# Explicit Scheme for Fixed Point Problem for Nonexpansive Semigroup and Split Equilibrium Problem in Hilbert Space

Authors: Pei Zhou; Gou-Jie Zhao
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We establish an iterative method for finding a common element of the set of fixed points of nonexpansive semigroup and the set of split equilibrium problems. Under suitable conditions, some strong convergence theorems are proved. Our works improve previous results for nonexpansive semigroup.


---

## Body

## 1. Introduction

LetH be a real Hilbert space whose inner product and norm are denoted by 〈
·
,
·
〉 and ∥
·
∥, respectively. Let C be a nonempty closed convex subset of H, and let F be a bifunction of C
×
C into R which is the set of real numbers. The equilibrium problem introduced by Blum and Oettli [1] for F
:
C
×
C
→
R is to find x
∈
C such that
(1)
F
(
x
,
y
)
≥
0
,
∀
y
∈
C
.
The set of solutions of (1) is denoted by EP
⁡
(
F
). Numerous problems in physics, optimization, and economics reduce to finding a solution of (1) (see [2–8]). The split equilibrium problem was introduced by Moudafi in [9]; he considers the following pair of equilibrium problems in different spaces: let H
1 and H
2 be two real Hilbert spaces, let F
1
:
C
×
C
→
R and F
2
:
Q
×
Q
→
R be nonlinear bifunctions, and let A
:
H
1
→
H
2 be a bounded linear operator, and consider the nonempty closed convex subsets C
⊆
H
1 and Q
⊆
H
2; then the split equilibrium problem (SEP) is to find x
*
∈
C such that
(2)
F
1
(
x
*
,
x
)
≥
0
,
∀
x
∈
C
,
and such that
(3)
y
*
∈
A
x
*
∈
Q
,
F
2
(
y
*
,
y
)
≥
0
,
∀
y
∈
Q
.
The solution set of SEP (2)-(3) is denoted by Ω
=
{
p
∈
EP
⁡
(
F
1
)
:
A
p
∈
EP
⁡
(
F
2
)
}.Recall that mappingT of C into itself is called nonexpansive if
(4)
∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥
,
∀
x
,
y
∈
C
.
Let a family S
=
(
T
(
s
)
)
s
≥
0 be a nonexpansive semigroup on H, if it satisfies the following conditions: (S1)
T
(
0
)
x
=
x  ∀
x
∈
H;
(S2)
T
(
s
+
t
)
=
T
(
s
)
T
(
t
) for each s
,
t
≥
0;
(S3)
∥
T
(
s
)
x
-
T
(
t
)
y
∥
≤
∥
x
-
y
∥ for each x
,
y
∈
H and s
≥
0;
(S4)
∀
x
∈
H, s
→
T
(
s
)
x is continuous.The set of all the common fixed points of family S is denoted by F
i
x
(
S
); that is,
(5)
F
i
x
(
S
)
=
{
x
∈
C
:
T
(
s
)
x
=
x
,
0
≤
s
<
∞
}
=
⋂
0
≤
s
<
∞
‍
F
i
x
(
T
(
s
)
)
,
where F
i
x
(
T
(
s
)
) is the set of fixed points of T
(
s
). It is well known that F
i
x
(
S
) is closed and convex.In 2010, Tian [10] introduced the following general iterative scheme for finding an element of set of solutions to the fixed point of nonexpansive mapping in a Hilbert space. Define the sequence {
x
n
} by
(6)
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
1
-
μ
α
n
B
)
T
x
n
,
where B is k-Lipschitzian and η-strongly monotone operator. Then he proved that if the sequence {
α
n
} satisfies appropriate conditions, the sequence {
x
n
} generated by (6) converges strongly to the unique solution x
*
∈
F
i
x
(
T
) of the variational inequality
(7)
〈
(
γ
f
-
μ
B
)
x
*
,
x
-
x
*
〉
≤
0
,
∀
x
∈
F
i
x
(
T
)
.In 2011, Ceng et al. [11] added the metric project to the method of Tian (6) and studied the following explicit iterative scheme to find fixed points:
(8)
x
n
+
1
=
P
C
[
α
n
γ
f
(
x
n
)
+
(
1
-
μ
α
n
B
)
T
x
n
]
.
They prove the strong convergence of x
n to a fixed point x
*
∈
F
i
x
(
T
) of the same variational inequality (7).In 2008, Plubtieng and Punpaeng [12] introduced the following implicit iterative algorithm to prove a strong convergence theorem for fixed point problem with nonexpansive semigroup:
(9)
x
n
=
α
n
f
(
x
n
)
+
(
1
-
α
n
)
1
s
n
∫
0
s
n
‍
T
(
s
)
x
n
d
s
,
where x
n is a continuous net and s
n is a positive real divergent net.In 2014, Kazmi and Rizvi [13] studied the following implicit iterative algorithm. Under some assumptions, they obtain some strong convergence theorem for EP (1) and the fixed point problem:
(10)
u
n
=
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
,
x
n
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
,
where s
n and r
n are the continuous nets in (
0,1
).Motivated and inspired by [10–13], we introduce an explicit iterative scheme for finding a common element of the set of solutions SEP and fixed point for a nonexpansive semigroup in real Hilbert spaces. Starting with an arbitrary x
1
∈
H, define sequences {
x
n
} and {
u
n
} by
(11)
u
n
=
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
,
x
n
+
1
=
P
C
[
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
]
.
Under suitable conditions, some strong convergence theorems for approximating to these common elements are proved.


## 2. Preliminaries

This section collects some results that will be used in the proofs.LetH be a real Hilbert space with the inner product 〈
·
,
·
〉 and the norm ∥
·
∥, respectively.It is well known that, for allx
,
y
∈
H and λ
∈
[
0,1
], the following holds:
(12)
∥
λ
x
+
(
1
-
λ
)
y
∥
2
=
λ
∥
x
∥
2
+
(
1
-
λ
)
∥
y
∥
2
-
λ
(
1
-
λ
)
∥
x
-
y
∥
2
.LetC be a nonempty closed convex subset of H. Then, for any x
∈
H, there exists a unique nearest point of C, denoted by P
C
x, such that ∥
x
-
P
C
x
∥
≤
∥
x
-
y
∥ for all y
∈
C. Such P
C is called the metric projection from H into C. We know that P
C is nonexpansive. It is also known that P
C
x
∈
C and
(13)
〈
x
-
P
C
x
,
P
C
x
-
z
〉
≥
0
,
∀
x
∈
H
,
z
∈
C
.
It is easy to see that (13) is equivalent to
(14)
∥
x
-
z
∥
2
≥
∥
x
-
P
C
x
∥
2
+
∥
P
C
x
-
z
∥
2
,
∀
x
∈
H
,
z
∈
C
.LetB
:
C
→
H be a nonlinear mapping. Recall the following definitions.Definition 1.
