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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2007, Article ID 68616, 10 pages
doi:10.1155/2007/68616
Research Article
Convergece Theorems for Finite Families of Asymptotically
Quasi-Nonexpansive Mappings
C. E. Chidume and Bashir Ali
Received 20 October 2006; Revised 30 January 2007; Accepted 31 January 2007
Recommended by Donal O’Regan
Let E be a real Banach space, K a closed convex nonempty subset of E,andT
1
,T
2
, ,T
m
:
K
→ K asymptotically quasi-nonexpansive mappings with sequences (resp.) {k
in
}
∞
n=1
sat-
isfying k
in
→ 1asn →∞,and

∞
n=1
(k

in
− 1) < ∞, i = 1,2, ,m.Let{α
n
}
∞
n=1
be a se-
quence in [
,1−

], 
∈
(0,1). Define a sequence {x
n
} by x
1
∈ K, x
n+1
= (1 − α
n
)x
n
+
α
n
T
n
1
y
n+m−2

, y
n+m−2
= (1 − α
n
)x
n
+ α
n
T
n
2
y
n+m−3
, , y
n
= (1 − α
n
)x
n
+ α
n
T
n
m
x
n
, n ≥ 1,
m
≥ 2. Let


m
i
=1
F(T
i
) = ∅. Necessary and sufficient conditions for a strong convergence
of the sequence
{x
n
} to a common fixed point of the family {T
i
}
m
i
=1
are proved. Under
some appropriate conditions, strong and weak convergence theorems are also proved.
Copyright © 2007 C. E. Chidume and B. Ali. This is an open access article distributed
under the Creative Commons Attribution License, which permits unrestricted use, dis-
tribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Let K be a nonempty subset of a real normed space E. A self-mapping T : K
→ K is
called nonexpansive if
Tx − Ty≤x − y for every x, y ∈ K,andquasi-nonexpansive
if F(T):
={x ∈ K : Tx = x} =∅and Tx− p≤x − p for every x ∈ K and p ∈ F(T).
The mapping T is called asymptotically nonex pansive if there exists a sequence
{k
n

}⊂
[1,∞)withk
n
→ 1asn →∞such that for every n ∈ N,


T
n
x − T
n
y


≤
k
n
x − y for every x, y ∈ K. (1.1)
If F(T)
=∅and there exists a sequence {k
n
}⊂[1,∞)withk
n
→ 1asn →∞such that for
2 Journal of Inequalities and Applications
n
∈ N,


T
n

x − p


≤
k
n
x − p for every x ∈ K, (1.2)
and p
∈ F(T), then T is called asymptotically quasi-nonexpansive mapping.
Iterative methods for approximating fixed points of nonexpansive mappings and their
generalisations have been studied by numerous authors (see, e.g., [1–9] and the references
contained therein).
Petryshyn and Williamson [4]provednecessaryandsufficient conditions for the Pi-
card and Mann [10] iterative sequences to strongly converge to a fixed point of a quasi-
nonexpansive map T in a real Banach space.
Ghosh and Debnath [3]extendedtheresultsin[4] and proved necessary and suf-
ficient conditions for strong convergence of Ishikawa-type [11] iteration process to a
fixed point of a quasi-nonexpansive mapping T in a real Banach space. Furthermore,
they proved strong convergence theorem of the Ishikawa-type iteration process for quasi-
nonexpansive mappings in a uniformly convex Banach space.
Qihou [5] extended the results of Ghosh and Debnath to asymptotically quasi-non-
expansive mappings. In some other papers, Qihou [6, 7] studied the convergence of
Ishikawa-type iteration process with errors for asymtotically quasi-nonexpansive map-
pings.
Recently, Sun [12] studied the convergence of an implicit iteration process (see [12]for
definition) to a common fixed point of finite family of asymptotically quasi-nonexpansive
mappings. He proved the following theorems.
Theorem 1.1 (see [12]). Let K be a nonempty close d convex subset of a Banach space E.
Let
{T

