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Petryshyn and Williamson [4] proved necessary and sufficient conditions for the Pi-card and Mann [10] iterative sequences to strongly converge to a fixed point of a quasi-nonexpansive map

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

LetE be a real Banach space, K a closed convex nonempty subset of E, and T1,T2, , T m:

K → K asymptotically quasi-nonexpansive mappings with sequences (resp.) { k in } ∞ n =1 sat-isfyingk in →1 as n → ∞, 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 }byx1∈ K, x n+1 =(1− α n)x n+

α n T1n y n+m −2, y n+m −2=(1− α n)x n+α n T2n y n+m −3, , y n =(1− α n)x n+α n T n

m x n,n ≥1,

m ≥2 Letm

i =1F(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 − T y  ≤  x − y for everyx, y ∈ K, and quasi-nonexpansive

ifF(T) : = { x ∈ K : Tx = x } = ∅and Tx − p  ≤  x − p for everyx ∈ K and p ∈ F(T).

The mappingT is called asymptotically nonexpansive if there exists a sequence { k n } ⊂

[1,∞) withk n →1 asn → ∞such that for everyn ∈ N,

T n x − T n y  ≤ k n  x − y  for everyx, y ∈ K. (1.1)

IfF(T) = ∅and there exists a sequence{ k n } ⊂[1,∞) withk n →1 asn → ∞such that for

n ∈ N,

T n x − p  ≤ k n  x − p  for everyx ∈ K, (1.2) andp ∈ 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] proved necessary and sufficient 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] extended the results in [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 mappingT 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 closed convex subset of a Banach space E Let { T i, ∈ I } be m asymptotically quasi-nonexpansive self-mappings of K with sequences

{1 +u in } n , =1, 2, , m, respectively Suppose that F : =m

i =1 F(T i)= ∅ and that x0∈ K,

{ α n } ⊂(s, 1 − s) for some s ∈ (0, 1),∞

n =1u 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 i k 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) =infx ∗ ∈ 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 } be m uniformly L-Lipschitzian asymp-totically quasi-nonexpansive self-mappings of K with sequences {1 +u in } n , =1, 2, , m, respectively Suppose that F : =m

i =1F(T i)= ∅ and that x0∈ K, { α n } ⊂(s, 1 − s) for some

s ∈(0, 1),∞

n =1u in < ∞ for all i ∈ I If there exists one member T ∈ { T i, ∈ I } which is semi-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 }

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

Theorem 1.3 (see [8]) Let E be a real Banach space and let K be a nonempty closed convex subset of E Let S, 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 <

∞ , and F : = F(S) ∩ F(T) = { x ∈ K : Sx = Tx = x } = ∅ Let { α n } and { β n } be sequences in [0, 1] From arbitrary x1∈ 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

Then, { x n } converges strongly to some common fixed point of S and T if and only if

lim infn →∞ 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 Let S, 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 < ∞ , and F : = F(S) ∩ F(T) = { x ∈ K : Sx = Tx = x } = ∅ Let { α n } and { β n } be se-quences in [ , 1−  ] for some  ∈ (0, 1) From arbitrary x1∈ 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

x1∈ K,

x n+1 = P 

1− α1n



x n+α1n T1



PT1

n −1

y n+m −2 ,

y n+m −2 = P 

1− α2n

x n+α2n T2



PT2

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 T1,T2, , T m:K → K be a

family of mappings Define a sequence{ x n }by

x1∈ K,

x n+1 =1− α n



x n+α n T1n y n+m −2,

y n+m −2=1− α n



x n+α n T2n y n+m −3,

y n =1− α n



x n+α n T m n 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 familyT1,T2, , T m of asymptotically quasi-nonexpansive mappings We also prove strong

and weak convergence theorems for the family in a uniformly convex Banach spaces Our results generalize and improve some recent important results (seeRemark 3.9)

2 Preliminaries

LetE be a real normed linear space The modulus of convexity of E is the function δ E: (0, 2]→[0, 1] defined by

δ E()=inf

1−

x + y2 : x  =  y  =1, =  x − y 

E is called uniformly convex if and only if δ E()> 0 ∀ ∈(0, 2]

A mappingT with domain D(T) and range R(T) in E is said to be demiclosed at p

if whenever{ x n }is a sequence inD(T) such that x n  x ∗ ∈ D(T) and Tx n → p then

Tx ∗ = p.

