
-
B
1
q
∥
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
+
2
∥
u
-
n
-
u
~
n
∥
∥
u
-
n
-
p
∥
≤
∥
x
n
-
p
∥
2
-
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
2
+
2
μ
2
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
∥
B
2
x
n
-
B
2
p
∥
-
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
x
~
n
-
B
1
q
∥
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
+
2
∥
u
-
n
-
u
~
n
∥
∥
u
-
n
-
p
∥
.
Thus from (17) and (76) it follows that
(77)
∥
x
n
+
1
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
∥
y
n
-
p
∥
2
=
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
[
σ
n
∥
Q
x
n
-
p
∥
2
+
(
1
-
σ
n
)
∥
u
-
n
-
p
∥
2
]
≤
β
n
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
u
-
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
1
-
β
n
)
[
{
1
2
}
∥
x
n
-
p
∥
2
-
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
2
+
2
μ
2
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
∥
B
2
x
n
-
B
2
p
∥
-
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
x
~
n
-
B
1
q
∥
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
+
2
∥
u
-
n
-
u
~
n
∥
∥
u
-
n
-
p
∥
{
1
2
}
]
,
which hence implies that
(78)
(
1

-

β
n
)
[
∥
x
n

-

x
~
n

-

(
p

-

q
)
∥
2

+

∥
x
~
n

-

u
n

+

(
p

-

q
)
∥
2
]
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
1
-
β
n
)
[
{
1
2
}
2
μ
2
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
∥
B
2
x
n
-
B
2
p
∥
+
(
1
-
β
n
)
+
2
μ
1
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
x
~
n
-
B
1
q
∥
+
(
1
-
β
n
)
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
+
(
1
-
β
n
)
+
2
∥
u
-
n
-
u
~
n
∥
∥
u
-
n
-
p
∥
{
1
2
}
]
≤
(
∥
x
n
-
p
∥
+
∥
x
n
+
1
-
p
∥
)
×
∥
x
n
-
x
n
+
1
∥
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
μ
2
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
∥
B
2
x
n
-
B
2
p
∥
+
2
μ
1
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
x
~
n
-
B
1
q
∥
+
2
λ
n
α
n
∥
p
∥
∥
u
~
n
-
p
∥
+
2
∥
u
-
n
-
u
~
n
∥
∥
u
-
n
-
p
∥
.
Since limsup
n
→
∞
β
n
<
1,  {
λ
n
}
⊂
[
a
,
b
],  α
n
→
0,  σ
n
→
0,  ∥
B
2
x
n
-
B
2
p
∥
→
0,  ∥
B
1
x
~
n
-
B
1
q
∥
→
0,  ∥
u
-
n
-
u
~
n
∥
→
0 and ∥
x
n
+
1
-
x
n
∥
→
0, it follows from the boundedness of {
x
n
}
,
{
x
~
n
}
,
{
u
n
}
,
{
u
~
n
}, and {
u
-
n
} that
(79)
lim
n
→
∞
∥
x
n
-
x
~
n
-
(
p
-
q
)
∥
=
0
,
lim
n
→
∞
∥
x
~
n
-
u
n
+
(
p
-
q
)
∥
=
0
.
Consequently, it immediately follows that
(80)
lim
n
→
∞
∥
x
n
-
u
n
∥
=
0
,
lim
n
→
∞
∥
x
n
-
u
-
n
∥
=
0
.
This together with ∥
y
n
-
u
-
n
∥
≤
σ
n
∥
Q
x
n
-
u
-
n
∥
→
0 implies that
(81)
lim
n
→
∞
∥
x
n
-
y
n
∥
=
0
.
Since
(82)
∥
δ
n
(
S
y
n
-
x
n
)
∥
≤
∥
x
n
+
1
-
x
n
∥
+
γ
n
∥
y
n
-
x
n
∥
,
it follows that
(83)
lim
n
→
∞
∥
S
y
n
-
x
n
∥
=
0
,
lim
n
→
∞
∥
S
y
n
-
y
n
∥
=
0
.
Step  5.
limsup
n
→
∞
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
≤
0 where x
-
=
P
Fix
(
S
)
∩
Ξ
∩
Γ
Q
x
-.
Indeed, since{
x
n
} is bounded, there exists a subsequence {
x
n
i
} of {
x
n
} such that
(84)
lim
sup
n
→
∞
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
=
lim
i
→
∞
〈
Q
x
-
-
x
-
,
x
n
i
-
x
-
〉
.
Also, since H is reflexive and {
y
n
} is bounded, without loss of generality we may assume that y
n
i
→
p
^ weakly for some p
^
∈
C. First, it is clear from Lemma 15 that p
^
∈
Fix
(
S
). Now let us show that p
^
∈
Ξ. We note that
(85)
∥
x
n
-
G
(
x
n
)
∥
=
∥
x
n
-
P
C
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
∥
=
∥
x
n
-
u
n
∥
⟶
0
(
n
⟶
∞
)
,
where G
:
C
→
C is defined as such that in Lemma 4. According to Lemma 15 we obtain p
^
∈
Ξ. Further, let us show that p
^
∈
Γ. As a matter of fact, since ∥
x
n
-
u
n
∥
→
0
,
∥
u
~
n
-
u
n
∥
→
0 and ∥
x
n
-
y
n
∥
→
0, we deduce that x
n
i
→
p
^ weakly and u
~
n
i
→
p
^ weakly. Let
(86)
T
v
=
{
∇
f
(
v
)
+
N
C
v
,
if
v
∈
C
,
∅
,
if
v
∉
C
,
where N
C
v
=
{
w
∈
ℋ
1
:
〈
v
-
u
,
w
〉
≥
0
,
∀
u
∈
C
}. Then, T is maximal monotone and 0
∈
T
v if and only if v
∈
VI
(
C
,
∇
f
); see [40] for more details. Let (
v
,
w
)
∈
Gph
(
T
). Then, we have
(87)
w
∈
T
v
=
∇
f
(
v
)
+
N
C
v
,
and hence
(88)
w
-
∇
f
(
v
)
∈
N
C
v
.
So, we have
(89)
〈
v
-
u
,
w
-
∇
f
(
v
)
〉
≥
0
,
∀
u
∈
C
.
On the other hand, from
(90)
u
~
n
=
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
n
)
)
,
v
∈
C
,
we have
(91)
〈
u
n
-
λ
n
∇
f
α
n
(
u
n
)
-
u
~
n
,
u
~
n
-
v
〉
≥
0
,
and, hence,
(92)
〈
v
-
u
~
n
,
u
~
n
-
u
n
λ
n
+
∇
f
α
n
(
u
n
)
〉
≥
0
.
Therefore, from
(93)
w
-
∇
f
(
v
)
∈
N
C
v
,
u
~
n
i
∈
C
,
we have
(94)
〈
v
-
u
~
n
i
,
w
〉
≥
〈
v
-
u
~
n
i
,
∇
f
(
v
)
〉
≥
〈
v
-
u
~
n
i
,
∇
f
(
v
)
〉
-
〈
v
-
u
~
n
i
,
u
~
n
i
-
u
n
i
λ
n
i
+
∇
f
α
n
i
(
u
n
i
)
〉
=
〈
v
-
u
~
n
i
,
∇
f
(
v
)
〉
-
〈
v
-
u
~
n
i
,
u
~
n
i
-
u
n
i
λ
n
i
+
∇
f
(
u
n
i
)
〉
-
α
n
i
〈
v
-
u
~
n
i
,
u
n
i
〉
=
〈
v
-
u
~
n
i
,
∇
f
(
v
)
-
∇
f
(
u
~
n
i
)
〉
+
〈
v
-
u
~
n
i
,
∇
f
(
u
~
n
i
)
-
∇
f
(
u
n
i
)
〉
-
〈
v
-
u
~
n
i
,
u
~
n
i
-
u
n
i
λ
n
i
〉
-
α
n
i
〈
v
-
u
~
n
i
,
u
n
i
〉
≥
〈
v
-
u
~
n
i
,
∇
f
(
u
~
n
i
)
-
∇
f
(
u
n
i
)
〉
-
〈
v
-
u
~
n
i
,
u
~
n
i
-
u
n
i
λ
n
i
〉
-
α
n
i
〈
v
-
u
~
n
i
,
u
n
i
〉
.
Hence, we get
(95)
〈
v
-
p
^
,
w
〉
≥
0
,
as
i
⟶
∞
.
Since T is maximal monotone, we have p
^
∈
T
-
1
0, and, hence, p
^
∈
VI
(
C
,
∇
f
). Thus it is clear that p
^
∈
Γ. Therefore, p
^
∈
Fix
(
S
)
∩
Ξ
∩
Γ. Consequently, in terms of Proposition 8(i) we obtain from (84) that
(96)
limsup
n
→
∞
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
=
lim
i
→
∞
〈
Q
x
-
-
x
-
,
x
n
i
-
x
-
〉
=
〈
Q
x
-
-
x
-
,
p
^
-
x
-
〉
≤
0
.
Step 6.
lim
n
→
∞
∥
x
n
-
x
-
∥
=
0.
Indeed, from (48) and (51) it follows that
(97)
∥
u
-
n
-
x
-
∥
2
≤
∥
u
n
-
x
-
∥
2
+
2
λ
n
α
n
∥
x
-
∥
∥
u
~
n
-
x
-
∥
≤
∥
x
n
-
x
-
∥
2
+
2
λ
n
α
n
∥
x
-
∥
∥
u
~
n
-
x
-
∥
.
Note that
(98)
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
=
〈
Q
x
n
-
x
-
,
x
n
-
x
-
〉
+
〈
Q
x
n
-
x
-
,
y
n
-
x
n
〉
=
〈
Q
x
n
-
Q
x
-
,
x
n
-
x
-
〉
+
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
〈
Q
x
n
-
x
-
,
y
n
-
x
n
〉
≤
ρ
∥
x
n
-
x
-
∥
2
+
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
∥
Q
x
n
-
x
-
∥
∥
y
n
-
x
n
∥
.
Utilizing Lemmas 17 and 18, we obtain from (48) and the convexity of ∥
·
∥
2
(99)
∥
x
n
+
1
-
x