i
, i ∈ I} be m asymptotically quasi-nonexpansive self-mappings of K with sequences
{1+u
in
}
n
, i = 1,2, ,m, respectively. Suppose that F :=

m
i
=1
F(T
i
) =∅and that x
0
∈ K,
{α
n
}⊂(s,1 − s) for some s ∈ (0,1),

∞
n=1
u
in
< ∞ for all i ∈ I. Then the implicit iterative
sequence
{x
n
} generated by
x

n
= α
n
x
n−1
+

1 − α
n

T
k
i
x
n
, n ≥ 1, n = (k − 1)m + i, i = 1,2, ,m, (1.3)
converges to a common fixed point in F if and only if lim inf
n→∞
d(x
n
,F) = 0,where
d(x
n
,F) = inf
x
∗
∈F
x
n
− x

∗
.
Theorem 1.2 (see [12]). Let K be a nonempty closed convex and bounded subset of a real
uniformly convex Banach space E.Let
{T
i
, i ∈ I} be m uniformly L-Lipschitzian asymp-
totically quasi-nonexpansive s elf-mappings of K with sequences
{1+u
in
}
n
, i = 1,2, ,m,
respectively. Suppose that F :
=

m
i
=1
F(T
i
) =∅and that x
0
∈ K, {α
n
}⊂(s,1− s) for some
s
∈ (0,1),

∞

n=1
u
in
< ∞ for all i ∈ I. If there exists one member T ∈{T
i
,i ∈ I} whichissemi-
compact, then the implicit iterative sequence
{x
n
} generated by (1.3) converges strongly to a
common fixed point of the mappings
{T
i
, i ∈ I}.
Very recently, Shahzad and Udomene [8] proved necessary and sufficient conditions
for the strong convergence of the Ishikawa-like iteration process to a common fixed point
of two uniformly continuous asymptotically quasi-nonexpansive mappings.
Their main results are the following theorems.
C. E. Chidume and B. Ali 3
Theorem 1.3 (see [8]). Let E be a real B anach space and let K beanonemptyclosedconvex
subset of E.LetS,T : K
→ K be two asymptotically quasi-nonexpansive mappings (S and T
need not be continuous) with sequences
{u
n
},{v
n
}⊂[0,∞) such that

u

n
< ∞ and

v
n
<
∞,andF := F(S) ∩ F(T) ={x ∈ K : Sx = Tx = x} =∅.Let{α
n
} and {β
n
} be sequences in
[0,1].Fromarbitraryx
1
∈ K define a sequence {x
n
} by
x
n+1
=

1 − α
n

x
n
+ α
n
S
n


1 − β
n

x
n
+ β
n
T
n
x
n

. (1.4)
Then,
{x
n
} converges strongly to some common fixed point of S and T if and only if
liminf
n→∞
d(x
n
,F) = 0.
Theorem 1.4 (see [8]). Let E be a real uniformly convex Banach space and let K be a
nonempty closed convex subset of E.LetS, T : K
→ K be two uniformly continuous asymptot-
ically quasi-nonexpansive mappings with sequences
{u
n
},{v
n

}⊂[0,∞) such that

u
n
< ∞,

v
n
< ∞,andF := F(S) ∩ F(T) ={x ∈ K : Sx = Tx = x} =∅.Let{α
n
} and {β
n
} be se-
quences in [
,1 −

] for some 
∈
(0,1).Fromarbitraryx
1
∈ K define a sequence {x
n
} by
(1.4). Assume, in addition, that either T or S is compact. Then,
{x
n
} converges strongly to
some common fixed point of S and T.
More recently, the authors [2] introduced a scheme defined by
x

