5_253356_dehghan.dvi


EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS

Vol. 5, No. 1, 2012, 45-54

ISSN 1307-5543 – www.ejpam.com

SPECIAL ISSUE FOR THE INTERNATIONAL CONFERENCE ON APPLIED ANALYSIS AND ALGEBRA

29 JUNE - 02 JULY 2011, ISTANBUL TURKEY

Weak and Strong Convergence of a Two-Step Iterations for a

Finite Family of Generalized Asymptotically Quasi-Nonexpansive

Mappings

Hossein Dehghan 1,∗, Amir Gharajelo 2

1 Department of Mathematics, Institute for Advanced Studies in Basic Sciences (IASBS), Zanjan,

Iran
2 Department of Mathematics, Roozbeh Institute of Higher Education, Zanjan, Iran

Abstract. In this paper, we introduce and study a new two-step iterative scheme to approximate com-

mon fixed points for a finite family of generalized asymptotically quasi-nonexpansive mappings. We

establish several strong and weak convergence results of the proposed algorithm in Banach spaces.

These results generalize and refine many known results in the current literature.

2000 Mathematics Subject Classifications: AMS 47H09, 47H10

Key Words and Phrases: Generalized asymptotically quasi-nonexpansive mapping, Mann-type itera-

tion, Ishikawa-type iteration, Uniformly convex Banach space, Common fixed point.

1. Introduction

In recent years, one-step and two-step iterative schemes( including Mann iteration and

Ishikawa iteration processes as the most important cases) have been studied extensively by

many authors to approximate fixed points of various classes of mappings (see for example

[1, 4, 5, 8, 10]).

Approximating common fixed points of a finite family of nonlinear mappings plays an

important role in solving systems of equations and inequalities that often arise in applied

mathematics. For a finite family of mappings {Ti : i = 1,2, . . . , m}, it is desirable to devise

a iteration scheme which extends the modified Mann iteration and the modified Ishikawa

∗Corresponding author.

Email addresses: h_dehghan�iasbs.a
.ir, hossein.dehgan�gmail.
om (H. Dehghan),amirgharajelo�yahoo.
om (A. Gharajelo)

http://www.ejpam.com 45 c© 2012 EJPAM All rights reserved.



H. Dehghan, A. Gharajelo / Eur. J. Pure Appl. Math, 5 (2012), 45-54 46

iteration from one mapping to a finite family of mappings. Thereby, to achieve this goal, we

introduce a new two-step iterative scheme for a finite family of mappings as follows:

Let C be a nonempty convex subset of a real Banach space X and {Ti : i = 1,2, . . . , m} be

a family of self-mappings of C . For a given x1 ∈ C , compute the sequences {xn} and {yn} by

the iterative schemes

yn =

m
∑

i=1

ainT n
i xn+ bn xn

xn+1 =

m
∑

i=1

�

αinT n
i yn +βinT n

i xn

�

+ γn xn, n≥ 1, (1)

where {ain}, {bn}, {αin}, {βin}, and {γn} are appropriate sequences in [0,1] for all

i ∈ {1,2, . . . , m} such that bn+
∑m

i=1 ain = γn+
∑m

i=1

�

αin + βin

�

= 1 for each n≥ 1.

If T1 = T2 = · · · = Tm, then the iteration process (1) reduces to

yn = anT n xn+ (1− an)xn

xn+1 = αnT n yn + βnT n xn+ γn xn, n≥ 1, (2)

where {an}, {αn}, {βn}, and {γn} are appropriate sequences in [0,1] such that αn+βn+γn = 1

for each n≥ 1.

Clearly, the iteration process (2) includes the modified Ishikawa iteration

yn = anT n xn+ (1− an)xn, (3)

xn+1 = αnT n yn + (1−αn)xn, n≥ 1,

where {an} and {αn} are appropriate sequences in [0,1], and the modified Mann iteration

xn+1 = anT n xn+ (1− an)xn, n≥ 1, (4)

where {an} is appropriate sequences in [0,1]. Therefore, (1) generalizes the modified Ishikawa

iteration and the modified Mann iteration from one mapping to the finite family of mappings

{T j : j = 1,2, . . . , m}.
The aim of this paper is to obtain some strong and weak convergence results for the itera-

tive process (1) of a finite family of generalized asymptotically quasi-nonexpansive mappings

in Banach spaces.

