# Strong Convergence Theorems of the General Iterative Methods for Nonexpansive Semigroups in Banach Spaces

Authors: Rattanaporn Wangkeeree
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

---

## Abstract

LetE be a real reflexive Banach space which admits a weakly sequentially continuous duality mapping from E to E
*. Let 𝒮
=
{
T
(
s
)
:
0
≤
s
<
∞
} be a nonexpansive semigroup on E such that Fix
(
𝒮
)
:
=
⋂
t
≥
0
Fix
(
T
(
t
)
)
≠
∅, and f is a contraction on E with coefficient 0
<
α
<
1. Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1 and γ a positive real number such that γ
<
1
/
α
(
1
-
1
-
δ
/
λ
). When the sequences of real numbers {
α
n
} and {
t
n
} satisfy some appropriate conditions, the three iterative processes given as follows: x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
F
)
T
(
t
n
)
x
n, n
≥
0, y
n
+
1
=
α
n
γ
f
(
T
(
t
n
)
y
n
)
+
(
I
-
α
n
F
)
T
(
t
n
)
y
n, n
≥
0, and z
n
+
1
=
T
(
t
n
)
(
α
n
γ
f
(
z
n
)
+
(
I
-
α
n
F
)
z
n
), n
≥
0 converge strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality 〈
(
F
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0,   x
∈
Fix
(
𝒮
). Our results extend and improve corresponding ones of Li et al. (2009) Chen and He (2007), and many others.


