# Hybrid Implicit Iteration Process for a Finite Family of Non-Self-Nonexpansive Mappings in Uniformly Convex Banach Spaces

Authors: Qiaohong Jiang; Jinghai Wang; Jianhua Huang
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

Weak and strong convergence theorems are established for hybrid implicit iteration for a finite family of non-self-nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper extend and improve some recent results.


---

## Body

## 1. Introduction

The convergence problem of an implicit (or nonimplicit) iterative process to a common fixed point for a finite family of nonexpansive mappings (or asymptotically nonexpansive mappings) in Hilbert spaces or uniformly convex Banach spaces has been considered by many authors (see [1–9]).In 2001, Xu and Ori [1] introduced the following implicit iteration scheme for common fixed points of a finite family of nonexpansive mappings {
T
i
}
i
=
1
N in Hilbert spaces:
(1)
x
n
=
α
n
x
n
-
1
+
(
1
-
α
n
)
T
n
x
n
,
n
≥
1
,
where T
n
=
T
n
(
mod
N
), and they proved the weak convergence theorem.In 2005, Zeng and Yao [2] introduced the following implicit iteration process with a perturbed mapping F in Hilbert space H.For an arbitrary initial pointx
0
∈
H, the sequence {
x
n
}
n
=
1
∞ is generated as follows:
(2)
x
n
=
α
n
x
n
-
1
+
(
1
-
α
n
)
[
T
n
x
n
-
λ
n
μ
F
(
T
n
x
n
)
]
,
n
≥
1
,
where T
n
=
T
n
(
mod
N
).Using this iteration process, they proved the following weak and strong convergence theorems for a family of nonexpansive mappings in Hilbert spaces.Theorem 1 (see [2]).
LetH be a real Hilbert space and let F
:
H
→
H be a mapping such that, for some constants k
,
η
>
0, F is k-Lipschitzian and η-strongly monotone. Let {
T
n
}
n
=
1
N be N nonexpansive self-mappings of H such that ⋂
n
=
1
N
F
i
x
(
T
n
)
≠
∅. Let μ
∈
(
0,2
η
/
k
2
) and x
0
∈
H. Let {
α
n
}
n
=
1
∞
⊂
(
0,1
) and {
λ
n
}
n
=
1
∞
⊂
[
0,1
) satisfying conditions ∑
n
=
1
∞
λ
n
<
∞ and α
≤
α
n
≤
β, n
≥
1, for some α
,
β
∈
(
0,1
). Then the sequence {
x
n
}
n
=
1
∞ defined by (2) converges weakly to a common fixed point of the mappings {
T
n
}
n
=
1
N.Theorem 2 (see [2]).
LetH be a real Hilbert space and let F
:
H
→
H be a mapping such that, for some constants k
,
η
>
0, F is k-Lipschitzian and η-strongly monotone. Let {
T
n
}
n
=
1
N be N nonexpansive self-mappings of H such that ⋂
i
=
1
N
F
i
x
(
T
i
)
≠
∅. Let μ
∈
(
0,2
η
/
k
2
) and x
0
∈
H. Let {
α
n
}
n
=
1
∞
⊂
(
0,1
) and {
λ
n
}
n
=
1
∞
⊂
[
0,1
) satisfying conditions ∑
n
=
1
∞
‍
λ
n
<
∞ and α
≤
α
n
≤
β, n
≥
1, for some α
,
β
∈
(
0,1
). Then the sequence {
x
n
}
n
=
1
∞ defined by (2) converges strongly to a common fixed point of the mappings {
T
n
}
n
=
1
N if and only if lim
⁡
inf
⁡
n
→
∞
d
(
x
n
,
⋂
n
=
1
N
F
i
x
(
T
n
)
)
=
∅.The purpose of this paper is to extend Theorems1 and 2 from Hilbert spaces to uniformly convex Banach spaces and from self-mappings to non-self-mappings. Our results are more general and applicable than the results of Zeng and Yao [2] because the strong monotonicity condition imposed on F by them is not required in this paper.


