# Convergence of Viscosity Iteration Process for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mappings

Authors: Zhiming Cheng
Journal: Journal of Applied Mathematics
 (2014)

---

## Abstract

We introduce a general iteration method for a finite family of generalized asymptotically quasi-nonexpansive mappings. The results presented in the paper extend and improve some recent results in the works by Shahzad and Udomene (2006); L. Qihou (2001); Khan et al. (2008).


---

## Body

## 1. Introduction and Preliminaries

LetC be a nonempty subset of a real Banach space E and T a self-mapping of C. The set of fixed points of T is denoted by F
(
T
) and we assume that F
(
T
)
≠
∅. The mapping T is said to be(i)
contractive mapping if there exists a constantα in (
0,1
) such that ∥
f
(
x
)
-
f
(
y
)
∥
≤
α
∥
x
-
y
∥, for all x
,
y
∈
C;
(ii)
asymptotically nonexpansive mapping if there exists a sequence{
u
n
} in [
0
,
∞
) with lim
n
→
∞
u
n
=
0 such that ∥
T
n
x
-
T
n
y
∥
≤
(
1
+
u
n
)
∥
x
-
y
∥, for all x
,
y
∈
C and n
=
1,2
,
3
,
…;
(iii)
asymptotically quasi-nonexpansive if there exists a sequence{
u
n
} in [
0
,
∞
) with lim
n
→
∞
u
n
=
0 such that ∥
T
n
x
-
p
∥
≤
(
1
+
u
n
)
∥
x
-
p
∥, for all x
∈
C, p
∈
F
(
T
) and n
=
1,2
,
3
,
…;
(iv)
generalized asymptotically quasi-nonexpansive [1] if there exist two sequences {
u
n
}, {
h
n
} in [
0
,
∞
) with lim
n
→
∞
u
n
=
0 and lim
n
→
∞
h
n
=
0 such that
(1)
∥
T
n
x
-
p
∥
≤
(
1
+
u
n
)
∥
x
-
p
∥
+
h
n
,
∀
x
∈
C
,
p
∈
F
(
T
)
,
where n
=
1,2
,
3
,
…;
(v)
uniformlyL-Lipschitzian if there exists a constant L
>
0 such that ∥
T
n
x
-
T
n
y
∥
≤
L
∥
x
-
y
∥, for all x
,
y
∈
C and n
=
1,2
,
3
,
…;
(vi)
(
L
-
γ
) uniform L-Lipschitz if there are constants L
>
0 and γ
>
0 such that ∥
T
n
x
-
T
n
y
∥
≤
L
∥
x
-
y
∥
γ, for all x
,
y
∈
C and n
=
1,2
,
3
,
…;
(vii)
semicompact if for a sequence{
x
n
} in C with lim
n
→
∞
∥
x
n
-
T
x
n
∥
=
0, there exists a subsequence {
x
n
i
} of {
x
n
} such that x
n
i
→
p
∈
C.In (1), if h
n
=
0 for all n
≥
1, then T becomes an asymptotically quasi-nonexpansive mapping; if u
n
=
0 and h
n
=
0 for all n
≥
1, then T becomes a quasi-nonexpansive mapping. It is known that an asymptotically nonexpansive mapping is an asymptotically quasi-nonexpansive and a uniformly L-Lipschitzian mapping is (
L
-
1
) uniform L-Lipschitz.The mappingT
:
C
→
E is said to be demiclosed at 0 if for each sequence {
x
n
}
⊂
C converging weakly to x
0 and{
T
x
n
} converging strongly to 0, we have T
x
0
=
0.A Banach spaceE is said to satisfy Opial’s property if for each x
∈
E and each sequence {
x
n
} weakly convergent to x, the following condition holds for all x
≠
y:
(2)
lim
inf
n
→
∞
∥
x
n
-
x
∥
<
lim
inf
n
→
∞
∥
x
n
-
y
∥
.LetC be a nonempty closed convex subset of a real Banach space E and {
T
i
:
i
=
1,2
,
…
k
} a finite family of asymptotically nonexpansive mappings of C into itself. Suppose that α
i
n
∈
[
0,1
], n
=
1,2
,
3
,
…, and i
=
1,2
,
…
k. Then we consider the following mapping of C into itself:
(3)
U
1
n
=
(
1
-
α
1
n
)
I
+
α
1
n
T
1
n
U
0
n
,
U
2
n
=
(
1
-
α
2
n
)
I
+
α
2
n
T
2
n
U
1
n
,
U
2
n
⋮
U
(
k
-
1
)
n
=
(
1
-
α
(
k
-
1
)
n
)
I
+
α
(
k
-
1
)
n
T
k
-
1
n
U
(
k
-
2
)
n
,
W
n
=
U
k
n
=
(
1
-
α
k
n
)
I
+
α
k
n
T
k
n
U
(
k
-
1
)
n
,
where U
0
n
=
I (identity mapping). Such a mapping W
n is called the modified W-mapping generated by T
1
,
T
2
,
…
,
T
k and α
1
n
,
α
2
n
,
…
,
α
k
n (see [2, 3]).In the sequel, we assume thatF
=
⋂
i
=
1
k
‍
F
(
T
i
).In 2008, Khan et al. [4] introduced the following iteration process for a family of asymptotically quasi-nonexpansive mappings, for an arbitrary x