---

## Body

## 1. Introduction

LetE be a real Banach space. A mapping T of E into itself is said to be nonexpansive if ∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥ for each x
,
y
∈
E. We denote by Fix
(
T
) the set of fixed points of T. A mapping f
:
E
→
E is called α-contraction if there exists a constant 0
<
α
<
1 such that ∥
f
(
x
)
-
f
(
y
)
∥
≤
α
∥
x
-
y
∥ for all x
,
y
∈
E. A family 𝒮
=
{
T
(
t
)
:
0
≤
t
<
∞
} of mappings of E into itself is called anonexpansive semigroup on E if it satisfies the following conditions: (i)
T
(
0
)
x
=
x for all x
∈
E;
(ii)
T
(
s
+
t
)
=
T
(
s
)
T
(
t
) for all s
,
t
≥
0;
(iii)
∥
T
(
t
)
x
-
T
(
t
)
y
∥
≤
∥
x
-
y
∥ for all x
,
y
∈
E and t
≥
0;
(iv)
for allx
∈
E, the mapping t
↦
T
(
t
)
x is continuous.We denote byFix
(
𝒮
) the set of all common fixed points of 𝒮, that is, (1.1)
Fix
⁡
(
S
)
∶
=
{
x
∈
E
:
T
(
t
)
x
=
x
,
0
≤
t
<
∞
}
=
⋂
t
≥
0
Fix
⁡
(
T
(
t
)
)
.In [1], Shioji and Takahashi introduced the following implicit iteration in a Hilbert space(1.2)
x
n
=
α
n
x
+
(
1
-
α
n
)
1
t
n
∫
0
t
n
T
(
s
)
x
n
d
s
,
∀
n
∈
N
,
where {
α
n
} is a sequence in (
0,1
) and {
t
n
} is a sequence of positive real numbers which diverges to ∞. Under certain restrictions on the sequence {
α
n
}, Shioji and Takahashi [1] proved strong convergence of the sequence {
x
n
} to a member of F
(
𝒮
). In [2], Shimizu and Takahashi studied the strong convergence of the sequence {
x
n
} defined by(1.3)
x
n
+
1
=
α
n
x
+
(
1
-
α
n
)
1
t
n
∫
0
t
n
T
(
s
)
x
n
d
s
,
∀
n
∈
N
in a real Hilbert space where {
T
(
t
)
:
t
≥
0
} is a strongly continuous semigroup of nonexpansive mappings on a closed convex subset C of a Banach space E and lim
n
→
∞
t
n
=
∞. Using viscosity method, Chen and Song [3] studied the strong convergence of the following iterative method for a nonexpansive semigroup {
T
(
t
)
:
t
≥
0
} with Fix
(
𝒮
)
≠
∅ in a Banach space:(1.4)
x
n
+
1
=
α
n
f
(
x
)
+
(
1
-
α
n
)
1
t
n
∫
0
t
n
T
(
s
)
x
n
d
s
,
∀
n
∈
N
,
where f is a contraction. Note however that their iterate x
n at step n is constructed through the average of the semigroup over the interval (
0
,
t
). Suzuki [4] was the first to introduce again in a Hilbert space the following implicit iteration process:(1.5)
x
n
=
α
n
u
+
(
1
-
α
n
)
T
(
t
n
)
x
n
,
∀
n
∈
N
,
for the nonexpansive semigroup case. In 2002, Benavides et al. [5], in a uniformly smooth Banach space, showed that if 𝒮 satisfies an asymptotic regularity condition and {
α
n
} fulfills the control conditions lim
n
→
∞
α
n
=
0, ∑
n
=
1
∞
α
n
=
∞, and lim
n
→
∞
α
n
/
α
n
+
1
=
0, then both the implicit iteration process (1.5) and the explicit iteration process (1.6),(1.6)
x
n
+
1
=
α
n
u
+
(
1
-
α
n
)
T
(
t
n
)
x
n
,
∀
n
∈
N
,
converge to a same point of F
(
𝒮
). In 2005, Xu [6] studied the strong convergence of the implicit iteration process (1.2) and (1.5) in a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping. Recently, Chen and He [7] introduced the viscosity approximation process:(1.7)
x
n
+
1
=
α
n
f
(
x
n
)
+
(
1
-
β
n
)
T
(
t
n
)
x
n
,
∀
n
∈
N
,
where f is a contraction and {
α
n
} is a sequence in (
0,1
) and a nonexpansive semigroup {
T
(
t
)
:
t
≥
0
}. The strong convergence theorem of {
x
n
} is proved in a reflexive Banach space which admits a weakly sequentially continuous duality mapping. In [8], Chen et al. introduced and studied modified Mann iteration for nonexpansive mapping in a uniformly convex Banach space.On the other hand, iterative approximation methods for nonexpansive mappings have recently been applied to solve convex minimization problems; see, for example, [9–11] and the references therein. Let H be a real Hilbert space, whose inner product and norm are denoted by 〈
·
,
·
〉 and ∥
·
∥, respectively. Let A be a strongly positive bounded linear operator on H; that is, there is a constant γ
¯
>
0 with property(1.8)
〈
A
x
,
x
〉
≥
γ
¯
‖
x
‖
2
∀
x
∈
H
.
A typical problem is to minimize a quadratic function over the set of the fixed points of a nonexpansive mapping on a real Hilbert space H:(1.9)
min
⁡
x
∈
C
1
2
〈
A
x
,
x
〉
-
〈
x
,
b
〉
,
