# Strong Convergence of an Iterative Algorithm for Hierarchical Problems

Authors: Poom Kumam; Thanyarat Jitpeera
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We introduce the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping. The strong convergence of the algorithm is proved under some mild conditions. Our results extend those of Yao et al., Iiduka, Ceng et al., and other authors.


---

## Body

## 1. Introduction

LetC be a closed convex subset of a real Hilbert space H with inner product 〈
·
,
·
〉 and norm ∥
·
∥. We denote weak convergence and strong convergence by notations ⇀ and →, respectively. Let A be a nonlinear mapping. TheHartman-Stampacchia variational inequality [1] is to find x
∈
C such that 〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
C. The set of solutions is denoted by VI
⁡
(
C
,
A
). f
:
C
→
C is said to be a ρ-contraction if there exists a constant ρ
∈
[
0,1
) such that ∥
f
(
x
)
-
f
(
y
)
∥
≤
ρ
∥
x
-
y
∥
,
∀
x
,
y
∈
C. A mapping A
:
H
→
H is said to bemonotone if 〈
A
x
-
A
y
,
x
-
y
〉
≥
0
,
∀
x
,
y
∈
H. A mapping A
:
H
→
H is said to be α-strongly monotone if there exists a positive real number α such that 〈
A
x
-
A
y
,
x
-
y
〉
≥
α
∥
x
-
y
∥
2
,
∀
x
,
y
∈
H. A mapping A
:
H
→
H is said to be β-inverse-strongly monotone if there exists a positive real number β such that 〈
A
x
-
A
y
,
x
-
y
〉
≥
β
∥
A
x
-
A
y
∥
2
,
∀
x
,
y
∈
H. A mapping A
:
H
→
H is said to be L-Lipschitz continuous if there exists a positive real number L such that ∥
A
x
-
A
y
∥
≤
L
∥
x
-
y
∥
,
∀
x
,
y
∈
H. A linear bounded operator A is said to bestrongly positive on H if there exists a constant γ
¯
>
0 with the property 〈
A
x
,
x
〉
≥
γ
¯
∥
x
∥
2
,
∀
x
∈
H. A mapping T
:
C
→
C is said to benonexpansive if ∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥
,
∀
x
,
y
∈
C.A pointx
∈
C is afixed point of T provided T
x
=
x. Denote by F
(
T
) the set of fixed points of T; that is, F
(
T
)
=
{
x
∈
C
:
T
x
=
x
}. If C is bounded closed convex and T is a nonexpansive mapping of C into itself, then F
(
T
) is nonempty (see [2]).We discuss the following variational inequality problem over the fixed point set of a nonexpansive mapping (see [3–16]), which is said to be thehierarchical problem. Let a monotone, continuous mapping A
:
H
→
H and a nonexpansive mapping T
:
H
→
H. Find x
∈
VI
⁡
(
F
(
T
)
,
A
)
=
{
x
∈
F
(
T
)
:
〈
A
x
,
y
-
x
〉
≥
0
,
∀
y
∈
F
(
T
)
}, where F
(
T
)
≠
∅. This solution set is denoted by Ξ.We introduce the following variational inequality problem over the solution set of variational inequality problem and the fixed point set of a nonexpansive mapping (see [17, 18]), which is said to be thetriple hierarchical problem. Let an inverse-strongly monotone A
:
H
→
H, a strongly monotone and Lipschitz continuous B
:
H
→
H, and a nonexpansive mapping T
:
H
→
H. Find x
∈
VI
⁡
(
Ξ
,
B
)
=
{
x
∈
Ξ
:
〈
B
x
,
y
-
x
〉
≥
0
,
∀
y
∈
Ξ
}, where Ξ
:
=
VI
⁡
(
F
(
T
)
,
A
)
≠
∅.In 2009, Yao et al. [19] considered the following two-step iterative algorithm with the initial guess x
0
∈
C which is chosen arbitrarily:
(1)
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
T
y
n
,
y
n
=
β
n
S
x
n
+
(
1
-
β
n
)
x
n
,
∀
n
≥
0
,
where α
n
,
β
n
∈
(
0,1
) satisfies certain assumptions. Let S
,
T be two nonexpansive mappings and let f
:
C
→
C be a contraction mapping. Then, they proved that the above iterative sequence {
x
n
} converges strongly to fixed point.Next, Iiduka [17] introduced a monotone variational inequality with variational inequality constraint over the fixed point set of a nonexpansive mapping; the sequence {
x
n
} defined by the iterative method below, with the initial guess x
1
∈
H, is chosen arbitrarily:
(2)
y
n
=
T
(
x
n
-
λ
n
A
1
x
n
)
,
x
n
+
1
=
y
n
-
μ
α
n
A
2
y
n
,
∀
n
≥
0
,
where α
n
∈
(
0,1
] and λ
n
∈
(
0,2
α
] satisfy certain conditions, A
1
:
H
→
H is an inverse-strongly monotone, A
2
:
H
→
H is a strongly monotone and Lipschitz continuous, and T
:
H
→
H is a nonexpansive mapping; then the strongly convergence analysis of the sequence generated by (2) is proved under some appropriate conditions.In 2011, Yao et al. [20] studied the hierarchical problem over the fixed point set. Let the sequences {
x
n
} be generated by these two following algorithms:implicit algorithmx