Now, we recall the well-known concepts and results. For convenience, we use the nota-

tions limn ≡ limn→∞, lim infn ≡ lim infn→∞ and lim supn ≡ lim supn→∞. Let C be a nonempty

subset of a real Banach space X and T be a self-mapping of C . The fixed point set of T is

denoted by F(T ) = {x ∈ C : T x = x}. The mapping T is called

(i) asymptotically nonexpansive if there exists a sequence {rn} in [0,∞) with limn rn = 0

such that ‖T n x − T n y‖ ≤ (1+ rn)‖x − y‖ for all x , y ∈ C and each n≥ 1;

(ii) asymptotically quasi-nonexpansive if F(T ) 6= ; and there exists a sequence {rn} in [0,∞)
with limn rn = 0 such that ‖T n x− p‖ ≤ (1+ rn)‖x− p‖ for all x ∈ C , p ∈ F(T ) and each

n≥ 1;



H. Dehghan, A. Gharajelo / Eur. J. Pure Appl. Math, 5 (2012), 45-54 47

(iii) generalized asymptotically quasi-nonexpansive [7] if F(T ) 6= ; and there exist two se-

quences {rn} and {sn} in [0,∞) with limn rn = limn sn = 0 such that

‖T n x − p‖ ≤ (1+ rn)‖x − p‖+ sn for all x ∈ C , p ∈ F(T ) and each n≥ 1;

(iv) uniformly L-Lipschitz if there exists constant L > 0 such that ‖T n x − T n y‖ ≤ L‖x − y‖
for all x , y ∈ C and each n≥ 1.

It is clear that a generalized asymptotically quasi-nonexpansive mapping is to unify various

classes of mappings associated with the class of asymptotically quasi-nonexpansive mapping,

asymptotically nonexpansive mappings and nonexpansive mappings. However, the converse

of each of above statement may be not true. The example, shows that a generalized asymp-

totically quasi-nonexpansive mapping is not an asymptotically quasi-nonexpansive mapping,

can be found in [7].

A family of self-mappings {Ti : i = 1,2, . . . , m} of C is said to satisfy Condition (A′′) [3] if

there exists a nondecreasing function f : [0,∞)→ [0,∞) with f (0) = 0 and f (r) > 0 for all

r ∈ (0,∞) such that f (d(x , F)) ≤ ‖xn − Ti xn‖ for some 1 ≤ i ≤ m and for all x ∈ C where

d(x , F) = inf{‖x− y‖ : y ∈ F =
⋂m

i=1 F(Ti)}. We recall that a Banach space X is said to satisfy

Opial’s condition [6] if xn converging to x weakly and x 6= y imply that

lim sup
n




xn− x



< lim sup
n




xn− y



 .

In the sequel, the following lemmas are needed to prove our main results.

Lemma 1 ([9, Lemma 1]). Let {an}, {bn} and {δn} be sequences of nonnegative real numbers

satisfying the inequality

an+1 ≤ (1+δn)an+ bn, ∀n≥ 1.

If
∑∞

n=1 δn <∞ and
∑∞

n=1 bn <∞, then limn an exists.

Lemma 2 ([2, Lemma 1.2] ). Let k ≥ 2 and {y(1)n }, . . . , {y(k)n } be sequences in a uniformly con-

vex Banach space X with lim supn ‖y
(i)
n ‖ ≤ a for each i = 1,2, . . . , k and for some a ≥ 0. Suppose

{α(1)n }, . . . , {α(k)n } be sequences in [0,1] such that
∑k

i=1α
(i)
n = 1 and limn ‖

∑k

i=1α
(i)
n y(i)n ‖ = a.

If lim infnα
(i)
n > 0 and lim infnα

( j)
n > 0 for some i, j ∈ {1,2, . . . , k}, then limn ‖y

(i)
n − y

( j)
n ‖= 0.