where C is the fixed point set of a nonexpansive mapping T on H and b is a given point in H. In 2003, Xu [10] proved that the sequence {
x
n
} defined by the iterative method below, with the initial guess x
0
∈
H chosen arbitrarily,(1.10)
x
n
+
1
=
(
I
-
α
n
A
)
T
x
n
+
α
n
u
,
n
≥
0
,
converges strongly to the unique solution of the minimization problem (1.9) provided the sequence {
α
n
} satisfies certain conditions. Using the viscosity approximation method, Moudafi [12] introduced the following iterative process for nonexpansive mappings (see [13] for further developments in both Hilbert and Banach spaces). Let f be a contraction on H. Starting with an arbitrary initial x
0
∈
H, define a sequence {
x
n
} recursively by(1.11)
x
n
+
1
=
(
1
-
α
n
)
T
x
n
+
α
n
f
(
x
n
)
,
n
≥
0
,
where {
α
n
} is a sequence in (
0,1
). It is proved [12, 13] that, under certain appropriate conditions imposed on {
α
n
}, the sequence {
x
n
} generated by (1.11) strongly converges to the unique solution x
* in C of the variational inequality(1.12)
〈
(
I
-
f
)
x
*
,
x
-
x
*
〉
≥
0
,
x
∈
H
.
Recently, Marino and Xu [14] mixed the iterative method (1.10) and the viscosity approximation method (1.11) and considered the following general iterative method:(1.13)
x
n
+
1
=
(
I
-
α
n
A
)
T
x
n
+
α
n
γ
f
(
x
n
)
,
n
≥
0
,
where A is a strongly positive bounded linear operator on H. They proved that if the sequence {
α
n
} of parameters satisfies the certain conditions, then the sequence {
x
n
} generated by (1.13) converges strongly to the unique solution x
* in H of the variational inequality(1.14)
〈
(
A
-
γ
f
)
x
*
,
x
-
x
*
〉
≥
0
,
x
∈
H
which is the optimality condition for the minimization problem, min
x
∈
C
(
1
/
2
)
〈
A
x
,
x
〉
-
h
(
x
), where h is a potential function for γ
f
(
i
.
e
.
,
h
′
(
x
)
=
γ
f
(
x
) for x
∈
H).Very recently, Li et al. [15] introduced the following iterative procedures for the approximation of common fixed points of a one-parameter nonexpansive semigroup on a Hilbert space H:(1.15)
x
0
=
x
∈
H
,
x
n
+
1
=
(
I
-
α
n
A
)
1
t
n
∫
0
t
n
T
(
s
)
x
n
d
s
+
α
n
γ
f
(
x
n
)
,
n
≥
0
,
where A is a strongly positive bounded linear operator on H.Letδ and λ be two positive real numbers such that δ
,
λ
<
1. Recall that a mapping F with domain D
(
F
) and range R
(
F
) in E is called  δ
-strongly accretive if, for each x
,
y
∈
D
(
F
), there exists j
(
x
-
y
)
∈
J
(
x
-
y
) such that(1.16)
〈
F
x
-
F
y
,
j
(
x
-
y
)
〉
≥
δ
‖
x
-
y
‖
2
,
where J is the normalized duality mapping from E into the dual space E
*. Recall also that a mapping F is called  λ
-strictly pseudocontractive if, for each x
,
y
∈
D
(
F
), there exists j
(
x
-
y
)
∈
J
(
x
-
y
) such that(1.17)
〈
F
x
-
F
y
,
j
(
x
-
y
)
〉
≤
‖
x
-
y
‖
2
-
λ
‖
(
x
-
y
)
-
(
F
x
-
F
y
)
‖
2
.
It is easy to see that (1.17) can be rewritten as(1.18)
〈
(
I
-
F
)
x
-
(
I
-
F
)
y
,
j
(
x
-
y
)
〉
≥
λ
‖
(
I
-
F
)
x
-
(
I
-
F
)
y
‖
2
,
see [16].In this paper, motivated by the above results, we introduce and study the strong convergence theorems of the general iterative scheme{
x
n
} defined by (1.19) in the framework of a reflexive Banach space E which admits a weakly sequentially continuous duality mapping:(1.19)
x
0
=
x
∈
E
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
F
)
T
(
t
n
)
x
n
,
n
≥
0
,
where F is δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, f is a contraction on E with coefficient 0
<
α
<
1, γ is a positive real number such that γ
<
(
1
/
α
)
(
1
-
(
1
-
δ
)
/
λ
), and 𝒮
=
{
T
(
t
)
:
0
≤
t
<
∞
} is a nonexpansive semigroup on E. The strong convergence theorems are proved under some appropriate control conditions on parameters {
α
n
} and {
t
n
}. Furthermore, by using these results, we obtain strong convergence theorems of the following new general iterative schemes {
y
n
} and {
z
n
} defined by(1.20)
y
0
=
y
∈
E
,
y
n
+
1
=
α
n
γ
f
(
T
(
t
n
)
y
n
)
+
(
I
-
α
n
F
)
T
(
t
n
)
y
n
,
n
≥
0
,
(1.21)
z
0
=
z
∈
E
,
z
n
+
1
=
T
(
t
n
)
(
α
n
γ
f
(
z
n
)
+
(
I
-
α
n
F
)
z
n
)
,
n
≥
0
.
The results presented in this paper extend and improve the main results in Li et al. [15], Chen and He [7], and many others.