-
∥
2
=
∥
β
n
(
x
n
-
x
-
)
+
γ
n
(
y
n
-
x
-
)
+
δ
n
(
S
y
n
-
x
-
)
∥
2
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
∥
1
γ
n
+
δ
n
[
γ
n
(
y
n
-
x
-
)
+
δ
n
(
S
y
n
-
x
-
)
]
∥
2
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
∥
y
n
-
x
-
∥
2
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
[
{
1
2
}
(
1
-
σ
n
)
2
∥
u
-
n
-
x
-
∥
2
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
+
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
{
1
2
}
]
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
[
{
1
2
}
(
1
-
σ
n
)
+
(
γ
n
+
δ
n
)
×
(
∥
x
n
-
x
-
∥
2
+
2
λ
n
α
n
∥
x
-
∥
∥
u
~
n
-
x
-
∥
)
+
(
γ
n
+
δ
n
)
+
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
{
1
2
}
]
=
(
1
-
(
γ
n
+
δ
n
)
σ
n
)
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
+
(
γ
n
+
δ
n
)
2
λ
n
α
n
∥
x
-
∥
∥
u
~
n
-
x
-
∥
≤
(
1
-
(
γ
n
+
δ
n
)
σ
n
)
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
+
2
λ
n
α
n
∥
x
-
∥
∥
u
~
n
-
x
-
∥
≤
(
1
-
(
γ
n
+
δ
n
)
σ
n
)
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
2
σ
n
×
[
{
1
2
}
ρ
∥
x
n
-
x
-
∥
2
+
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
∥
Q
x
n
-
x
-
∥
∥
y
n
-
x
n
∥
{
1
2
}
]
+
2
λ
n
α
n
∥
x
-
∥
∥
u
~
n
-
x
-
∥
=
[
1
-
(
1
-
2
ρ
)
(
γ
n
+
δ
n
)
σ
n
]
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
2
σ
n
×
[
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
∥
Q
x
n
-
x
-
∥
∥
y
n
-
x
n
∥
]
+
2
λ
n
α
n
∥
x
-
∥
∥
u
~
n
-
x
-
∥
=
[
1
-
(
1
-
2
ρ
)
(
γ
n
+
δ
n
)
σ
n
]
∥
x
n
-
x
-
∥
2
+
(
1
-
2
ρ
)
×
σ
n
2
[
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
∥
Q
x
n
-
x
-
∥
∥
y
n
-
x
n
∥
]
1
-
2
ρ
+
2
λ
n
α
n
∥
x
-
∥
∥
u
~
n
-
x
-
∥
.
Note that liminf
n
→
∞
(
1
-
2
ρ
)
(
γ
n
+
δ
n
)
>
0. It follows that ∑
n
=
0
∞
‍
(
1
-
2
ρ
)
(
γ
n
+
δ
n
)
σ
n
=
∞. It is clear that
(100)
lim
sup
n
→
∞
2
[
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
∥
Q
x
n
-
x
-
∥
∥
y
n
-
x
n
∥
]
1
-
2
ρ
≤
0
,
because limsup
n
→
∞
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
≤
0 and lim
n
→
∞
∥
x
n
-
y
n
∥
=
0. In addition, note also that {
λ
n
}
⊂
[
a
,
b
],  ∑
n
=
0
∞
‍
α
n
<
∞ and {
u
~
n
} is bounded. Hence we get ∑
n
=
0
∞
‍
2
λ
n
α
n
∥
x
-
∥
∥
u
~
n
-
x
-
∥
<
∞. Therefore, all conditions of Lemma 16 are satisfied. Consequently, we immediately deduce that ∥
x
n
-
x
-
∥
→
0 as n
→
∞. This completes the proof.Corollary 20.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ
1. Let A
∈
B
(
ℋ
1
,
ℋ
2
) and B
i
:
C
→
ℋ
1 be β
i-inverse strongly monotone for i
=
1,2. Let S
:
C
→
C be a k-strictly pseudo-contractive mapping such that Fix
(
S
)
∩
Ξ
∩
Γ
≠
∅. For fixed u
∈
C and given x
0
∈
C arbitrarily, let the sequences {
x
n
}
,
{
u
n
}
,
{
u
~
n
} be generated iteratively by
(101)
u
n
=
P
C
[
P
C
(
x
n
-
μ
2
B
2
x
n
)
-
μ
1
B
1
P
C
(
x
n
-
μ
2
B
2
x
n
)
]
,
u
~
n
=
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
n
)
)
,
y
n
=
σ
n
u
+
(
1
-
σ
n
)
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
,
x
n
+
1
=
β
n
x
n
+
γ
n
y
n
+
δ
n
S
y
n
,
∀
n
≥
0
,
where μ
i
∈
(
0,2
β
i
) for i
=
1,2, and the following conditions hold for six sequences {
α
n
}
⊂
(
0
,
∞
)
,
{
λ
n
}
⊂
(
0
,
1
/
∥
A
∥
2
) and {
σ
n
}
,
{
β
n
}
,
{
γ
n
}
,
{
δ
n
}
⊂
[
0,1
]:(i)
∑
n
=
0
∞
‍
α
n
<
∞;
(ii)
β
n
+
γ
n
+
δ
n
=
1 and (
γ
n
+
δ
n
)
k
≤
γ
n for all n
≥
0;
(iii)
lim
n
→
∞
σ
n
=
0 and ∑
n
=
0
∞
‍
σ
n
=
∞;
(iv)
0
<
liminf
n
→
∞
β
n
≤
limsup
n
→
∞
β
n
<
1 and liminf
n
→
∞
δ
n
>
0;
(v)
lim
n
→
∞
(
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
)
=
0;
(vi)
0
<
liminf
n
→
∞
λ
n
≤
limsup
n
→
∞
λ
n
<
1
/
∥
A
∥
2  and lim
n
→
∞
|
λ
n
+
1
-
λ
n
|
=
0. Then the sequences {
x
n
}
,
{
u
n
}
,
{
u
~
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Ξ
∩
Γ
u if and only if lim
n
→
∞
∥
u
n
+
1
-
u
n
∥
=
0. Furthermore, (
x
-
,
y
-
) is a solution of the GSVI (14), where y
-
=
P
C
(
x
-
-
μ
2
B
2
x
-
).Next, utilizing Corollary20 we give the following improvement and extension of the main result in [18] (i.e., [18, Theorem  3.1]).Corollary 21.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ
1. Let A
∈
B
(
ℋ
1
,
ℋ
2
) and S
:
C
→
C be a nonexpansive mapping such that Fix
(
S
)
∩
Γ
≠
∅. For fixed u
∈
C and given x
0
∈
C arbitrarily, let the sequences {
u
n
}
,
{
u
~
n
} be generated iteratively by
(102)
u
~
n
=
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
n
)
)
,
u
n
+
1
=
β
n
u
n
+
(
1
-
β
n
)
×
S
[
{
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
}
σ
n
u
+
(
1
-
σ
n
)
P
C
×
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
]
,
∀
n
≥
0
,
where the following conditions hold for four sequences {
α
n
}
⊂
(
0
,
∞
)
,
{
λ
n
}
⊂
(
0
,
1
/
∥
A
∥
2
) and {
σ
n
}
,
{
β
n
}
⊂
[
0,1
]:(i)
∑
n
=
0
∞
‍
α
n
<
∞;
(ii)
lim
n
→
∞
σ
n
=
0 and ∑
n
=
0
∞
‍
σ
n
=
∞;
(iii)
0
<
liminf
n
→
∞
β
n
≤
limsup
n
→
∞
β
n
<
1;
(iv)
0
<
liminf
n
→
∞
λ
n
≤
limsup
n
→
∞
λ
n
<
1
/
∥
A
∥
2 and lim
n
→
∞
|
λ
n
+
1
-
λ
n
|
=
0.
Then the sequences{
u
n
}
,
{
u
~
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Γ
u if and only if lim
n
→
∞
∥
u
n
+
1
-
u
n
∥
=
0.Proof.
In Corollary20, put B
1
=
B
2
=
0 and γ
n
=
0. Then, Ξ
=
C,  β
n
+
δ
n
=
1 for all n
≥
0, and the iterative scheme (101) is equivalent to
(103)
u
n
=
x
n
,
u
~
n
=
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
n
)
)
,
y
n
=
σ
n
u
+
(
1
-
σ
n
)
P
C
(
u
n
-
λ
n
∇
f
α
n
(
u
~
n
)
)
,
u
n
+
1
=
β
n
x
n
+
δ
n
S
y
n
,
∀
n
≥
0
.
This is is equivalent to (102). Since S is a nonexpansive mapping, S must be a k-strictly pseudo-contractive mapping with k
=
0. In this case, it is easy to see that conditions (i)–(vi) in Corollary 20 all are satisfied. Therefore, in terms of Corollary 20, we obtain the desired result.Now, we are in a position to prove the strong convergence of the sequences generated by the relaxed extragradient iterative algorithm (18) with regularization.Theorem 22.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ
1. Let A
∈
B
(
ℋ
1
,
ℋ
2
) and B
i
:
C
→
ℋ
1 be β
i-inverse strongly monotone for i
=
1,2. Let S
:
C
→
C be a k-strictly pseudocontractive mapping such that Fix
(
S
)
∩
Ξ
∩
Γ
≠
∅. Let Q
:
C
→