t
=
T
P
C
[
I
-
t
(
A
-
γ
f
)
]
x
t
,
∀
t
∈
(
0,1
)
explicit algorithmx
n
+
1
=
β
n
x
n
+
(
1
-
β
n
)
T
P
C
[
I
-
α
n
(
A
-
γ
f
)
]
x
n
,
∀
n
≥
0.They illustrated that these two algorithms converge strongly to the unique solution of the variational inequality which is to find x
*
∈
F
(
T
) such that
(3)
〈
(
A
-
γ
f
)
x
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
F
(
T
)
,
where A
:
C
→
H is a strongly positive linear bounded operator, f
:
C
→
H is a ρ-contraction, and T
:
C
→
C is a nonexpansive mapping satisfying some conditions.Very recently, Ceng et al. [21] studied the following new algorithms. For x
0
∈
C is chosen arbitrarily, they defined a sequence {
x
n
} by
(4)
x
n
+
1
=
P
C
[
λ
n
γ
(
α
n
f
(
x
n
)
+
(
1
-
α
n
)
S
x
n
)
+
(
I
-
λ
n
μ
F
)
T
x
n
]
,
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
i
∀
n
≥
0
,
where the mappings S, T are nonexpansive mappings with F
(
T
)
≠
∅. Let F
:
C
→
H be a Lipschitzian and strongly monotone operator and let f
:
C
→
H be a contraction mapping satisfying some appropriate conditions. They proved that the proposed algorithms strongly converge to the minimum norm fixed point of T.In this paper, we consider a new iterative algorithm for solving the triple hierarchical problem over the solution set of the variational inequality problem and the fixed point set of a nonexpansive mapping which contain algorithms (1) and (4) as follows:
(5)
y
n
=
P
C
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
,
x
n
+
1
=
γ
λ
n
ϕ
(
x
n
)
+
(
I
-
λ
n
μ
F
)
T
y
n
,
∀
n
≥
0
,
where the mappings S, T are nonexpansive mappings with F
(
T
)
≠
∅. Let F
:
C
→
H be a Lipschitzian and strongly monotone operator, and let ϕ
:
H
→
H be a contraction mapping satisfying some mild conditions. Find a point x
*
∈
F
(
T
) such that
(6)
〈
(
I
-
S
)
x
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
F
(
T
)
.
This solution set of (6) is denoted by Ω
:
=
V
I
(
F
(
T
)
,
S
). The strong convergence for the proposed algorithms to the solution is solved under some appropriate assumptions. Our results improve the results of Ceng et al. [21], Iiduka [17], Yao et al. [19], Yao et al. [20], and some authors.


## 2. Preliminaries

LetC be a nonempty closed convex subset of H. There holds the following inequality in an inner product space ∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
x
+
y
〉
,
∀
x
,
y
∈
H. For every point x
∈
H, there exists a unique nearest point in C, denoted by P
C
x, such that
(7)
∥
x
-
P
C
x
∥
≤
∥
x
-
y
∥
,
∀
y
∈
C
.
P
C is called the metric projection of H onto C. It is well known that P
C is a nonexpansive mapping of H onto C and satisfies
(8)
〈
x
-
y
,
P
C
x
-
P
C
y
〉
≥
∥
P
C
x
-
P
C
y
∥
2
,
for every x
,
y
∈
H. Moreover, P
C
x is characterized by the following properties: P
C
x
∈
C and
(9)
〈
x
-
P
C
x
,
y
-
P
C
x
〉
≤
0
,
(10)
∥
x
-
y
∥
2
≥
∥
x
-
P
C
x
∥
2
+
∥
y
-
P
C
x
∥
2
,
for all x
∈
H
,
y
∈
C. Let B be a monotone mapping of C into H. In the context of the variational inequality problem the characterization of projection (9) implies the following:
(11)
u
∈
V
I
(
C
,
B
)
⟺
u
=
P
C
(
u
-
λ
B
u
)
,
λ
>
0
.
It is also known that H satisfies the Opial’s condition [22]; that is, for any sequence {
x
n
}
⊂
H with x
n
⇀
x, the inequality liminf
⁡
n
→
∞
∥
x
n
-
x
∥
<
liminf
⁡
n
→
∞
∥
x
n
-
y
∥ holds for every y
∈
H with x
≠
y.Lemma 1 (see [23]).
LetC be a closed convex subset of a real Hilbert space H and let T
:
C
→
C be a nonexpansive mapping. Then I
-
T is demiclosed at zero; that is, x
n
⇀
x  and  x
n
-
T
x
n
→
0 imply x
=
T
x.Lemma 2 (see [24]).
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 3 (see [10]).
LetB
:
H
→
H be β-strongly monotone and L-Lipschitz continuous and let μ
∈
(
0
,
2
β
/
L
2
). For λ
∈
[
0,1
], define T
λ
:
H
→
H by T
λ
(
x
)
:
=
x
-
λ
μ
B
(
x
) for all x
∈
H. Then, for all x
,
y
∈
H, ∥
T
λ
(
x
)
-
T
λ
(
y
)
∥
≤
(
1
-
λ
τ
)
∥
x
-
y
∥ hold, where τ
:
=
1
-
1
-
μ
(
2
β
-
μ
L
2
)
∈
(
0,1
].Lemma 4 (see [25]).
Assume that{
a
n
} is a sequence of nonnegative real numbers such that
(12)
a
n
+
1
≤
(
1
-
γ
n
)
a
n
+
δ
n
,
∀
n
≥
0
,
where {
γ
n
}
⊂
(
0,1
) and {
δ
n
} is a sequence in R such that(i)
∑
n
=
1
∞
‍
γ
n
=
∞;
(ii)
limsup
⁡
n
→
∞
(
δ
n
/
γ
n
)
≤
0 or ∑
n
=
1
∞
‍
|
δ
n
|
<
∞.Then lim
⁡
n
→
∞
a
n
=
0.