## 2. Preliminaries

Throughout this paper, it is assumed thatE is a real Banach space with norm ∥
·
∥ and let J denote the normalized duality mapping from E into E
* given by (2.1)
J
(
x
)
=
{
f
∈
E
*
:
〈
x
,
f
〉
=
‖
x
‖
2
=
‖
f
‖
2
}
for each x
∈
E, where E
* denotes the dual space of E
,
〈
·
,
·
〉 denotes the generalized duality pairing, and ℕ denotes the set of all positive integers. In the sequel, we will denote the single-valued duality mapping by j, and consider F
(
T
)
=
{
x
∈
C
:
T
x
=
x
}. When {
x
n
} is a sequence in E, then x
n
→
x (resp., x
n
⇀
x, x
n
⇀
*
x ) will denote strong (resp., weak, weak*) convergence of the sequence {
x
n
} to x. In a Banach space E, the following result (the subdifferential inequality) is well known [17, Theorem  4.2.1]: for all x
,
y
∈
E, for all j
(
x
+
y
)
∈
J
(
x
+
y
), for all j
(
x
)
∈
J
(
x
),(2.2)
‖
x
‖
2
+
2
〈
y
,
j
(
x
)
〉
≤
‖
x
+
y
‖
2
≤
‖
x
‖
2
+
〈
y
,
j
(
x
+
y
)
〉
.
A real Banach space E is said to bestrictly convex if ∥
x
+
y
∥
/
2
<
1 for all x
,
y
∈
E with ∥
x
∥
=
∥
y
∥
=
1 and x
≠
y. It is said to be uniformly convex if, for all ϵ
∈
[
0,2
], there exits δ
ϵ
>
0 such that (2.3)
‖
x
‖
=
‖
y
‖
=
1
with
‖
x
-
y
‖
≥
ϵ
implies
‖
x
+
y
‖
2
<
1
-
δ
ϵ
.
The following results are well known and can be founded in [17]:(i)
a uniformly convex Banach spaceE is reflexive and strictly convex [17, Theorems  4.2.1 and 4.1.6],
(ii)
ifE is a strictly convex Banach space and T
:
E
→
E is a nonexpansive mapping, then fixed point set F
(
T
) of T is a closed convex subset of E [17, Theorem  4.5.3].If a Banach spaceE admits a sequentially continuous duality mapping J from weak topology to weak star topology, then from Lemma  1 of [18], it follows that the duality mapping J is single-valued and also E is smooth. In this case, duality mapping J is also said to beweakly sequentially continuous, that is, for each {
x
n
}
⊂
E with x
n
⇀
x, then J
(
x
n
)
⇀
*
J
(
x
) (see [18, 19]).In the sequel, we will denote the single-valued duality mapping byj. A Banach space E is said to satisfy Opial's condition if, for any sequence {
x
n
} in E, x
n
⇀
x as n
→
∞ implies(2.4)
limsup
⁡
n
→
∞
‖
x
n
-
x
‖
<
limsup
⁡
n
→
∞
‖
x
n
-
y
‖
∀
y
∈
E
with
x
≠
y
.
By Theorem  1 of [18], we know that if E admits a weakly sequentially continuous duality mapping, then E satisfies Opial’s condition and E is smooth; for the details, see [18].Now, we present the concept of uniformly asymptotically regular semigroup (also see [20, 21]). Let C be a nonempty closed convex subset of a Banach space E, 𝒮
=
{
T
(
t
)
:
0
≤
t
<
∞
} a continuous operator semigroup on C. Then, 𝒮 is said to be uniformly asymptotically regular(in short, u.a.r.) on C if, for all h
≥
0 and any bounded subset D of C,(2.5)
lim
⁡
t
→
∞
sup
⁡
x
∈
D
‖
T
(
h
)
(
T
(
t
)
x
)
-
T
(
t
)
x
‖
=
0
.
The nonexpansive semigroup {
σ
t
:
t
>
0
} defined by the following lemma is an example of u.a.r. operator semigroup. Other examples of u.a.r. operator semigroup can be found in [20, Examples  17 and 18].Lemma 2.1 (see [3, Lemma  2.7]).
LetC be a nonempty closed convex subset of a uniformly convex Banach space E, D a bounded closed convex subset of C, and 𝒮
=
{
T
(
s
)
:
0
≤
s
<
∞
} a nonexpansive semigroup on C such that F
(
𝒮
)
≠
∅. For each h
>
0, set σ
t
(
x
)
=
(
1
/
t
)
∫
0
t
T
(
s
)
x
d
s, then
(2.6)
lim
⁡
t
→
∞
sup
⁡
x
∈
D
‖
σ
t
(
x
)
-
T
(
h
)
σ
t
(
x
)
‖
=
0
.Example 2.2.
The set{
σ
t
:
t
>
0
} defined by Lemma 2.1 is u.a.r. nonexpansive semigroup. In fact, it is obvious that {
σ
t
:
t
>
0
} is a nonexpansive semigroup. For each h
>
0, we have
(2.7)
‖
σ
t
(
x
)
-
σ
h
σ
t
(
x
)
‖
=
‖
σ
t
(
x
)
-
1
h
∫
0
h
T
(
s
)
σ
t
(
x
)
d
s
‖
=
‖
1
h
∫
0
h
(
σ
t
(
x
)
-
T
(
s
)
σ
t
(
x
)
)
d
s
‖
≤
1
h
∫
0
h
‖
σ
t
(
x
)
-
T
(
s
)
σ
t
(
x
)
‖
d
s
.
Applying Lemma 2.1, we have
(2.8)
lim
⁡
t
→
∞
sup
⁡
x
∈
D
‖
σ
t
(
x
)
-
σ
h
σ
t
(
x
)
‖
≤
1
h
∫
0
h
lim
⁡
t
→
∞
sup
⁡
x
∈
D
‖
σ
t
(
x
)
-
T
(
s
)
σ
t
(
x
)
‖
d
s
=
0
.
LetC be a nonempty closed and convex subset of a Banach space E and D a nonempty subset of C. A mapping Q
:
C
→
D is said to be sunny if
(2.9)
Q
(
Q
x
+
t
(
x
-
Q
x
)
)
=
Q
x
,
whenever Q
x
+
t
(
x
-
Q
x
)
∈
C for x
∈
C and t
=
0. A mapping Q
:
C
→
D is called a retraction if Q
x
=
x for all x
∈
D. Furthermore, Q is a sunny nonexpansive retraction from C onto D if Q is a retraction from C onto D which is also sunny and nonexpansive. A subset D of C is called a sunny nonexpansive retraction of C if there exists a sunny nonexpansive retraction from C onto D. The following lemma concerns the sunny nonexpansive retraction.Lemma 2.3 (see [22, 23]).
LetC be a closed convex subset of a smooth Banach space E. Let D be a nonempty subset of C and Q
:
C
→
D be a retraction. Then, Q is sunny and nonexpansive if and only if
(2.10)
〈
u
-
Q
u
,
j
(
y
-
Q
u
)
〉
≤
0
for all u
∈
C and y
∈
D.Lemma 2.4 (see [24, Lemma  2.3]).
Let{
a
n
} be a sequence of nonnegative real numbers satisfying the property
(2.11)
a
n
+
1
≤
(
1
-
t
n
)
a
n
+
t
n
c
n
+
b
n
,
∀
n
≥
0
,
where {
t
n
}
,
{
b
n
}
,
and
{
c
n
} satisfy the restrictions  (i)
∑
n
=
1
∞
t
n
=
∞;
(ii)
∑
n
=
1
∞
b
n
<
∞;
(iii)
limsup
n
→
∞
c
n
≤
0.
Then,lim
n
→
∞
a
n
=
0.The following lemma will be frequently used throughout the paper and can be found in [25].Lemma 2.5 (see [25, Lemma  2.7]).
LetE be a real smooth Banach space and F
:
E
→
E a mapping.
(i)
IfF is δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, then I
-
F is contractive with constant (
1
-
δ
)
/
λ.
(i)
IfF is δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, then, for any fixed number τ
∈
(
0,1
), I
-
τ
F is contractive with constant 1
-
τ
(
1
-
(
1
-
δ
)
/
λ
).