1
∈ K,
x
n+1
= P


1 − α
1n

x
n
+ α
1n
T
1

PT
1

n−1
y
n+m−2

,
y
n+m−2
= P



1 − α
2n

x
n
+ α
2n
T
2

PT
2

n−1
y
n+m−3

,
.
.
.
y
n
= P


1 − α
mn

x

n
+ α
mn
T
m

PT
m

n−1
x
n

, n ≥ 1,
(1.5)
and studied the convergence of this sheme to a common fixed point of finite families of
nonself asymptotically nonexpansive mappings.
Let
{α
n
} be a real sequence in [,1 −

], 
∈
(0,1). Let T
1
,T
2
, ,T
m

: K → K be a
family of mappings. Define a sequence
{x
n
} by
x
1
∈ K,
x
n+1
=

1 − α
n

x
n
+ α
n
T
n
1
y
n+m−2
,
y
n+m−2
=

1 − α

n

x
n
+ α
n
T
n
2
y
n+m−3
,
.
.
.
y
n
=

1 − α
n

x
n
+ α
n
T
n
m
x

n
, n ≥ 1.
(1.6)
It is our purpose in this paper to prove necessary and sufficient conditions for the
strong convergence of the scheme defined by (1.6) to a common fixed point of finite
family T
1
,T
2
, ,T
m
of asymptotically quasi-nonexpansive mappings.Wealsoprovestrong
and weak convergence theorems for the family in a uniformly convex Banach spaces. Our
results generalize and improve some recent i mportant results (see Remark 3.9).
4 Journal of Inequalities and Applications
2. Preliminaries
Let E be a real nor med linear space. The modulus of convexity of E is the function δ
E
:
(0,2]
→ [0,1] defined by
δ
E
() = inf

1 −





x + y
2




: x=y=1,  =x − y

. (2.1)
E is called uniformly convex if and only if δ
E
() > 0 ∀

∈ (0,2].
AmappingT with domain D(T) and range R(T)inE is said to be demiclosed at p
if whenever
{x
n
} is a sequence i n D(T)suchthatx
n
 x
∗
∈ D(T)andTx
n
→ p then
Tx
∗
= p.
AmappingT : K
→ K is said to be s emicompact if, for any bounded sequence {x

n
} in
K such that
x
n
− Tx
n
→0asn →∞, there exists a subsequence say {x
n
j
} of {x
n
} such
that
{x
n
j
} converges strongly to some x
∗
in K.
ABanachspaceE is said to satisfy Opial’s condition if for any sequence
{x
n
} in E,
x
n
 x implies that
liminf
n→∞



x
n
− x


< liminf
n→∞


x
n
− y


∀
y ∈ E, y = x. (2.2)
We will say that a mapping T satisfies condition (P) if it satisfies the weak version of
demiclosedness at origin as defined in [4] (i.e., if
{x
n
j
} is any subsequence of a sequence
{x
n
} with x
n
j
 x
∗

and (I − T)x
n
j
→ 0asj →∞,thenx
∗
− Tx
∗
= 0).
In what follows we will use the following results.
Lemma 2.1 (see [9]). Let
{λ
n
} and {σ
n
} be sequences of nonnegative real numbers such
that λ
n+1
≤ λ
n
+ σ
n
for all n ≥ 1,and

∞
n=1
σ
n
< ∞, then lim
n→∞
λ

n
exists. Moreover, if there
exists a subsequence
{λ
n
j
} of {λ
n
} such that λ
n
j
→ 0 as j →∞, then λ
n
→ 0 as n →∞.
Lemma 2.2 (see [13]). Let p>1 and r>1 be two fixed numbers and E a Banach space.
Then E is uniformly convex if and only if there exists a continuous, strictly increasing, and
convex function g :[0,
∞) → [0,∞) with g(0) = 0 such that


λx +(1− λ)y


p
≤ λx
p
+(1− λ)y
p
− W
p

(λ)g


x − y

(2.3)
for all x, y
∈ B
r
(0) ={z ∈ E : z≤r}, λ ∈ [0,1] and W
p
(λ) = λ(1 − λ)
p
+ λ
p
(1 − λ).
3. Main results
In this section, we state and prove the main results of this paper. In the sequel, we desig-
nate the set
{1,2, ,m} by I and we always assume F :=

m
i
=1
F(T
i
) =∅.
Lemma 3.1. Let E be a real normed linear space and let K be a nonempty, closed convex
subset of E.LetT
1