1
∈
C:
(4)
y
1
n
=
(
1
-
α
1
n
)
x
n
+
α
1
n
T
1
n
y
0
n
,
y
2
n
=
(
1
-
α
2
n
)
x
n
+
α
2
n
T
2
n
y
1
n
,
y
2
n
⋮
y
(
k
-
1
)
n
=
(
1
-
α
(
k
-
1
)
n
)
x
n
+
α
(
k
-
1
)
n
T
k
-
1
n
y
(
k
-
2
)
n
,
x
n
+
1
=
(
1
-
α
k
n
)
x
n
+
α
k
n
T
k
n
y
(
k
-
1
)
n
,
where y
0
n
=
x
n, α
i
n
∈
[
0,1
], i
=
1,2
,
…
,
k, n
=
1,2
,
… and proved that the iterative sequence {
x
n
} defined by (4) converges strongly to a common fixed point of the family of mappings if and only if lim
inf
n
→
∞
d
(
x
n
,
F
)
=
0, where d
(
x
,
F
)
=
inf
p
∈
F
∥
x
-
p
∥. With the help of (3), we write (4) as
(5)
x
n
+
1
=
W
n
x
n
.Recently, Chang et al. [5] introduced the following iteration process of asymptotically nonexpansive mappings in Banach space:
(6)
x
n
+
1
=
λ
n
f
(
x
n
)
+
(
1
-
λ
n
)
T
n
y
n
,
y
n
=
β
n
x
n
+
(
1
-
β
n
)
T
n
x
n
,
where {
λ
n
}
,
{
β
n
}
⊂
[
0,1
] and f is a fixed contractive mapping, and necessary and sufficient conditions are given for the iterative sequence {
x
n
} to converge to the fixed points of T.For a family of mappings, it is quite significant to devise a general iteration scheme which extends the iteration processes (4) and (6), simultaneously. Thereby, to achieve this goal, we introduce a new iteration process for a family of mappings as follows.LetC be a nonempty closed convex subset of a real Banach space E, {
T
i
:
C
→
C
,
i
=
1,2
,
…
,
k
} a family of generalized asymptotically quasi-nonexpansive mappings, and f
:
C
→
C a fixed contractive mapping with contractive coefficient α
∈
(
0,1
). For a given x
1
∈
C, the iteration scheme is defined as follows:
(7)
x
n
+
1
=
λ
n
f
(
x
n
)
+
(
1
-
λ
n
)
W
n
x
n
,
where {
λ
n
}
∈
[
0,1
] and W
n is the modified W-mapping generated by T
1
,
T
2
,
…
,
T
k, and α
1
n
,
α
2
n
,
…
,
α
k
n for all positive integers n.The purpose of this paper is to study the convergence problem of the iterative sequences{
x
n
} defined by (7). The obtained results extend the corresponding results in [4–8], and Lemma 11 partly improves the method of proof of Lemma  3.1 in [4].In what follows, we need the following useful known lemmas.Lemma 1 (see [9]).
Let{
a
n
}, {
δ
n
}, and {
γ
n
} be nonnegative real sequences satisfying the following condition:
(8)
a
n
+
1
≤
(
1
+
δ
n
)
a
n
+
γ
n
,
where ∑
n
=
1
∞
‍
δ
n
<
∞ and ∑
n
=
1
∞
‍
γ
n
<
∞; then lim
n
→
∞
a
n exists.Moreover, if in addition,lim
inf
n
→
∞
a
n
=
0, then lim
n
→
∞
a
n
=
0.Lemma 2 (see [4]).
LetE be a uniformly convex Banach space, 0
<
b
≤
t
n
≤
c
<
1 for all n
≥
1, and let {
x
n
} and {
y
n
} be sequences in E. Assume that lim
sup
n
→
∞
∥
x
n
∥
≤
a, lim
sup
n
→
∞
∥
y
n
∥
≤
a, and lim
n
→
∞
∥
t
n
x
n
+
(
1
-
t
n
)
y
n
∥
=
a for some a
≥
0. Then lim
n
→
∞
∥
x
n
-
y
n
∥
=
0.


## 2. Main Results

Lemma 3.
LetC be a nonempty closed convex subset of a real Banach space E and T an asymptotically quasi-nonexpansive self-mapping of C with {
u
n
}
⊂
[
0
,
∞
) for all n
≥
1. Suppose F
(
T
)
≠
ϕ. Then F
(
T
) is a closed subset in C.Proof.
Let{
z
n
} be an arbitrary sequence of F
(
T
) and z
n
→
z
0 as n
→
∞. Since C is closed, we have z
0
∈
C. For any ϵ
>
0, there exists a natural number N such that
(9)
∥
z
n
-
z
0
∥
<
ϵ
2
+
u
1
,
∀
n
≥
N
.
Thus, we get(10)
∥
T
z
0
-
z
0
∥
≤
∥
T
z
0
-
z
N
∥
+
∥
z
N
-
z
0
∥
∥
T
z
0
-
z
0
∥
≤
(
1
+
u
1
)
∥
z
N
-
z
0
∥
+
∥
z
N
-
z
0
∥
∥
T
z
0
-
z
0
∥
=
(
2
+
u
1
)
∥
z
N
-
z
0
∥
<
ϵ
.
Sinceϵ is arbitrary, it follows that ∥
T
z
0
-
z
0
∥
=
0; that is, T
z
0
=
z
0. Hence z
0
∈
F
(
T
) and F
(
T
) is closed. This completes the proof.Lemma 4.
