# Iterative Schemes by a New Generalized Resolvent for a Monotone Mapping and a Relatively Weak Nonexpansive Mapping

Authors: Xian Wang; Jun-min Chen; Hui Tong
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We introduce a new generalized resolvent in a Banach space and discuss some of its properties. Using these properties, we obtain an iterative scheme for finding a point which is a fixed point of relatively weak nonexpansive mapping and a zero of monotone mapping. Furthermore, strong convergence of the scheme to a point which is a fixed point of relatively weak nonexpansive mapping and a zero of monotone mapping is proved.


---

## Body

## 1. Preliminaries

LetE be a real Banach space with dual E
*. We denote by J the normalized duality mapping from E into 2
E
*, defined by
(1)
J
x
:
=
{
f
*
∈
E
*
:
〈
x
,
f
*
〉
=
∥
x
∥
2
=
∥
f
*
∥
2
}
,
where 〈
·
,
·
〉 denotes the generalized duality pairing. It is well known that if E
* is strictly convex, then J is single valued and if E is uniformly smooth, then J is uniformly continuous on bounded subsets of E. Moreover, if E is a reflexive and strictly convex Banach space with a strictly convex dual, then J
-
1 is single valued, one-to-one, and surjective, and it is the duality mapping from E
* into E and thus J
J
-
1
=
I
E
*
=
I
* and J
-
1
J
=
I
E
=
I (see [1]). We note that, in a Hilbert space H, J is the identity mapping.LetE be a smooth, reflexive, and strictly convex Banach space. We define the function V
2
:
E
×
E
→
R by
(2)
V
2
(
y
,
x
)
=
∥
x
∥
2
-
2
〈
J
y
,
x
〉
+
∥
y
∥
2
,
for all x
∈
E
,
y
∈
E. Let C be a nonempty closed convex subset of E. For an arbitrary point x of E, consider the set {
z
∈
C
:
V
2
(
z
,
x
)
=
min
⁡
y
∈
C
V
2
(
y
,
x
)
}. In 1996, Alber [2] introduced generalized projection Π
C
:
E
→
C from Hilbert space to uniformly convex and uniformly smooth Banach space:
(3)
V
2
(
Π
C
x
,
x
)
=
min
⁡
y
∈
C
V
2
(
y
,
x
)
.
Such a mapping Π
C is called the generalized projection.Applying the definitions ofV
2 and J, a functional V
:
E
*
×
E
→
R is defined by the following formula:
(4)
V
(
x
*
,
y
)
=
V
2
(
J
-
1
x
*
,
y
)
,
∀
x
*
∈
E
*
,
y
∈
E
.In the following, we will make use of the following lemmas.Lemma 1 (see [3]).
LetE be a real smooth Banach space and let A
:
E
→
2
E
* be a maximal monotone mapping; then A
-
1
0 is a closed and convex subset of E and the graph of A, G
(
A
), is demiclosed in the following sense, for all x
n
∈
D
(
A
) with x
n
→
x in E and for all y
n
∈
A
x
n with y
n
→
y in E implying that x
∈
D
(
A
) and y
∈
A
x.Lemma 2 (see [2]).
LetC be a nonempty closed and convex subset of a real reflexive, strictly convex, and smooth Banach space E and let x
∈
E. Then, y
∈
C and
(5)
V
2
(
y
,
Π
C
x
)
+
≤
V
2
(
Π
C
x
,
x
)
≤
V
2
(
y
,
x
)
.Lemma 3 (see [2]).
LetC be a convex subset of a real smooth Banach space E. Let x
∈
E and x
0
∈
C. Then, V
2
(
x
0
,
x
)
=
inf
⁡
{
V
2
(
z
,
x
)
:
z
∈
C
} if and only if
(6)
〈
z
-
x
0
,
J
x
0
-
J
x
〉
≥
0
.Lemma 4 (see [4]).