## 3. Strong Convergence Theorem

In this section, we introduce an iterative algorithm of triple hierarchical for solving monotone variational inequality problems forκ-Lipschitzian and η-strongly monotone operators over the solution set of variational inequality problems and the fixed point set of a nonexpansive mapping.Theorem 5.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let F
:
C
→
C be κ-Lipschitzian and η-strongly monotone operators with constant κ and η
>
0, respectively, and let ϕ
:
C
→
C be a ρ-contraction with coefficient ρ
∈
[
0,1
). Let T
:
C
→
C be a nonexpansive mapping with F
(
T
)
≠
∅, and let S
:
H
→
H be a nonexpansive mapping. Let 0
<
μ
<
2
η
/
κ
2 and 0
<
γ
<
τ, where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
). Suppose that {
x
n
} is a sequence generated by the following algorithm where x
0
∈
C is chosen arbitrarily:
(13)
y
n
=
P
C
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
,
x
n
+
1
=
γ
λ
n
ϕ
(
x
n
)
+
(
I
-
λ
n
μ
F
)
T
y
n
,
∀
n
≥
0
,
where {
β
n
}
,
{
λ
n
}
,
⊂
(
0,1
) satisfy the following conditions:(C1):
β
n
≤
k
λ
n;
(C2):
lim
⁡
n
→
∞
λ
n
=
0, lim
⁡
n
→
∞
(
(
λ
n
-
λ
n
-
1
)
/
λ
n
)
=
0, ∑
n
=
0
∞
‍
λ
n
=
∞;
(C3):
lim
⁡
n
→
∞
(
(
β
n
-
β
n
-
1
)
/
β
n
)
=
0.Then {
x
n
} converges strongly to x
*
∈
Ω, which is the unique solution of another variational inequality:
(14)
〈
(
μ
F
-
γ
ϕ
)
x
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
Ω
,
where Ω
:
=
V
I
(
F
(
T
)
,
S
)
≠
∅.Proof.
We will divide the proof into four steps.
Step  1. We will show that {
x
n
} is bounded. Indeed, for any x
*
∈
F
(
T
), we have
(15)
∥
y
n
-
x
*
∥
=
∥
P
C
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
-
P
C
x
*
∥
≤
∥
β
n
S
x
n
+
(
1
-
β
n
)
x
n
-
x
*
∥
=
∥
β
n
(
S
x
n
-
S
x
*
)
+
(
1
-
β
n
)
(
x
n
-
x
*
)
+
β
n
(
S
x
*
-
x
*
)
∥
≤
β
n
∥
x
n
-
x
*
∥
+
(
1
-
β
n
)
∥
x
n
-
x
*
∥
+
β
n
∥
S
x
*
-
x
*
∥
≤
∥
x
n
-
x
*
∥
+
β
n
∥
S
x
*
-
x
*
∥
.
From (13), we deduce that
(16)
∥
x
n
+
1
-
x
*
∥
=
∥
γ
λ
n
ϕ
(
x
n
)
+
(
I
-
λ
n
μ
F
)
T
y
n
-
x
*
∥
=
∥
γ
λ
n
(
ϕ
(
x
n
)
-
ϕ
(
x
*
)
)
+
(
I
-
λ
n
μ
F
)
(
T
y
n
-
x
*
)
+
λ
n
(
γ
ϕ
(
x
*
)
-
μ
F
x
*
)
(
ϕ
(
x
n
)
-
ϕ
(
x
*
)
)
∥
≤
γ
λ
n
∥
ϕ
(
x
n
)
-
ϕ
(
x
*
)
∥
+
(
I
-
λ
n
μ
F
)
∥
T
y
n
-
x
*
∥
+
λ
n
∥
γ
ϕ
(
x
*
)
-
μ
F
x
*
∥
≤
γ
ρ
λ
n
∥
x
n
-
x
*
∥
+
(
1
-
λ
n
τ
)
∥
y
n
-
x
*
∥
+
λ
n
∥
γ
ϕ
(
x
*
)
-
μ
F
x
*
∥
.
Substituting (15) into (16), we obtain
(17)
∥
x
n
+
1
-
x
*
∥
≤
γ
ρ
λ
n
∥
x
n
-
x
*
∥
+
(
1
-
λ
n
τ
)
{
∥
x
n
-
x
*
∥
+
β
n
∥
S
x
*
-
x
*
∥
}
+
λ
n
∥
γ
ϕ
(
x
*
)
-
μ
F
x
*
∥
≤
γ
ρ
λ
n
∥
x
n
-
x
*
∥
+
(
1
-
λ
n
τ
)
∥
x
n
-
x
*
∥
+
β
n
∥
S
x
*
-
x
*
∥
+
λ
n
∥
γ
ϕ
(
x
*
)
-
μ
F
x
*
∥
≤
[
1
-
λ
n
(
τ
-
γ
ρ
)
]
∥
x
n
-
x
*
∥
+
k
λ
n
∥
S
x
*
-
x
*
∥
+
λ
n
∥
γ
ϕ
(
x
*
)
-
μ
F
x
*
∥
≤
[
1
-
λ
n
(
τ
-
γ
ρ
)
]
∥
x
n
-
x
*
∥
+
λ
n
(
k
∥
S
x
*
-
x
*
∥
+
∥
γ
ϕ
(
x
*
)
-
μ
F
x
*
∥
)
≤
max
⁡
{
∥
x
n
-
x
*
∥
+
1
τ
-
γ
ρ
×
(
k
∥
S
x
*
-
x
*
∥
+
∥
γ
ϕ
(
x
*
)
-
μ
F
x
*
∥
)
1
τ
-
γ
ρ
}
.
By induction, it follows that
(18)
∥
x
n
-
x
*
∥
≤
max
⁡
{
∥
x
0
-
x
*
∥
+
1
τ
-
γ
ρ
w
w
w
w
w
w
×
(
k
∥
S
x
*
-
x
*
∥
+
∥
γ
ϕ
(
x
*
)
-
μ
F
x
*
∥
)
1
τ
-
γ
ρ
}
,
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
n
≥
0
.