2. Convergence in Banach spaces

The aim of this section is to establish the strong convergence of the iterative process (1)

to converge to a common fixed point of a finite family of generalized asymptotically quasi-

nonexpansive mappings in a Banach space. To proceed in this direction, the following lemma

is needed.

Lemma 3. Let X be a real Banach space, C be a nonempty closed convex subset of X and

{Ti : i = 1,2, . . . , m} be a family of generalized asymptotically quasi-nonexpansive self-mappings

of C with the sequences {r(1)n }, . . . , {r(m)n } and {s(1)n }, . . . , {s(m)n } such that
∑∞

n=1 r(i)n < ∞ and
∑∞

n=1 s(i)n <∞ for each i = 1,2, . . . , m. Let {xn} be the sequence defined by (1). If

F =
⋂m

i=1 F(Ti) 6= ;, then we have the following conclusions.



H. Dehghan, A. Gharajelo / Eur. J. Pure Appl. Math, 5 (2012), 45-54 48

(i) limn ‖xn− p‖ exists for all p ∈ F.

(ii) limn d(xn, F) exists.

(iii) If lim infnαin > 0 for some i ∈ {1,2, . . . , m}, then limn ‖yn − p‖ = limn ‖xn − p‖ for all

p ∈ F.

Proof. Let p ∈ F , rn =max1≤i≤m rin and sn =max1≤i≤m sin. For each n≥ 1, we note that

‖yn − p‖ =











m
∑

i=1

ainT n
i xn+ bn xn− p











≤
m
∑

i=1

ain




T n
i xn− p




+ bn‖xn− p‖

≤
m
∑

i=1

ain

�

(1+ rin)‖xn− p‖+ sin)
�

+ bn(1+ rn)‖xn− p‖

≤ (1+ rn)‖xn− p‖+ sn. (5)

It follows from (5) that

‖xn+1− p‖ ≤
m
∑

i=1

�

αin‖T
n
i yn − p‖+ βin‖T

n
i xn− p‖

�

+ γn‖xn− p‖

≤
m
∑

i=1

�

αin[(1+ rin)‖yn− p‖+ sin] + βin[(1+ rin)‖xn− p‖+ sin]
�

+γn(1+ rn)
2‖xn− p‖

≤
m
∑

i=1

�

αin[(1+ rn)[(1+ rn)‖xn− p‖+ sn] + sn]

+βin[(1+ rn)
2‖xn − p‖+ sn]

�

+ γn(1+ rn)
2‖xn− p‖

≤ (1+ rn)
2‖xn− p‖+ (2+ rn)sn. (6)

Since {rn} and {sn} are independent of p, by taking infimum over all p ∈ F in both sides of

(6), we obtain

d(xn+1, F) ≤ (1+ rn)
2d(xn, F) + (2+ rn)sn. (7)

Using Lemma 1, the conclusions (i) and (ii) of lemma follow from (6) and (7), respectively.

(iii) Since limn ‖xn− p‖ exists, it follows from (5) that lim supn ‖yn − p‖ ≤ limn ‖xn− p‖.
Also, by (6)

‖xn+1− p‖ ≤
m
∑

i=1

�

αin[(1+ rn)‖yn− p‖+ sn] + βin[(1+ rn)‖xn− p‖+ sn]
�

+γn(1+ rn)
2‖xn− p‖



H. Dehghan, A. Gharajelo / Eur. J. Pure Appl. Math, 5 (2012), 45-54 49

≤ (1+ rn)
2

 

m
∑

i=1

αin‖yn − p‖+

 

1−
m
∑

i=1

αin

!

‖xn − p‖

!

+(2+ rn)sn.

for all n≥ 1. Since lim infn

∑m

i=1αin > 0, we have

‖xn+1− p‖ − (1+ rn)
2‖xn− p‖

(1+ rn)
2
∑m

i=1αin

+ ‖xn− p‖ ≤ ‖yn − p‖+
(2+ rn)sn

(1+ rn)
2
∑m

i=1αin

for sufficiently large numbers n. By taking lim infn in both sides, we obtain

lim
n
‖xn− p‖ ≤ lim inf

n
‖yn − p‖.