LetE be a real smooth and uniformly convex Banach space and let {
x
n
} and {
y
n
} be two sequences of E. If either {
x
n
} or {
y
n
} is bounded and V
2
(
x
n
,
y
n
)
→
0 as n
→
∞, then x
n
-
y
n
→
0, as n
→
∞.
LetE
* be a smooth Banach space and let D
* be a nonempty closed convex subset of E
*. A mapping R
*
:
D
*
→
D
* is called generalized nonexpansive if F
(
R
*
)
≠
Ø and
(7)
V
(
R
*
x
*
,
J
-
1
y
*
)
≤
V
(
x
*
,
J
-
1
y
*
)
,
vvvvvvv
∀
x
*
∈
D
*
,
y
*
∈
F
(
R
*
)
,
where F
(
R
*
) is the set of fixed points of R
*.
LetC be a nonempty closed convex subset of E, and let T be a mapping from C into itself. We denote by F
(
T
) the set of fixed points of T. A point of p in C is said to be a strong asymptotic fixed point of T if C contains a sequence {
x
n
} which converges strongly to p such that the strong lim
⁡
n
→
∞
(
T
x
n
-
x
n
)
=
0. The set of strong asymptotic fixed points of T will be denoted by F
~
(
T
). A mapping T from C into itself is called weak relatively nonexpansive if F
~
(
T
)
=
F
(
T
) and V
2
(
p
,
T
x
)
≤
V
2
(
p
,
x
) for all x
∈
C and p
∈
F
(
T
) (see [5]).
LetE be a smooth Banach space and let C be a nonempty closed convex subset of E. A mapping R
:
C
→
C is called generalized nonexpansive if F
(
R
)
≠
Ø and
(8)
V
2
(
R
x
,
y
)
≤
V
2
(
x
,
y
)
,
∀
x
∈
C
,
y
∈
F
(
R
)
,
where F
(
R
) is the set of fixed points of R. Let E be a reflexive and smooth Banach space and let B
⊂
E
*
×
E be a maximal monotone operator. For each λ
>
0 and x
∈
E, Ibaraki and Takahashi [6] considered the set
(9)
J
λ
x
:
=
{
z
∈
E
:
x
∈
z
+
λ
B
J
(
z
)
}
.
Such a J
λ is called the generalized resolvent and is denoted by
(10)
J
λ
=
(
I
+
λ
B
J
)
-
1
.
By sunny nonexpansive retractions, they discussed the existence of a retraction R
C of E onto C such that, for any x
∈
E,
(11)
〈
x
-
R
C
x
,
J
(
R
C
x
)
-
J
(
y
)
〉
≥
0
,
∀
y
∈
C
,
where E is a smooth Banach space and C is nonempty closed subset of E (see  [7]).
In [7], Zegeye and Shahzad studied the following iterative scheme for finding a zero point of a maximal strongly monotone mapping A in a real uniformly smooth and uniformly convex Banach space E. Then the sequence {
x
n
} generated by
(12)
x
0
∈
K
,
c
h
o
s
e
n
a
r
b
i
t
r
a
r
y
,
y
n
=
J
-
1
(
J
x
n
-
α
n
A
x
n
)
,
z
n
=
T
y
n
,
H
0
=
{
v
∈
K
:
ϕ
(
v
,
z
0
)
≤
ϕ
(
v
,
y
0
)
≤
ϕ
(
v
,
x
0
)
}
,
H
n
=
{
v
∈
H
n
-
1
∩
W
n
-
1
:
ϕ
(
v
,
z
n
)
h
≤
ϕ
(
v
,
y
n
)
≤
ϕ
(
v
,
x
n
)
}
,
W
0
=
E
,
W
n
=
{
v
∈
H
n
-
1
∩
W
n
-
1
:
g
〈
x
n
-
v
,
J
x
0
-
J
x
n
〉
≥
0
}
,
x
n
+
1
=
∏
H
n
∩
W
n
(
x
0
)
,
n
≥
1
converges strongly to Π
A
-
1
0
∩
F
(
T
)
(
x
0
), where Π
A
-
1
0
∩
F
(
T
) is the generalized projection from E onto A
-
1
0
∩
F
(
T
).In this paper, motivated by Alber [2], Ibaraki and Takahashi [6], and Zegeye and Shahzad [7], we first introduce the generalized resolvent and discuss its properties. Secondly, we give an iterative scheme for finding a point which is a fixed point of relatively weak nonexpansive mapping and a zero of monotone mapping. Finally, we show its convergence.