Therefore, {
x
n
} is bounded and so are {
y
n
}, {
T
y
n
}, {
S
x
n
}, {
ϕ
(
x
n
)
}, and {
F
T
(
y
n
)
}.
Step  2. We will show that lim
⁡
n
→
∞
∥
x
n
-
T
x
n
∥
=
0. Setting v
n
:
=
β
n
S
x
n
+
(
1
-
β
n
)
x
n, we obtain
(19)
∥
v
n
-
v
n
-
1
∥
=
∥
β
n
S
x
n
+
(
1
-
β
n
)
x
n
-
β
n
-
1
S
x
n
-
1
-
(
1
-
β
n
-
1
)
x
n
-
1
∥
=
∥
β
n
(
S
x
n
-
S
x
n
-
1
)
+
(
β
n
-
β
n
-
1
)
S
x
n
-
1
+
(
1
-
β
n
)
(
x
n
-
x
n
-
1
)
+
(
β
n
-
1
-
β
n
)
x
n
-
1
∥
≤
β
n
∥
x
n
-
x
n
-
1
∥
+
|
β
n
-
β
n
-
1
|
(
∥
S
x
n
-
1
∥
+
∥
x
n
-
1
∥
)
+
(
1
-
β
n
)
∥
x
n
-
x
n
-
1
∥
≤
∥
x
n
-
x
n
-
1
∥
+
|
β
n
-
β
n
-
1
|
(
∥
S
x
n
-
1
∥
+
∥
x
n
-
1
∥
)
,
which implies that
(20)
∥
y
n
-
y
n
-
1
∥
=
∥
P
C
v
n
-
P
C
v
n
-
1
∥
≤
∥
v
n
-
v
n
-
1
∥
≤
∥
x
n
-
x
n
-
1
∥
+
|
β
n
-
β
n
-
1
|
(
∥
S
x
n
-
1
∥
+
∥
x
n
-
1
∥
)
.
It follows from (13) that
(21)
∥
x
n
+
1
-
x
n
∥
=
∥
γ
λ
n
ϕ
(
x
n
)
+
(
I
-
λ
n
μ
F
)
T
y
n
-
γ
λ
n
-
1
ϕ
(
x
n
-
1
)
-
(
I
-
λ
n
-
1
μ
F
)
T
y
n
-
1
∥
=
∥
γ
λ
n
(
ϕ
(
x
n
)
-
ϕ
(
x
n
-
1
)
)
+
(
λ
n
-
λ
n
-
1
)
γ
ϕ
(
x
n
-
1
)
+
(
I
-
λ
n
μ
F
)
T
y
n
-
(
I
-
λ
n
-
1
μ
F
)
T
y
n
-
1
∥
≤
γ
ρ
λ
n
∥
x
n
-
x
n
-
1
∥
+
|
λ
n
-
λ
n
-
1
|
γ
∥
ϕ
(
x
n
-
1
)
∥
+
∥
(
I
-
λ
n
μ
F
)
T
y
n
-
(
I
-
λ
n
μ
F
)
T
y
n
-
1
+
(
I
-
λ
n
μ
F
)
T
y
n
-
1
-
(
I
-
λ
n
-
1
μ
F
)
T
y
n
-
1
∥
≤
γ
ρ
λ
n
∥
x
n
-
x
n
-
1
∥
+
|
λ
n
-
λ
n
-
1
|
γ
∥
ϕ
(
x
n
-
1
)
∥
+
(
1
-
λ
n
τ
)
∥
y
n
-
y
n
-
1
∥
+
|
λ
n
-
λ
n
-
1
|
μ
∥
F
T
y
n
-
1
∥
≤
γ
ρ
λ
n
∥
x
n
-
x
n
-
1
∥
+
|
λ
n
-
λ
n
-
1
|
×
(
γ
∥
ϕ
(
x
n
-
1
)
∥
+
μ
∥
F
T
y
n
-
1
∥
)
+
(
1
-
λ
n
τ
)
{
∥
x
n
-
x
n
-
1
∥
+
|
β
n
-
β
n
-
1
|
×
(
∥
S
x
n
-
1
∥
+
∥
x
n
-
1
∥
)
}
≤
[
1
-
λ
n
(
τ
-
γ
ρ
)
]
∥
x
n
-
x
n
-
1
∥
+
|
λ
n
-
λ
n
-
1
|
(
γ
∥
ϕ
(
x
n
-
1
)
∥
+
μ
∥
F
T
y
n
-
1
∥
)
+
|
β
n
-
β
n
-
1
|
(
∥
S
x
n
-
1
∥
+
∥
x
n
-
1
∥
)
=
[
1
-
λ
n
(
τ
-
γ
ρ
)
]
∥
x
n
-
x
n
-
1
∥
+
(
|
λ
n
-
λ
n
-
1
|
λ
n
+
|
β
n
-
β
n
-
1
|
λ
n
)
λ
n
M
1
≤
[
1
-
λ
n
(
τ
-
γ
ρ
)
]
∥
x
n
-
x
n
-
1
∥
+
(
|
λ
n
-
λ
n
-
1
|
λ
n
+
k
|
β
n
-
β
n
-
1
|
β
n
)
λ
n
M
1
,
where M
1 is a constant such that
(22)
sup
⁡
n
≥
0
{
(
γ
∥
ϕ
(
x
n
)
∥
+
μ
∥
F
T
y
n
∥
)
,
(
∥
S
x
n
∥
+
∥
x
n
∥
)
}
≤
M
1
.