Bis said to be(i)
monotone if(15)
〈
B
x
-
B
y
,
x
-
y
〉
≥
0
,
∀
x
,
y
∈
C
,
(ii)
strongly monotone if there exists a constantα
>
0 such that
(16)
〈
B
x
-
B
y
,
x
-
y
〉
≥
α
∥
x
-
y
∥
2
,
∀
x
,
y
∈
C
,
for such a case, B is said to be α-strongly monotone,
(iii)
α-inverse strongly monotone (α-ism) if there exists a constant α
>
0 such that
(17)
〈
B
x
-
B
y
,
x
-
y
〉
≥
α
∥
B
x
-
B
y
∥
2
,
∀
x
,
y
∈
C
,
(iv)
k-Lipschitz continuous if there exists a constant k
≥
0 such that
(18)
∥
B
x
-
B
y
∥
≤
k
∥
x
-
y
∥
,
∀
x
,
y
∈
C
.Remark 2.
LetF
=
μ
B
-
γ
f, where B is a k-Lipschitz and η-strongly monotone operator on H with k
>
0 and f is a Lipschitz mapping on H with coefficient L
>
0, 0
<
γ
≤
μ
η
/
L. It is a simple matter to see that the operator F is (
μ
η
-
γ
L
)-strongly monotone over H; that is,
(19)
〈
F
x
-
F
y
,
x
-
y
〉
≥
(
μ
η
-
γ
L
)
∥
x
-
y
∥
2
,
mm
∀
(
x
,
y
)
∈
H
×
H
.Lemma 3 (see [14]).
LetT be a nonexpansive mapping of a closed convex subset C of a Hilbert space H. If T has a fixed point, then I
-
T is demiclosed; that is, whenever the sequence of x
n is weakly convergent to x and (
I
-
T
)
x
n is strongly convergent to y, then (
I
-
T
)
x
=
y.Lemma 4 (see [15]).
LetC be a nonempty bounded closed convex subset of a Hilbert space H and let (
T
(
s
)
)
s
≥
0 be a nonexpansive semigroup on C. Then, for each h
≥
0,
(20)
lim
⁡
t
→
+
∞
sup
⁡
x
∈
C
∥
1
t
∫
0
t
‍
T
(
s
)
x
d
s
-
T
(
h
)
1
t
∫
0
t
‍
T
(
s
)
x
d
s
∥
=
0
.Definition 5 (see [9]).
A mappingT
:
H
→
H is said to be averaged if it can be written as the average of the identity mapping and a nonexpansive mapping; that is,
(21)
T
=
(
1
-
ε
)
I
+
ε
S
,
where ε
∈
(
0,1
), S
:
H
→
H is nonexpansive, and I is the identity operator on H.Proposition 6 (see [9]).
(i)
IfT
=
(
1
-
ε
)
S
+
ε
V, where S
:
H
→
H is averaged, V
:
H
→
H is nonexpansive, and ε
∈
(
0,1
), then T is averaged.
(ii)
The composite of finitely many averaged mappings is averaged.
(iii)
IfT is ν-ism, then, for γ
>
0, γ
T is (
ν
/
γ
)-ism.
(iv)
T is averaged if, and only if, its complement I
-
T is ν-ism for some ν
>
1
/
2.Assumption 7 (see [1]).
For solving the equilibrium problem for a bifunctionF
:
C
×
C
→
R, let us assume that F satisfies the following conditions:(A1)
F
(
x
,
x
)
=
0 for all x
∈
C;
(A2)
F is monotone, that is, F
(
x
,
y
)
+
F
(
y
,
x
)
≤
0 for all x
,
y
∈
C;
(A3)
for eachx
,
y
,
z
∈
C,
(22)
lim
⁡
t
→
0
F
(
t
z
+
(
1
-
t
)
x
,
y
)
≤
F
(
x
,
y
)
;
(A4)
for eachx
∈
C,  y
↦
F
(
x
,
y
) is convex and lower semicontinuous.Lemma 8 (see [2]).
LetC be a nonempty closed convex subset of H, and let F be a bifunction of  C
×
C into R satisfying (A1)–(A4). Let r
>
0 and x
∈
H. Then there exists z
∈
C such that
(23)
F
(
z
,
y
)
+
1
r
〈
y
-
z
,
z
-
x
〉
≥
0
,
∀
y
∈
C
.
Define a mapping T
r
:
H
→
C as follows:
(24)
T
r
F
(
x
)
=
{
z
∈
C
:
F
(
z
,
y
)
+
1
r
〈
y
-
z
,
z
-
x
〉
≥
0
,
∀
y
∈
C
}
for all x
∈
H. Then the following hold: (1)
T
r
F is single-valued;
(2)
T
r
F is firmly nonexpansive; that is, for any x
,
y
∈
H,
(25)
∥
T
r
F
x
-
T
r
F
y
∥
2
≤
〈
T
r
x
-
T
r
y
,
x
-
y
〉
;
(3)
F
i
x
(
T
r
F
)
=
E
P
(
F
);
(4)
E
P
(
F
) is closed and convex.Lemma 9 (see [16]).
LetC be a nonempty closed convex subset of a Hilbert space H, and let F
:
C
×
C
→
R be a bifunction. Let x
∈
C and r
1
,
r
2
∈
(
0
,
∞
). Then
(26)
∥
T
r
1
F
x
-
T
r
2
F
x
∥
≤
|
1
-
r
2
r
1
|
(
∥
T
r
1
F
x
∥
+
∥
x
∥
)
.Lemma 10 (see [17]).
Assume thatλ
∈
(
0,1
) and μ
>
0. Let B be a B
:
H
→
H which is η-strongly monotone and θ-Lipschitzian on H with θ
>
0, η
>
0. Let 0
<
μ
<
2
η
/
θ
2. Then the operator I
-
λ
μ
B is contraction; that is, ∥
I
-
λ
μ
B
∥
≤
(
1
-
λ
τ
), where τ
=
1
-
1
-
μ
(
2
η
-
μ
θ
2
)
∈
(
0,1
].Lemma 11 (see [10]).
LetH be a Hilbert space, f
:
H
→
H is a contractive mapping with constant β
∈
(
0,1
). B
:
H
→
H is θ-Lipschitzian and η-strongly monotone operator with θ
>
0, η
>
0. Then, for 0
<
σ
<
μ
η
/
β,
(27)
〈
x
-
y
,
(
μ
B
-
σ
f
)
x
-
(
μ
B
-
σ
f
)
y
〉
≥
(
μ
η
-
σ
β
)
∥
x
-
y
∥
2
,
m
m
m
h
m
m
m
x
,
y
∈
H
.
That is, μ
B
-
σ
f is strongly monotone with coefficient μ
η
-
σ
β.Lemma 12.
LetH be a real Hilbert space. Then the following well-known results hold:   ∀
x
,
y
∈
H,
(28)
∥
x
+
y
∥
2
≤
∥
x
∥
2
+
2
〈
y
,
x
+
y
〉
.Lemma 13 (see [18]).