C be a ρ-contraction with ρ
∈
[
0
,
1
/
2
). For given x
0
∈
C arbitrarily, let the sequences {
x
n
}
,
{
y
n
}
,
{
z
n
} be generated by the relaxed extragradient iterative algorithm (18) with regularization, where μ
i
∈
(
0,2
β
i
) for i
=
1,2
,
{
α
n
}
⊂
(
0
,
∞
)
,
{
λ
n
}
⊂
(
0
,
1
/
∥
A
∥
2
) and {
σ
n
}
,
{
τ
n
}
,
{
β
n
}
,
{
γ
n
}
,
{
δ
n
}
⊂
[
0,1
] such that(i)
∑
n
=
0
∞
‍
α
n
<
∞;
(ii)
σ
n
+
τ
n
≤
1, β
n
+
γ
n
+
δ
n
=
1 and (
γ
n
+
δ
n
)
k
≤
γ
n for all n
≥
0;
(iii)
lim
n
→
∞
σ
n
=
0 and ∑
n
=
0
∞
‍
σ
n
=
∞;
(iv)
0
<
liminf
n
→
∞
τ
n
≤
limsup
n
→
∞
τ
n
<
1 and lim
n
→
∞
|
τ
n
+
1
-
τ
n
|
=
0;
(v)
0
<
liminf
n
→
∞
β
n
≤
limsup
n
→
∞
β
n
<
1 and liminf
n
→
∞
δ
n
>
0;
(vi)
lim
n
→
∞
(
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
)
=
0;
(vii)
0
<
liminf
n
→
∞
λ
n
≤
limsup
n
→
∞
λ
n
<
1
/
∥
A
∥
2 and lim
n
→
∞
|
λ
n
+
1
-
λ
n
|
=
0. Then the sequences {
x
n
}
,
{
y
n
}
,
{
z
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Ξ
∩
Γ
Q
x
- if and only if lim
n
→
∞
∥
z
n
+
1
-
z
n
∥
=
0. Furthermore, (
x
-
,
y
-
) is a solution of the GSVI (14), where y
-
=
P
C
(
x
-
-
μ
2
B
2
x
-
).Proof.
First, taking into account0
<
liminf
n
→
∞
λ
n
≤
limsup
n
→
∞
λ
n
<
1
/
∥
A
∥
2, without loss of generality we may assume that {
λ
n
}
⊂
[
a
,
b
] for some a
,
b
∈
(
0
,
1
/
∥
A
∥
2
). Repeating the same argument as that in the proof of Theorem 19, we can show that P
C
(
I
-
λ
∇
f
α
) is ζ-averaged for each λ
∈
(
0
,
2
/
(
α
+
∥
A
∥
2
)
), where ζ
=
(
2
+
λ
(
α
+
∥
A
∥
2
)
)
/
4. Further, repeating the same argument as that in the proof of Theorem 19, we can also show that for each integer n
≥
0
,
P
C
(
I
-
λ
n
∇
f
α
n
) is ζ
n-averaged with ζ
n
=
(
2
+
λ
n
(
α
n
+
∥
A
∥
2
)
)
/
4
∈
(
0,1
).
Next we divide the remainder of the proof into several steps.
Step  1. {
x
n
} is bounded.
Indeed, takep
∈
Fix
(
S
)
∩
Ξ
∩
Γ arbitrarily. Then S
p
=
p
,
P
C
(
I
-
λ
∇
f
)
p
=
p for λ
∈
(
0
,
2
/
∥
A
∥
2
), and
(104)
p
=
P
C
[
P
C
(
p
-
μ
2
B
2
p
)
-
μ
1
B
1
P
C
(
p
-
μ
2
B
2
p
)
]
.
Utilizing the arguments similar to those of (45) and (46) in the proof of Theorem 19, from (18) we can obtain
(105)
∥
z
n
-
p
∥
≤
∥
x
n
-
p
∥
+
λ
n
α
n
∥
p
∥
,
(106)
∥
z
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
.
For simplicity, we write q
=
P
C
(
p
-
μ
2
B
2
p
),  z
~
n
=
P
C
(
z
n
-
μ
2
B
2
z
n
),
(107)
u
n
=
P
C
[
P
C
(
z
n
-
μ
2
B
2
z
n
)
-
μ
1
B
1
P
C
(
z
n
-
μ
2
B
2
z
n
)
]
,
u
-
n
=
P
C
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
,
for each n
≥
0. Then y
n
=
σ
n
Q
x
n
+
τ
n
u
-
n
+
(
1
-
σ
n
-
τ
n
)
u
n for each n
≥
0. Since B
i
:
C
→
ℋ
1 is β
i-inverse strongly monotone and 0
<
μ
i
<
2
β
i for i
=
1,2, utilizing the argument similar to that of (48) in the proof of Theorem 19, we can obtain that for all n
≥
0,
(108)
∥
u
n
-
p
∥
2
≤
∥
z
n
-
p
∥
2
-
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
z
n
-
B
2
p
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
z
~
n
-
B
1
q
∥
2
≤
∥
z
n
-
p
∥
2
.
Furthermore, utilizing Proposition 8(i)-(ii) and the argument similar to that of (51) in the proof of Theorem 19, from (105) we obtain
(109)
∥
u
-
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
+
(
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
-
1
)
∥
x
n
-
z
n
∥
2
≤
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
≤
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
[
∥
x
n
-
p
∥
+
λ
n
α
n
∥
p
∥
]
≤
∥
x
n
-
p
∥
2
+
4
λ
n
α
n
∥
p
∥
∥
x
n
-
p
∥
+
4
λ
n
2
α
n
2
∥
p
∥
2
=
(
∥
x
n
-
p
∥
+
2
λ
n
α
n
∥
p
∥
)
2
.
Hence it follows from (105), (108), and (109) that
(110)
∥
y
n
-
p
∥
=
∥
σ
n
(
Q
x
n
-
p
)
+
τ
n
(
u
-
n
-
p
)
+
(
1
-
σ
n
-
τ
n
)
(
u
n
-
p
)
∥
≤
σ
n
∥
Q
x
n
-
p
∥
+
τ
n
∥
u
-
n
-
p
∥
+
(
1
-
σ
n
-
τ
n
)
∥
u
n
-
p
∥
≤
σ
n
(
ρ
∥
x
n
-
p
∥
+
∥
Q
p
-
p
∥
)
+
τ
n
∥
u
-
n
-
p
∥
+
(
1
-
σ
n
-
τ
n
)
∥
z
n
-
p
∥
≤
σ
n
(
ρ
∥
x
n
-
p
∥
+
∥
Q
p
-
p
∥
)
+
τ
n
(
∥
x
n
-
p
∥
+
2
λ
n
α
n
∥
p
∥
)
+
(
1
-
σ
n
-
τ
n
)
(
∥
x
n
-
p
∥
+
λ
n
α
n
∥
p
∥
)
=
(
1
-
(
1
-
ρ
)
σ
n
)
∥
x
n
-
p
∥
+
σ
n
∥
Q
p
-
p
∥
+
[
2
τ
n
+
(
1
-
σ
n
-
τ
n
)
]
λ
n
α
n
∥
p
∥
≤
(
1
-
(
1
-
ρ
)
σ
n
)
∥
x
n
-
p
∥
+
σ
n
∥
Q
p
-
p
∥
+
2
λ
n
α
n
∥
p
∥
=
(
1
-
(
1
-
ρ
)
σ
n
)
∥
x
n
-
p
∥
+
(
1
-
ρ
)
σ
n
∥
Q
p
-
p
∥
1
-
ρ
+
2
λ
n
α
n
∥
p
∥
≤
max
{
∥
x
n
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
λ
n
α
n
∥
p
∥
.
Since (
γ
n
+
δ
n
)
k
≤
γ
n for all n
≥
0, utilizing Lemma 17 we obtain from (110)
(111)
∥
x
n
+
1
-
p
∥
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
∥
≤
β
n
∥
x
n
-
p
∥
+
∥
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
∥
≤
β
n
∥
x
n
-
p
∥
+
(
γ
n
+
δ
n
)
∥
y
n
-
p
∥
≤
β
n
∥
x
n
-
p
∥
+
(
γ
n
+
δ
n
)
[
max
{
∥
x
n
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
λ
n
α
n
∥
p
∥
]
≤
β
n
∥
x
n
-
p
∥
+
(
γ
n
+
δ
n
)
max
{
∥
x
n
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
λ
n
α
n
∥
p
∥
≤
max
{
∥
x
n
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
α
n
.
Repeating the same argument as that of (54) in the proof of Theorem 19, by induction we can prove that
(112)
∥
x
n
+
1
-
p
∥
≤
max
{
∥
x
0
-
p
∥
,
∥
Q
p
-
p
∥
1
-
ρ
}
+
2
b
∥
p
∥
∑
j
=
0
n
α
j
.
Thus, {
x
n
} is bounded. Since P
C
,
∇
f
α
n
,
B
1 and B
2 are Lipschitz continuous, it is easy to see that {
z
n
}
,
{
u
n
}
,
{
u
-
n
}
,
{
y
n
}, and {
z
~
n
} are bounded, where z
~
n
=
P
C
(
z
n
-
μ
2
B
2
z
n
) for all n
≥
0.
Step  2. lim
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0.
Indeed, definex
n
+
1
=
β
n
x
n
+
(
1
-
β
n
)
w
n for all n
≥
0. Then, utilizing the arguments similar to those of (58)–(61) in the proof of Theorem 19, we can obtain that
(113)
w
n
+
1
-
w
n
=
γ
n
+
1
(
y
n
+
1
-
y
n
)
+
δ
n
+
1
(
S
y
n
+
1
-
S
y
n
)
1
-
β
n
+
1
+
(
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
)
y
n
+
(
δ
n
+
1
1
-
β
n
+
1
-
δ
n
1
-
β
n
)
S
y
n
,
∥
γ
n
+
1
(
y
n
+
1
-
y
n
)
+
δ
n
+
1
(
S
y
n
+
1
-
S
y
n
)
∥
≤
(
γ
n
+
1
+
δ
n
+
1
)
∥
y
n
+
1
-
y
n
∥
,
(due to Lemma 17)
(114)
∥
z
n
+
1
-
z
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
x
n
)
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
x
n
∥
,
(115)
∥
u
n
+
1
-
u
n
∥
2
≤
∥
P
C
(
z
n
+
1