,T
2
, ,T
m
: K → K be asymptotically quasi-nonexpansive mappings with
sequence
{k
in
}
∞
n=1
satisfying k
in
→ 1 as n →∞and

∞
n=1
(k
in
− 1) < ∞, i ∈ I.Let{α
n
}
∞
n=1
be
C. E. Chidume and B. Ali 5
a sequences in [
,1−

], 

∈
(0,1).Let{x
n
} beasequencedefinediterativelyby
x
1
∈ K,
x
n+1
=

1 − α
n

x
n
+ α
n
T
n
1
y
n+m−2
,
y
n+m−2
=

1 − α
n


x
n
+ α
n
T
n
2
y
n+m−3
,
.
.
.
y
n
=

1 − α
n

x
n
+ α
n
T
n
m
x
n

, n ≥ 1, m ≥ 2.
(3.1)
Let x
∗
∈ F.Then,{x
n
} is bounded and the limits lim
n→∞
x
n
− x
∗
 and lim
n→∞
d(x
n
,F)
exist, where d(x
n
,F) = inf
x
∗
∈F
x
n
− x
∗
.
Proof. Set k
in

= 1+u
in
so that

∞
n=1
u
in
< ∞ for each i ∈ I.Letw
n
:=

m
i
=1
u
in
.Letx
∗
∈ F.
Then we have, for some positive integer h,2
≤ h<m,


x
n+1
− x
∗



=



1 − α
n

x
n
+ α
n
T
n
1
y
n+m−2
− x
∗


≤

1 − α
n



x
n
− x

∗


+ α
n

1+u
1n



y
n+m−2
− x
∗


≤

1 − α
n



x
n
− x
∗



+ α
n

1+u
1n



1 − α
n



x
n
− x
∗


+ α
n

1+u
2n



y
n+m−3
− x

∗



≤

1 − α
n



x
n
− x
∗


+ α
n

1 − α
n

1+u
1n



x
n

− x
∗


+ ···+

α
n

h−1

1 − α
n

1+u
1n

1+u
2n

···

1+u
h−1n



x
n
− x

∗


+ ···+

α
n

m

1+u
1n

1+u
2n

···

1+u
mn



x
n
− x
∗


≤



x
n
− x
∗



1+u
1n
+ u
2n

1+u
1n

+ u
3n

1+u
1n

1+u
2n

+ ···
+ u
mn


1+u
1n

1+u
2n

···

1+u
m−1n


≤


x
n
− x
∗



1+

m
1

w
n
+


m
2

w
2
n
+ ···+

m
m

w
m
n

≤


x
n
− x
∗



1+δ
m
w
n


≤


x
n
− x
∗


e
δ
m
w
n
≤


x
1
− x
∗


e
δ
m

∞
n=1

w
n
< ∞,
(3.2)
where δ
m
is a positive real number defined by δ
m
:=


m
1

+

m
2

+ ···+

m
m


.
This implies that
{x
n
} is bounded and so there exists a positive integer M such that



x
n+1
− x
∗


≤


x
n
− x
∗


+ δ
m
Mw
n
. (3.3)
6 Journal of Inequalities and Applications
Since (3.3)istrueforeachx
∗
in F,wehave
d

x
n+1

,F

≤
d

x
n
,F

+ δ
m
Mw
n
. (3.4)
By Lemma 2.1,lim
n→∞
x
n
− x
∗
 and lim
n→∞
d(x
n
,F) exist. This completes the proof of
Lemma 3.1.