Assume that{
α
n
} is a sequence of nonnegative real numbers such that
(29)
α
n
+
1
≤
(
1
-
γ
n
)
α
n
+
δ
n
,
n
≥
0
,
where {
α
n
} is a sequence in (
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
→
∞
α
n
=
0.


## 3. The General Explicit Iterative Method

Letf
:
H
1
→
H
1 be a contractive mapping with constant β
∈
(
0,1
) and let B
:
H
1
→
H
1 be η-strongly monotone and θ-Lipschitzian with θ
>
0, η
>
0. Let 0
<
μ
<
2
η
/
θ
2 and 0
<
γ
<
μ
(
η
-
μ
θ
2
/
2
)
/
β
=
τ
/
β. Let S
=
{
T
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on C such that Γ
=
F
i
x
(
S
)
⋂
‍
Ω
≠
∅. Assume {
r
n
} and {
s
n
} are the continuous nets of positive real numbers such that lim
⁡
n
→
0
r
n
=
r
>
0 and lim
⁡
n
→
0
s
n
=
+
∞.In this section, we introduce the following explicit iterative scheme that the nets{
u
n
} and {
x
n
} are generated by
(30)
u
n
=
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
,
x
n
+
1
=
P
C
[
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
]
,
where δ
∈
(
0,1
/
L
), L is the spectral radius of the operator A
*
A, and A
* is the adjoint of A.We prove the strong convergence of{
u
n
} and {
x
n
} to a fixed point x
* of S which solves the following variational inequality:
(31)
〈
(
μ
F
-
γ
f
)
x
*
,
x
*
-
x
-
〉
≤
0
,
∀
x
-
∈
Γ
=
F
i
x
(
S
)
⋂
‍
Ω
.
In the sequel, we denote by y
n the sequence defined by
(32)
y
n
=
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
.
Next, we will prove first a lemma and then some corollaries to be used in the proofs for the main result of this section.Theorem 14.
LetH
1 and H
2 be two real Hilbert spaces and let C
⊆
H
1 and Q
⊆
H
2 be nonempty closed subsets. Let A
:
H
1
→
H
2 be a bounded linear operator. Assume that F
1
:
C
×
C
→
R and F
2
:
Q
×
Q
→
R are the bifunctions satisfying Assumption 7 and F
2 is upper semicontinuous in the first argument. Let the sequences {
u
n
} and {
x
n
} be generated by (30), and suppose that the sequence {
α
n
} satisfies the following conditions:(a)
α
n
∈
(
0,1
) and lim
⁡
n
→
∞
α
n
=
0;
(b)
∑
n
=
0
∞
‍
α
n
=
0;
(c)
either∑
n
=
0
∞
‍
|
α
n
+
1
-
α
n
|
<
∞ or lim
⁡
n
⇒
∞
(
α
n
/
α
n
+
1
)
=
1.
Then the sequences{
u
n
} and {
x
n
} converge strongly to x
*
∈
Γ
=
F
i
x
(
S
)
⋂
‍
Ω, where x
*
=
P
Γ
(
I
-
μ
B
+
γ
f
)
x
*, which is the unique solution of the variational inequality (31).Proof.
We divide the proof into several steps.
(i){
x
n
} is well defined.
Forα
n
∈
(
0,1
) and ∀
x
∈
H
1, define a mapping G
:
H
1
→
H
2 by
(33)
G
x
=
P
C
[
α
n
γ
f
(
x
)
+
(
I
-
μ
α
n
B
)
1
s
n
m
h
m
×
∫
0
s
n
‍
T
(
s
)
T
r
n
F
1
(
x
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
)
d
s
]
.
According to Lemma 8, we can easily know that T
r
1
F
1 and T
r
2
F
2 both are firmly nonexpansive mappings and are averaged operators. From Proposition 6, we can obtain that the operator (
I
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
) is averaged and hence nonexpansive. Following Lemma 9 and ∀
x
,
y
∈
H
1, we get
(34)
∥
G
x
-
G
y
∥
=
∥
P
C
[
1
s
n
∫
0
s
n
‍
α
n
γ
f
+
(
I
-
μ
α
n
B
)
1
s
n
mmm
×
∫
0
s
n
‍
T
(
s
)
T
r
n
F
1
(
I
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
)
d
s
1
s
n
]
x
m
-
P
C
[
∫
0
s
n
‍
α
n
γ
f
+
(
I
-
μ
α
n
B
)
1
s
n
mmmm
×
∫
0
s
n
‍
T
(
s
)
T
r
n
F
1
(
I
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
)
d
s
]
y
∥
≤
∥
[
∫
0
s
n
‍
α
n
γ
f
+
(
I
-
μ
α
n
B
)
1
s
n
m
m
×
∫
0
s
n
‍
T
(
s
)
T
r
n
F
1
(
I
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
)
d
s
]
x
m
-
[
∫
0
s
n
‍
α
n
γ
f
+
(
I
-
μ
α
n
B
)
1
s
n
m
m
m
×
∫
0
s
n
‍
T
(
s
)
T
r
n
F
1
(
I
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
)
d
s
]
y
∥
≤
α
n
γ
∥
f
(
x
)
-
f
(
y
)
∥
+
(
1
-
α
n
τ
)
×
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
[
T
r
n
F
1
(
x
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
)
d
s
m
m
m
m
m
m
m
-
T
r
n
F
1
(
y
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
y
)
d
s
]
∫
0
s
n
‍
∥
≤
α
n
γ
β
∥
x
-
y
∥
+
(
1
-
α
n
τ
)
×
∥
T
r
n
F
1
(
x
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
)
d
s
m
-
T
r
n
F
1
(
y
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
y
)
d
s
∥
≤
α
n
γ
β
∥
x
-
y
∥
+
(
1
-
α
n
τ
)
∥
x
-
y
∥
=
(
1
-
α
n
(
τ
-
α
n
β
)
)
∥
x
-
y
∥
.
Since 0
<
1
-
α
n
(
τ
-
α
n
β
)
<
1, we can claim that the mapping G is a contraction mapping. Therefore, by Banach contraction principle, G has the unique fixed point x
*.
(ii) The sequences{
u
n
}, {
y
n
}, and {
x
n
} are bounded.
Lettingp
∈
Γ
=
F
i
x
(
S
)
⋂
‍
Ω, we obtain that p
=
T
r
n
F
1
p, A
p
=
T
r
n
F
2
A
p, and p
=
T
(
s
)
p.