## 2. Preliminaries

Throughout this paper, we assume thatE is a real Banach space. T
:
D
(
T
)
⊆
E
→
E is a mapping, where D
(
T
) is the domain of T. F
(
T
) denotes the set of fixed points of a mapping T.Recall thatE is said to satisfy Opial’s condition [10], if, for each sequence {
x
n
} in E, the condition that the sequence x
n
→
x weakly implies that
(3)
limsup
⁡
n
→
∞
∥
x
n
-
x
∥
<
limsup
⁡
n
→
∞
∥
x
n
-
y
∥
,
for all y
∈
E with y
≠
x.Definition 3.
LetK be a closed subset of E and let T
:
K
→
E, f
:
E
→
E be two mappings.(1)
Tis said to be demiclosed at the origin, if, for each sequence {
x
n
} in K, the condition x
n
→
x
0 weakly and T
x
n
→
0 strongly implies T
x
0
=
0.
(2)
T is said to be semicompact, if, for any bounded sequence {
x
n
} in K, such that ∥
x
n
-
T
x
n
∥
→
0
(
n
→
∞
), there exists a subsequence {
x
n
i
}
⊂
{
x
n
} converging to some x
* in K.
(3)
T is said to be nonexpansive, if ∥
T
x
-
T
y
∥
≤
∥
x
-
y
∥ for any x
,
y
∈
E.
(4)
f is said to be L-Lipschitzian if there exists constant L
>
0 such that ∥
f
x
-
f
y
∥
≤
L
∥
x
-
y
∥ for any x
,
y
∈
E.Definition 4.
A nonempty subsetK of E is said to be a retract of E, if there exists a continuous mapping r
:
E
→
K such that r
x
=
x, for any x
∈
K. And r is called the retraction of E onto K.Remark 5 (see [3]).
It is known that every nonempty closed convex subsetK of a uniformly convex Banach space E is a retract of E and the retraction r is a nonexpansive mapping.
Suppose thatK is a nonempty closed convex subset of E, which is also a retract of E. Let x
0
∈
K be any given point. Let {
T
1
,
T
2
,
…
,
T
N
}
:
K
→
E be N nonexpansive mappings with T
n
=
T
n
(
mod
N
). Let f
:
E
→
E be an L-Lipschitzian mapping. Assume that {
α
n
} is a sequence in (0,1) and {
λ
n
}
⊂
[
0,1
), given μ
>
0. Then the sequence {
x
n
} defined by
(4)
x
n
=
α
n
x
n
-
1
+
(
1
-
α
n
)
r
T
n
λ
n
x
n
≔
α
n
x
n
-
1
+
(
1
-
α
n
)
r
[
T
n
x
n
-
λ
n
μ
f
(
T
n
x
n
)
]
,
n
≥
1
,
is called hybrid implicit iteration for a finite family of nonexpansive mappings {
T
1
,
T
2
,
…
,
T
N
} in Banach spaces, where T
n
n
=
T
n
(
mod
N
)
n and μ is a fixed constant.The purpose of this paper is to study weak and strong convergence of hybrid implicit iteration{
x
n
} defined by (4) to a common fixed point of {
T
1
,
T
2
,
…
,
T
N
}
:
K
→
E in Banach spaces. The results we obtained in this paper extend and improve the corresponding results of Xu and Ori [1], Zeng and Yao [2], and others.In order to prove our main results of this paper, we need the following lemmas.Lemma 6 (see [4]).
Let{
a
n
}, {
b
n
}, and {
δ
n
} be three nonnegative sequences satisfying
(5)
a
n
+
1
≤
(
1
+
δ
n
)
a
n
+
b
n
,
∀
n
=
1,2
,
…
.