LetC be a nonempty closed convex subset of a real Banach space E. Let {
T
i
:
i
=
1,2
,
…
,
k
} be k generalized asymptotically quasi-nonexpansive self-mappings of C with {
u
i
n
}
,
{
h
i
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
u
i
n
<
∞ and ∑
n
=
1
∞
‍
h
i
n
<
∞ for all i
∈
{
1,2
,
3
,
…
,
k
}. Suppose F
≠
∅ and {
α
i
n
}
n
≥
1
⊂
[
0,1
] for all i
∈
{
1,2
,
3
,
…
,
k
}. Let W
n be the modified W-mapping generated by T
1
,
T
2
,
…
,
T
k and α
1
n
,
α
2
n
,
…
,
α
k
n. Let the sequence {
x
n
} be defined by (7) and assuming ∑
n
=
1
∞
λ
n
<
∞, then(1)
there exist two sequences{
ν
n
} and {
ξ
n
} in [
0
,
∞
) with ∑
n
=
1
∞
ν
n
<
∞
,
∑
n
=
1
∞
ξ
n
<
∞ such that
(11)
∥
x
n
+
1
-
p
∥
≤
(
1
+
ν
n
)
k
∥
x
n
-
p
∥
+
ξ
n
,
∀
p
∈
F
,
n
≥
1
;
(2)
there exists a constantM
1
>
0, such that
(12)
∥
x
n
+
m
-
p
∥
≤
M
1
∥
x
n
-
p
∥
+
M
1
∑
i
=
n
∞
‍
ξ
i
,
1111111
∀
p
∈
F
,
n
,
m
=
1
,
2
,
3
,
…
,
where {
ξ
n
}
∈
[
0
,
∞
) and Σ
n
=
1
∞
ξ
n
<
∞.Proof.
(
1
) Let ν
n
=
max
1
≤
i
≤
k
u
i
n, for all n. Since ∑
n
=
1
∞
‍
u
i
n
<
∞ for each i, we can get ∑
n
=
1
∞
‍
ν
n
<
∞. For all p
∈
F, it follows from (3) that
(13)
∥
U
1
n
x
n
-
p
∥
≤
(
1
-
α
1
n
)
∥
x
n
-
p
∥
+
α
1
n
∥
T
1
n
x
n
-
p
∥
∥
U
1
n
x
n
-
p
∥
≤
(
1
-
α
1
n
)
∥
x
n
-
p
∥
∥
U
1
n
x
n
-
p
∥
=
+
α
1
n
[
(
1
+
u
1
n
)
∥
x
n
-
p
∥
+
h
1
n
]
∥
U
1
n
x
n
-
p
∥
≤
(
1
+
u
1
n
)
∥
x
n
-
p
∥
+
h
1
n
∥
U
1
n
x
n
-
p
∥
≤
(
1
+
ν
n
)
∥
x
n
-
p
∥
+
h
1
n
.
Assume that∥
U
j
n
x
n
-
p
∥
≤
(
1
+
ν
n
)
j
∥
x
n
-
p
∥
+
(
1
+
ν
n
)
j
-
1
∑
i
=
1
j
‍
h
i
n for some 1
≤
j
≤
k
-
1. Then
(14)
∥
U
(
j
+
1
)
n
x
n
-
p
∥
≤
(
1
-
α
(
j
+
1
)
n
)
∥
x
n
-
p
∥
+
α
(
j
+
1
)
n
∥
T
j
+
1
n
U
j
n
x
n
-
p
∥
≤
(
1
-
α
(
j
+
1
)
n
)
∥
x
n
-
p
∥
+
α
(
j
+
1
)
n
(
(
1
+
u
(
j
+
1
)
n
)
∥
U
j
n
x
n
-
p
∥
+
h
(
j
+
1
)
n
)
≤
(
1
-
α
(
j
+
1
)
n
)
∥
x
n
-
p
∥
+
α
(
j
+
1
)
n
h
(
j
+
1
)
n
+
α
(
j
+
1
)
n
(
1
+
u
(
j
+
1
)
n
)
×
(
(
1
+
ν
n
)
j
∥
x
n
-
p
∥
+
(
1
+
ν
n
)
j
-
1
∑
i
=
1
j
‍
h
i
n
)
≤
(
(
1
-
α
(
j
+
1
)
n
)
+
α
(
j
+
1
)
n
(
1
+
ν
n
)
j
+
1
)
∥
x
n
-
p
∥
+
(
1
+
ν
n
)
j
∑
i
=
1
j
‍
h
i
n
+
h
(
j
+
1
)
n
≤
(
(
1
-
α
(
j
+
1
)
n
)
(
1
+
ν
n
)
j
+
1
+
α
(
j
+
1
)
n
(
1
+
ν
n
)
j
+
1
)
×
∥
x
n
-
p
∥
+
(
1
+
ν
n
)
j
∑
i
=
1
j
+
1
‍
h
i
n
≤
(
1
+
ν
n
)
j
+
1
∥
x
n
-
p
∥
+
(
1
+
ν
n
)
j
∑
i
=
1
j
+
1
‍
h
i
n
.