From (30), we obtain
(35)
∥
u
n
-
p
∥
2
=
∥
T
r
n
F
1
(
I
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
)
x
n
-
p
∥
2
=
∥
T
r
n
F
1
(
I
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
)
x
n
-
T
r
n
F
1
p
∥
2
≤
∥
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
+
∥
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
+
2
δ
〈
x
n
-
p
,
A
*
(
T
r
n
F
2
-
I
)
A
x
n
〉
≤
∥
x
n
-
p
∥
2
+
δ
2
〈
(
T
r
n
F
2
-
I
)
A
x
n
,
A
A
*
(
T
r
n
F
2
-
I
)
A
x
n
〉
+
2
δ
〈
A
(
x
n
-
p
)
,
(
T
r
n
F
2
-
I
)
A
x
n
〉
≤
∥
x
n
-
p
∥
2
+
L
δ
2
〈
(
T
r
n
F
2
-
I
)
A
x
n
,
(
T
r
n
F
2
-
I
)
A
x
n
〉
+
2
δ
〈
A
(
x
n
-
p
)
+
(
T
r
n
F
2
-
I
)
A
x
n
m
m
m
-
(
T
r
n
F
2
-
I
)
A
x
n
,
A
*
(
T
r
n
F
2
-
I
)
A
x
n
〉
≤
∥
x
n
-
p
∥
2
+
L
δ
2
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
+
2
δ
{
〈
T
r
n
F
2
A
x
n
-
A
p
,
(
T
r
n
F
2
-
I
)
A
x
n
〉
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
mmm
-
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
}
≤
∥
x
n
-
p
∥
2
+
L
δ
2
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
+
2
δ
{
1
2
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
-
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
}
≤
∥
x
n
-
p
∥
2
+
L
δ
2
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
-
δ
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
=
∥
x
n
-
p
∥
2
+
δ
(
L
δ
-
1
)
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
.
Since δ
∈
(
0,1
/
L
), we have
(36)
∥
u
n
-
p
∥
≤
∥
x
n
-
p
∥
.
From (32), we have
(37)
∥
y
n
-
p
∥
=
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
p
∥
≤
1
s
n
∥
∫
0
s
n
‍
(
T
(
s
)
u
n
-
T
(
s
)
p
)
d
s
∥
≤
∥
u
n
-
p
∥
≤
∥
x
n
-
p
∥
.
Further, using (31) again, we obtain
(38)
∥
x
n
-
p
∥
=
∥
P
C
[
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
]
-
p
∥
≤
∥
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
p
∥
=
∥
∫
0
s
n
‍
α
n
(
γ
f
(
x
n
)
-
μ
B
p
)
+
(
I
-
μ
α
n
B
)
1
s
n
m
×
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
(
I
-
μ
α
n
B
)
p
∥
≤
α
n
∥
γ
f
(
x
n
)
-
μ
B
p
∥
+
(
1
-
α
n
τ
)
×
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
p
∥
≤
α
n
∥
γ
f
(
x
n
)
-
μ
B
p
∥
+
(
1
-
α
n
τ
)
1
s
n
×
∫
0
s
n
‍
∥
T
(
s
)
u
n
-
T
(
s
)
p
∥
d
s
≤
α
n
γ
∥
f
(
x
n
)
-
f
(
p
)
∥
+
α
n
∥
γ
f
(
p
)
-
μ
B
p
∥
+
(
1
-
α
n
τ
)
∥
u
n
-
p
∥
=
[
1
-
α
n
(
τ
-
γ
β
)
]
∥
u
n
-
p
∥
+
α
n
∥
γ
f
(
p
)
-
μ
B
p
∥
.
It follows from (40) and induction that
(39)
∥
x
n
-
p
∥
≤
∥
γ
f
(
p
)
-
μ
B
p
∥
τ
-
γ
β
.
Hence, the sequence {
x
n
} is bounded and therefore {
u
n
}, {
y
n
}, and {
f
(
x
n
)
} are also bounded.
(iii) Consider thatlim
⁡
n
⇒
∞
∥
x
n
-
u
n
∥
=
0.
According to (35) and Lemma 12, we obtain
(40)
∥
x
n
-
p
∥
2
≤
(
1
-
α
n
τ
)
2
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
p
∥
2
+
2
α
n
〈
γ
f
(
x
n
)
-
γ
f
(
p
)
+
γ
f
(
p
)
-
μ
B
p
,
x
n
-
p
〉
≤
(
1
+
α
n
2
τ
2
)
∥
u
n
-
p
∥
2
+
2
α
n
γ
β
∥
x
n
-
p
∥
2
+
2
α
n
〈
γ
f
(
p
)
-
μ
B
p
,
x
n
-
p
〉
≤
∥
u
n
-
p
∥
2
+
α
n
τ
2
∥
x
n
-
p
∥
2
+
2
α
n
γ
β
∥
x
n
-
p
∥
2
+
2
α
n
∥
γ
f
(
p
)
-
μ
B
p
∥
∥
x
n
-
p
∥
≤
∥
x
n
-
p
∥
2
+
δ
(
L
δ
-
1
)
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
+
α
n
τ
2
∥
x
n
-
p
∥
2
+
2
α
n
γ
β
∥
x
n
-
p
∥
2
+
2
α
n
∥
γ
f
(
p
)
-
μ
B
p
∥
∥
x
n
-
p
∥
.
From (40), we obtain
(41)
δ
(
1
-
L
δ
)
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
≤
α
n
(
∥
γ
f
(
p
)
-
μ
B
p
∥
∥
x
n
-
p
∥
(
τ
2
+
2
γ
β
)
∥
x
n
-
p
∥
2
mm
+
2
∥
γ
f
(
p
)
-
μ
B
p
∥
∥
x
n
-
p
∥
(
τ
2
+
2
γ
β
)
∥
x
n
-
p
∥
2
)
.
Since x
n is bounded, lim
⁡
n
→
∞
α
n
=
0, and δ
(
1
-
L
δ
)
>
0, we obtain that
(42)
lim
⁡
n
→
∞
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
=
0
.
From (30), we have
(43)
∥
u
n
-
p
∥
2
=
∥
T
r
n
F
1
(
I
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
)
x
n
-
p
∥
2
=
∥
T
r
n
F
1
(
I
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
)
x
n
-
T
r
n
F
1
p
∥
2
≤
〈
u
n
-
p
,
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
-
p
〉
=
1
2
{
∥
u
n
-
p
∥
2
+
∥
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
-
p
∥
2
hh
-
∥
u
n
-
p
-
[
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
-
p
]
∥
2
}
≤
1
2
{
∥
u
n
-
x
n
-
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
∥
u
n
-
p
∥
2
+
∥
x
n
-
p
∥
2
hh
-
∥
u
n
-
x
n
-
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
}
≤
1
2
{
(
T
r
n
F
2
-
I
)
∥
u
n
-
p
∥
2
+
∥
x
n
-
p
∥
2
-
∥
u
n
-
x
n
∥
2
hh
-
δ
2
∥
A
*
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
hh
+
2
δ
∥
A
(
u
n
-
x
n
)
∥
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
}
.
Hence, we obtain
(44)
∥
u
n
-
p
∥
2
≤
∥
x
n
-
p
∥
2
-
∥
u
n
-
x
n
∥
2
-
δ
2
∥
A
*
(
T
r
n
F
2
-
I
)
A
x
n
∥
2
+
2
δ
∥
A
(
u
n
-
x
n
)
∥
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
≤
∥
x
n
-
p
∥
2
-
∥
u
n
-
x
n
∥
2
+
2
δ
∥
A
(
u
n
-
x
n
)
∥
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
.