-

μ
2
B
2
z
n
+
1
)

-

P
C
(
z
n

-

μ
2
B
2
z
n
)
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
P
C
(
z
n
+
1
-
μ
2
B
2
z
n
+
1
)
-
μ
1
(
2
β
1
-
μ
1
)
-
B
1
P
C
(
z
n
-
μ
2
B
2
z
n
)
∥
2
≤
∥
(
z
n
+
1
-
μ
2
B
2
z
n
+
1
)
-
(
z
n
-
μ
2
B
2
z
n
)
∥
2
≤
∥
z
n
+
1
-
z
n
∥
2
-
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
z
n
+
1
-
B
2
z
n
∥
2
≤
∥
z
n
+
1
-
z
n
∥
2
.
So, we have
(116)
∥
u
-
n
+
1
-
u
-
n
∥
=
∥
P
C
(
x
n
+
1
-
λ
n
+
1
∇
f
α
n
+
1
(
z
n
+
1
)
)
-
P
C
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
∥
≤
∥
(
x
n
+
1
-
λ
n
+
1
∇
f
α
n
+
1
(
z
n
+
1
)
)
-
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
∥
=
∥
x
n
+
1
-
x
n
-
λ
n
+
1
[
∇
f
α
n
+
1
(
z
n
+
1
)
-
∇
f
α
n
+
1
(
z
n
)
]
-
[
λ
n
+
1
∇
f
α
n
+
1
(
z
n
)
-
λ
n
∇
f
α
n
(
z
n
)
]
∥
≤
∥
x
n
+
1
-
x
n
∥
+
λ
n
+
1
∥
∇
f
α
n
+
1
(
z
n
+
1
)
-
∇
f
α
n
+
1
(
z
n
)
∥
+
∥
λ
n
+
1
∇
f
α
n
+
1
(
z
n
)
-
λ
n
∇
f
α
n
(
z
n
)
∥
≤
∥
x
n
+
1
-
x
n
∥
+
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
∥
z
n
+
1
-
z
n
∥
+
∥
λ
n
+
1
(
α
n
+
1
I
+
∇
f
)
z
n
-
λ
n
(
α
n
I
+
∇
f
)
z
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
λ
n
+
1
(
α
n
+
1
+
∥
A
∥
2
)
∥
z
n
+
1
-
z
n
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
z
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
z
n
)
∥
≤
∥
x
n
+
1
-
x
n
∥
+
∥
z
n
+
1
-
z
n
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
z
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
z
n
)
∥
.
This together with (114) implies that
(117)
∥
y
n
+
1
-
y
n
∥
=
∥
σ
n
+
1
(
Q
x
n
+
1
-
u
n
+
1
)
+
τ
n
+
1
u
-
n
+
1
+
(
1
-
τ
n
+
1
)
u
n
+
1
-
σ
n
(
Q
x
n
-
u
n
)
-
τ
n
u
-
n
-
(
1
-
τ
n
)
u
n
∥
≤
∥
τ
n
+
1
u
-
n
+
1
-
τ
n
u
-
n
∥
+
∥
(
1
-
τ
n
+
1
)
u
n
+
1
-
(
1
-
τ
n
)
u
n
∥
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
n
∥
≤
|
τ
n
+
1
-
τ
n
|
∥
u
-
n
+
1
∥
+
τ
n
∥
u
-
n
+
1
-
u
-
n
∥
+
|
τ
n
+
1
-
τ
n
|
∥
u
n
+
1
∥
+
(
1
-
τ
n
)
∥
u
n
+
1
-
u
n
∥
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
n
∥
≤
τ
n
[
{
1
2
}
∥
x
n
+
1
-
x
n
∥
+
∥
z
n
+
1
-
z
n
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
z
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
z
n
)
∥
{
1
2
}
]
+
(
1
-
τ
n
)
∥
z
n
+
1
-
z
n
∥
+
|
τ
n
+
1
-
τ
n
|
(
∥
u
-
n
+
1
∥
+
∥
u
n
+
1
∥
)
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
∥
z
n
+
1
-
z
n
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
z
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
z
n
)
∥
+
|
τ
n
+
1
-
τ
n
|
(
∥
u
-
n
+
1
∥
+
∥
u
n
+
1
∥
)
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
n
∥
.
Hence it follows from (113), and (117) that
(118)
∥
w
n
+
1
-
w
n
∥
≤
∥
γ
n
+
1
(
y
n
+
1
-
y
n
)
+
δ
n
+
1
(
S
y
n
+
1
-
S
y
n
)
∥
1
-
β
n
+
1
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
y
n
∥
+
|
δ
n
+
1
1
-
β
n
+
1
-
δ
n
1
-
β
n
|
∥
S
y
n
∥
≤
γ
n
+
1
+
δ
n
+
1
1
-
β
n
+
1
∥
y
n
+
1
-
y
n
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
(
∥
y
n
∥
+
∥
S
y
n
∥
)
=
∥
y
n
+
1
-
y
n
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
(
∥
y
n
∥
+
∥
S
y
n
∥
)
≤
∥
x
n
+
1
-
x
n
∥
+
∥
z
n
+
1
-
z
n
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
z
n
∥
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
z
n
)
∥
+
|
τ
n
+
1
-
τ
n
|
(
∥
u
-
n
+
1
∥
+
∥
u
n
+
1
∥
)
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
n
∥
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
(
∥
y
n
∥
+
∥
S
y
n
∥
)
.
Since {
x
n
}
,
{
y
n
}
,
{
z
n
}
,
{
u
n
}, and {
u
-
n
} are bounded, it follows from conditions (i), (iii), (iv), (vi), and (vii) that
(119)
lim
sup
n
→
∞
(
∥
w
n
+
1
-
w
n
∥
-
∥
x
n
+
1
-
x
n
∥
)
≤
lim
sup
n
→
∞
{
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
∥
z
n
+
1
-
z
n
∥
+
|
λ
n
+
1
α
n
+
1
-
λ
n
α
n
|
∥
z
n
∥
≤
lim
sup
n
→
∞
+
|
λ
n
+
1
-
λ
n
|
∥
∇
f
(
z
n
)
∥
≤
lim
sup
n
→
∞
+
|
τ
n
+
1
-
τ
n
|
(
∥
u
-
n
+
1
∥
+
∥
u
n
+
1
∥
)
≤
lim
sup
n
→
∞
+
σ
n
+
1
∥
Q
x
n
+
1
-
u
n
+
1
∥
+
σ
n
∥
Q
x
n
-
u
n
∥
≤
lim
sup
n
→
∞
+
|
γ
n
+
1
1
-
β
n
+
1
-
γ
n
1
-
β
n
|
(
∥
y
n
∥
+
∥
S
y
n
∥
)
{
1
2
}
}
=
0
.
Hence by Lemma 14 we get lim
n
→
∞
∥
w
n
-
x
n
∥
=
0. Thus,
(120)
lim
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
lim
n
→
∞
(
1
-
β
n
)
∥
w
n
-
x
n
∥
=
0
.
Step  3.
lim
n
→
∞
∥
B
2
z
n
-
B
2
p
∥
=
0,  lim
n
→
∞
∥
B
1
z
~
n
-
B
1
q
∥
=
0, and lim
n
→
∞
∥
x
n
-
z
n
∥
=
0, where q
=
P
C
(
p
-
μ
2
B
2
p
).
Indeed, utilizing Lemma17 and the convexity of ∥
·
∥
2, we obtain from (18) and (106)–(109) that
(121)
∥
x
n
+
1
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
×
∥
1
γ
n
+
δ
n
[
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
]
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
×
[
σ
n
∥
Q
x
n