Hence, conditions (C2) and (C3) allow us to apply Lemma 4; then we get
(23)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
By (21), we get
(24)
∥
x
n
+
1
-
x
n
∥
λ
n
≤
[
1
-
λ
n
(
τ
-
γ
ρ
)
]
∥
x
n
-
x
n
-
1
∥
λ
n
+
|
λ
n
-
λ
n
-
1
|
+
|
β
n
-
β
n
-
1
|
λ
n
M
1
=
[
1
-
λ
n
(
τ
-
γ
ρ
)
]
∥
x
n
-
x
n
-
1
∥
λ
n
-
1
+
[
1
-
λ
n
(
τ
-
γ
ρ
)
]
(
∥
x
n
-
x
n
-
1
∥
λ
n
-
∥
x
n
-
x
n
-
1
∥
λ
n
-
1
)
+
|
λ
n
-
λ
n
-
1
|
+
|
β
n
-
β
n
-
1
|
λ
n
M
1
≤
[
1
-
λ
n
(
τ
-
γ
ρ
)
]
∥
x
n
-
x
n
-
1
∥
λ
n
-
1
+
λ
n
∥
x
n
-
x
n
-
1
∥
1
λ
n
|
1
λ
n
-
1
λ
n
-
1
|
+
M
1
λ
n
|
λ
n
-
λ
n
-
1
|
+
|
β
n
-
β
n
-
1
|
λ
n
2
.
Using the conditions (C2) and (C3), we can apply Lemma 4 to conclude that
(25)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
λ
n
=
0
.
By (13), we compute
(26)
∥
x
n
+
1
-
T
y
n
∥
=
∥
γ
λ
n
ϕ
(
x
n
)
+
(
I
-
λ
n
μ
F
)
T
y
n
-
T
y
n
∥
=
∥
γ
λ
n
ϕ
(
x
n
)
+
T
y
n
-
λ
n
μ
F
T
y
n
-
T
y
n
∥
≤
λ
n
∥
γ
ϕ
(
x
n
)
-
μ
F
T
y
n
∥
.
From the condition (C2), we note that lim
⁡
n
→
∞
∥
x
n
+
1
-
T
y
n
∥
=
0. At the same time, from (13), we also have
(27)
∥
y
n
-
x
n
∥
=
∥
P
C
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
-
P
C
x
n
∥
≤
∥
β
n
S
x
n
+
(
1
-
β
n
)
x
n
-
x
n
∥
≤
β
n
∥
S
x
n
-
x
n
∥
.
By the conditions (C1) and (C2), we note that lim
⁡
n
→
∞
∥
y
n
-
x
n
∥
=
0. Consider
(28)
∥
y
n
-
T
y
n
∥
≤
∥
y
n
-
x
n
∥
+
∥
x
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
T
y
n
∥
⟶
0
.
From (23), (26), and (27), we obtain
(29)
lim
⁡
n
→
∞
∥
y
n
-
T
y
n
∥
=
0
.
We set v
n
=
β
n
S
x
n
+
(
1
-
β
n
)
x
n; then we get
(30)
∥
y
n
-
v
n
∥
=
∥
P
C
v
n
-
v
n
∥
≤
∥
v
n
-
v
n
∥
⟶
0
,
as
n
⟶
∞
.
From (13), we have
(31)
∥
T
y
n
-
T
x
n
∥
=
∥
T
P
C
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
-
T
P
C
x
n
∥
≤
∥
β
n
S
x
n
+
(
1
-
β
n
)
x
n
-
x
n
∥
≤
β
n
∥
S
x
n
-
x
n
∥
.
By the conditions (C1) and (C2) again, we note that lim
⁡
n
→
∞
∥
T
y
n
-
T
x
n
∥
=
0. Consider(32)
∥
x
n
-
T
x
n
∥
≤
∥
x
n
-
y
n
∥
+
∥
y
n
-
T
y
n
∥
+
∥
T
y
n
-
T
x
n
∥
⟶
0
.
From (29), lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0, and lim
⁡
n
→
∞
∥
T
y
n
-
T
x
n
∥
=
0, we obtain
(33)
lim
⁡
n
→
∞
∥
x
n
-
T
x
n
∥
=
0
.
Step  3. We will show that limsup
⁡
n
→
∞
〈
μ
F
x
*
-
γ
ϕ
(
x
*
)
,
x
n
-
x
*
〉
≤
0. Rewrite (13) as
(34)
x
n
+
1
=
γ
λ
n
ϕ
(
x
n
)
+
(
I
-
μ
λ
n
F
)
T
y
n
-
v
n
+
β
n
S
x
n
+
(
1
-
β
n
)
x
n
.
We observe that
(35)
x
n
-
x
n
+
1
=
x
n
-
γ
λ
n
ϕ
(
x
n
)
-
(
I
-
μ
λ
n
F
)
T
y
n
+
v
n
-
β
n
S
x
n
-
x
n
+
β
n
x
n
=
λ
n
(
μ
F
-
γ
ϕ
)
x
n
-
λ
n
μ
F
x
n
-
(
I
-
μ
λ
n
F
)
T
y
n
+
(
I
-
μ
λ
n
F
)
y
n
-
(
I
-
μ
λ
n
F
)
y
n
+
v
n
+
β
n
(
I
-
S
)
x
n
=
λ
n
(
μ
F
-
γ
ϕ
)
x
n
+
λ
n
μ
(
F
y
n
-
F
x
n
)
+
(
y
n
-
T
y
n
)
-
μ
λ
n
F
(
y
n
-
T
y
n
)
+
(
v
n
-
y
n
)
+
β
n
(
I
-
S
)
x
n
=
λ
n
(
μ
F
-
γ
ϕ
)
x
n
+
λ
n
μ
(
F
y
n
-
F
x
n
)
+
(
y
n
-
T
y
n
)
-
μ
λ
n
F
(
y
n
-
T
y
n
)
+
λ
n
(
y
n
-
T
y
n
)
-
λ
n
(
y
n
-
T
y
n
)
+
(
v
n
-
y
n
)
+
β
n
(
I
-
S
)
x
n
=
λ
n
(
μ
F
-
γ
ϕ
)
x
n
+
λ
n
μ
(
F
y
n
-
F
x
n
)
+
λ
n
(
I
-
μ
F
)
(
y
n
-
T
y
n
)
+
(
1
-
λ
n
)
(
y
n
-
T
y
n
)
+
(
v
n
-
y
n
)
+
β
n
(
I
-
S
)
x
n
.