Thus, by induction, we have(15)
∥
U
j
n
x
n
-
p
∥
≤
(
1
+
ν
n
)
j
∥
x
n
-
p
∥
+
(
1
+
ν
n
)
j
-
1
∑
i
=
1
j
‍
h
i
n
,
for all j
=
1,2
,
…
,
k. Hence,
(16)
∥
W
n
x
n
-
p
∥
=
∥
U
k
n
x
n
-
p
∥
≤
(
1
+
ν
n
)
k
∥
x
n
-
p
∥
∥
W
n
x
n
-
p
∥
=
+
(
1
+
ν
n
)
k
-
1
∑
i
=
1
k
‍
h
i
n
.
By (7) and (16), we obtain
(17)
∥
x
n
+
1
-
p
∥
≤
λ
n
∥
f
(
x
n
)
-
p
∥
+
(
1
-
λ
n
)
∥
W
n
x
n
-
p
∥
∥
x
n
+
1
-
p
∥
≤
λ
n
∥
f
(
x
n
)
-
f
(
p
)
∥
+
λ
n
∥
f
(
p
)
-
p
∥
∥
x
n
+
1
-
p
∥
=
+
(
1
-
λ
n
)
∥
W
n
x
n
-
p
∥
∥
x
n
+
1
-
p
∥
≤
λ
n
α
∥
x
n
-
p
∥
+
λ
n
∥
f
(
p
)
-
p
∥
+
(
1
-
λ
n
)
∥
x
n
+
1
-
p
∥
=
×
[
(
1
+
ν
n
)
k
∥
x
n
-
p
∥
+
(
1
+
ν
n
)
k
-
1
∑
i
=
1
k
‍
h
i
n
]
∥
x
n
+
1
-
p
∥
≤
(
1
+
ν
n
)
k
∥
x
n
-
p
∥
∥
x
n
+
1
-
p
∥
=
+
(
1
-
λ
n
)
(
1
+
ν
n
)
k
-
1
∑
i
=
1
k
‍
h
i
n
∥
x
n
+
1
-
p
∥
=
+
λ
n
∥
f
(
p
)
-
p
∥
.
Since∑
n
=
1
∞
‍
ν
n
<
∞, {
ν
n
}
n
=
1
∞ is bounded. Setting M
=
max
{
sup
n
(
1
+
ν
n
)
k
-
1
,
∥
f
(
p
)
-
p
∥
}, we get that
(18)
∥
x
n
+
1
-
p
∥
≤
(
1
+
ν
n
)
k
∥
x
n
-
p
∥
+
ξ
n
,
∀
p
∈
F
,
n
≥
1
,
where ξ
n
=
M
(
∑
i
=
1
k
‍
h
i
n
+
λ
n
) and ∑
n
=
1
∞
‍
ξ
n
<
∞. This completes the proof of (
1
).
(
2
) If t
≥
0, then 1
+
t
≤
e
t and consequently, (
1
+
t
)
k
≤
e
k
t, k
=
1,2
,
…. Thus, from part (
1
), we get
(19)
∥
x
n
+
m
-
p
∥
≤
(
1
+
ν
n
+
m
-
1
)
k
∥
x
n
+
m
-
1
-
p
∥
+
ξ
n
+
m
-
1
∥
x
n
+
m
-
p
∥
≤
exp
{
k
ν
n
+
m
-
1
}
∥
x
n
+
m
-
1
-
p
∥
+
ξ
n
+
m
-
1
∥
x
n
+
m
-
p
∥
≤
exp
{
k
ν
n
+
m
-
1
}
∥
x
n
+
m
-
p
∥
=
×
(
exp
{
k
ν
n
+
m
-
2
}
∥
x
n
+
m
-
2
-
p
∥
+
ξ
n
+
m
-
2
)
1111111111111111
+
ξ
n
+
m
-
1
∥
x
n
+
m
-
p
∥
≤
exp
{
k
(
ν
n
+
m
-
1
+
ν
n
+
m
-
2
)
}
∥
x
n
+
m
-
2
-
p
∥
∥
x
n
+
m
-
p
∥
=
+
exp
{
k
ν
n
+
m
-
1
}
(
ξ
n
+
m
-
2
+
ξ
n
+
m
-
1
)
∥
x
n
+
m
-
p
∥
⋮
∥
x
n
+
m
-
p
∥
≤
exp
{
k
∑
i
=
n
n
+
m
-
1
‍
ν
i
}
∥
x
n
-
p
∥
∥
x
n
+
m
-
p
∥
=
+
exp
{
k
∑
i
=
n
+
1
n
+
m
-
1
‍
ν
i
}
∑
i
=
n
n
+
m
-
1
‍
ξ
i
∥
x
n
+
m
-
p
∥
≤
M
1
∥
x
n
-
p
∥
+
M
1
∑
i
=
n
∞
‍
ξ
i
,
for any positive integers m, n, where M
1
=
exp
{
k
∑
i
=
1
∞
‍
ν
i
}, ∑
i
=
1
∞
‍
ξ
i
<
∞. This completes the proof of (
2
).Remark 5.
Lemma4 generalizes Lemma  2.1 in [4].Theorem 6.
LetC be a nonempty closed convex subset of a real Banach space E. Let {
T
i
:
i
=
1,2
,
…
,
k
} be k generalized asymptotically quasi-nonexpansive self-mappings of C with {
u
i
n
}
,
{
h
i
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
u
i
n
<
∞ and ∑
n
=
1
∞
‍
h
i
n
<
∞ for all i
∈
{
1,2
,
3
,
…
,
k
}. Let {
α
i
n
}
n
≥
1
⊂
[
0,1
] for all i
∈
{
1,2
,
3
,
…
,
k
} and let W
n be a modified W-mapping generated by T
1
,
T
2
,
…
,
T
k and α
1
n
,
α
2
n
,
…
,
α
k
n. Suppose that F
≠
∅ is closed and ∑
n
=
1
∞
‍
λ
n
<
∞. Starting from arbitrary x
1
∈
C, define the sequence {
x
n
} by the recursion (7); then the sequence {
x
n
} converges strongly to p
∈
F if and only if lim
inf
n
→
∞
d
(
x
n
,
F
)
=
0.Proof.