If ∑
n
=
1
∞
δ
n
<
∞ and ∑
n
=
1
∞
b
n
<
∞, then lim
⁡
n
→
∞
⁡
a
n exists.Lemma 7 (see [5]).
LetE be a uniformly convex Banach space. Let b, c be two constants with 0
<
b
<
c
<
1. Suppose that {
t
n
} is a sequence in [b, c] and {
x
n
}, {
y
n
} are two sequences in E. Then the conditions
(6)
lim
⁡
n
→
∞
∥
t
n
x
n
+
(
1
-
t
n
)
y
n
∥
=
d
,
limsup
⁡
n
→
∞
∥
x
n
∥
≤
d
,
limsup
⁡
n
→
∞
∥
y
n
∥
≤
d
imply that lim
⁡
n
→
∞
∥
x
n
-
y
n
∥
=
0, where d
≥
0 is some constant.Lemma 8 (see [6]).
LetK be a nonempty closed convex subset of real Banach space E and T
:
K
→
E a nonexpansive mapping. If T has a fixed point, then I
-
T is demiclosed at zero, where I is the identity mapping of E.


## 3. Main Results

Theorem 9.
Suppose thatE is a real uniformly convex Banach space satisfying Opial’s condition and K is a nonempty closed convex subset of E with a nonexpansive retraction r
:
E
→
K. Let {
T
1
,
T
2
,
…
,
T
N
}
:
K
→
E be N nonexpansive mappings with F
=
⋂
n
=
1
N
F
(
T
n
)
≠
∅ and let f
:
E
→
E be an L-Lipschitzian mapping. Assume that {
α
n
} is a sequence in (
0,1
) and {
λ
n
}
⊂
[
0,1
) satisfying the following conditions: (i)
∑
n
=
1
∞
λ
n
<
∞;
(ii)
there exist constantsτ
1
,
τ
2
∈
(
0,1
) such that
(7)
τ
1
≤
(
1
-
α
n
)
≤
τ
2
,
∀
n
≥
1
.Then, the implicit iterative process {
x
n
} defined by (4) converges weakly to a common fixed point of {
T
1
,
T
2
,
…
,
T
N
} in E.Proof.
SinceF
=
⋂
n
=
1
N
‍
F
(
T
i
)
≠
∅, for each q
∈
F, we have
(8)
∥
x
n
-
q
∥
=
∥
α
n
(
x
n
-
1
-
q
)
+
(
1
-
α
n
)
(
r
T
n
λ
n
x
n
-
q
)
∥
=
∥
α
n
(
x
n
-
1
-
q
)
+
(
1
-
α
n
)
(
r
T
n
λ
n
x
n
-
r
q
)
∥
≤
α
n
∥
x
n
-
1
-
q
∥
+
(
1
-
α
n
)
∥
T
n
λ
n
x
n
-
q
∥
≤
α
n
∥
x
n
-
1
-
q
∥
+
(
1
-
α
n
)
∥
T
n
x
n
-
q
∥
+
(
1
-
α
n
)
λ
n
μ
∥
f
(
T
n
x
n
)
∥
≤
α
n
∥
x
n
-
1
-
q
∥
+
(
1
-
α
n
)
∥
x
n
-
q
∥
+
(
1
-
α
n
)
λ
n
μ
∥
f
(
T
n
x
n
)
-
f
(
q
)
∥
+
(
1
-
α
n
)
λ
n
μ
∥
f
(
q
)
∥
≤
α
n
∥
x
n
-
1
-
q
∥
+
(
1
-
α
n
)
∥
x
n
-
q
∥
+
λ
n
μ
L
∥
x
n
-
q
∥
+
λ
n
μ
∥
f
(
q
)
∥
.
Simplifying we have
(9)
∥
x
n
-
q
∥
≤
∥
x
n
-
1
-
q
∥
+
λ
n
μ
L
α
n
∥
x
n
-
q
∥
+
λ
n
μ
α
n
∥
f
(
q
)
∥
.