Theorem 3.2. Let K be a nonempty closed convex subset of a Banach space E.LetT
1
,T

2
, ,
T
m
: K → K be asymptotically quasi-nonexpansive mappings with sequences {k
in
}
∞
n=1
and
{α
n
}
∞
n=1
as in Lemma 3.1.Let{x
n
} be defined by (3.1). Then, {x
n
} convergestoacommon
fixed point of the family T
1
,T
2
, ,T
m
if and only if liminf
n→∞
d(x
n

,F) = 0.
Proof. The necessity is trivial. We prove the sufficiency. Let liminf
n→∞
d(x
n
,F) = 0. Since
lim
n→∞
d(x
n
,F) exists by Lemma 3.1, we have that lim
n→∞
d(x
n
,F) = 0. Thus, given  > 0
there exist a positive integer N
0
and b
∗
∈ F such that for all n ≥ N
0
x
n
− b
∗
 < /2.
Then, for any k
∈ N,wehaveforn ≥ N
0
,



x
n+k
− x
n


≤


x
n+k
− b
∗


+


b
∗
− x
n


<

2
+


2
= , (3.5)
and so
{x
n
} is Cauchy. Let lim
n→∞
x
n
= b. We need to show that b ∈ F.LetT
i
∈{T
1
,T
2
, ,
T
m
}.Sincelim
n→∞
d(x
n
,F) = 0, there exists N ∈ N sufficiently large and b
∗
∈ F such that
n
≥ N implies b − x
n
 < /6(1 + w

1
), b
∗
− x
n
 < /6(1 + w
1
). Then, b
∗
− b < /3(1+
w
1
). Thus, we have the following estimates, for n ≥ N and ar bitrary T
i
, i = 1,2, ,m,


b − T
i
b


≤


b − x
n


+



x
n
− b
∗


+


b
∗
− T
i
b


≤


b − x
n


+


x
n

− b
∗


+

1+w
1



b
∗
− b


<

3

1+w
1

+

3

1+w
1


+

3
≤

.
(3.6)
This implies that b
∈ Fix(T
i
)foralli = 1,2, ,m and thus b ∈ F. This completes the
proof.

Corollary 3.3. LetKbeanonemptyclosedconvexsubsetofaBanachspaceE.LetT
1
,
T
2
, ,T
m
: K → K be quasi-nonexpansive mappings. Let the sequence {α
n
}
∞
n=1
be as in
Lemma 3.1.Let
{x
n
} be defined by

x
1
∈ K,
x
n+1
=

1 − α
n

x
n
+ α
n
T
1
y
n+m−2
,
y
n+m−2
=

1 − α
n

x
n
+ α
n

T
2
y
n+m−3
,
.
.
.
y
n
=

1 − α
n

x
n
+ α
n
T
m
x
n
, n ≥ 1.
(3.7)
Then,
{x
n
} converges to a common fixed point of the family T
1

,T
2
, ,T
m
if and only if
liminf
n→∞
d(x
n
,F) = 0.
C. E. Chidume and B. Ali 7
For our next theorems, we start by proving the following lemma which will be needed
in the sequel.
Lemma 3.4. Let E be a real uniformly convex Banach space and let K b e a closed convex
nonempty subset of E.LetT
1
,T
2
, ,T
m
: K → K be uniformly continuous asymptotically
quasi-nonexpansive mappings with sequences
{k
in
}
∞
n=1
satisfying k
in
→ 1 as n →∞ and


∞
n=1
(k
in
− 1) < ∞, i = 1,2, ,m.Let{α
n
}
∞
n=1
beasequencein[,1 −

], 
∈
(0,1).Let
{x
n
} be a sequence defined iteratively by (3.1). Then,
lim
n→∞


x
n
− T
1
x
n



=
lim
n→∞


x
n
− T
2
x
n


=···=
lim
n→∞


x
n
− T
m
x
n


=
0. (3.8)
Proof. Since
{x

n
} is bounded, for some x
∗
∈ F, there exists a positive real number γ such
that
x
n
− x
∗