We will only prove the sufficiency; the necessity is obvious. From Lemma4
(
1
), we have
(20)
∥
x
n
+
1
-
p
∥
≤
(
1
+
ν
n
)
k
∥
x
n
-
p
∥
+
ξ
n
,
for all p
∈
F and all n. Therefore,
(21)
d
(
x
n
+
1
,
F
)
≤
(
1
+
ν
n
)
k
d
(
x
n
,
F
)
+
ξ
n
d
(
x
n
+
1
,
F
)
=
(
1
+
∑
r
=
1
k
‍
k
(
k
-
1
)
⋯
(
k
-
r
+
1
)
r
!
ν
n
r
)
d
(
x
n
+
1
,
F
)
=
×
d
(
x
n
,
F
)
+
ξ
n
.
As∑
n
=
1
∞
‍
ν
n
<
∞, so ∑
r
=
1
k
‍
(
k
(
k
-
1
)
⋯
(
k
-
r
+
1
)
/
r
!
)
ν
n
r
<
∞. By Lemma 1 and lim
inf
n
→
∞
d
(
x
n
,
F
)
=
0, we get that lim
n
→
∞
d
(
x
n
,
F
)
=
0. Next, we prove that {
x
n
} is a Cauchy sequence. From Lemma 4
(
2
), we have
(22)
∥
x
n
+
m
-
p
∥
≤
M
1
∥
x
n
-
p
∥
+
M
1
∑
i
=
n
∞
‍
ξ
i
111111111111111
∀
p
∈
F
,
n
,
m
≥
1
.
Hence, for all integersm
≥
1 and all p
∈
F,
(23)
∥
x
n
+
m
-
x
n
∥
≤
∥
x
n
+
m
-
p
∥
+
∥
x
n
-
p
∥
∥
x
n
+
m
-
x
n
∥
≤
(
M
1
+
1
)
∥
x
n
-
p
∥
+
M
1
∑
j
=
n
∞
‍
ξ
j
.
Taking infimum overp
∈
F in (23) gives
(24)
∥
x
n
+
m
-
x
n
∥
≤
(
M
1
+
1
)
d
(
x
n
,
F
)
+
M
1
∑
j
=
n
∞
‍
ξ
j
.
Now, sincelim
n
→
∞
d
(
x
n
,
F
)
=
0 and ∑
j
=
1
∞
‍
ξ
j
<
∞, given ϵ
>
0, there exists an integer N
1
>
0 such that for all n
≥
N
1, d
(
x
n
,
F
)
<
ϵ
/
(
2
(
M
1
+
2
)
) and ∑
j
=
n
∞
‍
ξ
n
<
ϵ
/
(
2
(
M
1
+
1
)
). So for all integers n
≥
N
1, m
≥
1, we obtain from (24) that
(25)
∥
x
n
+
m
-
x
n
∥
<
ϵ
,
∀
n
≥
N
1
,
m
≥
1
.
Hence,{
x
n
} is a Cauchy sequence in E. Since E is complete, there exists q
∈
E such that lim
n
→
∞
x
n
=
q. We now show that q
∈
F. Since d
(
x
n
,
F
)
→
0 and x
n
→
q as n
→
∞, for each ϵ
-
>
0, there exists an integer N
2
>
0 such that, d
(
x
n
,
F
)
=
inf
p
∈
F
∥
x
n
-
p
∥
<
ϵ
-
/
3 and ∥
x
n
-
q
∥
<
ϵ
-
/
2 for all n
≥
N
2. In particular, we have d
(
x
N
2
,
F
)
=
inf
p
∈
F
∥
x
N
2
-
p
∥
<
ϵ
-
/
3; that is, there exists a p
-
∈
F such that ∥
x
N
2
-
p
-
∥
<
ϵ
-
/
2; hence
(26)
∥
q
-
p
-
∥
≤
∥
x
N
2
-
q
∥
+
∥
x
N
2
-
p
-
∥
<
ϵ
-
.
SinceF is a closed subset of E, we obtain q
∈
F. This completes the proof.Remark 7.
Theorem6 generalizes and extends Theorem 2.2 of Khan et al. [4], Theorem 3.1 of Ghosh and Debnath [8], Theorem 3.2 of Shahzad and Udomene [6], and Theorem 1 of Qihou [7] together with its Corollaries 1 and 2.Asymptotically nonexpansive mappings and asymptotically quasi-nonexpansive mappings are all generalized asymptotically quasi-nonexpansive, by Theorem6 and Lemma 3, so we haveCorollary 8.