A mappingT : K → K is said to be semicompact if, for any bounded sequence { x n }in

K such that  x n − Tx n  →0 asn → ∞, there exists a subsequence say{ x n j }of{ x n }such that{ x n j }converges strongly to somex ∗inK.

A Banach spaceE is said to satisfy Opial’s condition if for any sequence { x n } inE,

x n  x implies that

lim inf

n →∞ x n − x< lim inf

n →∞ x n − y  ∀ y ∈ E, y = x. (2.2)

We will say that a mappingT 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 }withx n j  x ∗and (I − T)x n j →0 asj → ∞, 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 =1σ 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)yp

≤ 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 }byI and we always assume F : =m

i =1F(T i)= ∅

Lemma 3.1 Let E be a real normed linear space and let K be a nonempty, closed convex subset of E Let T1,T2, , T m:K → K be asymptotically quasi-nonexpansive mappings with sequence { k in } ∞

n =1satisfying k in → 1 as n → ∞ and∞

n =1(k in −1)< ∞ , ∈ I Let { α n } ∞

n =1be

a sequences in [ , 1−  ],  ∈ (0, 1) Let { x n } be a sequence defined iteratively by

x1∈ K,

x n+1 =1− α n

x n+α n T n

1y n+m −2,

y n+m −2=1− α n



x n+α n T2n y n+m −3,

.

y n =1− α n



x n+α n T m n 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) =infx ∗ ∈ F  x n − x ∗ 

Proof Set k in =1 +u inso that∞

n =1u in < ∞for eachi ∈ I Let w n:=m

i =1u in Letx ∗ ∈ F.

Then we have, for some positive integerh, 2 ≤ h < m,

x n+1 − x ∗  = 1− α n



x n+α n T1n y n+m −2− x ∗

≤1− α nx n − x ∗+α n

1 +u1ny n+m −2 − x ∗

≤1− α nx n − x ∗

+α n

1 +u1n

 

1− α nx n − x ∗+α n

1 +u2ny n+m −3 − x ∗

≤1− α nx n − x ∗+α n

1− α n

1 +u1nx n − x ∗

+···+

α nh −1

1− α n

1 +u1n



1 +u2n



···1 +u h −1 nx n − x ∗

+···+

α nm

1 +u1n

1 +u2n

···1 +u mnx n − x ∗

≤x n − x ∗ 1 +u1n+u2n

1 +u1n

+u3n

1 +u1n

1 +u2n

+···

+u mn

1 +u1n



1 +u2n



···1 +u m −1 n

≤x n − x ∗ 1 +m

1



w n+



m

2



w2

n+···+



m m



w m n



≤x n − x ∗1 +δ m w n

≤x n − x ∗e δ m w n

≤x1− x ∗e δ m∞

n =1w n < ∞,

(3.2)

whereδ mis a positive real number defined byδ m:=m

1



+

m

2



+···+

m m



This implies that{ x n }is bounded and so there exists a positive integerM such that

x n+1 − x ∗  ≤  x n − x ∗+δ m Mw n . (3.3)

Since (3.3) is true for eachx ∗inF, we have

d

x n+1,F

≤ d

x n,F

Theorem 3.2 Let K be a nonempty closed convex subset of a Banach space E Let T1,T2, ,

T m:K → K be asymptotically quasi-nonexpansive mappings with sequences { k in } ∞

n =1 and

{ α n } ∞

n =1as in Lemma 3.1 Let { x n } be defined by ( 3.1 ) Then, { x n } converges to a common fixed point of the family T1,T2, , T m if and only if lim inf n →∞ d(x n,F) = 0.