## 2. The Generalized ResolventJ
λ
* and Some of Its Properties

LetE
* be a reflexive and smooth Banach space and let B
⊂
E
×
E
* be a maximal monotone operator. For each λ
>
0 and x
∈
E, consider the set:
(13)
J
λ
*
x
*
:
=
{
z
*
∈
E
*
:
x
*
∈
z
*
+
λ
B
J
-
1
(
z
*
)
}
.
If z
1
*
+
λ
w
1
*
=
x
*, z
2
*
+
λ
w
2
*
=
x
*, w
1
*
∈
B
J
-
1
(
z
1
*
), w
2
*
∈
B
J
-
1
(
z
2
*
), then we have from the monotonicity of B that
(14)
〈
w
1
*
-
w
2
*
,
J
-
1
(
z
1
*
)
-
J
-
1
(
z
2
*
)
〉
≥
0
,
and hence
(15)
〈
x
*
-
z
1
*
λ
-
x
*
-
z
2
*
λ
,
J
-
1
(
z
1
*
)
-
J
-
1
(
z
2
*
)
〉
≥
0
.
So, we obtain
(16)
〈
x
*
-
z
1
*
-
(
x
*
-
z
2
*
)
,
J
-
1
(
z
1
*
)
-
J
-
1
(
z
2
*
)
〉
≥
0
,
and hence
(17)
〈
z
2
*
-
z
1
*
,
J
-
1
(
z
1
*
)
-
J
-
1
(
z
2
*
)
〉
≥
0
.
This implies z
1
*
=
z
2
*. Then, J
λ
*
x
* consists of one point. We also denote the domain and the range of J
λ
*
x
* by D
(
J
λ
*
)
=
R
(
I
*
+
λ
B
J
-
1
) and R
(
J
λ
*
)
=
D
(
B
J
-
1
), respectively, where I
* is the identity on E
*. Such a J
λ
*
:
E
*
→
E
* is called the generalized resolvent of B and is denoted by
(18)
J
λ
*
=
(
I
*
+
λ
B
J
-
1
)
-
1
.
We get some properties of J
λ
* and (
B
J
-
1
)
-
1
0.Proposition 5.
LetE
* be a reflexive and strictly convex Banach space with a Fréchet differentiable norm and let B
⊂
E
×
E
* be a maximal monotone operator with B
-
1
0
≠
Ø. Then, the following hold:(1)
D
(
J
λ
*
)
=
E
*
for each
λ
>
0;
(2)
(
B
J
-
1
)
-
1
0
=
F
(
J
λ
*
)
for each
λ
>
0
, where
F
(
J
λ
*
)
is the set of fixed points of
J
λ
*;
(3)
(
B
J
-
1
)
-
1
0
is closed;
(4)
J
λ
*
:
E
*
→
E
*
is generalized nonexpansive for each
λ
>
0.Proof.
(1) From the maximality ofB, we have
(19)
R
(
J
+
λ
B
)
=
E
*
,
∀
λ
>
0
.
Hence, for each x
*
∈
E
*, there exists x
∈
E such that x
*
∈
J
x
+
λ
B
x. Since E is reflexive and strictly convex, J is bijective. Therefore, there exists z
*
∈
E
* such that x
=
J
-
1
(
z
*
). Therefore, we have
(20)
x
*
∈
J
J
-
1
(
z
*
)
+
λ
B
J
-
1
(
z
*
)
hh
=
z
*
+
λ
B
J
-
1
(
z
*
)
⊂
R
(
I
*
+
λ
B
J
-
1
)
=
D
(
J
λ
*
)
.
This implies E
*
⊂
D
(
J
λ
*
). D
(
J
λ
*
)
⊂
E
* is clear. So, we have D
(
J
λ
*
)
=
E
*.
(2) Letλ
>
0. Then, we have
(21)
x
*
∈
F
(
J
λ
)
⟺
J
λ
*
x
*
=
x
*
⟺
x
*
∈
x
*
+
λ
B
J
-
1
(
x
*
)
⟺
0
∈
λ
B
J
-
1
(
x
*
)
⟺
0
∈
B
J
-
1
(
x
*
)
⟺
x
*
∈
(
B
J
-
1
)
-
1
0
.
(3) Let{
x
n
*
}
⊂
(
B
J
-
1
)
-
1
0 with x
n
*
→
x
*. From x
n
*
∈
(
B
J
-
1
)
-
1
0, we have J
-
1
(
x
n
*
)
∈
B
-
1
0. Since J
-
1 is norm to norm continuous and B
-
1
0 is closed, we have that J
-
1
(
x
n
*
)
→
J
-
1
(
x
*
)
∈
B
-
1
0. This implies x
*
∈
(
B
J
-
1
)
-
1
0. That is, (
B
J
-
1
)
-
1
0 is closed.
(4) Letx
*
∈
E
*, y
*
∈
E
*
,
z
*
∈
E
*, and λ
>
0. By Definition (2) and calculating that
(22)
V
(
x
*
,
J
-
1
z
*
)
+
V
(
z
*
,
J
-
1
y
*
)
hhhh
=
∥
x
*
∥
2
+
∥
z
*
∥
2
-
2
〈
x
*
,
J
-
1
z
*
〉
hhhhhh