LetC be a nonempty closed convex subset of a real Banach space E. Let {
T
i
:
i
=
1,2
,
…
,
k
} be k asymptotically quasi-nonexpansive self-mappings of C with {
u
i
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
u
i
n
<
∞ for all i
∈
{
1,2
,
3
,
…
,
k
}. Let {
α
i
n
}
n
≥
1
⊂
[
0,1
] for all i
∈
{
1,2
,
3
,
…
,
k
} and let W
n be a modified W-mapping generated by T
1
,
T
2
,
…
,
T
k and α
1
n
,
α
2
n
,
…
,
α
k
n. Suppose F
≠
∅ and ∑
n
=
1
∞
‍
λ
n
<
∞. Starting from arbitrary x
1
∈
C, define the sequence {
x
n
} by the recursion (7). Then the sequence {
x
n
} converges strongly to p
∈
F if and only if lim
inf
n
→
∞
d
(
x
n
,
F
)
=
0.Corollary 9.
LetC be a nonempty closed convex subset of a real Banach space E. Let {
T
i
:
i
=
1,2
,
…
,
k
} be k asymptotically nonexpansive self-mappings of C with {
u
i
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
u
i
n
<
∞ for all i
∈
{
1,2
,
3
,
…
,
k
}. Let {
α
i
n
}
n
≥
1
⊂
[
0,1
] for all i
∈
{
1,2
,
3
,
…
,
k
} and let W
n be a modified W-mapping generated by T
1
,
T
2
,
…
,
T
k and α
1
n
,
α
2
n
,
…
,
α
k
n. Suppose F
≠
∅ and ∑
n
=
1
∞
‍
λ
n
<
∞. Starting from arbitrary x
1
∈
C, define the sequence {
x
n
} by the recursion (7). Then the sequence {
x
n
} converges strongly to p
∈
F if and only if lim
inf
n
→
∞
d
(
x
n
,
F
)
=
0.Corollary 10.
LetC be a nonempty closed convex subset of a real Banach space E. Let {
T
i
:
i
=
1,2
,
…
,
k
} be k generalized asymptotically quasi-nonexpansive self-mappings of C with {
u
i
n
}
,
{
h
i
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
u
i
n
<
∞ and ∑
n
=
1
∞
‍
h
i
n
<
∞ for all i
∈
{
1,2
,
3
,
…
,
k
}. Let {
α
i
n
}
n
≥
1
⊂
[
0,1
] for all i
∈
{
1,2
,
3
,
…
,
k
} and let W
n be a modified W-mapping generated by T
1
,
T
2
,
…
,
T
k and α
1
n
,
α
2
n
,
…
,
α
k
n. Suppose that F
≠
∅ is closed and ∑
n
=
1
∞
‍
λ
n
<
∞. Starting from arbitrary x
1
∈
C, define the sequence {
x
n
} by the recursion (7). Then the sequence {
x
n
} converges strongly to p
∈
F if and only if there exists a subsequence {
x
n
j
} of {
x
n
} which converges to p.


## 3. Results in Uniformly Convex Banach Spaces

Lemma 11.
LetC be a nonempty closed convex subset of a uniformly convex Banach space E. Let {
T
i
:
i
=
1,2
,
…
,
k
} be k
(
L
-
γ
) uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of C with {
u
i
n
}
,
{
h
i
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
u
i
n
<
∞ and ∑
n
=
1
∞
‍
h
i
n
<
∞ for all i
∈
{
1,2
,
3
,
…
,
k
}. Let α
i
n
∈
[
δ
,
1
-
δ
] for some δ
∈
(
0,1
/
2
) and let W
n be a modified W-mapping generated by T
1
,
T
2
,
…
,
T
k and α
1
n
,
α
2
n
,
…
,
α
k
n. Suppose F
=
⋂
i
=
1
k
‍
F
(
T
i
)
≠
∅ and ∑
n
=
1
∞
‍
λ
n
<
∞. Starting from arbitrary x
1
∈
C, define the sequence {
x
n
} by the recursion (7). Then lim
n
→
∞
∥
x
n
-
T
j
x
n
∥
=
0 for each j
∈
{
1,2
,
3
,
…
k
}.Proof.
Letp
∈
F and ν
n
=
max
1
≤
i
≤
k
u
i
n, for all n. By Lemma 1 and Lemma 4
(
1
), it follows that lim
n
→
∞
∥
x
n
-
p
∥ exists for all p
∈
F. Assume that
(27)
lim
n
→
∞
∥
x
n
-
p
∥
=
c
.
From (15) and (27) we obtain that
(28)
lim
sup
n
→
∞
∥
U
j
n
x
n
-
p
∥
≤
c
,
∀
1
≤
j
≤
k
.
From (7), we have
(29)
∥
x
n
+
1
-
p
∥
=
∥
λ
n
f
(
x
n
)
+
(
1
-
λ
n
)
W
n
x
n
-
p
∥
∥
x
n
+
1
-
p
∥
≤
λ
n
α
∥
x
n
-
p
∥
+
λ
n
∥
f
(
p
)
-
p
∥
∥
x
n
+
1
-
p
∥
=
+
(
1
-
λ
n
)
∥
U
k
n
x
n
-
p
∥
;
therefore,
(30)
lim
inf
n
→
∞
∥
U
k
n
x
n
-
p
∥
≥
c
.
From (28) and (30) we can obtain that
(31)
lim
n
→
∞
∥
U
k
n
x
n
-
p
∥
=
c
.