-

p
∥
2

+

τ
n
∥
u
-
n

-

p
∥
2

+

(
1

-

σ
n

-

τ
n
)
∥
u
n

-

p
∥
2
]
≤
β
n
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
[
τ
n
∥
u
-
n
-
p
∥
2
+
(
1
-
τ
n
)
∥
u
n
-
p
∥
2
]
≤
β
n
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
×
{
{
1
2
}
τ
n
[
{
1
2
}
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
+
(
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
-
1
)
∥
x
n
-
z
n
∥
2
{
2
1
}
]
+
(
1

-

τ
n
)
[
∥
z
n

-

p
∥
2

-

μ
2
(
2
β
2

-

μ
2
)
∥
B
2
z
n

-

B
2
p
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
z
~
n
-
B
1
q
∥
2
]
{
1
2
}
}
≤
β
n
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
×
{
{
1
2
}
τ
n
[
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
+
(
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
-
1
)
∥
x
n
-
z
n
∥
2
]
+
(
1
-
τ
n
)
[
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
-
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
z
n
-
B
2
p
∥
2
-
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
z
~
n
-
B
1
q
∥
2
]
{
1
2
}
}
=
β
n
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
×
{
{
1
2
}
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
-
τ
n
(
1
-
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
)
∥
x
n
-
z
n
∥
2
-
(
1
-
τ
n
)
×
[
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
z
n
-
B
2
p
∥
2
×
+
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
z
~
n
-
B
1
q
∥
2
]
{
1
2
}
}
≤
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
-
(
γ
n
+
δ
n
)
{
{
1
2
}
τ
n
(
1
-
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
)
∥
x
n
-
z
n
∥
2
-
(
γ
n
+
δ
n
)
+
(
1
-
τ
n
)
[
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
z
n
-
B
2
p
∥
2
-
(
γ
n
+
δ
n
)
+
(
1
-
τ
n
)
+
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
z
~
n
-
B
1
q
∥
2
]
{
1
1
}
}
.
Therefore,
(122)
(
γ
n
+
δ
n
)
{
{
2
1
}
τ
n
(
1
-
λ
n
2
(
α
n
+
∥
A
∥
2
)
2
)
∥
x
n
-
z
n
∥
2
+
(
1
-
τ
n
)
[
μ
2
(
2
β
2
-
μ
2
)
∥
B
2
z
n
-
B
2
p
∥
2
+
(
1
-
τ
n
)
+
μ
1
(
2
β
1
-
μ
1
)
∥
B
1
z
~
n
-
B
1
q
∥
2
]
{
1
2
}
}
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
≤
(
∥
x
n
-
p
∥
+
∥
x
n
+
1
-
p
∥
)
∥
x
n
-
x
n
+
1
∥
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
.
Since α
n
→
0
,
σ
n
→
0
,
∥
x
n
-
x
n
+
1
∥
→
0
,
liminf
n
→
∞
(
γ
n
+
δ
n
)
>
0
,
{
λ
n
}
⊂
[
a
,
b
], and 0
<
liminf
n
→
∞
τ
n
≤
limsup
n
→
∞
τ
n
<
1, it follows that
(123)
lim
n
→
∞
∥
x
n
-
z
n
∥
=
0
,
lim
n
→
∞
∥
B
1
z
~
n
-
B
1
q
∥
=
0
,
lim
n
→
∞
∥
B
2
z
n
-
B
2
p
∥
=
0
.
Step  4. lim
n
→
∞
∥
S
y
n
-
y
n
∥
=
0.
Indeed, observe that(124)
∥
u
-
n
-
z
n
∥
=
∥
P
C
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
-
P
C
(
x
n
-
λ
n
∇
f
α
n
(
x
n
)
)
∥
≤
∥
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
-
(
x
n
-
λ
n
∇
f
α
n
(
x
n
)
)
∥
=
λ
n
∥
∇
f
α
n
(
z
n
)
-
∇
f
α
n
(
x
n
)
∥
≤
λ
n
(
α
n
+
∥
A
∥
2
)
∥
z
n
-
x
n
∥
.
This together with ∥
z
n
-
x
n
∥
→
0 implies that lim
n
→
∞
∥
u
-
n
-
z
n
∥
=
0 and hence lim
n
→
∞
∥
u
-
n
-
x
n
∥
=
0. Utilizing the arguments similar to those of (73) and (75) in the proof of Theorem 19 we can prove that
(125)
∥
z
~
n
-
q
∥
2
≤
∥
z
n
-
p
∥
2
-
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
2
+
2
μ
2
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
∥
B
2
z
n
-
B
2
p
∥
,
∥
u
n
-
p
∥
2
≤
∥
z
~
n
-
q
∥
2
-
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
z
~
n
-
B
1
q
∥
.
Utilizing (106), (109), (125), we have
(126)
∥
y
n
-
p
∥
2
=
∥
σ
n
(
Q
x
n
-
p
)
+
τ
n
(
u
-
n
-
p
)
+
(
1
-
σ
n
-
τ
n
)
(
u
n
-
p
)
∥
2
≤
σ
n
∥
Q
x
n
-
p
∥
2
+
τ
n
∥
u
-
n
-
p
∥
2
+
(
1
-
σ
n
-
τ
n
)
∥
u
n
-
p
∥
2
≤
σ
n
∥
Q
x
n
-
p
∥
2
+
τ
n
∥
u
-
n
-
p
∥
2
+
(
1
-
τ
n
)
∥
u
n
-
p
∥
2
≤
σ
n
∥
Q
x
n
-
p
∥
2
+
τ
n
(
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
)
+
(
1
-
τ
n
)
×
[
∥
z
~
n
-
q
∥
2
-
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
z
~
n
-
B
1
q
∥
]
≤
σ
n
∥
Q
x
n
-
p
∥
2
+
τ
n
(
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
)
+
(
1
-
τ
n
)
×
{
∥
z
n
-
p
∥
2
-
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
2
+
2
μ
2
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
∥
B
2
z
n
-
B
2
p
∥
-
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
z
~
n
-
B
1
q
∥
{
∥
z
n
-
p
∥
2
}
}
≤
σ
n
∥
Q
x
n
-
p
∥
2
+
τ
n
(
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
)
+
(
1
-
τ
n
)
×
{
∥
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
-
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
2
+
2
μ
2
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
∥
B
2
z
n
-
B
2
p
∥
-
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
2
+
2
μ
1
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
z
~
n
-
B
1
q
∥
}
≤
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
+
2
μ
2
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
∥
B
2
z
n
-
B
2
p
∥
+
2
μ
1
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
z
~
n
-
B
1
q
∥
-
(
1