+
∥
y
*
∥
2
+
∥
z
*
∥
2
-
2
〈
z
*
,
J
-
1
y
*
〉
hhhh
=
V
(
x
*
,
J
-
1
y
*
)
+
2
〈
z
*
-
x
*
,
J
-
1
z
*
-
J
-
1
y
*
〉
,
we have that
(23)
V
(
x
*
,
J
-
1
y
*
)
=
V
(
x
*
,
J
-
1
z
*
)
+
V
(
z
*
,
J
-
1
y
*
)
+
2
〈
x
*
-
z
*
,
J
-
1
z
*
-
J
-
1
y
*
〉
.
Let x
*
∈
E
*
,
y
*
∈
F
(
J
λ
), and λ
>
0. From the above formula, we have
(24)
V
(
x
*
,
J
-
1
y
*
)
=
V
(
x
*
,
J
-
1
J
λ
*
x
*
)
+
V
(
J
λ
*
x
*
,
J
-
1
y
*
)
+
2
〈
x
*
-
J
λ
*
x
*
,
J
-
1
J
λ
x
*
-
J
-
1
y
*
〉
.
Since (
(
x
*
-
J
λ
*
x
*
)
/
λ
)
∈
B
J
-
1
(
J
λ
*
x
*
) and 0
∈
B
J
-
1
(
y
*
), we have
(25)
〈
x
*
-
J
λ
*
x
*
,
J
-
1
J
λ
*
x
*
-
J
-
1
y
*
〉
≥
0
.
Therefore, we get
(26)
V
(
x
*
,
J
-
1
y
*
)
≥
V
(
x
*
,
J
-
1
J
λ
*
x
*
)
+
V
(
J
λ
*
x
*
,
J
-
1
y
*
)
≥
V
(
J
λ
*
x
*
,
J
-
1
y
*
)
.
That is, J
λ
* is generalized nonexpansive on E
*.Theorem 6 (see [8]).
LetE be a Banach space and let A
⊂
E
×
E
* be a maximal monotone operator with A
-
1
0
≠
Ø. IfE* is strictly convex and has a Fréchet differentiable norm, then, for each x
∈
E, lim
⁡
λ
→
∞
(
J
+
λ
A
)
-
1
J
(
x
) exists and belongs to A
-
1
0.
Using Theorem6, we get the following result.Theorem 7.
LetE
* be a uniformly convex Banach space with a Fréchet differentiable norm and let B
⊂
E
×
E
* be a maximal monotone operator with B
-
1
0
≠
Ø. Then the following hold:(1)
for eachx
*
∈
E
*, lim
⁡
λ
→
∞
J
λ
*
x
* exists and belongs to (
B
J
-
1
)
-
1
0;
(2)
ifR
*
x
*
:
=
lim
⁡
λ
→
∞
J
λ
*
x
* for each x
*
∈
E
*, then R
* is a sunny generalized nonexpansive retraction of E
* onto (
B
J
-
1
)
-
1
0.Proof.
(1) By defining a mappingQ
λ from E to E by
(27)
Q
λ
x
:
=
(
I
+
λ
J
-
1
B
)
x
,
∀
x
∈
E
,
λ
>
0
,
we have, for all x
*
∈
E
*
,
λ
>
0, J
λ
*
x
*
=
J
Q
λ
J
-
1
(
x
*
). In fact, define
(28)
x
λ
*
:
=
J
Q
λ
J
-
1
(
x
*
)
=
[
J
(
I
+
λ
J
-
1
B
)
J
-
1
]
-
1
(
x
*
)
.
Then, we have
(29)
x
*
∈
J
(
I
+
λ
J
-
1
B
)
J
-
1
(
x
λ
*
)
=
(
I
*
+
λ
B
J
-
1
)
x
λ
*
,
and hence x
λ
*
=
J
λ
*
x
*. From Theorem 6, we get
(30)
lim
⁡
λ
→
∞
Q
λ
J
-
1
(
x
*
)
=
u
∈
B
-
1
0
.
If E
* is uniformly convex, then E has a Fréchet differentiable norm. So, J is norm to norm continuous. Since B
-
1
0 is closed, we have
(31)
lim
⁡
λ
→
∞
J
λ
*
x
*
=
lim
⁡
λ
→
∞
J
Q
λ
J
-
1
(
x
*
)
=
J
u
∈
J
B
-
1
0
=
(
B
J
-
1
)
-
1
0
.
(2) We define a mappingR
* from E
* to E
* by
(32)
R
*
x
*
:
=
lim
⁡
λ
→
∞
J
λ
*
x
*
,
∀
x
*
∈
E
*
.
Let u
*
∈
(
B
J
-
1
)
-
1
0
=
F
(
J
λ
*
x
*
). Then, R
*
u
*
=
lim
⁡
λ
→
∞
J
λ
*
u
*
=
lim
⁡
λ
→
∞
u
*
=
u
*. Therefore, R
* is a retraction of E
* onto (
B
J
-
1
)
-
1
0. Since x
*
∈
J
λ
*
x
*
+
λ
B
J
-
1
(
J
λ
*
x
*
), we have
(33)
〈
x
*
-
J
λ
*
x
*
λ
,
J
-
1
(
J
λ
*
x
*
)
-
J
-
1
(
z
*
)
〉
≥
0
,
vvvvvvvvvvvvvvvvvvvv
∀
z
*
∈
(
B
J
-
1
)
-
1
0
,
and hence
(34)
〈
x
*
-
J
λ
*
x
*
,
J
-
1
(
J
λ
*
x
*
)
-
J
-
1
(
z
*
)
〉
≥
0
.
Letting λ
→
0, we get
(35)
〈
x
*