Suppose thatlim
n
→
∞
∥
U
(
j
+
1
)
n
x
n
-
p
∥
=
c for some 1
≤
j
≤
k
-
1. Since
(32)
∥
U
(
j
+
1
)
n
x
n
-
p
∥
≤
(
1
-
α
(
j
+
1
)
n
)
∥
x
n
-
p
∥
∥
U
(
j
+
1
)
n
x
n
-
p
∥
=
+
α
(
j
+
1
)
n
∥
T
j
+
1
n
U
j
n
x
n
-
p
∥
∥
U
(
j
+
1
)
n
x
n
-
p
∥
≤
(
1
-
α
(
j
+
1
)
n
)
∥
x
n
-
p
∥
+
α
(
j
+
1
)
n
∥
U
(
j
+
1
)
n
x
n
-
p
∥
=
×
[
(
1
+
u
(
j
+
1
)
n
)
∥
U
j
n
x
n
-
p
∥
+
h
(
j
+
1
)
n
]
,
so we obtain that
(33)
lim
inf
n
→
∞
∥
U
j
n
x
n
-
p
∥
≥
c
.
From (28) and (33), we have that
(34)
lim
n
→
∞
∥
U
j
n
x
n
-
p
∥
=
c
.
Thus, by induction, we have(35)
lim
n
→
∞
∥
U
j
n
x
n
-
p
∥
=
c
,
for each j
=
1,2
,
3
,
…
,
k. That is,
(36)
lim
n
→
∞
∥
(
1
-
α
j
n
)
(
x
n
-
p
)
+
α
j
n
(
T
j
n
U
(
j
-
1
)
n
x
n
-
p
)
∥
=
c
,
for each j
=
1,2
,
3
,
…
,
k. From (28), we obtain
(37)
lim
sup
n
→
∞
∥
T
j
n
U
(
j
-
1
)
n
x
n
-
p
∥
≤
c
,
for each j
=
1,2
,
3
,
…
,
k. By Lemma 2, we get
(38)
lim
n
→
∞
∥
T
j
n
U
(
j
-
1
)
n
x
n
-
x
n
∥
=
0
,
∀
1
≤
j
≤
k
.
Ifj
=
1, from (38), we have
(39)
lim
n
→
∞
∥
T
1
n
x
n
-
x
n
∥
=
0
.
Ifj
=
2,3
,
…
,
k, then we have
(40)
∥
T
j
n
x
n
-
x
n
∥
≤
∥
T
j
n
x
n
-
T
j
n
U
(
j
-
1
)
n
x
n
∥
+
∥
T
j
n
U
(
j
-
1
)
n
x
n
-
x
n
∥
∥
T
j
n
x
n
-
x
n
∥
≤
L
∥
x
n
-
U
(
j
-
1
)
n
x
n
∥
γ
∥
T
j
n
x
n
-
x
n
∥
=
+
∥
T
j
n
U
(
j
-
1
)
n
x
n
-
x
n
∥
∥
T
j
n
x
n
-
x
n
∥
=
L
(
α
(
j
-
1
)
n
∥
x
n
-
T
j
-
1
n
U
(
j
-
2
)
n
x
n
∥
)
γ
∥
T
j
n
x
n
-
x
n
∥
=
+
∥
T
j
n
U
(
j
-
1
)
n
x
n
-
x
n
∥
.
Hence,(41)
lim
n
→
∞
∥
T
j
n
x
n
-
x
n
∥
=
0
,
∀
1
≤
j
≤
k
.
Note that(42)
∥
x
n
+
1
-
x
n
∥
=
∥
λ
n
f
(
x
n
)
+
(
1
-
λ
n
)
W
n
x
n
-
x
n
∥
∥
x
n
+
1
-
x
n
∥
≤
λ
n
(
α
∥
x
n
-
p
∥
+
∥
f
(
p
)
-
p
∥
+
∥
x
n
-
p
∥
)
∥
x
n
+
1
-
x
n
∥
=
+
(
1
-
λ
n
)
∥
W
n
x
n
-
x
n
∥
∥
x
n
+
1
-
x
n
∥
=
λ
n
(
α
∥
x
n
-
p
∥
+
∥
f
(
p
)
-
p
∥
+
∥
x
n
-
p
∥
)
∥
x
n
+
1
-
x
n
∥
=
+
(
1
-
λ
n
)
α
k
n
∥
T
k
n
U
(
k
-
1
)
n
x
n
-
x
n
∥
;
therefore, we have
(43)
lim
n
→
∞
∥
x
n
+
1
-
x
n
∥
=
0
.
Now, we observe that(44)
∥
x
n
-
T
j
x
n
∥
≤
∥
x
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
T
j
n
+
1
x
n
+
1
∥
∥
x
n
-
T
j
x
n
∥
=
+
∥
T
j
n
+
1
x
n
+
1
-
T
j
n
+
1
x
n
∥
+
∥
T
j
n
+
1
x
n
-
T
j
x
n
∥
∥
x
n
-
T
j
x
n
∥
≤
∥
x
n
-
x
n
+
1
∥
+
∥
x
n
+
1
-
T
j
n
+
1
x
n
+
1
∥
∥
x
n
-
T
j
x
n
∥
=
+
L
∥
x
n
+
1
-
x
n
∥
γ
+
L
∥
T
j
n
x
n
-
x
n
∥
γ
.
By (41) and (43), we have
(45)
lim
n
→
∞
∥
x
n
-
T
j
x
n
∥
=
0
,
for j
=
1,2
,
3
,
…
,
k. This completes the proof.Theorem 12.