2
≤ γ for all n ≥ 1. By using Lemma 2.2 and the recursion formula (3.1),
we have


y
n
− x
∗


2
=



1 − α
n

x

n
− x
∗

+ α
n

T
n
m
x
n
− x
∗



2
≤

1 − α
n



x
n
− x
∗



2
+ α
n

1+u
mn

2


x
n
− x
∗


2
− α
n

1 − α
n

g



x
n

− T
n
m
x
n



≤


x
n
− x
∗


2
+ α
n

2u
mn
+ u
2
mn



x

n
− x
∗


2
−

2
g



x
n
− T
n
m
x
n



≤


x
n
− x
∗



2
+3w
n
γ −

2
g



x
n
− T
n
m
x
n



.
(3.9)
Also


y
n+1
− x

∗


2
=



1 − α
n

x
n
− x
∗

+ α
n

T
n
m
−1
y
n
− x
∗




2
≤

1 − α
n



x
n
− x
∗


2
+ α
n

1+u
m−1n

2


y
n
− x
∗



2
− α
n

1 − α
n

g



x
n
− T
n
m
−1
y
n



≤

1 − α
n



x

n
− x
∗


2
+ α
n

1+2u
m−1n
+ u
2
m
−1n



y
n
− x
∗


2
−

2
g




x
n
− T
n
m
−1
y
n



≤

1 − α
n



x
n
− x
∗


2
+ α
n


1+3u
m−1n



x
n
− x
∗


2
+3w
n
γ −

2
g



x
n
− T
n
m
x
n




−

2
g



x
n
− T
n
m
−1
y
n



≤


x
n
− x
∗


2
+3w

n
γ −

3
g



x
n
− T
n
m
x
n



+3w
n
γ +

3w
n

2
γ
− 3w
n


3
g



x
n
− T
n
m
x
n



−

2
g



x
n
− T
n
m
−1
y
n




≤


x
n
− x
∗


2
+3
3
w
n
γ −

3

g



x
n
− T
n
m

x
n



+ g



x
n
− T
n
m
−1
y
n



.
(3.10)
Continuing in this fashion we get, using x
n+1
= (1 − α
n
)x
n
+ α
n

T
1
y
n+m−2
,that


x
n+1
− x
∗


2
≤


x
n
− x
∗


2
+3
2m−1
w
n
γ
− 

m+1

g



x
n
− T
n
m
x
n



+
m−1

k=1
g



x
n
− T
n
m
−k

y
n+k−1




,
(3.11)
8 Journal of Inequalities and Applications
so that

m+1

g



x
n
− T
n
m
x
n



+
m−1


k=1
g



x
n
− T
n
m
−k
y
n+k−1




≤


x
n
− x
∗


2
−



x
n+1
− x
∗


2
+3
2m−1
w
n
γ.
(3.12)
This implies that

m+1
∞

n=1

g



x
n
− T
n
m
x

n



+
m−1

k=1
g



x
n
− T
n
m
−k
y
n+k−1




< ∞, (3.13)
and by the property of g,wehave
lim
n→∞



x
n
− T
n
m
x
n


=
lim
n→∞


x
n
− T
n
m
−1
y
n


.
.
.
= lim
n→∞



x
n
− T
n
h
y
n+m−h−1


.
.
.
= lim
n→∞


x
n
− T
n
1
y
n+m−2


=
0
(3.14)
for 2

≤ h<m.
Now,


x
n
− T
h
x
n


≤


x
n
− T
n
h
y
n+m−h−1


+


T
n
h

y
n+m−h−1
− T
h
x
n


, (3.15)
but ( T
n−1
h
y
n+m−h−1
− x
n
) → 0asn →∞, and since T
h
is uniformly continuous we have
that (T
n
h
y
n+m−1
− T
h
x
n
) → 0asn →∞. So, from inequality (3.15), we get lim
n→∞

x
n
−
T
h
x
n
=0. Also for h = m,from(3.14)wehave
lim
n−→ ∞


x
n
− T
n
m
x
n


=
0. (3.16)
Moreover ,


x
n
− T
m

x
n


≤


x
n
− T
n
m
x
n


+


T
n
m
x
n
− T
m
x
n



. (3.17)
Similarly, since
T
n−1
m
x
n
− x
n
→0asn →∞and T
m
is uniformly continuous, we have
(T
n
m
x
n
− T
m
x
n
) → 0asn →∞hence from (3.17)wegetlim
n→∞
x
n
− T
m
x
n
=0, and this

completes the proof.