-

τ
n
)
(
∥
z
n

-

z
~
n

-

(
p
-
q
)
∥
2

+

∥
z
~
n

-

u
n

+

(
p

-

q
)
∥
2
)
.
Thus from (18) and (126) it follows that
(127)
∥
x
n
+
1
-
p
∥
2
=
∥
β
n
(
x
n
-
p
)
+
γ
n
(
y
n
-
p
)
+
δ
n
(
S
y
n
-
p
)
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
γ
n
+
δ
n
)
∥
y
n
-
p
∥
2
=
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
∥
y
n
-
p
∥
2
≤
β
n
∥
x
n
-
p
∥
2
+
(
1
-
β
n
)
×
{
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
+
2
μ
2
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
∥
B
2
z
n
-
B
2
p
∥
+
2
μ
1
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
z
~
n
-
B
1
q
∥
-
(
1
-
τ
n
)
(
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
2
-
(
1
-
τ
n
)
+
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
2
)
}
≤
∥
x
n
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
+
2
μ
2
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
∥
B
2
z
n
-
B
2
p
∥
+
2
μ
1
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
z
~
n
-
B
1
q
∥
-
(
1
-
β
n
)
(
1
-
τ
n
)
×
(
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
2
+
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
2
)
,
which hence implies that
(128)
(
1
-
β
n
)
(
1
-
τ
n
)
×
(
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
2
+
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
2
)
≤
∥
x
n
-
p
∥
2
-
∥
x
n
+
1
-
p
∥
2
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
+
2
μ
2
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
∥
B
2
z
n
-
B
2
p
∥
+
2
μ
1
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
z
~
n
-
B
1
q
∥
≤
(
∥
x
n
-
p
∥
+
∥
x
n
+
1
-
p
∥
)
∥
x
n
-
x
n
+
1
∥
+
σ
n
∥
Q
x
n
-
p
∥
2
+
2
λ
n
α
n
∥
p
∥
∥
z
n
-
p
∥
+
2
μ
2
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
∥
B
2
z
n
-
B
2
p
∥
+
2
μ
1
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
∥
B
1
z
~
n
-
B
1
q
∥
.
Since limsup
n
→
∞
β
n
<
1,  limsup
n
→
∞
τ
n
<
1,  {
λ
n
}
⊂
[
a
,
b
],  α
n
→
0,  σ
n
→
0,  ∥
B
2
z
n
-
B
2
p
∥
→
0,  ∥
B
1
z
~
n
-
B
1
q
∥
→
0 and ∥
x
n
+
1
-
x
n
∥
→
0, it follows from the boundedness of {
x
n
}
,
{
u
n
}
,
{
z
n
}, and {
z
~
n
} that
(129)
lim
n
→
∞
∥
z
n
-
z
~
n
-
(
p
-
q
)
∥
=
0
,
lim
n
→
∞
∥
z
~
n
-
u
n
+
(
p
-
q
)
∥
=
0
.
Consequently, it immediately follows that
(130)
lim
n
→
∞
∥
z
n
-
u
n
∥
=
0
,
lim
n
→
∞
∥
u
n
-
u
-
n
∥
=
0
.
Also, note that
(131)
∥
y
n
-
u
-
n
∥
≤
σ
n
∥
Q
x
n
-
u
-
n
∥
+
(
1
-
σ
n
-
τ
n
)
∥
u
n
-
u
-
n
∥
⟶
0
.
This together with ∥
x
n
-
u
-
n
∥
→
0 implies that
(132)
lim
n
→
∞
∥
x
n
-
y
n
∥
=
0
.
Since
(133)
∥
δ
n
(
S
y
n
-
x
n
)
∥
≤
∥
x
n
+
1
-
x
n
∥
+
γ
n
∥
y
n
-
x
n
∥
,
it follows that
(134)
lim
n
→
∞
∥
S
y
n
-
x
n
∥
=
0
,
lim
n
→
∞
∥
S
y
n
-
y
n
∥
=
0
.
Step  5.
limsup
n
→
∞
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
≤
0 where x
-
=
P
Fix
(
S
)
∩
Ξ
∩
Γ
Q
x
-.
Indeed, since{
x
n
} is bounded, there exists a subsequence {
x
n
i
} of {
x
n
} such that
(135)
lim
sup
n
→
∞
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
=
lim
i
→
∞
〈
Q
x
-
-
x
-
,
x
n
i
-
x
-
〉
.
Also, since H is reflexive and {
x
n
} is bounded, without loss of generality we may assume that x
n
i
→
p
^ weakly for some p
^
∈
C. Taking into account that ∥
x
n
-
y
n
∥
→
0 and ∥
x
n
-
z
n
∥
→
0 as n
→
∞, we deduce that y
n
i
→
p
^ weakly and z
n
i
→
p
^ weakly.
First, it is clear from Lemma15 and ∥
S
y
n
-
y
n
∥
→
0 that p
^
∈
Fix
(
S
). Now let us show that p
^
∈
Ξ. Note that
(136)
∥
z
n
-
G
(
z
n
)
∥
=
∥
z
n
-
P
C
[
P
C
(
z
n
-
μ
2
B
2
z
n
)
-
μ
1
B
1
P
C
(
z
n
-
μ
2
B
2
z
n
)
]
∥
=
∥
z
n
-
u
n
∥
⟶
0
(
n
⟶
∞
)
,
where G
:
C
→
C is defined as that in Lemma 4. According to Lemma 15 we get p
^
∈
Ξ. Further, let us show that p
^
∈
Γ. As a matter of fact, define
(137)
T
v
=
{
∇
f
(
v
)
+
N
C
v
,
if
v
∈
C
,
∅
,
if
v
∉
C
,
where N
C
v
=
{
w
∈
ℋ
1
:
〈
v
-
u
,
w
〉
≥
0
,
∀
u
∈
C
}. Then, T is maximal monotone and 0
∈
T
v if and only if v
∈
VI
(
C
,
∇
f
); see [40] for more details. Let (
v
,
w
)
∈
Gph
(
T
). Then, we have
(138)
〈
v
-
u
,
w
-
∇
f
(
v
)
〉
≥
0
,
∀
u
∈
C
.
Observe that
(139)
z
n
=
P
C
(
x
n
-
λ
n
∇
f
α
n
(
x
n
)
)
,
v
∈
C
.
Utilizing the arguments similar to those of Step  5 in the proof of Theorem 19 we can prove that
(140)
〈
v
-
p
^
,
w
〉
≥
0
.
Since T is maximal monotone, we have p
^
∈
T
-
1
0, and, hence, p
^
∈
VI
(
C
,
∇
f
). Thus it is clear that p
^
∈
Γ. Therefore, p
^
∈
Fix
(
S
)
∩
Ξ
∩
Γ. Consequently, in terms of Proposition 8 (i) we obtain from (135) that
(141)
lim
sup
n
→
∞
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
=
lim
i
→
∞
〈
Q
x
-
-
x
-
,
x
n
i
-
x
-
〉
=
〈
Q
x
-
-
x
-
,
p
^
-
x
-
〉
≤
0
.
Step  6.
lim
n
→
∞
∥
x
n
-
x
-
∥
=
0.
Indeed, observe that(142)
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
=
〈
Q
x
n
-
x
-
,
x
n
-
x
-
〉
+
〈
Q
x
n
-
x
-
,
y
n
-
x
n
〉
=
〈
Q
x
n
-
Q
x
-
,
x
n
-
x
-
〉
+
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
〈
Q
x
n