LetC be a nonempty closed convex subset of a uniformly convex Banach space E. Let {
T
i
:
i
=
1,2
,
…
,
k
} be k
(
L
-
γ
) uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of C with {
u
i
n
}
,
{
h
i
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
u
i
n
<
∞ and ∑
n
=
1
∞
‍
h
i
n
<
∞ for all i
∈
{
1,2
,
3
,
…
,
k
}. Let α
i
n
∈
[
δ
,
1
-
δ
] for some δ
∈
(
0,1
/
2
) and let W
n be a modified W-mapping generated by T
1
,
T
2
,
…
,
T
k and α
1
n
,
α
2
n
,
…
,
α
k
n. Suppose F
=
⋂
i
=
1
k
‍
F
(
T
i
)
≠
∅, ∑
n
=
1
∞
‍
λ
n
<
∞ and there exists one member in {
T
i
m
:
i
=
1,2
,
…
,
k
} which is semicompact for some positive integer m. Starting from arbitrary x
1
∈
C, define the sequence {
x
n
} by the recursion (7). Then {
x
n
} converges strongly to some common fixed point of the family {
T
i
:
i
=
1,2
,
…
,
k
}.Proof.
By Lemma11, we have
(46)
lim
n
→
∞
∥
x
n
-
T
j
x
n
∥
=
0
,
for each j
=
1,2
,
3
,
…
k. Without loss of generality, we may assume that T
1
m is semicompact for some m
≥
1; then we have
(47)
∥
T
1
m
x
n
-
x
n
∥
≤
∥
T
1
m
x
n
-
T
1
m
-
1
x
n
∥
+
∥
T
1
m
-
1
x
n
-
T
1
m
-
2
x
n
∥
∥
T
1
m
x
n
-
x
n
∥
=
+
⋯
+
∥
T
1
x
n
-
x
n
∥
∥
T
1
m
x
n
-
x
n
∥
≤
∥
T
1
x
n
-
x
n
∥
+
(
m
-
1
)
L
∥
T
1
x
n
-
x
n
∥
γ
⟶
0
.
SinceT
1
m is semicompact, then there exists a subsequence {
x
n
j
} of {
x
n
} such that x
n
j
→
q
∈
C. Hence, we have
(48)
∥
q
-
T
i
q
∥
=
lim
j
→
∞
∥
x
n
j
-
T
j
x
n
j
∥
=
0
,
for each i
=
1,2
,
3
,
…
,
k. This implies that q
∈
F. By Corollary 10, {
x
n
} converges strongly to some common fixed point of the family {
T
i
:
i
=
1,2
,
…
,
k
}.Theorem 13.
LetC be a nonempty closed convex subset of a uniformly convex Banach space E. Let {
T
i
:
i
=
1,2
,
…
,
k
} be k
(
L
-
γ
) uniform Lipschitz and generalized asymptotically quasi-nonexpansive self-mappings of C with {
u
i
n
}
,
{
h
i
n
}
⊂
[
0
,
∞
) such that ∑
n
=
1
∞
‍
u
i
n
<
∞ and ∑
n
=
1
∞
‍
h
i
n
<
∞ for all i
∈
{
1,2
,
3
,
…
,
k
}. Let α
i
n
∈
[
δ
,
1
-
δ
] for some δ
∈
(
0,1
/
2
) and let W
n be a modified W-mapping generated by T
1
,
T
2
,
…
,
T
k and α
1
n
,
α
2
n
,
…
,
α
k
n. Suppose F
=
⋂
i
=
1
k
‍
F
(
T
i
)
≠
∅, ∑
n
=
1
∞
‍
λ
n
<
∞ and each I
-
T
i, i
=
1,2
,
…
,
k, is demiclosed at 0. If E satisfies Opial’s condition, then the sequence {
x
n
} defined by (7) converges weakly to a common fixed point of the family {
T
i
:
i
=
1,2
,
…
,
k
}.Proof.
From the proof of Lemma11, we know that {
x
n
} is a bounded sequence in C. Since E is uniformly convex, it must be reflexive. Therefore, there exists a subsequence {
x
n
j
} in {
x
n
} converging weakly to u
∈
C. By Lemma 11, lim
j
→
∞
∥
x
n
j
-
T
i
x
n
j
∥
=
0 and I
-
T
i is demiclosed at 0 for i
=
1,2
,
…
k, so we obtain T
i
u
=
u. That is, u
∈
F. Suppose that there exists another subsequence {
x
n
k
} of {
x
n
} converging weakly to v
∈
C. As above, we can prove v
∈
F. By (27) we know that lim
n
→
∞
∥
x
n
-
u
∥ and lim
n
→
∞
∥
x
n
-
v
∥ exist. Assume v
≠
u. Then by the Opial’s condition, we have
(49)
lim
n
→
∞
∥
x
n
-
u
∥
=
lim
n
j
→
∞
∥
x
n
j
-
u
∥
<
lim
n
j
→
∞
∥
x
n
j
-
v
∥
lim
n
→
∞
∥
x
n
-
u
∥
=
lim
n
→
∞
∥
x
n
-
v
∥
=
lim
n
k
→
∞
∥
x
n
k
-
v
∥
lim
n
→
∞
∥
x
n
-
u
∥
<
lim
n
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 the family {
T
i
:
i
=
1,2
,
…
,
k
}.Remark 14.
Lemma11, Theorem 12, and Theorem 13 extend Lemma 3.1, Theorem 3.3, and Theorem 3.2 of Khan et al. [4], respectively.



---

*Source: 642167-2014-01-02.xml*