It follows from (40) and (44) that
(45)
∥
x
n
-
p
∥
2
≤
∥
u
n
-
p
∥
2
+
α
n
τ
2
∥
x
n
-
p
∥
2
+
2
α
n
γ
β
∥
x
n
-
p
∥
2
+
2
α
n
∥
γ
f
(
p
)
-
μ
B
p
∥
∥
x
n
-
p
∥
≤
∥
x
n
-
p
∥
2
-
∥
u
n
-
x
n
∥
2
+
2
δ
∥
A
(
u
n
-
x
n
)
∥
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
+
α
n
τ
2
∥
x
n
-
p
∥
2
+
2
α
n
γ
β
∥
x
n
-
p
∥
2
+
2
α
n
∥
γ
f
(
p
)
-
μ
B
p
∥
∥
x
n
-
p
∥
=
∥
x
n
-
p
∥
2
-
∥
u
n
-
x
n
∥
2
+
2
δ
∥
A
(
u
n
-
x
n
)
∥
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
+
α
n
M
1
,
where M
1
=
τ
2
∥
x
n
-
p
∥
2
+
2
γ
β
∥
x
n
-
p
∥
2
+
2
∥
γ
f
(
p
)
-
μ
B
p
∥
∥
x
n
-
p
∥. From (46), we obtain
(46)
∥
u
n
-
x
n
∥
2
≤
2
δ
∥
A
(
u
n
-
x
n
)
∥
∥
(
T
r
n
F
2
-
I
)
A
x
n
∥
+
α
n
M
1
.
Since x
n is bounded, lim
⁡
n
→
∞
α
n
=
0, and δ
>
0 and considering (42), we can claim that
(47)
lim
⁡
n
→
∞
∥
u
n
-
x
n
∥
=
0
.
(iv) Consider thatlim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0.
From (30) and Lemma 13, we have
(48)
∥
u
n
-
u
n
-
1
∥
=
∥
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
m
-
T
r
n
-
1
F
1
(
x
n
-
1
+
δ
A
*
(
T
r
n
-
1
F
2
-
I
)
A
x
n
-
1
)
∥
≤
∥
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
m
-
(
x
n
-
1
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
-
1
)
∥
+
|
1
-
r
n
-
1
r
n
|
∥
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
mmmmmm
-
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
∥
≤
∥
x
n
-
x
n
-
1
-
δ
A
*
A
(
x
n
-
x
n
-
1
)
∥
+
δ
∥
A
∥
∥
T
r
n
F
2
A
x
n
-
T
r
n
-
1
F
2
A
x
n
-
1
∥
+
|
1
-
r
n
-
1
r
n
|
∥
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
mmmmmmm
-
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
∥
≤
(
∥
x
n
-
x
n
-
1
∥
2
-
2
δ
∥
A
(
x
n
-
x
n
-
1
)
∥
2
mm
+
δ
2
∥
A
∥
4
∥
x
n
-
x
n
-
1
∥
2
)
1
/
2
+
δ
∥
A
∥
(
|
1
-
r
n
-
1
r
n
|
∥
A
(
x
n
-
x
n
-
1
)
∥
mmmmm
+
|
1
-
r
n
-
1
r
n
|
∥
T
r
n
F
2
A
x
n
-
A
x
n
-
1
∥
)
+
|
1
-
r
n
-
1
r
n
|
∥
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
m
m
m
m
m
m
m
m
-
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
∥
≤
(
1
-
2
δ
∥
A
∥
2
+
δ
2
∥
A
∥
4
)
1
/
2
∥
x
n
-
x
n
-
1
∥
+
δ
∥
A
∥
2
(
|
1
-
r
n
-
1
r
n
|
∥
x
n
-
x
n
-
1
∥
mmmmm
+
|
1
-
r
n
-
1
r
n
|
∥
T
r
n
F
2
A
x
n
-
A
x
n
-
1
∥
)
+
|
1
-
r
n
-
1
r
n
|
∥
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
mmmmmmmm
-
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
∥
≤
(
1
-
δ
∥
A
∥
2
)
∥
x
n
-
x
n
-
1
∥
+
δ
∥
A
∥
2
(
∥
x
n
-
x
n
-
1
∥
+
|
1
-
δ
∥
A
∥
r
n
-
1
r
n
|
mmmmmm
×
∥
T
r
n
F
2
A
x
n
-
A
x
n
-
1
∥
|
1
-
δ
∥
A
∥
r
n
-
1
r
n
|
)
+
|
1
-
r
n
-
1
r
n
|
∥
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
mmmmmmm
-
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
∥
=
∥
x
n
-
x
n
-
1
∥
+
δ
∥
A
∥
|
1
-
r
n
-
1
r
n
|
×
∥
T
r
n
F
2
A
x
n
-
A
x
n
-
1
∥
+
|
1
-
r
n
-
1
r
n
|
∥
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
mmmmmmm
-
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
∥
=
∥
x
n
-
x
n
-
1
∥
+
|
1
-
r
n
-
1
r
n
|
(
δ
∥
A
∥
ε
n
+
ξ
n
)
,
where
(49)
ε
n
=
∥
T
r
n
F
2
A
x
n
-
A
x
n
-
1
∥
,
ξ
n
=
∥
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
m
m
-
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
∥
.
From (32), we obtain
(50)
∥
y
n
-
y
n
-
1
∥
=
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
1
s
n
-
1
∫
0
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
≤
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
-
1
d
s
∥
+
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
-
1
d
s
-
1
s
n
-
1
∫
0
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
≤
1
s
n
∫
0
s
n
‍
∥
T
(
s
)
(
u
n
-
u
n
-
1
)
∥
d
s
+
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
-
1
d
s
-
1
s
n
-
1
∫
0
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
≤
∥
u
n
-
u
n
-
1
∥
+
|
1
s
n
-
1
s
n
-
1
|
∥
∫
0
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
+
1
s
n
∥
∫
s
n
-
1
s
n
‍
T
(
s
)
u
n
-
1
d
s
∥
.
Following (48) and (50), we obtain
(51)
∥
y
n
-
y
n
-
1
∥
≤
(
∥
x
n
-
x
n
-
1
∥
+
|
1
-
r
n
-
1
r
n
|
(
δ
∥
A
∥
ε
n
+
ξ
n
)
)
+
|
1
s
n
-
1
s
n
-
1
|
∥
∫
0
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
+
1
s
n
∥
∫
s
n
-
1
s
n
‍
T
(
s
)
u
n
-
1
d
s
∥
.