By condition (ii), 1
-
τ
2
≤
α
n; hence from (9) we have
(10)
∥
x
n
-
q
∥
≤
∥
x
n
-
1
-
q
∥
+
λ
n
μ
L
1
-
τ
2
∥
x
n
-
q
∥
+
λ
n
μ
1
-
τ
2
∥
f
(
q
)
∥
.
By condition (i), we know that λ
n
→
0 and λ
n
μ
L
→
0 as n
→
∞; therefore there exists a positive integer n
0 such that λ
n
μ
L
≤
(
1
-
τ
2
)
/
2, for all n
≥
n
0; then we have
(11)
∥
x
n
-
q
∥
≤
1
-
τ
2
1
-
τ
2
-
λ
n
μ
L
∥
x
n
-
1
-
q
∥
+
λ
n
μ
1
-
τ
2
-
λ
n
μ
L
∥
f
(
q
)
∥
=
(
1
+
λ
n
μ
L
1
-
τ
2
-
λ
n
μ
L
)
∥
x
n
-
1
-
q
∥
+
λ
n
μ
1
-
τ
2
-
λ
n
μ
L
∥
f
(
q
)
∥
.
It follows from (11) that
(12)
∥
x
n
-
q
∥
≤
(
1
+
2
λ
n
μ
L
1
-
τ
2
)
∥
x
n
-
1
-
q
∥
+
2
λ
n
μ
1
-
τ
2
∥
f
(
q
)
∥
,
kkkkkkkkkkkkkkkkkkkkkkkkkkkkk
∀
n
≥
n
0
.
Taking a
n
=
∥
x
n
-
q
∥, δ
n
=
2
λ
n
μ
L
/
(
1
-
τ
2
), and b
n
=
(
2
λ
n
μ
/
(
1
-
τ
2
)
)
∥
f
(
q
)
∥ and by using ∑
n
=
1
∞
λ
n
<
∞, it is easy to see that
(13)
∑
n
=
1
∞
δ
n
<
∞
,
∑
n
=
1
∞
b
n
<
∞
.
It follows from Lemma 6 that lim
⁡
n
→
∞
∥
x
n
-
q
∥ exists for each q
∈
F. Hence, there exists M
>
0, such that
(14)
∥
x
n
-
q
∥
≤
M
,
n
≥
0
.
We can assume that
(15)
lim
⁡
n
→
∞
∥
x
n
-
q
∥
=
d
,
where d
≥
0 is some number. Since {
∥
x
n
-
q
∥
} is a convergent sequence, {
x
n
} is a bounded sequence in K. Since
(16)
∥
x
n
-
q
∥
=
∥
α
n
(
x
n
-
1
-
q
)
+
(
1
-
α
n
)
(
r
T
n
λ
n
x
n
-
q
)
∥
,
by condition (i) and (8) and (15), that
(17)
limsup
⁡
n
→
∞
∥
r
T
n
λ
n
x
n
-
q
∥
=
limsup
⁡
n
→
∞
∥
r
T
n
λ
n
x
n
-
r
q
∥
≤
limsup
⁡
n
→
∞
∥
T
n
λ
n
x
n
-
q
∥
≤
limsup
⁡
n
→
∞
∥
T
n
x
n
-
T
n
q
∥
+
limsup
⁡
n
→
∞
λ
n
μ
∥
f
(
T
n
x
n
)
-
f
(
T
n
q
)
∥
+
limsup
⁡
n
→
∞
λ
n
μ
∥
f
(
T
n
q
)
∥
≤
limsup
⁡
n
→
∞
∥
x
n
-
q
∥
+
limsup
⁡
n
→
∞
λ
n
μ
L
∥
x
n
-
q
∥
+
limsup
⁡
n
→
∞
λ
n
μ
∥
f
(
q
)
∥
≤
d
.