## 3. Main Results

Now, we are in a position to state and prove our main results.Theorem 3.1.
LetE be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮
=
{
T
(
t
)
:
0
≤
t
<
∞
} be a u.a.r. nonexpansive semigroup on E such that Fix
(
𝒮
)
≠
∅. Suppose that the real sequences {
α
n
}
⊂
[
0,1
], {
t
n
}
⊂
(
0
,
∞
) satisfy the conditions
(3.1)
lim
⁡
n
→
∞
α
n
=
0
,
∑
n
=
0
∞
α
n
=
∞
,
lim
⁡
n
→
∞
t
n
=
∞
.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, f
:
E
→
E a contraction mapping with coefficient α
∈
(
0,1
), and γ a positive real number such that γ
<
(
1
/
α
)
(
1
-
(
1
-
δ
)
/
λ
). Then, the sequence {
x
n
} defined by (1.19) converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.2)
〈
(
F
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
I
-
F
+
γ
f
)
x
̃, where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).Proof.
Note thatFix
(
𝒮
) is a nonempty closed convex set. We first show that {
x
n
} is bounded. Let q
∈
Fix
(
𝒮
). Thus, by Lemma 2.5, we have
(3.3)
‖
x
n
+
1
-
q
‖
=
‖
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
F
)
T
(
t
n
)
x
n
-
(
I
-
α
n
F
)
q
-
α
n
F
q
‖
≤
α
n
‖
γ
f
(
x
n
)
-
F
q
‖
+
‖
I
-
α
n
F
‖
‖
T
(
t
n
)
x
n
-
q
‖
≤
α
n
γ
‖
f
(
x
n
)
-
f
(
q
)
‖
+
α
n
‖
γ
f
(
q
)
-
F
q
‖
+
‖
I
-
α
n
F
‖
‖
x
n
-
q
‖
≤
α
n
α
γ
‖
x
n
-
q
‖
+
α
n
‖
γ
f
(
q
)
-
F
q
‖
+
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
x
n
-
q
‖
=
(
1
-
α
n
(
1
-
1
-
δ
λ
-
α
γ
)
)
‖
x
n
-
q
‖
+
α
n
(
1
-
1
-
δ
λ
-
α
γ
)
‖
γ
f
(
q
)
-
F
q
‖
1
-
(
1
-
δ
)
/
λ
-
α
γ
≤
max
⁡
{
‖
x
n
-
q
‖
,
1
1
-
(
1
-
δ
)
/
λ
-
α
γ
‖
γ
f
(
q
)
-
F
q
‖
}
,
∀
n
≥
0
.
By induction, we get
(3.4)
‖
x
n
-
q
‖
≤
max
⁡
{
‖
x
0
-
q
‖
,
1
1
-
(
1
-
δ
)
/
λ
-
α
γ
‖
γ
f
(
q
)
-
F
q
‖
}
,
n
≥
0
.
This implies that {
x
n
} is bounded and, hence, so are {
f
(
x
n
)
} and {
F
T
(
t
n
)
x
n
}. This implies that
(3.5)
lim
⁡
n
→
∞
‖
x
n
+
1
-
T
(
t
n
)
x
n
‖
=
lim
⁡
n
→
∞
α
n
‖
γ
f
(
x
n
)
-
F
T
(
t
n
)
x
n
‖
=
0
.
Since {
T
(
t
)
} is a u.a.r. nonexpansive semigroup and lim
n
→
∞
t
n
=
∞, we have, for all h
>
0,
(3.6)
lim
⁡
n
→
∞
‖
T
(
h
)
(
T
(
t
n
)
x
n
)
-
T
(
t
n
)
x
n
‖
≤
lim
⁡
n
→
∞
sup
⁡
x
∈
{
x
n
}
‖
T
(
h
)
(
T
(
t
n
)
x
)
-
T
(
t
n
)
x
‖
=
0
.
Hence, for all h
>
0,
(3.7)
‖
x
n
+
1
-
T
(
h
)
x
n
+
1
‖
≤
‖
x
n
+
1
-
T
(
t
n
)
x
n
‖
+
‖
T
(
t
n
)
x
n
-
T
(
h
)
T
(
t
n
)
x
n
‖
+
‖
T
(
h
)
T
(
t
n
)
x
n
-
T
(
h
)
x
n
+
1
‖
≤
2
‖
x
n
+
1
-
T
(
t
n
)
x
n
‖
+
‖
T
(
t
n
)
x
n
-
T
(
h
)
T
(
t
n
)
x
n
‖
⟶
0
.
That is, for all h
>
0,
(3.8)
lim
⁡
n
→
∞
‖
x
n
-
T
(
h
)
x
n
‖
=
0
.
Let Φ
=
Q
Fix
(
𝒮
). Then, Φ
(
I
-
F
-
γ
f
) is a contraction on E. In fact, from Lemma 2.5(i), we have
(3.9)
‖
Φ
(
I
-
F
-
γ
f
)
x
-
Φ
(
I
-
F
-
γ
f
)
y
‖
≤
‖
(
I
-
F
-
γ
f
)
x
-
(
I
-
F
-
γ
f
)
y
‖
≤
‖
(
I
-
F
)
x
-
(
I
-
F
)
y
‖
+
γ
‖
f
(
x
)
-
f
(
y
)
‖
≤
1
-
δ
λ
‖
x
-
y
‖
+
α
γ
‖
x
-
y
‖
=
(
1
-
δ
λ
+
α
γ
)
‖
x
-
y
‖
,
∀
x
,
y
∈
E
.
Therefore, Φ
(
I
-
F
-
γ
f
) is a contraction on E due to (
(
1
-
δ
)
/
λ
+
α
γ
)
∈
(
0,1
). Thus, by Banach contraction principle, Q
Fix
(
𝒮
)
(
I
-
F
-
γ
f
) has a unique fixed point x
̃. Then, using Lemma 2.3, x
̃ is the unique solution in Fix
(
S
) of the variational inequality (3.2). Next, we show that
(3.10)
limsup
⁡
n
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
-
x
̃
)
〉
≤
0
.
Indeed, we can take a subsequence {
x
n
k
} of {
x
n
} such that
(3.11)
limsup
⁡
n
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
-
x
̃
)
〉
=
lim
⁡
k
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
k
-
x
̃
)
〉
.
We may assume that x
n
k
⇀
p
∈
E as k
→
∞, since a Banach space E has a weakly sequentially continuous duality mapping J satisfying Opial's condition [13]. We will prove that p
∈
Fix
(
𝒮
). Suppose the contrary, p
∉
Fix
(
𝒮
), that is, T
(
h
0
)
p
≠
p for some h
0
>
0. It follows from (3.8) and Opial’s condition that
(3.12)
lim
⁡
inf
⁡
k
→
∞
‖
x
n
k
-
p
‖
<
lim
⁡
inf
⁡
k
→
∞
‖
x
n
k
-
T
(
h
0
)
p
‖
≤
lim
⁡
inf
⁡
k
→
∞
{
‖
x
n
k
-
T
(
h
0
)
x
n
k
‖
+
‖
T
(
h
0
)
x
n
k
-
T
(
h
0
)
p
‖
}
≤
lim
⁡
inf
⁡
k
→
∞
{
‖
x
n
k
-
T
(
h
0
)
x
n
k
‖
+
‖
x
n
k
-
p
‖
}
=
lim
⁡
inf
⁡
k
→
∞
‖
x
n
k
-
p
‖
.
This is a contradiction, which shows that p
∈
F
(
T
(
h
)
) for all h
>
0, that is, p
∈
Fix
(
𝒮
). In view of the variational inequality (3.2) and the assumption that duality mapping J is weakly sequentially continuous, we conclude
(3.13)
limsup
⁡
n
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
-
x
̃
)
〉
=
lim
⁡
k
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
k
-
x
̃
)
〉
≤
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
p
-
x
̃
)
〉
≤
0
.
Finally, we will show that x
n
→
x
̃. For each n
≥
0, we have
(3.14)
‖
x
n
+
1
-
x
̃
‖
2
=
‖
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
F
)
T
(
t
n
)
x
n
-
(
I
-
α
n
F
)
x
̃
-
α
n
F
x
̃
‖
2
≤
‖
α
n
γ
f
(
x
n
)
-
α
n
F
x
̃
+
(
I
-
α
n
F
)
T
(
t
n
)
x
n
-
(
I
-
α
n
F
)
x
̃
‖
2
=
‖
(
I
-
α
n
F
)
T
(
t
n
)
x
n
-
(
I
-
α
n
F
)
x
̃
‖
2
+
2
α
n
〈
γ
f
(
x
n
)
-
F
x
̃
,
j
(
x
n
+
1
-
x
̃
)
〉
≤
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
2
‖
x
n
-
x
̃
‖
2
+
2
α
n
〈
γ
f
(
x
n
)
-
γ
f
(
x
̃
)
,
j
(
x
n
+
1
-
x
̃
)
〉
+
2
α
n
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
+
1
-
x
̃
)
〉
.
On the other hand,
(3.15)
〈
γ
f
(
x
n
)
-
γ
f
(
x
̃
)
,
j
(
x
n
+
1
-
x
̃
)
〉
≤
γ
α
‖
x
n
-
x
̃
‖
‖
x
n
+
1
-
x
̃
‖
≤
γ
α
‖
x
n
-
x
̃
‖