Proof The necessity is trivial We prove the sufficiency Let liminfn →∞ d(x n,F) =0 Since limn →∞ d(x n,F) exists byLemma 3.1, we have that limn →∞ d(x n,F) =0 Thus, given > 0

there exist a positive integerN0 and b ∗ ∈ F such that for all n ≥ N0  x n − b ∗  <  /2.

Then, for anyk ∈ N, we have forn ≥ N0,

x n+k − x n  ≤  x n+k − b ∗+b ∗ − x n< 

2+



2= , (3.5) and so{ x n }is Cauchy Let limn →∞ x n = b We need to show that b ∈ F Let T i ∈ { T1,T2, ,

T m } Since limn →∞ d(x n,F) =0, there existsN ∈ Nsufficiently large and b∗ ∈ F such that

n ≥ N implies  b − x n  <  /6(1 + w1), b ∗ − x n  <  /6(1 + w1) Then, b ∗ − b  <  /3(1 +

w1) Thus, we have the following estimates, forn ≥ N and arbitrary T 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 +w1 b ∗ − b

< 

3

1 +w1

3

1 +w1

+

3 ≤ 

(3.6)

This implies thatb ∈Fix(T i) for all i =1, 2, , m and thus b ∈ F This completes the

Corollary 3.3 Let K be a nonempty closed convex subset of a Banach space E Let T1,

T2, , 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

x1∈ K,

x n+1 =1− α n

x n+α n T1y n+m −2,

y n+m −2=1− α n



x n+α n T2y 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 T1,T2, , T m if and only if

lim infn →∞ d(x n,F) = 0.

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 be a closed convex nonempty subset of E Let T1,T2, , 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)< ∞ , =1, 2, , m Let { α n } ∞

n =1 be a sequence in [ , 1−  ],  ∈ (0, 1) Let

{ x n } be a sequence defined iteratively by ( 3.1 ) Then,

lim

n →∞x n − T1x n  =lim

n →∞x n − T2x n  = ··· =lim

n →∞x n − T m x n  =0. (3.8)

Proof Since { x n }is bounded, for somex ∗ ∈ F, there exists a positive real number γ such

that x n − x ∗ 2≤ γ for all n ≥1 By usingLemma 2.2and 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 nx n − x ∗ 2

+α n

1 +u mn2 x n − x ∗ 2

− α n

1− α n

gx n − T n

m x n

≤x n − x ∗ 2

+α n



2u mn+u2mnx n − x ∗ 2

− 2gx n − T n

m x n

≤x n − x ∗ 2

+ 3w n γ − 2gx n − T n

m x n.

(3.9) Also

y n+1 − x ∗ 2

=1− α n

x n − x ∗

+α n

T m n −1 y n − x ∗ 2

≤1− α nx n − x ∗ 2

+α n



1 +u m −1n

2 y n − x ∗ 2

− α n



1− α n



gx n − T n

m −1y n

≤1− α nx n − x ∗2

+α n

1 + 2u m −1 n+u2

m −1ny n − x ∗2

− 2gx n − T n

m −1y n  ≤ 1− α nx n − x ∗ 2

+α n

1 + 3u m −1 nx n − x ∗ 2

+ 3w n γ − 2gx n − T n

m x n

− 2gx n − T n

m −1 y n

≤x n − x ∗ 2

+ 3w n γ − 3gx n − T n

m x n+ 3w n γ +

3w n2

γ

−3w n 3gx n − T n

m x n −2gx n − T n

m −1 y n

≤x n − x ∗ 2

+ 33w n γ − 3 

gx n − T n

m x n+gx n − T n

m −1 y n.