Since E is a uniformly convex Banach space, from (15)–(17) and Lemma 7 we know that
(18)
lim
⁡
n
→
∞
∥
x
n
-
1
-
r
T
n
λ
n
x
n
∥
=
0
.
By (18), we have that
(19)
∥
x
n
-
x
n
-
1
∥
=
∥
(
α
n
-
1
)
x
n
-
1
+
(
1
-
α
n
)
r
T
n
λ
n
x
n
∥
≤
(
1
-
α
n
)
∥
x
n
-
1
-
r
T
n
λ
n
x
n
∥
⟶
0
,
(
n
⟶
∞
)
.
It follows from (18) and (19) that
(20)
lim
⁡
n
→
∞
∥
x
n
-
r
T
n
λ
n
x
n
∥
≤
lim
⁡
n
→
∞
(
∥
T
n
λ
n
x
n
-
x
n
-
1
∥
kkkkk
+
∥
x
n
-
1
-
r
T
n
λ
n
x
n
∥
)
=
0
,
(21)
lim
⁡
n
→
∞
∥
x
n
-
x
n
+
j
∥
=
0
,
∀
j
=
1,2
,
…
,
N
.
By (14), we know
(22)
∥
f
(
T
n
x
n
)
∥
≤
∥
f
(
T
n
x
n
)
-
f
(
q
)
∥
+
∥
f
(
q
)
∥
≤
L
∥
x
n
-
q
∥
+
∥
f
(
q
)
∥
≤
L
M
+
∥
f
(
q
)
∥
.
From (20), (22), and condition (i) we have
(23)
lim
⁡
n
→
∞
∥
x
n
-
T
n
x
n
∥
≤
lim
⁡
n
→
∞
(
∥
x
n
-
T
n
λ
n
x
n
∥
+
∥
T
n
λ
n
x
n
-
T
n
x
n
∥
)
≤
lim
⁡
n
→
∞
(
∥
x
n
-
T
n
λ
n
x
n
∥
+
λ
n
μ
(
L
M
+
∥
f
(
q
)
∥
)
)
=
0
.
Consequently, for any j
=
1,2
,
…
,
N, from (21) and (23) we have
(24)
∥
x
n
-
T
n
+
j
x
n
∥
≤
∥
x
n
-
x
n
+
j
∥
+
∥
x
n
+
j
-
T
n
+
j
x
n
+
j
∥
+
∥
T
n
+
j
x
n
+
j
-
T
n
+
j
x
n
∥
≤
2
∥
x
n
-
x
n
+
j
∥
+
∥
x
n
+
j
-
T
n
+
j
x
n
+
j
∥
⟶
0
,
kkkkkkkkkkkkkkkkkkkkkkkkk
(
n
⟶
∞
)
.
This implies that the sequence
(25)
⋃
j
=
1
N
{
∥
x
n
-
T
n
+
j
x
n
∥
}
n
=
1
∞
⟶
0
(
n
⟶
∞
)
.
Since, for each l
=
1,2
,
…
,
N, {
∥
x
n
-
T
l
x
n
∥
}
n
=
1
∞ is a subsequence of ⋃
j
=
1
N
{
∥
x
n
-
T
n
+
j
x
n
∥
}
n
=
1
∞, therefore we have
(26)
lim
⁡
n
→
∞
∥
x
n
-
T
l
x
n
∥
=
0
,
∀
l
=
1,2
,
…
N
.
Since E is uniformly convex, every bounded subset of E is weakly compact. Since {
x
n
} is a bounded sequence in E, there exists a subsequence {
x
n
j
}
⊂
{
x
n
} such that {
x
n
j
} converges weakly to u
∈
E. From (26) we have
(27)
lim
⁡
n
→
∞
∥
x
n
j
-
T
l
x
n
j
∥
=
0
,
∀
l
=
1,2
,
…
N
.