Using (30) again, we obtain
(52)
∥
x
n
+
1
-
x
n
∥
=
∥
P
C
[
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
y
n
]
m
-
P
C
[
α
n
-
1
γ
f
(
x
n
-
1
)
+
(
I
-
μ
α
n
-
1
B
)
y
n
-
1
]
∥
≤
∥
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
y
n
-
α
n
-
1
γ
f
(
x
n
-
1
)
mm
-
(
I
-
μ
α
n
-
1
B
)
y
n
-
1
∥
=
∥
α
n
γ
(
f
(
x
n
)
-
f
(
x
n
-
1
)
)
+
γ
(
α
n
-
α
n
-
1
)
f
(
x
n
-
1
)
m
+
(
I
-
μ
α
n
B
)
(
y
n
-
y
n
-
1
)
+
μ
(
α
n
-
α
n
-
1
)
y
n
-
1
∥
≤
α
n
γ
β
∥
x
n
-
x
n
-
1
∥
+
γ
|
α
n
-
α
n
-
1
|
∥
f
(
x
n
-
1
)
∥
+
(
1
-
α
n
τ
)
∥
y
n
-
y
n
-
1
∥
+
μ
|
α
n
-
α
n
-
1
|
∥
y
n
-
1
∥
≤
α
n
γ
β
∥
x
n
-
x
n
-
1
∥
+
γ
|
α
n
-
α
n
-
1
|
∥
f
(
x
n
-
1
)
∥
+
(
1
-
α
n
τ
)
(
∥
x
n
-
x
n
-
1
∥
+
|
1
-
r
n
-
1
r
n
|
(
δ
∥
A
∥
ε
n
+
ξ
n
)
)
+
|
1
s
n
-
1
s
n
-
1
|
∥
∫
0
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
+
1
s
n
∥
∫
s
n
-
1
s
n
‍
T
(
s
)
u
n
-
1
d
s
∥
+
μ
|
α
n
-
α
n
-
1
|
∥
y
n
-
1
∥
=
(
1
-
α
n
(
τ
-
γ
β
)
)
×
(
|
1
-
r
n
-
1
r
n
|
∥
x
n
-
x
n
-
1
∥
+
γ
|
α
n
-
α
n
-
1
|
∥
f
(
x
n
-
1
)
∥
hmm
+
|
1
-
r
n
-
1
r
n
|
(
δ
∥
A
∥
ε
n
+
ξ
n
)
)
+
|
1
s
n
-
1
s
n
-
1
|
∥
∫
0
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
+
1
s
n
∥
∫
s
n
-
1
s
n
‍
T
(
s
)
u
n
-
1
d
s
∥
+
μ
|
α
n
-
α
n
-
1
|
∥
y
n
-
1
∥
≤
(
1
-
α
n
(
τ
-
γ
β
)
)
∥
x
n
-
x
n
-
1
∥
+
M
2
(
γ
|
α
n
-
α
n
-
1
|
+
|
1
-
r
n
-
1
r
n
|
+
|
1
s
n
-
1
s
n
-
1
|
mmmm
+
1
s
n
-
1
+
μ
|
α
n
-
α
n
-
1
|
)
,
where
(53)
M
2
=
max
⁡
{
(
∥
∫
s
n
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
)
sup
⁡
n
≥
1
(
δ
∥
A
∥
ε
n
+
ξ
n
)
,
m
m
m
m
sup
⁡
n
≥
1
(
∥
∫
s
n
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
)
,
sup
⁡
n
≥
1
⁡
∥
y
n
-
1
∥
}
.
Since {
x
n
}, {
u
n
}, and {
y
n
} are bounded, we can claim that {
A
x
n
} and {
T
(
s
)
u
n
-
1
} are bounded. We can deduce that sup
⁡
n
≥
1
(
δ
∥
A
∥
ε
n
+
ξ
n
)
<
∞, sup
⁡
n
≥
1
(
∥
∫
s
n
s
n
-
1
‍
T
(
s
)
u
n
-
1
d
s
∥
)
<
∞ and sup
⁡
n
≥
1
∥
y
n
-
1
∥
<
∞, and M
2
<
∞.
Following conditions (a)–(c),lim
⁡
n
→
0
r
n
=
r
>
0, lim
⁡
n
→
0
s
n
=
+
∞, and Lemma 13, we obtain that
(54)
lim
⁡
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
(v) Consider thatlim
⁡
n
→
∞
∥
T
(
s
)
x
n
-
x
n
∥
=
0.
From (30) and (32), we obtain
(55)
∥
x
n
+
1
-
y
n
∥
≤
∥
P
C
[
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
]
m
-
P
C
y
n
[
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
]
∥
≤
∥
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
y
n
∥
≤
α
n
∥
γ
f
(
x
n
)
-
μ
B
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
∥
.
Since lim
⁡
n
→
∞
α
n
=
0 and x
n and u
n are bounded, we obtain
(56)
lim
⁡
n
→
∞
∥
x
n
+
1
-
y
n
∥
=
0
.
By (54) and (56), we get
(57)
∥
x
n
-
y
n
∥
≤
∥
x
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
y
n
∥
.
Next, we have
(58)
lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0
.
On the other hand, by (30), we have
(59)
∥
T
(
s
)
x
n
-
x
n
∥
≤
∥
T
(
s
)
x
n
-
T
(
s
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
∥
+
∥
T
(
s
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
∥
+
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
x
n
∥
≤
∥
x
n
-
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
∥
+
∥
T
(
s
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
∥
+
∥
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
x
n
∥
≤
2
∥
x
n
-
y
n
∥
+
∥
T
(
s
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
mmmmmmim
-
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
∥
.
So without loss of generality, we may assume that S
=
{
T
(
s
)
:
0
≤
s
<
+
∞
} is nonexpansive semigroup on K, and, by Lemma 4, we obtain
(60)
lim
⁡
n
→
∞
∥
T
(
s
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
-
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
∥
.
From (58), (59), and (60), we have
(61)
lim
⁡
n
→
∞
∥
T
(
s
)
x
n
-
x
n
∥
=
0
.
(vi) Consider thatω
∈
EP
⁡
(
F
1
)
⋂
‍
EP
⁡
(
F
2
).
Since{
x
n
} is bounded, there exists a subsequence {
x
n
i
} of {
x
n
} which converges weakly to ω. From (47), we obtain {
u
n
i
} which converges weakly to ω. From (58), y
n
i
⇀
ω follows. We show ω
∈
EP
⁡
(
F
1
). According to (30) and (A2),
(62)
1
r
n
〈
y
-
u
n
,
u
n
-
x
n
〉
≥
F
1
(
y
,
u
n
)
and hence
(63)
〈
y
-
u
n
i
,
u
n
i
-
x
n
i
r
n
i
〉
≥
F
1
(
y
,
u
n
i
)
.
Since (
u
n
i
-
x
n
i
)
/
r
n
i
→
0 and u
n
i
⇀
ω, from (A4), it follows that 0
≥
F
1
(
y
,
ω
) for all y
∈
H. For t with 0
<
t
≤
1 and y
∈
H, let y
t
=
t
y
+
(
1
-
t
)
ω; then we get 0
≥
F
1
(
y
t
,
ω
). So, from (A1) and (A4) we have
(64)
0
=
F
1
(
y
t
,
y
t
)
≤
t
F
1
(
y
t
,
y
)
+
(
1
-
t
)
F
(
y
t
,
ω
)
≤
t
F
1
(
y
t
,
y
)
and hence 0
≤
F
(
y
t
,
y
). From (A3), we have 0
≤
F
1
(
ω
,
y
) for all y
∈
H. Therefore, ω
∈
EP
⁡
(
F
1
).