Set
(36)
z
n
=
x
n
-
x
n
+
1
λ
n
,
∀
n
≥
0
.
We note from (35) that
(37)
z
n
=
(
μ
F
-
γ
ϕ
)
x
n
+
μ
(
F
y
n
-
F
x
n
)
+
(
I
-
μ
F
)
(
y
n
-
T
y
n
)
+
1
-
λ
n
λ
n
(
y
n
-
T
y
n
)
+
1
λ
n
(
v
n
-
y
n
)
+
β
n
λ
n
(
I
-
S
)
x
n
.
This yields that, for each x
*
∈
F
(
T
),
(38)
〈
z
n
,
x
n
-
x
*
〉
=
〈
(
μ
F
-
γ
ϕ
)
x
n
,
x
n
-
x
*
〉
+
μ
〈
(
F
y
n
-
F
x
n
)
,
x
n
-
x
*
〉
+
〈
(
I
-
μ
F
)
y
n
-
(
I
-
μ
F
)
T
y
n
,
x
n
-
x
*
〉
+
1
-
λ
n
λ
n
〈
y
n
-
T
y
n
,
x
n
-
x
*
〉
+
1
λ
n
〈
v
n
-
y
n
,
x
n
-
x
*
〉
+
β
n
λ
n
〈
(
I
-
S
)
x
n
,
x
n
-
x
*
〉
=
〈
(
μ
F
-
γ
ϕ
)
x
*
,
x
n
-
x
*
〉
+
〈
(
μ
F
-
γ
ϕ
)
x
n
-
(
μ
F
-
γ
ϕ
)
x
*
,
x
n
-
x
*
〉
+
μ
〈
(
F
y
n
-
F
x
n
)
,
x
n
-
x
*
〉
+
〈
(
I
-
μ
F
)
y
n
-
(
I
-
μ
F
)
T
y
n
,
x
n
-
x
*
〉
+
1
-
λ
n
λ
n
〈
y
n
-
T
y
n
,
x
n
-
x
*
〉
+
1
λ
n
〈
v
n
-
y
n
,
x
n
-
x
*
〉
+
β
n
λ
n
〈
(
I
-
S
)
x
n
,
x
n
-
x
*
〉
.
In view of (38), 〈
(
μ
F
-
γ
ϕ
)
x
n
-
(
μ
F
-
γ
ϕ
)
x
*
,
x
n
-
x
*
〉 is nonnegative due to the monotonicity of μ
F
-
γ
ϕ. From (38), we derive that
(39)
〈
z
n
,
x
n
-
x
*
〉
≥
〈
(
μ
F
-
γ
ϕ
)
x
*
,
x
n
-
x
*
〉
+
μ
〈
(
F
y
n
-
F
x
n
)
,
x
n
-
x
*
〉
+
〈
(
I
-
μ
F
)
y
n
-
(
I
-
μ
F
)
T
y
n
,
x
n
-
x
*
〉
+
1
-
λ
n
λ
n
〈
y
n
-
T
y
n
,
x
n
-
x
*
〉
+
1
λ
n
〈
v
n
-
y
n
,
x
n
-
x
*
〉
+
β
n
λ
n
〈
(
I
-
S
)
x
n
,
x
n
-
x
*
〉
.
Since (29) implies ∥
(
I
-
μ
F
)
y
n
-
(
I
-
μ
F
)
T
y
n
∥
→
0, as n
→
∞, from (25), then we get z
n
→
0. Using (C1) and (30), ∥
y
n
-
x
n
∥
→
0, as n
→
∞ and {
x
n
} is bounded. We obtain from (39) that
(40)
limsup
⁡
n
→
∞
〈
(
μ
F
-
γ
ϕ
)
x
*
,
x
n
-
x
*
〉
≤
0
,
∀
x
*
∈
F
(
T
)
.
Since the sequence {
x
n
} is bounded, we can take a subsequence {
x
n
j
} of {
x
n
} such that
(41)
limsup
⁡
n
→
∞
〈
(
μ
F
-
γ
ϕ
)
x
*
,
x
n
-
x
*
〉
=
limsup
⁡
j
→
∞
〈
(
μ
F
-
γ
ϕ
)
x
*
,
x
n
j
-
x
*
〉
and x
n
j
⇀
x
~. From (33), by the demiclosed principle of the nonexpansive mapping, it follows that x
~
∈
F
(
T
). Then
(42)
limsup
⁡
j
→
∞
〈
(
μ
F
-
γ
ϕ
)
x
*
,
x
n
j
-
x
*
〉
=
〈
(
μ
F
-
γ
ϕ
)
x
*
,
x
~
-
x
*
〉
≤
0
.
Step  4. Finally, we will prove x
n
+
1
→
x
*. From (13), we note that
(43)
∥
y
n
-
x
*
∥
2
=
∥
P
C
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
-
P
C
x
*
∥
2
≤
∥
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
-
x
*
∥
2
≤
∥
β
n
(
S
x
n
-
S
x
*
)
+
(
1
-
β
n
)
(
x
n
-
x
*
)
+
β
n
(
S
x
*
-
x
*
)
∥
2
≤
∥
β
n
(
S
x
n
-
S
x
*
)
+
(
1
-
β
n
)
(
x
n
-
x
*
)
∥
2
+
2
β
n
〈
S
x
*
-
x
*
,
y
n
-
x
*
〉
≤
β
n
∥
x
n
-
x
*
∥
2
+
(
1
-
β
n
)
∥
x
n
-
x
*
∥
2
+
2
β
n
〈
S
x
*
-
x
*
,
y
n
-
x
*
〉
≤
∥
x
n
-
x
*
∥
2
+
2
β
n
∥
S
x
*
-
x
*
∥
∥
y
n
-
x
*
∥
.