By Lemma 8, we know that u
∈
F
(
T
l
). By the arbitrariness of l
∈
{
1,2
,
…
N
}, we have that u
∈
F
=
⋂
l
=
1
N
‍
F
(
T
i
).
Suppose that there exists some subsequence{
x
n
k
}
⊂
{
x
n
} such that x
n
k
→
v
∈
E weakly and v
≠
u. From Lemma 8, v
∈
F. By (12) we know that lim
⁡
n
→
∞
∥
x
n
-
u
∥ and lim
⁡
n
→
∞
∥
x
n
-
v
∥ exist. Since E satisfies Opial’s condition, we have
(28)
lim
⁡
n
→
∞
∥
x
n
-
u
∥
=
lim
⁡
j
→
∞
∥
x
n
j
-
u
∥
<
lim
⁡
j
→
∞
∥
x
n
j
-
v
∥
=
lim
⁡
n
→
∞
∥
x
n
-
v
∥
=
lim
⁡
k
→
∞
∥
x
n
k
-
v
∥
<
lim
⁡
k
→
∞
∥
x
n
k
-
u
∥
=
lim
⁡
n
→
∞
∥
x
n
-
u
∥
,
which is a contradiction. Hence u
=
v. This implies that {
x
n
} converges weakly to a common fixed point of {
T
1
,
T
2
,
…
,
T
N
} in E.Theorem 10.
Suppose thatE is a real uniformly convex Banach space and K is a nonempty closed convex nonexpansive retract of E with r
:
E
→
K as a nonexpansive retraction. Let {
T
1
,
T
2
,
…
,
T
N
}
:
K
→
E be N nonexpansive mappings with F
=
⋂
n
=
1
N
F
(
T
n
)
≠
∅ and let f
:
E
→
E be an L-Lipschitzian mapping. Assume that {
α
n
} is a sequence in (
0,1
) and {
λ
n
}
⊂
[
0,1
) satisfying the following conditions: (i)
∑
n
=
1
∞
λ
n
<
∞;
(ii)
there exist constantsτ
1
,
τ
2
∈
(
0,1
) such that
(29)
τ
1
≤
(
1
-
α
n
)
≤
τ
2
,
∀
n
≥
1
.Then, the implicit iterative process {
x
n
} defined by (4) converges strongly to a common fixed point of {
T
1
,
T
2
,
…
,
T
N
} if and only if liminf
⁡
n
→
∞
d
(
x
n
,
F
(
T
l
)
)
=
0 (for all l
=
1,2
,
…
,
N).Proof.
From (12) and (14) in the proof of Theorem 9, we have
(30)
∥
x
n
-
q
∥
≤
(
1
+
δ
n
)
∥
x
n
-
1
-
q
∥
+
b
n
≤
∥
x
n
-
1
-
q
∥
+
M
δ
n
+
b
n
=
∥
x
n
-
1
-
q
∥
+
β
n
,
where δ
n
=
2
λ
n
μ
L
/
(
1
-
τ
2
), b
n
=
(
2
λ
n
μ
/
(
1
-
τ
2
)
)
∥
f
(
q
)
∥, and β
n
=
M
δ
n
+
b
n. Hence, d
(
x
n
,
F
)
≤
d
(
x
n
-
1
,
F
)
+
β
n. Since ∑
n
=
1
∞
β
n
<
∞, it follows from Lemma 6 that lim
⁡
n
→
∞
d
(
x
n
,
F
) exists.
If{
x
n
}
n
=
1
∞ converges strongly to a common fixed point p of {
T
1
,
T
2
,
…
,
T
N
}, then lim
⁡
n
→
∞
∥
x
n
-
p
∥
=
0. Since
(31)
0
≤
d
(
x
n
,
F
)
≤
∥
x
n
-
p
∥
,
we know that liminf
⁡
n
→
∞
d
(
x
n
,
F
)
=
0.