Theorem 3.5. Let E be a real uniformly c onvex Banach space and let K be a closed convex
nonempty subset of E.LetT
1
,T
2
, ,T
m
: K → K be uniformly continuous asymptotically
quasi-nonexpansive mappings with sequences
{k
in
}
∞
n=1
and {α
n
}
∞
n=1
as in Lemma 3.4.Ifat
C. E. Chidume and B. Ali 9
least one member of
{T
i
}
m
i
=1

is semicompact, then {x
n
} converges strongly to a common fixed
point of the family
{T
i
}
m
i
=1
.
Proof. Assume T
d
∈{T
i
}
m
i
=1
is semicompact. Since {x
n
} is bounded and by Lemma 3.4
x
n
− T
d
x
n
→0asn →∞, there exists a subsequence say {x
n

j
} of {x
n
} con verging
strongly to say x
∈ K. By the uniform continuity of T
d
, x = T
d
x. Using x
n
j
→ x, x
n
j
−
T
i
x
n
j
→0as j →∞, and the continuity of T
i
for each i ∈{1,2, ,m},wehavethat
x
∈

m
i
=1

Fix(T
i
). By Lemma 3.1,limx
n
− x exists, hence, {x
n
} converges strongly to a
common fixed point of the family
{T
i
}
m
i
=1
. 
Corollary 3.6. Let E be a real uniformly convex Banach space and let K be a closed
convex nonempty subset of E.LetT
1
,T
2
, ,T
m
: K → K be uniformly continuous quasi-
nonexpansive mappings. Let
{α
n
}
∞
n=1
be a sequence as in Corollary 3.3.Ifoneof{T

i
}
m
i
=1
is
semicompact, then
{x
n
} defined by (3.7) converges strongly to a common fixed point of the
family
{T
i
}
m
i
=1
.
We now prove weak convergence theorems.
Theorem 3.7. Let E be a real uniformly c onvex Banach space and let K be a closed convex
nonempty subset of E.LetT
1
,T
2
, ,T
m
: K → K be uniformly continuous asymptotically
quasi-nonexpansive mappings with sequences
{k
in

}
∞
n=1
and {α
n
}
∞
n=1
as in Lemma 3.4.IfE
satisfies Opial’s condition and each T
i
, i ∈ I, satisfies condition P,thenthesequence{x
n
}
defined by (3.1) converges weakly to a common fixed point of {T
i
}
m
i
=1
.
Proof. Since
{x
n
} is bounded and E is reflexive, there exists a subsequence say {x
n
k
} of
{x
n

}, converging weakly to some point say p ∈ K.ByLemma 3.4, x
n
k
− T
i
x
n
k
→0as
k
→∞. Condition (P)ofeachT
i
guarantees that p ∈ ω({x
n
})

m
i
=1
Fix(T
i
). If we have
another subsequence of
{x
n
} converging to another point say x

∈ K, by similar argument
we can easily show that x


∈ ω({x
n
})

m
i
=1
Fix(T
i
). Since E satisfies Opial’s condition,
using standard argument we get that x