This completes the proof.

Theorem 1. Let X , C, T1, . . . , Tm and {xn} be as in Lemma 3 with the restriction that

F =
⋂m

i=1 F(Ti) be nonempty and closed. Then {xn} converges strongly to a common fixed point

of the family of mappings if and only if lim infn d(xn, F) = 0.

Proof. The necessity is obvious and then we prove only the sufficiency. Let p ∈ F . From

Lemma 3(i), we know that limn ‖xn − p‖ exists and hence {‖xn − p‖} is bounded. We put

M = supn≥1 ‖xn − p‖. It follows from (6) that

‖xn+1 − p‖ ≤ (1+δn)‖xn− p‖+ dn, (8)

where δn = (1+ rn)
2 − 1 and dn = (2+ rn)sn so that

∑∞
n=1 δn <∞ and

∑∞
n=1 dn <∞. Thus,

for positive integers k and n, we have

‖xn+k − p‖ ≤ ‖xn+k−1− p‖+Mδn+k−1 + dn+k−1

≤ ‖xn+k−2− p‖+M(δn+k−2 + δn+k−1) + dn+k−2+ dn+k−1

...

≤ ‖xn− p‖+M

n+k−1
∑

i=n

δi +

n+k−1
∑

i=n

di. (9)

It follows from Lemma 3(ii) that limn d(xn, F) exists. Thus limn d(xn, F) = 0. Now, we show

that {xn} is a Cauchy sequence. By limn d(xn, F) = 0,
∑∞

n=1 δn < ∞ and
∑∞

n=1 dn < ∞, we

get that for any ε > 0, there exists a positive integer n0 such that

d(xn0
, F) <

ε

6
,

∞
∑

i=n0

δi <
ε

3M
and

∞
∑

i=n0

di <
ε

3
.

Therefore, there exists p0 ∈ F such that ‖xn0
− p0‖ < ε/6. It follows from (9) that

‖xn0+k − xn0
‖ ≤ ‖xn0+k − p0‖+ ‖xn0

− p0‖



H. Dehghan, A. Gharajelo / Eur. J. Pure Appl. Math, 5 (2012), 45-54 50

≤ 2‖xn0
− p0‖+M

n0+k−1
∑

i=n0

δi +

n0+k−1
∑

i=n0

di

< 2

�ε

6

�

+M

� ε

3M

�

+
ε

3
= ε

for all k ≥ 1. Thus {xn} is a Cauchy sequence and hence xn→ q for some q ∈ C . Moreover,

d(q, F) ≤ ‖xn− q‖+ d(xn, F)→ 0 as n→∞.

Since F is closed, then q ∈ F . This completes the proof.

3. Convergence in Uniformly Convex Banach Spaces

In this section, some weak and strong convergence results are established for iterative

scheme (1) in uniformly convex Banach spaces without using the condition

lim infn d(xn, F) = 0 appearing in the preceding section. For this we have to consider Condi-

tion (A′′) and Opial property.

The following lemma has the important ingredients for proving our main results.

Lemma 4. Let X be a uniformly convex Banach space, C be a nonempty closed convex subset of

X and {Ti : i = 1,2, . . . , m} be a family of uniformly L-Lipschitz and generalized asymptotically

quasi-nonexpansive self-mappings of C with the sequences {r(1)n }, . . . , {r(m)n } and {s(1)n }, . . . , {s(m)n }
such that

∑∞
n=1 r(i)n <∞ and

∑∞
n=1 s(i)n <∞ for each i = 1,2, . . . , m. Let {xn} be the sequence

defined by (1). Then we have the following conclusions.

(i) If 0< lim infnαin ≤ lim supn(1− γn)< 1, then limn ‖T
n
i

yn − xn‖ = 0.

(ii) If 0< lim infnβin ≤ lim supn(1− γn)< 1, then limn ‖T
n
i xn− xn‖= 0.