-
R
*
x
*
,
J
-
1
(
R
*
x
*
)

-
J
-
1
(
z
*
)
〉
≥
0
,

∀
z
*
∈
(
B
J
-
1
)
-
1
0
.
From Proposition 5, R
* is sunny and generalized nonexpansive. This implies that R
* is a sunny generalized nonexpansive retraction of E
* onto (
B
J
-
1
)
-
1
0.


## 3. An Iterative Scheme for Finding a Zero Point of a Monotone Mapping byJ
λ
*

Now we construct an iterative scheme which converges strongly to a point which is a fixed point of relatively weak nonexpansive mapping and a zero of monotone mapping.Theorem 8.
LetE
* be a uniformly convex Banach space and uniformly smooth Banach space. Let A
⊂
E
×
E
* be a maximal monotone operator. Let C be a nonempty closed convex subset of E. Let T
:
C
→
C be a relatively weak nonexpansive mapping with A
-
1
0
∩
F
(
T
)
≠
Ø. Assume that 0
≤
α
n
<
a
<
1 is a sequence of real numbers. Then, the sequence {
x
n
} generated by(36)
x
0
∈
C
,
λ
n
⟶
+
∞
,
y
n
=
J
-
1
(
α
n
J
x
n
+
(
1
-
α
n
)
J
λ
n
*
J
x
n
)
,
v
v
v
v
v
v
v
v
v
v
v
J
λ
n
*
=
(
I
*
+
λ
n
A
J
-
1
)
-
1
,
z
n
=
T
y
n
,
H
0
=
{
v
∈
C
:
V
2
(
v
,
z
0
)
≤
V
2
(
v
,
y
0
)
≤
V
2
(
v
,
x
0
)
}
,
H
n
=

{
v
∈
H
n
-
1

∩
W
n
-
1
:
V
2
(
v
,
z
n
)