[

(

1
-
α
n
(

1
-
1
-
δ
λ

)

)
2
‖
x
n
-
x
̃
‖
2
+
2
α
n
|
〈
γ
f
(
x
n
)
-
F
x
̃
,
j
(
x
n
+
1
-
x
̃
)
〉
|
]
≤
γ
α
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
x
n
-
x
̃
‖
2
+
γ
α
‖
x
n
-
x
̃
‖
2
|
〈
γ
f
(
x
n
)
-
F
x
̃
,
j
(
x
n
+
1
-
x
̃
)
〉
|
α
n
≤
γ
α
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
x
n
-
x
̃
‖
2
+
α
n
M
0
,
where M
0 is a constant satisfying M
0
≥
γ
α
∥
x
n
-
x
̃
∥
2
|
〈
γ
f
(
x
n
)
-
F
x
̃
,
j
(
x
n
+
1
-
x
̃
)
〉
|. Substituting (3.15) in (3.14), we obtain
(3.16)
‖
x
n
+
1
-
x
̃
‖
2
≤
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
2
‖
x
n
-
x
̃
‖
2
+
2
α
n
γ
α
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
×
‖
x
n
-
x
̃
‖
2
+
2
α
n
α
n
M
0
+
2
α
n
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
+
1
-
x
̃
)
〉
=
(
1
-
2
α
n
(
1
-
1
-
δ
λ
)
+
α
n
2
(
1
-
1
-
δ
λ
)
2
)
‖
x
n
-
x
̃
‖
2
+
2
α
n
γ
α
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
x
n
-
x
̃
‖
2
+
2
α
n
α
n
M
0
+
2
α
n
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
+
1
-
x
̃
)
〉
=
(
1
-
2
α
n
[
(
1
-
1
-
δ
λ
)
-
α
γ
+
α
n
γ
α
(
1
-
1
-
δ
λ
)
]
)
‖
x
n
-
x
̃
‖
2
+
α
n
[
α
n
(
1
-
1
-
δ
λ
)
2
‖
x
n
-
x
̃
‖
2
+
2
M
0
α
n
+
2
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
+
1
-
x
̃
)
〉
]
=
(
1
-
α
n
γ
n
)
‖
x
n
-
x
̃
‖
2
+
α
n
γ
n
β
n
γ
n
,
where
(3.17)
γ
n
=
2
[
(
1
-
1
-
δ
λ
)
-
α
γ
+
α
n
γ
α
(
1
-
1
-
δ
λ
)
]
,
β
n
=
[
α
n
(
1
-
1
-
δ
λ
)
2
‖
x
n
-
x
̃
‖
2
+
2
M
0
α
n
+
2
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
+
1
-
x
̃
)
〉
]
.
It is easily seen that ∑
n
=
1
∞
α
n
γ
n
=
∞. Since {
x
n
} is bounded and lim
n
→
∞
α
n
=
0, by (3.46), we obtain limsup
n
→
∞
β
n
/
γ
n
≤
0, applying Lemma 2.4 to (3.16) to conclude x
n
→
x
̃ as n
→
∞. This completes the proof.Using Theorem3.1, we obtain the following two strong convergence theorems of new iterative approximation methods for a nonexpansive semigroup {
T
(
t
)
:
0
≤
t
<
∞
}.Corollary 3.2.
LetE be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮
=
{
T
(
t
)
:
0
≤
t
<
∞
} be a u.a.r. nonexpansive semigroup on E such that Fix
(
𝒮
)
≠
∅. Suppose that the real sequences {
α
n
}
⊂
[
0,1
], {
t
n
}
⊂
(
0
,
∞
) satisfy the conditions
(3.18)
lim
⁡
n
→
∞
α
n
=
0
,
∑
n
=
0
∞
α
n
=
∞
,
lim
⁡
n
→
∞
t
n
=
∞
.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, f
:
E
→
E a contraction mapping with coefficient α
∈
(
0,1
), and γ a positive real number such that γ
<
(
1
/
α
)
(
1
-
(
1
-
δ
)
/
λ
). Then, the sequence {
y
n
} defined by (1.20) converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.19)
〈
(
F
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
I
-
F
+
γ
f
)
x
̃, where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).Proof.
Let{
x
n
} be the sequence given by x
0
=
y
0 and
(3.20)
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
F
)
T
(
t
n
)
x
n
,
∀
n
≥
0
.
Form Theorem 3.1, x
n
→
x
̃. We claim that y
n
→
x
̃. Indeed, we estimate
(3.21)
‖
x
n
+
1
-
y
n
+
1
‖
≤
α
n
γ
‖
f
(
T
(
t
n
)
y
n
)
-
f
(
x
n
)
‖
+
‖
I
-
α
n
F
‖
‖
T
(
t
n
)
x
n
-
T
(
t
n
)
y
n
‖
≤
α
n
γ
α
‖
T
(
t
n
)
y
n
-
x
n
‖
+
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
x
n
-
y
n
‖
≤
α
n
γ
α
‖
T
(
t
n
)
y
n
-
T
(
t
n
)
x
̃
‖
+
α
n
γ
α
‖
T
(
t
n
)
x
̃
-
x
n
‖
+
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
x
n
-
y
n
‖
≤
α
n
γ
α
‖
y
n
-
x
̃
‖
+
α
n
γ
α
‖
x
̃
-
x
n
‖
+
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
x
n
-
y
n
‖
≤
α
n
γ
α
‖
y
n
-
x
n
‖
+
α
n
γ
α
‖
x
n
-
x
̃
‖
+
α
n
γ
α
‖
x
̃
-
x
n
‖
+
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
x
n
-
y
n
‖
=
(
1
-
α
n
(
1
-
1
-
δ
λ
-
γ
α
)
)
‖
x
n
-
y
n
‖
+
α
n
(
1
-
1
-
δ
λ
-
γ
α
)
2
α
γ
(
1
-
(
1
-
δ
)
/
λ
-
γ
α
)
‖
x
̃
-
x
n
‖
.
It follows from ∑
n
=
1
∞
α
n
=
∞, lim
n
→
∞
∥
x
n
-
x
̃
∥
=
0, and Lemma 2.4 that ∥
x
n
-
y
n
∥
→
0. Consequently, y
n
→
x
̃ as required.Corollary 3.3.
LetE be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮
=
{
T
(
t
)
:
0
≤
t
<
∞
} be a u.a.r. nonexpansive semigroup on E such that Fix
(
𝒮
)
≠
∅. Suppose that the real sequences {
α
n
}
⊂
[
0,1
], {
t
n
}
⊂
(
0
,
∞
) satisfy the conditions
(3.22)
lim
⁡
n
→
∞
α
n
=
0
,
∑
n
=
0
∞
α
n
=
∞
,
lim
⁡
n
→
∞
t
n
=
∞
.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, f
:
E
→
E a contraction mapping with coefficient α
∈
(
0,1
), and γ a positive real number such that γ
<
(
1
/
α
)
(
1
-
(
1
-
δ
)
/
λ
). Then, the sequence {
z
n
} defined by (1.21) converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.23)
〈
(
F
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
I
-
F
+
γ
f
)
x
̃, where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).Proof.
Define the sequences{
y
n
} and {
β
n
} by
(3.24)
y
n
=
α
n
γ
f
(
z
n
)
+
(
I
-
α
n
F
)
z
n
,
β
n
=
α
n
+
1
∀
n
∈
N
.
Taking p
∈
Fix
(
𝒮
), we have
(3.25)
‖
z
n
+
1
-
p
‖
=
‖
T
(
t
n
)
y
n
-
T
(
t
n
)
p
‖
≤
‖
y
n
-
p
‖
=
‖
α
n
γ
f
(
z
n
)
+
(
I
-
α
n
F
)
z
n
-
(
I
-
α
n
F
)
p
-
α
n
F
p
‖
≤
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
z
n
-
p
‖
+
α
n
‖
γ
f
(
z
n
)
-
F
(
p
)
‖
=
(
1
-
α
n
(
1
-
1
-
δ
λ
)
)
‖
z
n
-
p
‖
+
α
n
(
1
-
1
-
δ
λ
)
‖
γ
f
(
z
n
)
-
F
(
p
)
‖
(
1
-
(
1
-
δ
)
/
λ
)
.
It follows from induction that