-
x
-
,
y
n
-
x
n
〉
≤
ρ
∥
x
n
-
x
-
∥
2
+
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
∥
Q
x
n
-
x
-
∥
∥
y
n
-
x
n
∥
.
Utilizing Lemmas 17 and 18, we obtain from (106)–(109) and the convexity of ∥
·
∥
2 that
(143)
∥
x
n
+
1
-
x
-
∥
2
=
∥
β
n
(
x
n
-
x
-
)
+
γ
n
(
y
n
-
x
-
)
+
δ
n
(
S
y
n
-
x
-
)
∥
2
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
∥
1
γ
n
+
δ
n
[
γ
n
(
y
n
-
x
-
)
+
δ
n
(
S
y
n
-
x
-
)
]
∥
2
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
∥
y
n
-
x
-
∥
2
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
×
[
∥
τ
n
(
u
-
n
-
x
-
)
+
(
1
-
σ
n
-
τ
n
)
(
u
n
-
x
-
)
∥
2
+
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
{
∥
τ
n
(
u
-
n
-
x
-
)
+
(
1
-
σ
n
-
τ
n
)
(
u
n
-
x
-
)
∥
2
}
]
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
×
[
τ
n
∥
u
-
n
-
x
-
∥
2
+
(
1
-
σ
n
-
τ
n
)
∥
u
n
-
x
-
∥
2
+
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
{
∥
u
n
-
x
-
∥
2
}
]
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
×
[
τ
n
∥
u
-
n
-
x
-
∥
2
+
(
1
-
σ
n
-
τ
n
)
∥
z
n
-
x
-
∥
2
+
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
]
≤
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
×
[
{
1
2
}
τ
n
(
∥
x
n
-
x
-
∥
2
+
2
λ
n
α
n
∥
x
-
∥
∥
z
n
-
x
-
∥
)
+
(
1
-
σ
n
-
τ
n
)
×
(
∥
x
n
-
x
-
∥
2
+
2
λ
n
α
n
∥
x
-
∥
∥
z
n
-
x
-
∥
)
+
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
{
1
2
}
]
=
β
n
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
×
[
(
1
-
σ
n
)
(
∥
x
n
-
x
-
∥
2
+
2
λ
n
α
n
∥
x
-
∥
∥
z
n
-
x
-
∥
)
+
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
{
(
∥
x
n
-
x
-
∥
2
+
2
λ
n
α
n
∥
x
-
∥
∥
z
n
-
x
-
∥
)
}
]
=
(
1
-
(
γ
n
+
δ
n
)
σ
n
)
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
(
1
-
σ
n
)
2
λ
n
α
n
∥
x
-
∥
∥
z
n
-
x
-
∥
+
(
γ
n
+
δ
n
)
2
σ
n
〈
Q
x
n
-
x
-
,
y
n
-
x
-
〉
≤
(
1
-
(
γ
n
+
δ
n
)
σ
n
)
∥
x
n
-
x
-
∥
2
+
(
γ
n
+
δ
n
)
2
σ
n
×
[
ρ
∥
x
n
-
x
-
∥
2
+
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
∥
Q
x
n
-
x
-
∥
∥
y
n
-
x
n
∥
{
∥
x
n
-
x
-
∥
2
}
]
+
2
λ
n
α
n
∥
x
-
∥
∥
z
n
-
x
-
∥
=
[
1
-
(
1
-
2
ρ
)
(
γ
n
+
δ
n
)
σ
n
]
∥
x
n
-
x
-
∥
2
+
(
1
-
2
ρ
)
(
γ
n
+
δ
n
)
σ
n
×
2
[
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
∥
Q
x
n
-
x
-
∥
∥
y
n
-
x
n
∥
]
1
-
2
ρ
+
2
λ
n
α
n
∥
x
-
∥
∥
z
n
-
x
-
∥
.
Note that liminf
n
→
∞
(
1
-
2
ρ
)
(
γ
n
+
δ
n
)
>
0. It follows that ∑
n
=
0
∞
‍
(
1
-
2
ρ
)
(
γ
n
+
δ
n
)
σ
n
=
∞. It is clear that
(144)
lim
sup
n
→
∞
2
[
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
+
∥
Q
x
n
-
x
-
∥
∥
y
n
-
x
n
∥
]
1
-
2
ρ
≤
0
,
because limsup
n
→
∞
〈
Q
x
-
-
x
-
,
x
n
-
x
-
〉
≤
0 and lim
n
→
∞
∥
x
n
-
y
n
∥
=
0. In addition, note also that {
λ
n
}
⊂
[
a
,
b
]
,
∑
n
=
0
∞
‍
α
n
<
∞ and {
z
n
} is bounded. Hence we get ∑
n
=
0
∞
‍
2
λ
n
α
n
∥
x
-
∥
∥
z
n
-
x
-
∥
<
∞. Therefore, all conditions of Lemma 16 are satisfied. Consequently, we immediately deduce that ∥
x
n
-
x
-
∥
→
0 as n
→
∞. In the meantime, taking into account that ∥
x
n
-
y
n
∥
→
0 and ∥
x
n
-
z
n
∥
→
0 as n
→
∞, we infer that
(145)
lim
n
→
∞
∥
y
n
-
x
-
∥
=
lim
n
→
∞
∥
z
n
-
x
-
∥
=
0
.
This completes the proof.Corollary 23.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ
1. Let A
∈
B
(
ℋ
1
,
ℋ
2
) and let B
i
:
C
→
ℋ
1 be β
i-inverse strongly monotone for i
=
1,2. Let S
:
C
→
C be a k-strictly pseudocontractive mapping such that Fix
(
S
)
∩
Ξ
∩
Γ
≠
∅. For fixed u
∈
C and given x
0
∈
C arbitrarily, let the sequences {
x
n
}
,
{
y
n
}
,
{
z
n
} be generated iteratively by
(146)
z
n
=
P
C
(
x
n
-
λ
n
∇
f
α
n
(
x
n
)
)
,
y
n
=
σ
n
u
+
τ
n
P
C
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
+
(
1
-
σ
n
-
τ
n
)
×
P
C
[
P
C
(
z
n
-
μ
2
B
2
z
n
)
-
μ
1
B
1
P
C
(
z
n
-
μ
2
B
2
z
n
)
]
,
x
n
+
1
=
β
n
x
n
+
γ
n
y
n
+
δ
n
S
y
n
,
∀
n
≥
0
,
where μ
i
∈
(
0,2
β
i
) for i
=
1,2
,
{
α
n
}
⊂
(
0
,
∞
)
,
{
λ
n
}
⊂
(
0
,
1
/
∥
A
∥
2
) and {
σ
n
}
,
{
τ
n
}
,
{
β
n
}
,
{
γ
n
}, {
δ
n
}
⊂
[
0,1
] such that(i)
∑
n
=
0
∞
‍
α
n
<
∞;
(ii)
σ
n
+
τ
n
≤
1, β
n
+
γ
n
+
δ
n
=
1 and (
γ
n
+
δ
n
)
k
≤
γ
n for all n
≥
0;
(iii)
lim
n
→
∞
σ
n
=
0 and ∑
n
=
0
∞
‍
σ
n
=
∞;
(iv)
0
<
liminf
n
→
∞
τ
n
≤
limsup
n
→
∞
τ
n
<
1 and lim
n
→
∞
|
τ
n
+
1
-
τ
n
|
=
0;
(v)
0
<
liminf
n
→
∞
β
n
≤
limsup
n
→
∞
β
n
<
1 and liminf
n
→
∞
δ
n
>
0;
(vi)
lim
n
→
∞
(
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
)
=
0;
(vii)
0
<
liminf
n
→
∞
λ
n
≤
limsup
n
→
∞
λ
n
<
1
/
∥
A
∥
2 and lim
n
→
∞
|
λ
n
+
1
-
λ
n
|
=
0. Then the sequences {
x
n
}
,
{
y
n
}
,
{
z
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Ξ
∩
Γ
u if and only if lim
n
→
∞
∥
z
n
+
1
-
z
n
∥
=
0. Furthermore, (
x
-
,
y
-
) is a solution of the GSVI (14), where y
-
=
P
C
(
x
-
-
μ
2
B
2
x
-
).Corollary 24.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ
1. Let A
∈
B
(
ℋ
1
,
ℋ
2
) and let B
i
:
C
→
ℋ
1 be β
i-inverse strongly monotone for i
=
1,2. Let S
:
C
→