Using (43), we compute
(44)
∥
x
n
+
1
-
x
*
∥
2
=
∥
γ
λ
n
ϕ
(
x
n
)
+
(
I
-
λ
n
μ
F
)
T
y
n
-
x
*
∥
2
=
∥
γ
λ
n
(
ϕ
(
x
n
)
-
ϕ
(
x
*
)
)
+
(
I
-
λ
n
μ
F
)
T
y
n
-
(
I
-
λ
n
μ
F
)
x
*
+
(
I
-
λ
n
μ
F
)
x
*
-
x
*
+
γ
λ
n
ϕ
(
x
*
)
∥
2
=
∥
γ
λ
n
(
ϕ
(
x
n
)
-
ϕ
(
x
*
)
)
+
(
I
-
λ
n
μ
F
)
(
T
y
n
-
x
*
)
+
λ
n
(
γ
ϕ
(
x
*
)
-
μ
F
x
*
)
∥
2
≤
∥
γ
λ
n
(
ϕ
(
x
n
)
-
ϕ
(
x
*
)
)
+
(
I
-
λ
n
μ
F
)
(
T
y
n
-
x
*
)
∥
2
+
2
λ
n
〈
γ
ϕ
(
x
*
)
-
μ
F
x
*
,
x
n
+
1
-
x
*
〉
≤
γ
2
λ
n
2
∥
ϕ
(
x
n
)
-
ϕ
(
x
*
)
∥
2
+
(
1
-
λ
n
τ
)
2
∥
T
y
n
-
x
*
∥
2
+
2
λ
n
〈
γ
ϕ
(
x
*
)
-
μ
F
x
*
,
x
n
+
1
-
x
*
〉
+
2
〈
γ
λ
n
(
ϕ
(
x
n
)
-
ϕ
(
x
*
)
)
,
(
I
-
μ
λ
n
F
)
(
T
y
n
-
x
*
)
〉
≤
γ
2
ρ
2
λ
n
2
∥
x
n
-
x
*
∥
2
+
(
1
-
2
λ
n
τ
+
λ
n
2
τ
2
)
∥
y
n
-
x
*
∥
2
+
2
λ
n
〈
γ
ϕ
(
x
*
)
-
μ
F
x
*
,
x
n
+
1
-
x
*
〉
+
2
γ
λ
n
〈
ϕ
(
x
n
)
-
ϕ
(
x
*
)
,
(
I
-
μ
λ
n
F
)
T
y
n
-
(
I
-
μ
λ
n
F
)
x
*
〉
=
γ
2
ρ
2
λ
n
2
∥
x
n
-
x
*
∥
2
+
(
1
-
2
λ
n
τ
+
λ
n
2
τ
2
)
∥
y
n
-
x
*
∥
2
+
2
λ
n
〈
γ
ϕ
(
x
*
)
-
μ
F
x
*
,
x
n
+
1
-
x
*
〉
+
2
γ
λ
n
〈
ϕ
(
x
n
)
-
ϕ
(
x
*
)
,
(
T
y
n
-
x
*
)
-
μ
λ
n
F
(
T
y
n
-
x
*
)
〉
=
γ
2
ρ
2
λ
n
2
∥
x
n
-
x
*
∥
2
+
(
1
-
2
λ
n
τ
+
λ
n
2
τ
2
)
∥
y
n
-
x
*
∥
2
+
2
λ
n
〈
γ
ϕ
(
x
*
)
-
μ
F
x
*
,
x
n
+
1
-
x
*
〉
+
2
γ
λ
n
〈
ϕ
(
x
n
)
-
ϕ
(
x
*
)
,
T
y
n
-
x
*
〉
-
2
γ
λ
n
〈
ϕ
(
x
n
)
-
ϕ
(
x
*
)
,
μ
λ
n
F
(
T
y
n
-
x
*
)
〉
≤
γ
2
ρ
2
λ
n
2
∥
x
n
-
x
*
∥
2
+
(
1
-
2
λ
n
τ
+
λ
n
2
τ
2
)
×
{
∥
x
n
-
x
*
∥
2
+
2
β
n
∥
S
x
*
-
x
*
∥
∥
y
n
-
x
*
∥
}
+
2
λ
n
〈
γ
ϕ
(
x
*
)
-
μ
F
x
*
,
x
n
+
1
-
x
*
〉
+
2
γ
ρ
λ
n
∥
x
n
-
x
*
∥
∥
T
y
n
-
x
*
∥
-
2
γ
ρ
μ
λ
n
2
∥
x
n
-
x
*
∥
∥
F
(
T
y
n
-
x
*
)
∥
≤
[
1
-
λ
n
(
2
τ
-
λ
n
τ
2
-
λ
n
γ
2
ρ
2
)
]
∥
x
n
-
x
*
∥
2
+
2
ε
n
λ
n
∥
S
x
*
-
x
*
∥
∥
y
n
-
x
*
∥
+
2
λ
n
〈
γ
ϕ
(
x
*
)
-
μ
F
x
*
,
x
n
+
1
-
x
*
〉
+
2
γ
ρ
λ
n
∥
x
n
-
x
*
∥
∥
T
y
n
-
x
*
∥
-
2
γ
ρ
μ
λ
n
2
∥
x
n
-
x
*
∥
∥
F
(
T
y
n
-
x
*
)
∥
.