Sincex
n
i
⇀
ω and A is a bounded linear operator, we obtain A
x
n
i
⇀
A
ω. Let v
n
j
=
A
x
n
j
-
T
r
n
j
F
2
x
n
j. Following (42), we obtain lim
⁡
n
→
∞
v
n
j
=
0 and A
x
n
j
-
v
n
j
=
T
r
n
j
F
2
x
n
j. Then from Lemma 8, we get
(65)
F
2
(
A
x
n
j
-
v
n
j
,
y
)
+
1
r
n
j
〈
y
-
(
A
x
n
j
-
v
n
j
)
,
(
A
x
n
j
-
v
n
j
)
m
m
m
m
m
m
m
m
m
m
m
m
-
A
x
n
j
〉
≥
0
,
∀
y
∈
Q
.
Since F
2 is upper semicontinuous in the first argument, taking limsup
⁡ to above inequality as j
→
∞ and using limsup
⁡
n
→
∞
r
n
=
r
>
0, we obtain that
(66)
F
2
(
A
ω
,
y
)
≥
0
,
∀
y
∈
Q
,
which means that A
ω
∈
EP
⁡
(
F
2
) and hence ω
∈
Ω.
(vii) Consider that〈
(
μ
F
-
γ
f
)
x
*
,
x
*
-
x
~
〉
≤
0
,
∀
x
-
∈
Γ
=
F
i
x
(
S
)
⋂
‍
Ω.
According to (30), letting
(67)
z
n
=
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
,
we can observe that
(68)
x
n
+
1
=
P
C
z
n
=
P
C
z
n
-
z
n
+
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
.
Observe that
(69)
(
μ
B
-
γ
f
)
x
n
=
1
α
n
(
P
C
z
n
-
z
n
)
+
1
α
n
(
x
n
-
x
n
+
1
)
+
(
I
-
μ
α
n
B
)
(
y
n
-
x
n
)
.
Hence, for each p
∈
Γ
=
F
i
x
(
S
)
⋂
‍
Ω, we can obtain
(70)
〈
(
μ
B
-
γ
f
)
x
n
,
x
n
-
p
〉
=
1
α
n
〈
P
C
z
n
-
z
n
,
x
n
-
p
〉
,
+
1
α
n
〈
x
n
-
x
n
+
1
,
x
n
-
p
〉
+
1
α
n
〈
(
I
-
μ
α
n
B
)
(
y
n
-
x
n
)
,
x
n
-
p
〉
=
1
α
n
〈
P
C
z
n
-
z
n
,
x
n
-
p
〉
,
+
1
α
n
〈
x
n
-
x
n
+
1
,
x
n
-
p
〉
+
1
α
n
〈
y
n
-
x
n
,
x
n
-
p
〉
-
μ
〈
B
y
n
-
B
x
n
,
x
n
-
p
〉
.
Taking limit n
→
∞ in (70) and noticing that B
y
n
-
B
x
n
→
B
x
*
-
B
x
*
=
0, y
n
-
x
n
→
∞, and P
C
z
n
-
z
n
→
P
C
x
*
-
x
*
=
0, we obtain that
(71)
〈
(
μ
B
-
γ
f
)
ω
,
ω
-
p
〉
≤
0
,
which implies ω
=
P
Γ
(
I
-
μ
B
+
γ
f
).
(viii) Consider thatω
∈
Γ
=
F
i
x
(
S
)
⋂
‍
Ω.
From (30), we have x
n
+
1
=
P
C
z
n, and, for any given x
*
∈
Γ,
(72)
x
n
+
1
-
x
*
=
P
C
z
n
-
z
n
+
z
n
-
x
*
=
P
C
z
n
-
z
n
+
α
n
(
γ
f
(
x
n
)
-
μ
B
x
*
)
+
(
I
-
μ
α
n
B
)
y
n
-
(
I
-
μ
α
n
B
)
x
*
.
Since P
C is the metric projection from H
1 onto C, we obtain
(73)
〈
P
C
z
n
-
z
n
,
P
C
z
n
-
x
*
〉
≤
0
.
It follows from (72) and (37) that
(74)
∥
x
n
+
1
-
x
∥
2
=
〈
P
C
z
n
-
z
n
,
x
n
+
1
-
x
*
〉
+
α
n
〈
γ
f
(
x
n
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
+
〈
(
I
-
μ
α
n
B
)
(
y
n
-
x
*
)
,
x
n
+
1
-
x
*
〉
≤
α
n
〈
γ
f
(
x
n
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
+
〈
(
I
-
μ
α
n
B
)
(
y
n
-
x
*
)
,
x
n
+
1
-
x
*
〉
≤
α
n
γ
〈
f
(
x
n
)
-
f
(
x
*
)
,
x
n
+
1
-
x
*
〉
+
α
n
〈
γ
f
(
x
*
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
+
〈
(
I
-
μ
α
n
B
)
(
y
n
-
x
*
)
,
x
n
+
1
-
x
*
〉
≤
α
n
γ
β
∥
x
n
-
x
*
∥
∥
x
n
+
1
-
x
*
∥
+
α
n
〈
γ
f
(
x
*
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
+
(
1
-
α
n
τ
)
∥
y
n
-
x
*
∥
∥
x
n
+
1
-
x
*
∥
≤
α
n
γ
β
∥
x
n
-
x
*
∥
∥
x
n
+
1
-
x
*
∥
+
α
n
〈
γ
f
(
x
*
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
+
(
1
-
α
n
τ
)
∥
x
n
-
x
*
∥
∥
x
n
+
1
-
x
*
∥
≤
(
1
-
α
n
(
τ
-
γ
β
)
)
∥
x
n
-
x
*
∥
∥
x
n
+
1
-
x
*
∥
+
α
n
〈
γ
f
(
x
*
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
≤
α
n
γ
β
∥
x
n
-
x
*
∥
∥
x
n
+
1
-
x
*
∥
+
α
n
〈
γ
f
(
x
*
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
+
(
1
-
α
n
τ
)
∥
x
n
-
x
*
∥
∥
x
n
+
1
-
x
*
∥
≤
(
1
-
α
n
(
τ
-
γ
β
)
)
2
(
∥
x
n
-
x
*
∥
2
+
∥
x
n
+
1
-
x
*
∥
2
)
+
α
n
〈
γ
f
(
x
*
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
,
which hence implies that
(75)
∥
x
n
+
1
-
x
∥
2
≤
(
1
-
α
n
(
τ
-
γ
β
)
)
(
1
+
α
n
(
τ
-
γ
β
)
)
∥
x
n
-
x
*
∥
2
+
2
α
n
1
+
α
n
(
τ
-
γ
β
)
〈
γ
f
(
x
*
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
≤
(
1
-
α
n
(
τ
-
γ
β
)
)
∥
x
n
-
x
*
∥
2
+
2
α
n
1
+
α
n
(
τ
-
γ
β
)
〈
γ
f
(
x
*
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
≤
(
1
-
a
n
)
∥
x
n
-
x
*
∥
2
+
α
n
b
n
,
where
(76)
a
n
=
α
n
(
τ
-
γ
β
)
,
b
n
=
2
1
+
α
n
(
τ
-
γ
β
)
〈
γ
f
(
x
*
)
-
μ
B
x
*
,
x
n
+
1
-
x
*
〉
.