≤
V
2
(
v
,
y
n
)
≤

V
2
(
v
,
x
n
)
}

,
W
0
=
C
,
W
n
=
{
v
∈
H
n
-
1
∩
W
n
-
1
:
〈
v
-
x
n
,
J
x
0
-
J
x
n
〉
≤
0
}
,
x
n
+
1
=
∏
H
n
∩
W
n
(
x
0
)
,
n
≥
1
converges strongly to Π
A
-
1
0
∩
F
(
T
)
(
x
0
), where Π
A
-
1
0
∩
F
(
T
) is the generalized projection from E onto A
-
1
0
∩
F
(
T
).Proof.
We first show thatH
n and W
n are closed and convex for each n
≥
0. From the definition of H
n and W
n, it is obvious that H
n is closed and W
n is closed and convex for each n
≥
0. We show that H
n is convex. Since
(37)
H
n
=
{
v
∈
H
n
-
1
∩
W
n
-
1
:
V
2
(
v
,
z
n
)
≤
V
2
(
v
,
y
n
)
}
∩
{
v
∈
H
n
-
1
∩
W
n
-
1
:
V
2
(
v
,
y
n
)
≤
V
2
(
v
,
x
n
)
}
,
V
2
(
v
,
y
n
)
≤
V
2
(
v
,
x
n
) is equivalent to
(38)
2
〈
v
,
J
x
n
-
J
y
n
〉
+
∥
y
n
∥
2
+
∥
x
n
∥
2
≤
0
,
and V
2
(
v
,
z
n
)
≤
V
2
(
v
,
y
n
) is equivalent to
(39)
2
〈
v
,
J
y
n
-
J
z
n
〉
+
∥
z
n
∥
2
+
∥
x
n
∥
2
≤
0
,
it follows that H
n is convex.
Next, we show thatF
=
:
A
-
1
0
∩
F
(
T
)
⊂
H
n
∩
W
n for each n
≥
0. Let p
∈
F; then relatively weak nonexpansiveness of T and generalized nonexpansiveness of J
λ
* give that
(40)
V
2
(
p
,
z
0
)
=
V
2
(
p
,
T
y
0
)
≤
V
2
(
p
,
y
0
)
=
V
2
(
p
,
J
-
1
(
α
0
J
x
0
+
(
1
-
α
0
)
J
λ
0
*
J
x
0
)
)
=
∥
p
∥
2
+
∥
α
0
J
x
0
+
(
1
-
α
0
)
J
λ
0
*
J
x
0
∥
2
-
2
〈
p
,
α
0
J
x
0
+
(
1
-
α
0
)
J
λ
0
*
J
x
0
〉
≤
∥
p
∥
2
-
2
α
0
〈
p
,
J
x
0
〉
-
2
(
1
-
α
0
)
〈
p
,
J
λ
0
*
J
x
0
〉
+
α
0
∥
J
x
0
∥
2
+
(
1
-
α
0
)
∥
J
λ
0
*
J
x
0
∥
2
=
α
0
(
∥
p
∥
2
-
2
α
0
〈
p
,
J
x
0
〉
+
∥
x
0
∥
2
)
+
(
1
-
α
0
)
(
∥
p
∥
2
-
2
〈
p
,
J
λ
0
*
J
x
0
〉
+
∥
J
λ
0
*
J
x
0
∥
2
)
=
α
0
V
2
(
p
,
x
0
)
+
(
1
-
α
0
)
V
2
(
p
,
J
-
1
J
λ
0
*
J
x
0
)
=
α
0
V
2
(
p
,
x
0
)
+
(
1
-
α
0
)
V
(
p
,
J
λ
0
*
J
x
0
)
≤
α
0
V
2
(
p
,
x
0
)
+
(
1
-
α
0
)
V
(
p
,
J
x
0
)
≤
α
0
V
2
(
p
,
x
0
)
+
(
1
-
α
0
)
V
2
(
p
,
x
0
)
=
V
2
(
p
,
x
0
)
.
Thus, we give that p
∈
H
0. On the other hand, it is clear that p
∈
C. Thus, F
⊂
H
0
∩
W
0 and, therefore, x
1
=
Π
H
0
∩
W
0 is well defined. Suppose that F
⊂
H
n
-
1
∩
W
n
-
1 and {
x
n
} is well defined. Then, the methods in (40) imply that V
2
(
p
,
z
n
)
≤
V
2
(
p
,
y
n
)
≤
V
2
(
p
,
x
n
) and p
∈
H
n. Moreover, it follows from Lemma 3 that
(41)
〈
p
-
x
n
,
J
x
n
-
J
x
0
〉
≥
0
,
which implies that p
∈
W
n. Hence F
⊂
H
n
∩
W
n and x
n
+
1
=
Π
H
n
∩
W
n is well defined. Then, by induction, F
⊂
H
n
∩
W
n and the sequence generated by (36) is well defined for each n
≥
0.
Now, we show that{
x
n
} is a bounded sequence and converges to a point of F. Let p