Conversely, supposeliminf
⁡
n
→
∞
d
(
x
n
,
F
)
=
0; then lim
⁡
n
→
∞
d
(
x
n
,
F
)
=
0. Moreover, we have ∑
n
=
1
∞
β
n
<
∞; thus for arbitrary ϵ
>
0, there exists a positive integer N such that d
(
x
n
,
F
)
<
ϵ
/
4 and ∑
j
=
n
∞
β
j
<
ϵ
/
4 for all n
≥
N. It follows from (30) that, for all n
,
m
≥
N and for all p
∈
F, we have
(32)
∥
x
n
-
x
m
∥
≤
∥
x
n
-
p
∥
+
∥
x
m
-
p
∥
≤
∥
x
N
-
p
∥
+
∑
j
=
N
+
1
n
‍
β
j
+
∥
x
N
-
p
∥
+
∑
j
=
N
+
1
m
β
j
≤
2
∥
x
N
-
p
∥
+
2
∑
j
=
N
∞
β
j
.
Taking infimum over all p
∈
F, we obtain
(33)
∥
x
n
-
x
m
∥
≤
2
d
(
x
N
,
F
)
+
2
∑
j
=
N
∞
β
j
<
ϵ
,
∀
n
,
m
≥
N
.
Thus, {
x
n
}
n
=
1
∞ is a Cauchy sequence. Letting li
m
n
→
∞
x
n
=
u, then, from Lemma 8, we have u
∈
F. This completes the proof of the theorem.Theorem 11.
Suppose thatE is a real uniformly convex Banach space and K is a nonempty closed convex nonexpansive retract of E with r
:
E
→
K as a nonexpansive retraction. Let {
T
1
,
T
2
,
…
,
T
N
}
:
K
→
E be N nonexpansive mappings with F
=
⋂
n
=
1
N
F
(
T
n
)
≠
∅ and at least there exists a T
l, 1
≤
l
≤
N, which is semicompact. Let f
:
E
→
E be L-Lipschitzian mapping. Assume that {
α
n
} is a sequence in (
0,1
) and {
λ
n
}
⊂
[
0,1
) satisfying the following conditions: (i)
∑
n
=
1
∞
λ
n
<
∞;
(ii)
there exist constantsτ
1
,
τ
2
∈
(
0,1
) such that
(34)
τ
1
≤
(
1
-
α
n
)
≤
τ
2
,
∀
n
≥
1
.Then, the implicit iterative process {
x
n
} defined by (4) converges strongly to a common fixed point of {
T
1
,
T
2
,
…
,
T
N
} in E.Proof.
From the proof of Theorem9, {
x
n
} is bounded, and lim
⁡
n
→
∞
∥
x
n
-
T
l
x
n
∥
=
0, for all l
=
1,2
,
…
N. We especially have
(35)
lim
⁡
n
→
∞
∥
x
n
-
T
1
x
n
∥
=
0
.
By the assumption of Theorem 11, we may assume that T
1 is semicompact, without loss of generality. Then, it follows from (35) that there exists a subsequence {
x
n
k
} of {
x
n
} such that {
x
n
k
} converges strongly to p
∈
K. Thus from (26) we have
(36)
∥
p
-
T
l
p
∥
=
lim
⁡
k
→
∞
∥
x
n
k
-
T
l
x
n
k
∥
=
lim
⁡
n
→
∞
∥
x
n
-
T
l
x
n
∥
=
0
,
kkkkkkkkkkkkkkkkkkkkkkkk
∀
l
=
1,2
,
…
N
.
This implies that p
∈
F. In addition, since lim
⁡
n
→
∞
∥
x
n
-
p
∥ exists, therefore lim
⁡
n
→
∞
∥
x
n
-
p
∥
=
0; that is, {
x
n
} converges strongly to a fixed point of {
T
1
,
T
2
,
…
,
T
N
} in E. The proof is completed.



---

*Source: 238053-2014-07-10.xml*