(3.26)
‖
z
n
+
1
-
p
‖
≤
max
⁡
{
‖
z
0
-
p
‖
,
‖
γ
f
(
z
0
)
-
F
(
p
)
‖
1
-
(
1
-
δ
)
/
λ
}
,
n
≥
0
.
Thus, both {
z
n
} and {
y
n
} are bounded. We observe that
(3.27)
y
n
+
1
=
α
n
+
1
γ
f
(
z
n
+
1
)
+
(
I
-
α
n
+
1
F
)
z
n
+
1
=
β
n
γ
f
(
T
(
t
n
)
y
n
)
+
(
I
-
β
n
F
)
T
(
t
n
)
y
n
.
Thus, Corollary 3.2 implies that {
y
n
} converges strongly to some point x
̃. In this case, we also have
(3.28)
‖
z
n
-
x
̃
‖
≤
‖
z
n
-
y
n
‖
+
‖
y
n
-
x
̃
‖
=
α
n
‖
γ
f
(
z
n
)
-
F
z
n
‖
+
‖
y
n
-
x
̃
‖
⟶
0
.
Hence, the sequence {
z
n
} converges strongly to some point x
̃. This complete the proof.Using Theorem3.1, Lemma 2.1, and Example 2.2, we have the following result.Corollary 3.4.
LetE be a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮
=
{
T
(
t
)
:
0
≤
t
<
∞
} be a nonexpansive semigroup on E such that Fix
(
𝒮
)
≠
∅. Suppose that the real sequences {
α
n
}
⊂
[
0,1
], {
t
n
}
⊂
(
0
,
∞
) satisfy the conditions
(3.29)
lim
⁡
n
→
∞
α
n
=
0
,
∑
n
=
0
∞
α
n
=
∞
,
lim
⁡
n
→
∞
t
n
=
∞
.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, f
:
E
→
E a contraction mapping with coefficient α
∈
(
0,1
), and γ a positive real number such that γ
<
(
1
/
α
)
(
1
-
(
1
-
δ
)
/
λ
). Then, the sequence {
x
n
} defined by
(3.30)
x
0
=
x
∈
E
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
F
)
1
t
n
∫
0
t
n
T
(
t
)
x
n
d
s
,
n
≥
0
converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.31)
〈
(
F
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
(
I
-
F
+
γ
f
)
x
̃
), where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).Corollary 3.5.
LetH be a real Hilbert space. Let 𝒮
=
{
T
(
t
)
:
0
≤
t
<
∞
} be a nonexpansive semigroup on H such that Fix
(
𝒮
)
≠
∅. Suppose that the real sequences {
α
n
}
⊂
[
0,1
], {
t
n
}
⊂
(
0
,
∞
) satisfy the conditions
(3.32)
lim
⁡
n
→
∞
α
n
=
0
,
∑
n
=
0
∞
α
n
=
∞
,
lim
⁡
n
→
∞
t
n
=
∞
.
Let f
:
E
→
E be a contraction mapping with coefficient α
∈
(
0,1
) and A a strongly positive bounded linear operator with coefficient γ
¯
>
1
/
2 and 0
<
γ
<
(
1
-
2
-
2
γ
¯
)
/
α. Then, the sequence {
x
n
} defined by
(3.33)
x
0
=
x
∈
E
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
A
)
1
t
n
∫
0
t
n
T
(
t
)
x
n
d
s
,
n
≥
0
converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.34)
〈
(
A
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
(
I
-
A
+
γ
f
)
x
̃
), where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).Proof.
SinceA is a strongly positive bounded linear operator with coefficient γ
¯, we have
(3.35)
〈
A
x
-
A
y
,
x
-
y
〉
≥
γ
¯
‖
x
-
y
‖
2
.
Therefore, A is γ
¯-strongly accretive. On the other hand,
(3.36)
‖
(
I
-
A
)
x
-
(
I
-
A
)
y
‖
2
=
〈
(
x
-
y
)
-
(
A
x
-
A
y
)
,
(
x
-
y
)
-
(
A
x
-
A
y
)
〉
=
〈
x
-
y
,
x
-
y
〉
-
2
〈
A
x
-
A
y
,
x
-
y
〉
+
〈
A
x
-
A
y
,
A
x
-
A
y
〉
=
‖
x
-
y
‖
2
-
2
〈
A
x
-
A
y
,
x
-
y
〉
+
‖
A
x
-
A
y
‖
2
≤
‖
x
-
y
‖
2
-
2
〈
A
x
-
A
y
,
x
-
y
〉
+
‖
A
‖
2
‖
x
-
y
‖
2
.
Since A is strongly positive if and only if (
1
/
∥
A
∥
)
A is strongly positive, we may assume, without loss of generality, that ∥
A
∥
=
1, so that
(3.37)
〈
A
x
-
A
y
,
x
-
y
〉
≤
‖
x
-
y
‖
2
-
1
2
‖
(
I
-
A
)
x
-
(
I
-
A
)
y
‖
2
=
‖
x
-
y
‖
2
-
1
2
‖
(
x
-
y
)
-
(
A
x
-
A
y
)
‖
2
.
Hence, A is 1⁄2-strongly pseudocontractive. Applying Corollary 3.4, we conclude the result.Theorem 3.6.
LetE be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮
=
{
T
(
t
)
:
0
<
t
<
∞
} be a u.a.r. nonexpansive semigroup on E such that Fix
(
𝒮
)
≠
∅. Let {
α
n
} and {
t
n
} be sequences of real number satisfying
(3.38)
0
<
α
n
<
1
,
∑
n
=
0
∞
α
n
=
∞
,
t
n
>
0
,
lim
⁡
n
→
∞
α
n
=
lim
⁡
n
→
∞
α
n
t
n
=
0
.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, f
:
E
→
E a contraction mapping with coefficient α
∈
(
0,1
), and γ a positive real number such that γ
<
(
1
/
α
)
(
1
-
(
1
-
δ
)
/
λ
). Then, the sequence {
x
n
} defined by
(3.39)
x
0
=
x
∈
E
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
F
)
T
(
t
n
)
x
n
,
n
≥
0
converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.40)
〈
(
F
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
I
-
F
+
γ
f
)
x
̃, where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).Proof.
By the same argument as in the proof of Theorem3.1, we can obtain that {
x
n
}, {
f
(
x
n
)
}, and {
F
T
(
t
n
)
x
n
} are bounded and Q
Fix
(
𝒮
)
(
I
-
F
-
γ
f
) is a contraction on E. Thus, by Banach contraction principle, Q
Fix
(
𝒮
)
(
I
-
F
-
γ
f
) has a unique fixed point x
̃. Then, using Lemma 2.3, x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality (3.40). Next, we show that
(3.41)
limsup
⁡
n
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
-
x
̃
)
〉
≤
0
.
Indeed, we can take a subsequence {
x
n
k
} of {
x
n
} such that
(3.42)
limsup
⁡
n
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
-
x
̃
)
〉
=
lim
⁡
k
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
k
-
x
̃
)
〉
.
We may assume that x
n
k
⇀
p
∈
E as k
→
∞. Now, we show that p
∈
Fix
(
𝒮
). Put
(3.43)
x
k
=
x
n
k
,
α
k
=
α
n
k
s
k
=
t
n
k
∀
k
∈
N
.
Fix t
>
0,  then we have
(3.44)
‖
x
k
-
T
(
t
)
p
‖
=
∑
i
=
0
[
t
/
s
i
]
-
1
‖
T
(
(
i
+
1
)
s
k
)
x
k
-
T
(
i
s
k
)
x
k
‖
+