It is easily seen that ∑
n
=
0
∞
‍
a
n
=
∞ and limsup
⁡
n
→
∞
b
n
≤
0 (due to (a), (b), (c), and (72)). According to Lemma 13, we can conclude that x
n
→
x
*. This completes the proof.Corollary 15.
LetH
1 and H
2 be two real Hilbert spaces and let C
⊆
H
1 and Q
⊆
H
2 be nonempty closed subsets. Let A
:
H
1
→
H
2 be a bounded linear operator. Assume that F
1
:
C
×
C
→
R and F
2
:
Q
×
Q
→
R are the bifunctions satisfying Assumption 7 and F
2 is upper semicontinuous in the first argument. Let the sequences {
u
n
} and {
x
n
} be generated by
(77)
u
n
=
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
μ
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
.
Suppose that the sequence {
α
n
} satisfies the following conditions: (a), (b), and (c) (the same in Theorem 14). Then the sequences {
u
n
} and {
x
n
} converge strongly to x
*
∈
Γ
=
F
i
x
(
S
)
⋂
‍
Ω, where x
*
=
P
Γ
(
I
-
μ
B
+
γ
f
)
x
*, which is the unique solution of the variational inequality (31).IfB is a strongly positive bounded linear self-adjoint operator on H
1 with constant γ
-
>
0 such that 0
<
γ
<
γ
-
/
α
<
γ
+
1
/
α, we can easily obtain the following corollary.Corollary 16.
LetH
1 and H
2 be two real Hilbert spaces and let C
⊆
H
1 and Q
⊆
H
2 be nonempty closed subsets. Let A
:
H
1
→
H
2 be a bounded linear operator. Assume that F
1
:
C
×
C
→
R and F
2
:
Q
×
Q
→
R are the bifunctions satisfying Assumption 7 and F
2 is upper semicontinuous in the first argument. Let the sequences {
u
n
} and {
x
n
} be generated by
(78)
u
n
=
T
r
n
F
1
(
x
n
+
δ
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
B
)
1
s
n
∫
0
s
n
‍
T
(
s
)
u
n
d
s
.
Suppose that the sequence {
α
n
} satisfies the following conditions: (a), (b), and (c) (the same in Theorem 14). Then the sequences {
u
n
} and {
x
n
} converge strongly to x
*
∈
Γ
=
F
i
x
(
S
)
⋂
‍
Ω, where x
*
=
P
Γ
(
I
-
μ
B
+
γ
f
)
x
*, which is the unique solution of the variational inequality
(79)
〈
(
γ
f
-
B
)
x
*
,
x
-
-
x
*
〉
≤
0
,
∀
x
-
∈
Γ
.


## 4. Application

In this section, we introduce an example of numerical test to illustrate the algorithm given in Corollary16.Example 17.
LetC
,
Q
=
[
0
,
+
∞
)
∈
H
1
=
H
2
=
R, the set of all real numbers, with the inner product defined by 〈
x
,
y
〉
=
x
y, ∀
x
,
y
∈
R, and induced usual norm ∥
·
∥. Assume that F
1
:
C
×
C
→
R and F
2
:
Q
×
Q
→
R are defined by F
1
(
x
,
y
)
=
(
x
-
4
)
(
y
-
x
), ∀
x
,
y
∈
C, and F
2
(
u
,
v
)
=
(
u
+
2
)
(
v
-
u
), ∀
u
,
v
∈
Q. It is easy to claim that the bifunctions F
1 and F
2 satisfy Assumption 7 and the bifunction F
2 is upper semicontinuous. Next, we can find the formula of T
r
n
F
1
x. From Lemma 8, we can claim that T
r
n
F
1
x is single-valued; for any y
∈
C
,
r
>
0,
(80)
F
1
(
x
,
y
)
+
1
r
〈
x
-
z
,
y
-
x
〉
≥
0
⟺
-
(
r
+
1
)
x
2
+
(
r
x
+
x
-
4
r
-
z
)
y
+
(
4
r
+
z
)
x
≥
0
.
Letting H
(
y
)
=
-
(
r
+
1
)
x
2
+
(
r
x
+
x
-
4
r
-
z
)
y
+
(
4
r
+
z
)
x, we obtain that
(81)
z
=
(
r
+
1
)
x
-
4
r
,
and so
(82)
T
r
n
F
1
x
=
(
r
n
+
1
)
x
-
4
r
n
.
Similarly, we can also obtain that
(83)
T
r
n
F
2
u
=
(
r
n
+
1
)
u
+
2
r
n
.
For allx
∈
R, we can define mapping f
(
x
)
=
(
1
/
4
)
x, A
(
x
)
=
-
(
1
/
2
)
x,   B
(
x
)
=
2
x, and the nonexpansive mapping T is satisfied as T
x
=
x
∀
x
∈
C. It is claimed that the mapping f is contraction with constant β
=
1
/
2, A is a bounded linear operator on R with adjoint operator A
* and ∥
A
∥
=
∥
A
*
∥
=
1
/
2, and B is a strongly positive bounded linear self-adjoint operator with constant γ
-
=
1 on R. On the other hand, we can take γ
=
2 which satisfies 0
<
γ
<
γ
-
/
α
<
γ
+
1
/
α. Hence, it is easy to observe that F
i
x
(
T
)
=
(
0
,
∞
), EP
⁡
(
F
1
)
=
{
4
}, and EP
⁡
(
F
2
)
=
{
-
2
}. Furthermore, we can obtain that Ω
=
{
p
∈
EP
⁡
(
F
1
)
:
A
p
∈
EP
⁡
(
F
2
)
}
=
{
-
2
}. Consequently, Γ
=
F
i
x
(
S
)
⋂
‍
Ω
=
{
-
2
}. Therefore, all the assumptions in Corollary 16 are satisfied. We can obtain the following numerical algorithm:
(84)
u
n
=
T
r
n
F
1
(
x
n
+
1
8
A
*
(
T
r
n
F
2
-
I
)
A
x
n
)
,
x
n
+
1
=
2
n
+
1
(
1
4
x
n
)
+
(
1
-
2
n
+
1
)
u
n
,
where α
n
=
1
/
(
n
+
1
) and r
n
=
1. Then, by Corollary 16, the sequence {
x
n
} converges to a solution of Example 17. For a number ε
=
1
0
-
5, using the MATLAB, we generated a sequence {
x
n
} as in Figure 1.Figure 1
Convergence of iterative sequence{
x
n
}.



---

*Source: 858679-2014-08-19.xml*