(3.10)

Continuing in this fashion we get, usingx n+1 =(1− α n)x n+α n T1y n+m −2, that

x n+1 − x ∗ 2

≤x n − x ∗ 2

+ 32m −1 w n γ

−  m+1



gx n − T n

m x n+m−1

k =1

gx n − T n

m − k y n+k −1 , (3.11)

so that

 m+1



gx n − T n

m x n+m−1

k =1

gx n − T n

m − k y n+k −1 

≤x n − x ∗ 2

−x n+1 − x ∗ 2

+ 32m −1 w n γ.

(3.12)

This implies that

 m+1∞

n =1



gx n − T n

m x n+m−1

k =1

gx n − T n

m − k y n+k −1< ∞, (3.13)

and by the property ofg, we have

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

1y n+m −2 =0

(3.14)

for 2≤ h < m.

Now,

x n − T h x n  ≤  x n − T h n y n+m − h −1 +T n

h y n+m − h −1− T h x n, (3.15)

but (T h n −1y n+m − h −1− x n)→0 as n → ∞, and sinceT h is uniformly continuous we have that (T n

h y n+m −1 − T h x n)→0 as n → ∞ So, from inequality (3.15), we get limn →∞  x n −

T h x n  =0 Also forh = m, from (3.14) we have

lim

n −→∞x n − T n

m x n  =0. (3.16) Moreover,

x n − T m x n  ≤  x n − T m n x n+T n

m x n − T m x n. (3.17)

Similarly, since T n −1

m x n − x n  →0 asn → ∞andT m is uniformly continuous, we have (T n

m x n − T m x n)→0 asn → ∞hence from (3.17) we get limn →∞  x n − T m x n  =0, and this

Theorem 3.5 Let E be a real uniformly convex Banach space and let K be a closed convex nonempty subset of E Let T1,T2, , T m:K → K be uniformly continuous asymptotically quasi-nonexpansive mappings with sequences { k in } ∞

n =1and { α n } ∞

n =1as in Lemma 3.4 If at

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 =1is semicompact Since{ x n }is bounded and byLemma 3.4

 x n − T d x n  →0 as n → ∞, there exists a subsequence say { x n j } of { x n } converging strongly to sayx ∈ 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  →0 as j → ∞, and the continuity of T i for each i ∈ {1, 2, , m }, we have that

x ∈m

i =1Fix(T i) ByLemma 3.1, lim x n − x exists, hence,{ x n }converges strongly to a common fixed point of the family{ T i } m

Corollary 3.6 Let E be a real uniformly convex Banach space and let K be a closed convex nonempty subset of E Let T1,T2, , T m:K → K be uniformly continuous quasi-nonexpansive mappings Let { α n } ∞

n =1be a sequence as in Corollary 3.3 If one of { T i } m

i =1is 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 convex Banach space and let K be a closed convex nonempty subset of E Let T1,T2, , 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 If E satisfies Opial’s condition and each T i , ∈ I, satisfies condition P, then the sequence { 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 andE 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  →0 as

k → ∞ Condition (P) of each T iguarantees thatp ∈ ω( { x n })m

i =1Fix(T i) If we have another subsequence of{ x n }converging to another point sayx  ∈ K, by similar argument

we can easily show thatx  ∈ ω( { x n })m

i =1Fix(T i) SinceE satisfies Opial’s condition,

using standard argument we get thatx  = p, completing the proof. 

The following corollary follows fromTheorem 3.7

Corollary 3.8 Let K be a nonempty closed convex subset of a real uniformly convex Ba-nach space E Let T1,T2, , T m:K → K be uniformly continuous quasi-nonexpansive map-pings Let the sequence { α n } ∞

n =1 be as in Corollary 3.3 If E 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 3.2extends [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, and includes as a special case [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

con-ditionP The scheme studied in [12] is implicit and not 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

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 asymptotically 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 in 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:chidume@ictp.trieste.it

Bashir Ali: Department of Mathematical Sciences, Bayero University, Kano, Nigeria

Email address:bashiralik@yahoo.com

For our next theorems, we start by proving the following lemma... . (3.3)

Since (3.3) is true for eachx ∗inF, we have

d... in Lemma 3.4 If at

least one member of { T i } m

i