‖
T
(
[
t
s
k
]
s
k
)
x
k
-
T
(
[
t
s
k
]
s
k
)
p
‖
+
‖
T
(
[
t
s
k
]
s
k
)
p
-
T
(
t
)
p
‖
≤
[
t
s
k
]
‖
T
(
s
k
)
x
k
-
x
k
+
1
‖
+
‖
x
k
+
1
-
p
‖
+
‖
T
(
t
-
[
t
s
k
]
s
k
)
p
-
p
‖
≤
[
t
s
k
]
α
k
‖
F
T
(
s
k
)
x
k
-
f
(
x
k
)
‖
+
‖
x
k
+
1
-
p
‖
+
‖
T
(
t
-
[
t
s
k
]
s
k
)
p
-
p
‖
≤
(
t
α
k
s
k
)
‖
F
T
(
s
k
)
x
k
-
f
(
x
k
)
‖
+
‖
x
k
+
1
-
p
‖
+
max
⁡
{
‖
T
(
s
)
p
-
p
‖
:
0
≤
s
≤
s
k
}
.
Thus, for all k
∈
ℕ, we obtain
(3.45)
limsup
⁡
k
→
∞
‖
x
k
-
T
(
t
)
p
‖
≤
limsup
⁡
k
→
∞
‖
x
k
+
1
-
p
‖
=
limsup
⁡
k
→
∞
‖
x
k
-
p
‖
.
Since Banach space E has a weakly sequentially continuous duality mapping satisfying Opial's condition [13], we can conclude that T
(
t
)
p
=
p for all t
>
0, that is, p
∈
Fix
(
𝒮
). In view of the variational inequality (3.2) and the assumption that duality mapping J is weakly sequentially continuous, we conclude
(3.46)
limsup
⁡
n
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
-
x
̃
)
〉
=
lim
⁡
k
→
∞
〈
γ
f
(
x
̃
)
-
F
x
̃
,
j
(
x
n
k
-
x
̃
)
〉
≤
〈
γ
f
(
x
̃
)
-
F
x
̃
,
J
(
p
-
x
̃
)
〉
≤
0
.
By the same argument as in the proof of Theorem 3.1, we conclude that x
n
→
x
̃ as n
→
∞. This completes the proof.Using Theorem3.6 and the method as in the proof of Corollary 3.7, we have the following result.Corollary 3.7.
LetE be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮
=
{
T
(
t
)
:
0
<
t
<
∞
} be a u.a.r. nonexpansive semigroup on E such that Fix
(
𝒮
)
≠
∅. Let {
α
n
} and {
t
n
} be sequences of real number satisfying
(3.47)
0
<
α
n
<
1
,
∑
n
=
0
∞
α
n
=
∞
,
t
n
>
0
,
lim
⁡
n
→
∞
α
n
=
lim
⁡
n
→
∞
α
n
t
n
=
0
.
Let F be a δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, f
:
E
→
E a contraction mapping with coefficient α
∈
(
0,1
), and γ is a positive real number such that γ
<
1
/
α
(
1
-
(
1
-
δ
)
/
λ
). Then, the sequence {
y
n
} defined by
(3.48)
y
0
=
y
∈
E
,
y
n
+
1
=
α
n
γ
f
(
T
(
t
n
)
y
n
)
+
(
I
-
α
n
F
)
T
(
t
n
)
y
n
,
n
≥
0
converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.49)
〈
(
F
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
I
-
F
+
γ
f
)
x
̃, where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).Using Theorem3.6 and the method as in the proof of Corollary 3.8, we have the following result.Corollary 3.8.
LetE be a reflexive Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮
=
{
T
(
t
)
:
0
<
t
<
∞
} be a u.a.r. nonexpansive semigroup on E such that Fix
(
𝒮
)
≠
∅. Let {
α
n
} and {
t
n
} be sequences of real number satisfying
(3.50)
0
<
α
n
<
1
,
∑
n
=
0
∞
α
n
=
∞
,
t
n
>
0
,
lim
⁡
n
→
∞
α
n
=
lim
⁡
n
→
∞
α
n
t
n
=
0
.
Let F be a δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, f
:
E
→
E a contraction mapping with coefficient α
∈
(
0,1
), and γ is a positive real number such that γ
<
(
1
/
α
)
(
1
-
(
1
-
δ
)
/
λ
). Then, the sequence {
z
n
} defined by
(3.51)
z
0
=
z
∈
E
,
z
n
+
1
=
T
(
t
n
)
(
α
n
γ
f
(
z
n
)
+
(
I
-
α
n
F
)
z
n
)
,
n
≥
0
converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.52)
〈
(
F
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
I
-
F
+
γ
f
)
x
̃, where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).Using Theorem3.6, Lemma 2.1, and Example 2.2, we have the following result.Corollary 3.9.
LetE be a uniformly convex Banach space which admits a weakly sequentially continuous duality mapping J. Let 𝒮
=
{
T
(
t
)
:
0
<
t
<
∞
} be a nonexpansive semigroup on E such that Fix
(
𝒮
)
≠
∅. Let {
α
n
} and {
t
n
} be sequences of real numbers satisfying
(3.53)
0
<
α
n
<
1
,
∑
n
=
0
∞
α
n
=
∞
,
t
n
>
0
,
lim
⁡
n
→
∞
α
n
=
lim
⁡
n
→
∞
α
n
t
n
=
0
.
Let F be δ-strongly accretive and λ-strictly pseudocontractive with δ
+
λ
>
1, f
:
E
→
E a contraction mapping with coefficient α
∈
(
0,1
), and γ a positive real number such that γ
<
(
1
/
α
)
(
1
-
(
1
-
δ
)
/
λ
). Then, the sequence {
x
n
} defined by
(3.54)
x
0
=
x
∈
E
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
F
)
1
t
n
∫
0
t
n
T
(
t
)
x
n
d
s
,
n
≥
0
converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.55)
〈
(
F
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
I
-
F
+
γ
f
)
x
̃, where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).Corollary 3.10.
LetH be a real Hilbert space. Let 𝒮
=
{
T
(
t
)
:
0
≤
t
<
∞
} be a nonexpansive semigroup on H such that Fix
(
𝒮
)
≠
∅. Suppose that the real sequences {
α
n
}
⊂
[
0,1
], {
t
n
}
⊂
(
0
,
∞
) satisfy the conditions
(3.56)
0
<
α
n
<
1
,
∑
n
=
0
∞
α
n
=
∞
,
t
n
>
0
,
lim
⁡
n
→
∞
α
n
=
lim
⁡
n
→
∞
α
n
t
n
=
0
.
Let f
:
E
→
E be a contraction mapping with coefficient α
∈
(
0,1
) and A a strongly positive bounded linear operator with coefficient γ
¯
>
1
/
2 and 0
<
γ
<
(
1
-
2
-
2
γ
¯
)
/
α. Then, the sequence {
x
n
} defined by
(3.57)
x
0
=
x
∈
E
,
x
n
+
1
=
α
n
γ
f
(
x
n
)
+
(
I
-
α
n
A
)
1
t
n
∫
0
t
n
T
(
t
)
x
n
d
s
,
n
≥
0
converges strongly to x
̃, where x
̃ is the unique solution in Fix
(
𝒮
) of the variational inequality
(3.58)
〈
(
A
-
γ
f
)
x
̃
,
j
(
x
-
x
̃
)
〉
≥
0
,
x
∈
Fix
⁡
(
S
)
or equivalently x
̃
=
Q
Fix
(
𝒮
)
(
(
I
-
A
+
γ
f
)
x
̃
), where Q
Fix
(
𝒮
) is the sunny nonexpansive retraction of E onto Fix
(
𝒮
).



---

*Source: 643740-2011-05-30.xml*