∈
F. Since x
n
+
1
=
Π
H
n
∩
W
n
(
x
0
) and H
n
∩
W
n
⊂
H
n
-
1
∩
W
n
-
1 for all n
≥
1, we have
(42)
V
2
(
x
n
,
x
0
)
≤
V
2
(
x
n
+
1
,
x
0
)
for all n
≥
0. Therefore, {
V
2
(
x
n
,
x
0
)
} is nondecreasing. In addition, it follows from definition of W
n and Lemma 3 that x
n
=
Π
W
n
(
x
0
). Therefore, by Lemma 2 we have
(43)
V
2
(
x
n
,
x
0
)
=
V
2
(
∏
W
n
(
x
0
)
,
x
0
)
≤
V
2
(
p
,
x
0
)
-
V
2
(
p
,
x
n
)
≤
V
2
(
p
,
x
0
)
,
for each p
∈
F
(
T
)
⊂
W
n for all n
≥
0. Therefore, {
V
2
(
x
n
,
x
0
)
} is bounded. This together with (40) implies that the limit of {
V
2
(
x
n
,
x
0
)
} exists. Put lim
⁡
n
→
∞
V
2
(
x
n
,
x
0
)
=
d. From Lemma 2, we have, for any positive integer m, that
(44)
V
2
(
x
n
+
m
,
x
n
)
=
V
2
(
x
n
+
m
,
∏
W
n
(
x
0
)
)
≤
V
2
(
x
n
+
m
,
x
0
)
-
V
2
(
∏
W
n
(
x
0
)
,
x
0
)
=
V
2
(
x
n
+
m
,
x
0
)
-
V
2
(
x
n
,
x
0
)
,
for all n
≥
0. The existence of lim
⁡
n
→
∞
V
2
(
x
n
,
x
0
) implies that lim
⁡
n
→
∞
V
2
(
x
m
+
n
,
x
n
)
=
0. Thus, Lemma 4 implies that
(45)
x
m
+
n
-
x
n
⟶
0
as
n
⟶
∞
,
and hence {
x
n
} is a Cauchy sequence. Therefore, there exists a point q
∈
E such that x
n
→
q as n
→
∞. Since x
n
+
1
∈
H
n, we have V
2
(
x
n
+
1
,
z
n
)
≤
V
2
(
x
n
+
1
,
y
n
)
≤
V
2
(
x
n
+
1
,
x
n
). Thus by Lemma 4 and (45) we get that
(46)
x
n
+
1
-
z
n
⟶
0
,
x
n
+
1
-
y
n
⟶
0
as
n
⟶
∞
,
and hence ∥
x
n
-
y
n
∥
≤
∥
x
n
+
1
-
x
n
∥
+
∥
x
n
+
1
-
y
n
∥
→
0 as n
→
∞. Furthermore, since J is uniformly continuous on bounded sets, we have
(47)
lim
⁡
n
→
∞
∥
J
x
n
+
1
-
J
z
n
∥
=
lim
⁡
n
→
∞
∥
J
x
n
-
J
y
n
∥
=
0
,
which implies that
(48)
∥
J
x
n
+
1
-
J
T
y
n
∥
⟶
as
n
⟶
∞
.
Since J
-
1 is also uniformly norm-continuous on bounded sets, we obtain
(49)
lim
⁡
n
→
∞
∥
x
n
+
1
-
T
y
n
∥
=
lim
⁡
n
→
∞
∥
J
-
1
J
x
n
+
1
-
J
-
1
J
T
y
n
∥
=
0
.
Therefore, from (46), (49), and ∥
y
n
-
T
y
n
∥
≤
∥
x
n
+
1
-
T
y
n
∥
+
∥
x
n
-
y
n
∥, we obtain that lim
⁡
n
→
∞
∥
y
n
-
T
y
n
∥
=
0. This together with the fact that {
x
n
} (and hence {
y
n
}) converges strongly to q
∈
E and the definition of relatively weak nonexpansive mapping implies that q
∈
F
(
T
). Furthermore, from (36) and (47), we have that (
1
-
α
n
)
∥
J
λ
n
*
J
x
n
-
J
x
n
∥
=
∥
J
x
n
-
J
y
n
∥
→
0 as n
→
∞. Thus, from lim
⁡
n
→
∞
J
λ
n
*
J
x
n
=
lim
⁡
n
→
∞
J
x
n
=
J
q
∈
J
A
-
1
0
=
(
A
J
-
1
)
-
1
0, we obtain that q
∈
A
-
1
0.
Finally, we show thatq
=
Π
A
-
1
0
∩
F
(
T
)
(
x
0
) as n
→
∞. From Lemma 2, we have
(50)
V
2
(
q
,
∏
A
-
1
0
∩
F
(
T
)
(
x
0
)
)