(iii) If lim infnα jn > 0 for some j = 1,2, . . . , m and 0 < lim infn ain ≤ lim supn(1− bn) < 1,

then limn ‖T
n
i xn− xn‖ = 0.

(iv) If limn ‖T
n
i xn− xn‖ = 0 for all i = 1,2, . . . , m, then limn ‖Ti xn− xn‖ = 0 for all

i = 1,2, . . . , m.

Proof. (i) Let p ∈ F . By Lemma 3(i), limn ‖xn − p‖ exists. Let limn ‖xn − p‖ = a for some

a ≥ 0. Then,

lim sup
n
‖T n

i xn− p‖ ≤ lim sup
n
((1+ rin)‖xn− p‖+ sin)≤ a (10)

for all i = 1,2, . . . , m. Also, by taking lim supn in both sides of (5), we obtain that

lim sup
n
‖yn − p‖ ≤ lim sup

n
‖xn− p‖ = a

and so

lim sup
n
‖T n

i yn − p‖ ≤ lim sup
n
((1+ rin)‖yn − p‖+ sin) ≤ a (11)



H. Dehghan, A. Gharajelo / Eur. J. Pure Appl. Math, 5 (2012), 45-54 51

for all i = 1,2, . . . , m. Moreover, we note that

a = lim
n
‖xn+1 − p‖ = lim

n











m
∑

i=1

�

αinT n
i yn + βinT n

i xn

�

+ γn xn− p











= lim
n











m
∑

i=1

�

αin(T
n
i yn− p) + βin(T

n
i xn− p)

�

+ γn(xn− p)











.

This together with (10), (11) and Lemma 2 implies that the conclusions (i) and (ii) of lemma

are satisfied.

Next, we shall prove (iii). Since lim infnα jn > 0, it follows from Lemma 3(iii) that

limn ‖yn − p‖ = a. Therefore,

a = lim
n




yn − p



 = lim
n











m
∑

i=1

ainT n
i xn+ bn xn− p











= lim
n











m
∑

i=1

ain(T
n
i xn− p) + bn(xn− p)











.

This together with (10) and Lemma 2 implies that limn ‖T
n
i xn− xn‖= 0.

(v) Using (1), we have

‖yn − xn‖ ≤
m
∑

i=1

ain‖T
n
i xn− xn‖ → 0,

‖T n
i yn − xn‖ ≤ ‖T

n
i yn − T n

i xn‖+ ‖T
n
i xn− xn‖

≤ L‖yn − xn‖+ ‖T
n
i xn− xn‖ → 0.

Then,

‖xn+1 − xn‖ ≤
m
∑

i=1

�

αin‖T
n
i yn − xn‖+ βin‖T

n
i xn− xn‖)

�

→ 0. (12)

For each i = 1,2, · · · , m, we have

‖Ti xn− xn‖ ≤ ‖Ti xn− T n+1
i

xn‖+ ‖T
n+1
i

xn− T n+1
i

xn+1‖+ ‖T
n+1
i

xn+1 − xn+1‖

+‖xn+1− xn‖

≤ L‖xn− T n
i xn‖+ L‖xn − xn+1‖+ ‖T

n+1
i

xn+1− xn+1‖

+‖xn+1− xn‖

which together with (12) implies that

lim
n
‖T1 xn− xn‖ = lim

n
‖T2 xn− xn‖ = · · ·= lim

n
‖Tm xn− xn‖= 0.

This completes the proof of lemma.



H. Dehghan, A. Gharajelo / Eur. J. Pure Appl. Math, 5 (2012), 45-54 52

Lemma 5. Let X , C and T1, T2, . . . , Tm be as in Lemma 4 and {xn} be the sequence defined by

(1) such that the parameters satisfy one of the following control conditions:

(C1) 0< lim infnαin ≤ lim supn(1− γn)< 1 for all i = 1,2, . . . , m and lim supn(1− bn)L < 1;

(C2) 0< lim infnβin ≤ lim supn(1− γn)< 1 for all i = 1,2, . . . , m;

(C3) lim infnα jn > 0 for some j ∈ {1,2, . . . , m} and 0< lim infn ain ≤ lim supn(1− bn)< 1 for

all i = 1,2, . . . , m.

