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.
Proof.
We first show that
H
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 that
F
=
:
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 normcontinuous 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 that
q
=
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 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 resolvent
J
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 retraction
R
*
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.