+
V
2
(
∏
A
-
1
0
∩
F
(
T
)

(
x
0
)
,
x
0
)

≤

V
2
(
q
,
x
0
)
.
On the other hand, since x
n
+
1
=
Π
H
n
∩
W
n
(
x
0
) and F
⊂
H
n
∩
W
n for all n
≥
0, we have by Lemma 2 that
(51)
V
2
(
∏
A
-
1
0
∩
F
(
T
)
(
x
0
)
,
x
n
+
1
)
+
V
2
(
x
n
+
1
,
x
0
)
≤
V
2
(
∏
A
-
1
0
∩
F
(
T
)
(
x
0
)
,
x
0
)
.
Moreover, by the definition of V
2
(
x
,
y
), we get that
(52)
lim
⁡
n
→
∞
V
2
(
x
n
+
1
,
x
0
)
=
V
2
(
q
,
x
0
)
.
By combining (50) and (52), we obtain that V
2
(
q
,
x
0
)
=
V
2
(
Π
A
-
1
0
∩
F
(
T
)
(
x
0
)
,
x
0
). Therefore, it follows from the uniqueness of Π
A
-
1
0
∩
F
(
T
)
(
x
0
) that q
=
Π
A
-
1
0
∩
F
(
T
)
(
x
0
). This completes the proof.Remark 9.
If in Theorem8 we have that T
=
I, the identity map on E, then we get the following.Corollary 10.
LetE
* be a uniformly convex Banach space and uniformly smooth Banach space. Let A
⊂
E
×
E
* be a maximal monotone operator. Let C be a nonempty closed convex subset of E with A
-
1
0
≠
Ø. Assume that 0
≤
α
n
<
a
<
1 is a sequence of real numbers. Then, the sequence {
x
n
} generated by
(53)
x
0
∈
C
,
λ
n
⟶
+
∞
,
y
n

=
J
-
1

(
α
n
J
x
n

+
(
1

-
α
n
)
J
λ
n
*
J
x
n
)
,
J
λ
n
*

=
(
I
*

+
λ
n
A
J
-
1
)
-
1
,
H
0
=
{
v
∈
C
:
V
2
(
v
,
z
0
)
≤
V
2
(
v
,
y
0
)
≤
V
2
(
v
,
x
0
)
}
,
H
n

=
{
v
∈
H
n
-
1
∩
W
n
-
1
:
V
2
(
v
,
z
n
)

≤
V
2
(
v
,
y
n
)

≤
V
2
(
v
,
x
n
)
}
,
W
0
=
C
,
W
n
=
{
v
∈
H
n
-
1
∩
W
n
-
1
:
〈
v
-
x
n
,
J
x
0
-
J
x
n
〉
≤
0
}
,
x
n
+
1
=
∏
H
n
∩
W
n
(
x
0
)
,
n
≥
1
converges strongly to Π
A
-
1
0, where Π
A
-
1
0 is the generalized projection from E onto A
-
1
0.Remark 11.
We have compared the results of [2, 6, 7] with the result in this paper.
(1) In [6], Ibaraki and Takahashi introduced the generalized resolvent J
λ
:
E
→
E, which was denoted by
(54)
J
λ
=
(
I
+
λ
B
J
)
-
1
.
In this paper, we introduce the generalized resolventJ
λ
*
:
E
*
→
E
*, which is denoted by
(55)
J
λ
*
=
(
I
*
+
λ
B
J
-
1
)
-
1
.
(2) In [6], Ibaraki and Takahashi defined a sunny generalized nonexpansive retraction R
C of E onto B
J
-
1
0
:
(56)
R
x
:
=
lim
⁡
λ
→
∞
J
λ
x
,
∀
x
∈
E
.
In this paper, we define a sunny generalized nonexpansive retractionR
* of E
* onto (
B
J
-
1
)
-
1
0
:
(57)
R
*
x
*
:
=
lim
⁡
λ
→
∞
J
λ
*
x
*
,
∀
x
∈
E
*
.
(3) In [7], Zegeye and Shahzad proved the strong convergence theorem of the sequence {
x
n
} generated by (12). Using J
λ
*, in this paper, we construct an iterative scheme in E
*, which converges strongly to a point which is a fixed point of a relatively weak nonexpansive mapping and a zero of a monotone mapping.The results we have obtained in this paper are studied inE
*, which is different from others.



---

*Source: 603186-2014-04-07.xml*