Then limn ‖T
n
i xn− xn‖= 0 for all i = 1,2, . . . , m, and so by Lemma 4(iv), limn ‖Ti xn− xn‖ = 0

for all i = 1,2, . . . , m.

Proof. (C1) It follows from Lemma 4(i) that limn ‖T
n
i

yn − xn‖ = 0 for all i = 1,2, . . . , m.

Using (1) we have

‖yn − xn‖ ≤
m
∑

i=1

ain‖T
n
i xn− xn‖

≤
m
∑

i=1

ain(‖T
n
i xn− T n

i yn‖+ ‖T
n
i yn − xn‖)

≤
m
∑

i=1

ain(L‖yn − xn‖+ ‖T
n
i yn − xn‖)

= (1− bn)L‖yn − xn‖+
m
∑

i=1

ain‖T
n
i yn − xn‖.

Thus, limn(1− (1− bn)L)‖yn − xn‖= 0. Since lim supn(1− bn)L < 1, then

lim
n
‖yn − xn‖ = 0. (13)

Next, we observe that

‖T n
i xn− xn‖ ≤ ‖T

n
i xn− T n

i yn‖+ ‖T
n
i yn − xn‖

≤ L‖yn − xn‖+ ‖T
n
i yn − xn‖.

This together with (13) implies that limn ‖T
n
i

xn−xn‖= 0 for all i = 1,2, . . . , m. This completes

the proof of (C1). (C2) and (C3) follow from (ii) and (iii) of Lemma 4, respectively.

Now, we state and prove the weak and strong convergence theorems of (1).

Theorem 2. Let X , C, T1, . . . , Tm and {xn} be as in Lemma 5. Then we have the followings.

(i) If {Ti : i = 1,2, . . . , m} satisfies Condition (A′′), then {xn} converges strongly to a common

fixed point of the family.

(ii) If X satisfies Opial’s condition and I − Ti is demiclosed at 0 for all i = 1,2, . . . , m, then

{xn} converges weakly to a common fixed point of the family.



REFERENCES 53

Proof. (i) It follows from Lemma 5 that limn ‖xn − Ti xn‖ = 0 for all i = 1,2, . . . , m.

Therefore, by using Condition (A′′), there exists a nondecreasing function f : [0,∞)→ [0,∞)
with f (0) = 0 and f (r)> 0 for all r ∈ (0,∞) such that

lim
n

f (d(xn, F)) ≤ lim
n
‖xn− Ti xn‖ = 0

for some i = 1,2, . . . , m. That is limn d(xn, F) = 0. By Theorem 1, we conclude that {xn}
converges strongly to a point p ∈ F .

(ii) Let p ∈ F . It follows from Lemma 3 that limn ‖xn − p‖ exists and hence {xn} is

bounded. Since X is uniformly convex, there exists a subsequence {xnk
} of {xn} converging

weakly to some u ∈ C . By Lemma 4, limn ‖xn− Ti xn‖= 0 for all i = 1,2, . . . , m. Since I − Ti is

demiclosed for all i = 1,2, . . . , m, we obtain u ∈ F . Suppose that subsequences {xnk
} and {xnl

}
of {xn} converge weakly to u and v, respectively. As proved above u, v ∈ F . Again by Lemma

3, limn ‖xn− u‖ and limn ‖xn − v‖ exist. Assume that u 6= v. Then by the Opial property

lim
n
‖xn− u‖ = lim

k
‖xnk
− u‖ < lim

k
‖xnk
− v‖= lim

n
‖xn − v‖

= lim
l
‖xnl
− v‖< lim

l
‖xnl
− u‖ = lim

n
‖xn − u‖

This contradiction proves that {xn} converges weakly to a common fixed point of the family

{Ti : i = 1,2, . . . , m} and the proof is completed.

Remark 1. If T1 = T2 = · · · = Tm in Theorem 2 we obtain weak and strong convergence of the

modified Mann iteration (4) and the modified Ishikawa iteration.

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