Since {
x
n
}, {
T
y
n
}, and {
F
T
y
n
} are all bounded, we can choose a constant M
2
>
0 such that
(45)
sup
⁡
n
≥
0
1
2
τ
-
λ
n
τ
2
-
λ
n
γ
2
ρ
2
×
{
2
γ
ρ
μ
∥
x
n
-
x
*
∥
∥
F
(
T
y
n
-
x
*
)
∥
}
≤
M
2
.
It follows that
(46)
∥
x
n
+
1
-
x
*
∥
2
≤
[
1
-
λ
n
(
2
τ
-
λ
n
τ
2
-
λ
n
γ
2
ρ
2
)
]
∥
x
n
-
x
*
∥
2
+
λ
n
(
2
τ
-
λ
n
τ
2
-
λ
n
γ
2
ρ
2
)
δ
n
,
where
(47)
δ
n
=
2
ε
n
2
τ
-
λ
n
τ
2
-
λ
n
γ
2
ρ
2
∥
S
x
*
-
x
*
∥
∥
y
n
-
x
*
∥
+
2
2
τ
-
λ
n
τ
2
-
λ
n
γ
2
ρ
2
〈
γ
ϕ
(
x
*
)
-
μ
F
x
*
,
x
n
+
1
-
x
*
〉
+
2
2
τ
-
λ
n
τ
2
-
λ
n
γ
2
ρ
2
γ
ρ
∥
x
n
-
x
*
∥
∥
T
y
n
-
x
*
∥
-
λ
n
M
2
.
Now, applying Lemma 4 and (35), we conclude that x
n
→
x
*. This completes the proof.Corollary 6.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let F
:
C
→
C be κ-Lipschitzian and η-strongly monotone operators with constant κ and η
>
0, respectively. Let T
:
C
→
C be a nonexpansive mapping with F
(
T
)
≠
∅, and let S
:
H
→
H be a nonexpansive mapping. Let 0
<
μ
<
2
η
/
κ
2 and 0
<
γ
<
τ, where τ
=
1
-
1
-
μ
(
2
η
-
μ
κ
2
). Suppose {
x
n
} is a sequence generated by the following algorithm x
0
∈
C arbitrarily:
(48)
x
n
+
1
=
(
I
-
λ
n
μ
F
)
T
P
C
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
,
∀
n
≥
0
,
where {
β
n
}
,
{
λ
n
}
⊂
(
0,1
) satisfy the following conditions (C1)–(C3). Then {
x
n
} converges strongly to x
*
∈
Ω, which is the unique solution of variational inequality:
(49)
〈
(
I
-
μ
F
)
x
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
Ω
,
where Ω
:
=
V
I
(
F
(
T
)
,
S
)
≠
∅.Proof.
Puttingϕ
≡
0 in Theorem 5, we can obtain the desired conclusion immediately.Corollary 7.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let ϕ
:
H
→
H be a ρ-contraction with coefficient ρ
∈
[
0,1
), and let T
:
C
→
C be a nonexpansive mapping with F
(
T
)
≠
∅ and S
:
H
→
H a nonexpansive mapping. Suppose {
x
n
} is a sequence generated by the following algorithm, x
0
∈
C, arbitrarily:
(50)
y
n
=
P
C
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
,
x
n
+
1
=
λ
n
ϕ
(
x
n
)
+
(
1
-
λ
n
)
T
y
n
,
∀
n
≥
0
,
where {
β
n
}
,
{
λ
n
}
⊂
(
0,1
) satisfy the following conditions (C1)–(C3). Then {
x
n
} converges strongly to x
*
∈
Ω, which is the unique solution of variational inequality:
(51)
〈
(
I
-
ϕ
)
x
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
Ω
,
where Ω
:
=
V
I
(
F
(
T
)
,
S
)
≠
∅.Proof.
Puttingγ
=
1, μ
=
2, and F
≡
I
/
2 in Theorem 5, we can obtain the desired conclusion immediately.Corollary 8.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let T
:
C
→
C be a nonexpansive mapping with F
(
T
)
≠
∅ and let S
:
H
→
H be a nonexpansive mapping. Suppose {
x
n
} is a sequence generated by the following algorithm, x
0
∈
C, arbitrarily:
(52)
x
n
+
1
=
(
1
-
λ
n
)
T
P
C
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
,
∀
n
≥
0
,
where {
β
n
}
,
{
λ
n
}
⊂
(
0,1
) satisfy the following conditions (C1)–(C3). Then {
x
n
} converges strongly to x
*
∈
F
(
T
), which is the unique solution of variational inequality:
(53)
〈
(
I
-
S
)
x
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
F
(
T
)
.Proof.
Puttingϕ
≡
0 in Corollary 7, we can obtain the desired conclusion immediately.Corollary 9.
LetC be a nonempty closed and convex subset of a real Hilbert space H. Let ϕ
:
H
→
H be a ρ-contraction with coefficient ρ
∈
[
0,1
), and let T
:
C
→
C be a nonexpansive mapping with F
(
T
)
≠
∅ and S
:
C
→
C a nonexpansive mapping. Suppose {
x
n
} is a sequence generated by the following algorithm, x
0
∈
C, arbitrarily:
(54)
x
n
+
1
=
λ
n
x
n
+
(
1
-
λ
n
)
T
[
β
n
S
x
n
+
(
1
-
β
n
)
x
n
]
,
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
w
i
i
∀
n
≥
0
,
where {
β
n
}
,
{
λ
n
}
⊂
(
0,1
) satisfy the following conditions (C1)–(C3). Then {
x
n
} converges strongly to x
*
∈
F
(
T
), which is the unique solution of variational inequality:
(55)
〈
(
I
-
S
)
x
*
,
x
-
x
*
〉
≥
0
,
∀
x
∈
F
(
T
)
.Proof.
PuttingP
C
≡
I in Corollary 7, we can obtain the desired conclusion immediately.



---

*Source: 678147-2014-07-20.xml*