C be a nonexpansive mapping such that Fix
(
S
)
∩
Ξ
∩
Γ
≠
∅. Let Q
:
C
→
C be a ρ-contraction with ρ
∈
[
0
,
1
/
2
). For given x
0
∈
C arbitrarily, let the sequences {
x
n
}
,
{
y
n
}
,
{
z
n
} be generated iteratively by
(147)
z
n
=
P
C
(
x
n
-
λ
n
∇
f
α
n
(
x
n
)
)
,
y
n
=
σ
n
Q
x
n
+
τ
n
P
C
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
+
(
1
-
σ
n
-
τ
n
)
×
P
C
[
P
C
(
z
n
-
μ
2
B
2
z
n
)
-
μ
1
B
1
P
C
(
z
n
-
μ
2
B
2
z
n
)
]
,
x
n
+
1
=
β
n
x
n
+
γ
n
y
n
+
δ
n
S
y
n
,
∀
n
≥
0
,
where μ
i
∈
(
0,2
β
i
) for i
=
1,2
,
{
α
n
}
⊂
(
0
,
∞
)
,
{
λ
n
}
⊂
(
0
,
1
/
∥
A
∥
2
) and {
σ
n
}
,
{
τ
n
}
,
{
β
n
}
,
{
γ
n
}, {
δ
n
}
⊂
[
0,1
] such that(i)
∑
n
=
0
∞
‍
α
n
<
∞;
(ii)
σ
n
+
τ
n
≤
1, β
n
+
γ
n
+
δ
n
=
1 and (
γ
n
+
δ
n
)
k
≤
γ
n for all n
≥
0;
(iii)
lim
n
→
∞
σ
n
=
0 and ∑
n
=
0
∞
‍
σ
n
=
∞;
(iv)
0
<
liminf
n
→
∞
τ
n
≤
limsup
n
→
∞
τ
n
<
1 and lim
n
→
∞
|
τ
n
+
1
-
τ
n
|
=
0;
(v)
0
<
liminf
n
→
∞
β
n
≤
limsup
n
→
∞
β
n
<
1 and liminf
n
→
∞
δ
n
>
0;
(vi)
lim
n
→
∞
(
γ
n
+
1
/
(
1
-
β
n
+
1
)
-
γ
n
/
(
1
-
β
n
)
)
=
0;
(vii)
0
<
liminf
n
→
∞
λ
n
≤
limsup
n
→
∞
λ
n
<
1
/
∥
A
∥
2 and lim
n
→
∞
|
λ
n
+
1
-
λ
n
|
=
0. Then the sequences {
x
n
}
,
{
y
n
}
,
{
z
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Ξ
∩
Γ
Q
x
- if and only if lim
n
→
∞
∥
z
n
+
1
-
z
n
∥
=
0. Furthermore, (
x
-
,
y
-
) is a solution of the GSVI (14), where y
-
=
P
C
(
x
-
-
μ
2
B
2
x
-
).Next, utilizing Corollary23 one gives the following improvement and extension of the main result in [18] (i.e., [18, Theorem  3.1]).Corollary 25.
LetC be a nonempty closed convex subset of a real Hilbert space ℋ
1. Let A
∈
B
(
ℋ
1
,
ℋ
2
) and let S
:
C
→
C be a nonexpansive mapping such that Fix
(
S
)
∩
Γ
≠
∅. For fixed u
∈
C and given x
0
∈
C arbitrarily, let the sequences {
x
n
}
,
{
z
n
} be generated iteratively by
(148)
z
n
=
P
C
(
x
n
-
λ
n
∇
f
α
n
(
x
n
)
)
,
x
n
+
1
=
β
n
x
n
+
(
1
-
β
n
)
×
S
[
σ
n
u
+
τ
n
P
C
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
+
(
1
-
σ
n
-
τ
n
)
z
n
{
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
}
]
,
∀
n
≥
0
,
where {
α
n
}
⊂
(
0
,
∞
)
,
{
λ
n
}
⊂
(
0
,
1
/
∥
A
∥
2
) and {
σ
n
}
,
{
τ
n
}
,
{
β
n
}
⊂
[
0,1
] such that(i)
∑
n
=
0
∞
‍
α
n
<
∞;
(ii)
σ
n
+
τ
n
≤
1 for all n
≥
0;
(iii)
lim
n
→
∞
σ
n
=
0 and ∑
n
=
0
∞
‍
σ
n
=
∞;
(iv)
0
<
liminf
n
→
∞
τ
n
≤
limsup
n
→
∞
τ
n
<
1 and lim
n
→
∞
|
τ
n
+
1
-
τ
n
|
=
0;
(v)
0
<
liminf
n
→
∞
β
n
≤
limsup
n
→
∞
β
n
<
1;
(vi)
0
<
liminf
n
→
∞
λ
n
≤
limsup
n
→
∞
λ
n
<
1
/
∥
A
∥
2 and lim
n
→
∞
|
λ
n
+
1
-
λ
n
|
=
0. Then the sequences {
x
n
}
,
{
z
n
} converge strongly to the same point x
-
=
P
Fix
(
S
)
∩
Γ
u if and only if lim
n
→
∞
∥
z
n
+
1
-
z
n
∥
=
0.Proof.
In Corollary23, put B
1
=
B
2
=
0 and γ
n
=
0. Then, Ξ
=
C
,
β
n
+
δ
n
=
1, P
C
[
P
C
(
z
n
-
μ
2
B
2
z
n
)
-
μ
1
B
1
P
C
(
z
n
-
μ
2
B
2
z
n
)
]
=
z
n, and the iterative scheme (146) is equivalent to
(149)
z
n
=
P
C
(
x
n
-
λ
n
∇
f
α
n
(
x
n
)
)
,
y
n
=
σ
n
u
+
τ
n
P
C
(
x
n
-
λ
n
∇
f
α
n
(
z
n
)
)
+
(
1
-
σ
n
-
τ
n
)
z
n
,
x
n
+
1
=
β
n
x
n
+
δ
n
S
y
n
,
∀
n
≥
0
.
This is equivalent to (148). Since S is a nonexpansive mapping, S must be a k-strictly pseudocontractive mapping with k
=
0. In this case, it is easy to see that conditions (i)–(vii) in Corollary 23 all are satisfied. Therefore, in terms of Corollary 23, we obtain the desired result.Remark 26.
Our Theorems19 and 22 improve, extend, and develop [6, Theorem  5.7], [18, Theorem  3.1], and [33, Theorem  3.1] in the following aspects.(i)
Because both [6, Theorem  5.7] and [18, Theorem  3.1] are weak convergence results for solving the SFP, beyond question, our Theorems 19 and 22 as strong convergence results are very interesting and quite valuable.
(ii)
The problem of finding an element ofFix
(
S
)
∩
Ξ
∩
Γ in our Theorems 19 and 22 is more general than the corresponding problems in [6, Theorem  5.7] and [18, Theorem  3.1], respectively.
(iii)
The relaxed extragradient iterative method for finding an element ofFix
(
S
)
∩
Ξ
∩
VI
(
C
,
A
) in [33, Theorem  3.1] is extended to develop the relaxed extragradient method with regularization for finding an element of Fix
(
S
)
∩
Ξ
∩
Γ in our Theorem 19.
(iv)
The proof of our Theorems19 and 22 is very different from that of [33, Theorem  3.1] because our argument technique depends on Lemma 16, the restriction on the regularization parameter sequence {
α
n
}, and the properties of the averaged mappings P
C
(
I
-
λ
n
∇
f
α
n
) to a great extent.
(v)
Because our iterative schemes (17) and (18) involve a contractive self-mapping Q, a k-strictly pseudo-contractive self-mapping S, and several parameter sequences, they are more flexible and more subtle than the corresponding ones in [6, Theorem  5.7] and [18, Theorem  3.1], respectively.



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*Source: 891232-2013-03-17.xml*