= p, completing the proof. 
The following corollary follows from Theorem 3.7.
Corollary 3.8. Let K be a nonempty closed convex subset of a real uniformly convex Ba-
nach space E.LetT
1
,T
2
, ,T
m
: K → K be uniformly continuous quasi-nonexpansive map-
pings. Let the sequence
{α
n
}
∞
n=1
be as in Corollary 3.3.IfE satisfies Opial’s condition and
at least one of the T

i
’s i ∈ I satisfies condition P, then the sequence {x
n
} defined by (3.7)
converges weakly to a common fixed point of
{T
i
}
m
i
=1
.
Remark 3.9. Theorem 3.2 extends [8, Theorem 3.2]. In the same way, Theorem 3.5 ex-
tends [8, Theorem 3.4] to finite family of asymptotically quasi-nonexpansive mappings,
andincludesasaspecialcase[8, Theorem 3.7]. In addition, the condition of compactness
on the operators imposed in [8, Theorem 3.4] is weaken, replacing it by semicompactness
in Theorem 3.5. It is clear that if T is compact, then it is semicompact and satisfies con-
dition P. The scheme studied in [12]isimplicitandnot iterative. Our scheme is iterative.
Remark 3.10. Addition of bounded error terms to any of the recurrence relations in our
iteration methods leads to no further generalization.
10 Journal of Inequalities and Applications
Acknowledgments
The authors thank the referee for the very useful comments which helped to improve this
work. The research of the second author was supported by the Japanese Mori Fellowship
of UNESCO at The Abdus Salam International Centre for Theoretical Physics, Trieste,
Italy.
References
[1] C. E. Chidume, H. Zegeye, and N. Shahzad, “Convergence theorems for a common fixed point
of a finite family of nonself nonexpansive mappings,” Fixed Point Theory and Applications,
vol. 2005, no. 2, pp. 233–241, 2005.

[2] C. E. Chidume and B. Ali, “Approximation of common fixed points for finite families of nonself
asymptotically nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and
Applications, vol. 326, no. 2, pp. 960–973, 2007.
[3] M. K. Ghosh and L. Debnath, “Convergence of Ishikawa iterates of quasi-nonexpansive map-
pings,” Journal of Mathematical Analysis and Applications, vol. 207, no. 1, pp. 96–103, 1997.
[4] W. V. Petryshyn and T. E. Williamson Jr., “Strong and weak convergence of the sequence of
successive approximations for quasi-nonexpansive mappings,” Journal of Mathematical Analysis
and Applications, vol. 43, no. 2, pp. 459–497, 1973.
[5] L. Qihou, “Iterative sequences for asymptotically quasi-nonexpansive mappings,” Journal of
Mathematical Analysis and Applications, vol. 259, no. 1, pp. 1–7, 2001.
[6] L. Qihou, “Iterative sequences for asym ptotically quasi-nonexpansive mappings with error
member,” Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 18–24, 2001.
[7] L. Qihou, “Iteration sequences for asymptotically quasi-nonexpansive mapping with an error
member of uniform convex Banach space,” Journal of Mathematical Analysis and Applications,
vol. 266, no. 2, pp. 468–471, 2002.
[8] N. Shahzad and A. Udomene, “Approximating common fixed points of two asymptotically
quasi-nonexpansive mappings i n Banach spaces,” Fixed Point Theory and Applications, vol. 2006,
Article ID 18909, 10 pages, 2006.
[9] K. K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa
iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–
308, 1993.
[10] W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical So-
ciety, vol. 4, no. 3, pp. 506–510, 1953.
[11] S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical
Society, vol. 44, no. 1, pp. 147–150, 1974.
[12] Z. H. Sun, “Strong convergence of an implicit iteration process for a finite family of asymptoti-
cally quasi-nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 286,
no. 1, pp. 351–358, 2003.
[13] Z B. Xu and G. F. Roach, “Characteristic inequalities of uniformly convex and uniformly
smooth Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 157, no. 1, pp.

189–210, 1991.
C. E. Chidume: Mathematics Section, The Abdus Salam International Centre for Theoretical Physics,
34014 Trieste, Italy
Email address:
Bashir Ali: Department of Mathematical Sciences, Bayero University, Kano, Nigeria
Email address: