Open Access

The convergence of the modified Mann and Ishikawa iterations in Banach spaces

Journal of Inequalities and Applications20132013:188

https://doi.org/10.1186/1029-242X-2013-188

Received: 19 August 2012

Accepted: 8 January 2013

Published: 19 April 2013

Abstract

In this paper, under the new condition we show that the convergence of the modified Mann and Ishikawa iterations is equivalent for uniformly L-Lipschitz asymptotically pseudocontractive mappings in real Banach spaces. Our results extend and improve the corresponding results of Zeng (Acta Math. Sin. 47:219-228, 2004).

MSC:47H09, 47H10.

Keywords

uniformly L-Lipschitzasymptotically pseudocontractive mappingfixed pointreal Banach space

1 Introduction and preliminaries

Throughout the paper, we assume that E is an arbitrary real Banach space, D is a nonempty closed convex subset of E, T : D D is a self-mapping and F ( T ) is the fixed point set of T, i.e., F ( T ) = { x D : T x = x } . Let J denote the normalized duality mapping from E to 2 E defined by
J ( x ) = { f E : x , f = x 2 = f 2 } , x E ,
(1.1)

where E denotes the dual space of E and , denotes the generalized duality pairing. The single-valued normalized duality mapping is denoted by j.

Definition 1.1 (see [1])
  1. (1)
    A mapping T is said to be uniformly L-Lipschitz if there exists a constant L > 0 such that, for all x , y D ,
    T n x T n y L x y , n 1 .
    (1.2)
     
  2. (2)
    The mapping T is said to be asymptotically nonexpansive with a sequence { k n } [ 1 , + ) and lim n k n = 1 if, for all x , y D ,
    T n x T n y k n x y , n 1 .
    (1.3)
     
  3. (3)
    The mapping T is said to be asymptotically pseudocontractive with a sequence { k n } [ 1 , + ) and lim n k n = 1 if, for all x , y D , there exists j ( x y ) J ( x y ) such that
    T n x T n y , j ( x y ) k n x y 2 , n 1 .
    (1.4)
     

Obviously, an asymptotically nonexpansive mapping is both asymptotically pseudocontractive and uniformly L-Lipschitz, but the converse is not true in general. For more details on uniformly L-Lipschitz asymptotically nonexpansive and asymptotically pseudocontractive mappings, see [26] and [711].

Definition 1.2 (see [1])

For any u 1 , x 1 D , the sequences { u n } and { x n } in D defined by
u n + 1 = ( 1 a n ) u n + a n T n u n , n 1 ,
(1.5)
and
{ y n = ( 1 b n ) x n + b n T n x n , x n + 1 = ( 1 a n ) x n + a n T n y n , n 1 ,
(1.6)

are called the modified Mann and Ishikawa iterations, respectively, where { a n } , { b n } are two real sequences in [ 0 , 1 ] satisfying some conditions. For more details on the Mann and Ishikawa iterations, see [4, 12] and [11].

In 2001, Chidume and Mutangadura [13] constructed an example for every nontrivial Mann iteration failing to converge while Ishikawa iteration converges. Therefore, there exist some differences between convergence of two kinds of the iterative sequences. Since then, many authors have shown that the Mann (modified Mann) and Ishikawa (modified Ishikawa) iterations (with errors) converge strongly to fixed points of pseudocontractive mappings and others under appropriate conditions.

Especially, Chang [1] proved the following.

Theorem 1.3 [[1], Theorem 2.1]

Let D be a nonempty closed convex subset of E and T : D D be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence { k n } [ 1 , + ) such that lim n k n = 1 and L 1 . Let { a n } and { b n } be two real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (a)

    a n , b n 0 as n ;

     
  2. (b)

    n = 0 a n = .

     
For any x 0 D , let { x n } be the modified Ishikawa iteration defined by (1.2). If F ( T ) , q F ( T ) and there exists a strictly increasing function Φ : [ 0 , + ) [ 0 , + ) with Φ ( 0 ) = 0 such that
T n x n + 1 q , j ( x n + 1 q ) k n x n + 1 q 2 Φ ( x n + 1 q ) , n 0 ,

where j ( x n + 1 q ) J ( x n + 1 q ) , then { x n } converges strongly to a fixed point q of T.

Theorem 1.4 [[1], Theorem 2.3]

Let D be a nonempty closed convex subset of E and T : D D be a uniformly L-Lipschitzian asymptotically pseudocontractive mapping with a sequence { k n } [ 1 , + ) such that lim n k n = 1 and L 1 . Let { a n } be the real sequence in [ 0 , 1 ] satisfying the following conditions:
  1. (a)

    a n 0 as n ;

     
  2. (b)

    n = 0 a n = .

     

For any u 0 D , let { u n } be the modified Mann iteration defined by (1.1). If F ( T ) , q F ( T ) and there exists a strictly increasing function Φ : [ 0 , + ) [ 0 , + ) with Φ ( 0 ) = 0 satisfying the condition (2.1) of [[1], Theorem 2.1], then { u n } converges strongly to a fixed point q of T.

Motivated by Theorems 1.3 and 1.4, Zeng [14] gave another interesting results as follows.

Theorem 1.5 [[14], Theorem 2.1]

Let D be a nonempty closed convex subset of E and T : D D be a uniformly L-Lipschitz asymptotically pseudocontractive mapping with a sequence { k n } [ 1 , + ) such that lim n k n = 1 and L 1 . Let { a n } and { b n } be two real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (a)

    a n 0 as n and n = 0 a n = ;

     
  2. (b)

    n = 0 a n 2 < and n = 0 a n ( k n 1 ) < ;

     
  3. (c)

    n = 0 a n b n < .

     
For arbitrary x 0 D , let { x n } be the modified Ishikawa iteration defined by (1.2). If F ( T ) , q F ( T ) and there exists a strictly increasing function Φ : [ 0 , + ) [ 0 , + ) with Φ ( 0 ) = 0 such that
T n x n + 1 q , j ( x n + 1 q ) k n x n + 1 q 2 Φ ( x n + 1 q ) , n 0 ,

where j ( x n + 1 q ) J ( x n + 1 q ) , then { x n } converges strongly to the fixed point q of T.

Theorem 1.6 [[14], Theorem 2.3]

Let D be a nonempty closed convex subset of E and T : D D be a uniformly L-Lipschitz asymptotically pseudocontractive mapping with a sequence { k n } [ 1 , + ) such that lim n k n = 1 and L 1 . Let { a n } be a real sequence in [ 0 , 1 ] satisfying the following conditions:
  1. (a)

    a n 0 as n and n = 0 a n = ;

     
  2. (b)

    n = 0 a n 2 < and n = 0 a n ( k n 1 ) < .

     

For arbitrary u 0 D , let { u n } be the modified Mann iteration defined by (1.1). If F ( T ) , q F ( T ) and there exists a strictly increasing function Φ : [ 0 , + ) [ 0 , + ) with Φ ( 0 ) = 0 satisfying the condition (2.1) of [[1], Theorem 2.1], then { u n } converges strongly to the fixed point q of T.

It is worth mentioning that the result of Chang [1] is different from that of Zeng [14]. This can be seen from the following example.

Example 1.7 Set
a n = { 0 , n = 2 i , 1 n , n = 2 i 1 , b n = { 1 2 , n = 2 i , 1 n , n = 2 i 1 , k n = 1 + 1 n , i 1 , n 1 .
Then a n 0 as n , n = 1 a n = and n = 1 a n 2 < , n = 1 a n b n < , n = 1 a n ( k n 1 ) < , but b n 0 as n does not hold. On the other hand, let
a n = { 0 , n = 2 i , 1 n , n = 2 i 1 , b n = { 0 , n = 2 i , 1 n , n = 2 i 1 , k n = 1 + 1 n , i 1 , n 1 .

Then a n , b n 0 as n and n = 1 a n = , but n = 1 a n 2 = , n = 1 a n b n = and n = 1 a n ( k n 1 ) = .

The aim of this paper is to extend and improve Theorem 1.5 and Theorem 1.6.

For this, we need to use the following lemmas.

Lemma 1.8 [1]

Let E be a real Banach space and J : E 2 E be a normalized duality mapping. Then, for all x , y E and j ( x + y ) J ( x + y ) ,
x + y 2 x 2 + 2 y , j ( x + y ) .
(1.7)

Lemma 1.9 [14]

Let { a n } , { b n } and { c n } be three nonnegative real sequences satisfying
a n + 1 ( 1 + b n ) a n + c n , n 0 .
(1.8)

If n = 0 b n < , n = 0 c n < , then lim n a n exists.

2 Main results

Now, we give the main results in this paper.

Theorem 2.1 Let D be a nonempty closed convex subset of E and T : D D be a uniformly L-Lipschitz asymptotically pseudocontractive mapping with a sequence { k n } [ 1 , + ) such that lim n k n = 1 and L 1 . Let { a n } be a real sequence in [ 0 , 1 ] satisfying the following conditions:
  1. (a)

    a n 0 as n and n = 1 a n = ;

     
  2. (b)

    n = 1 a n 2 < and n = 1 a n ( k n 1 ) < .

     
For arbitrary u 1 D , let { u n } be the modified Mann iteration defined by (1.5). If F ( T ) , q F ( T ) and there exists a strictly increasing continuous function Φ : [ 0 , + ) [ 0 , + ) with Φ ( 0 ) = 0 such that
T n u n + 1 q , j ( u n + 1 q ) k n u n + 1 q 2 Φ ( u n + 1 q ) 1 + Φ ( u n + 1 q ) + u n + 1 q 2 , n 1 ,

where j ( u n + 1 q ) J ( u n + 1 q ) , then { u n } converges strongly to a fixed point q of T.

Proof Applying (1.5) and Lemma 1.8, we have
u n + 1 q 2 = ( 1 a n ) ( u n q ) + a n ( T n u n q ) 2 ( 1 a n ) 2 u n q 2 + 2 a n T n u n q , j ( u n + 1 q ) = ( 1 a n ) 2 u n q 2 + 2 a n T n u n T n u n + 1 , j ( u n + 1 q ) + 2 a n T n u n + 1 T n q , j ( u n + 1 q ) ( 1 a n ) 2 u n q 2 + 2 a n L u n u n + 1 u n + 1 q + 2 a n [ k n u n + 1 q 2 Φ ( u n + 1 q ) 1 + Φ ( u n + 1 q ) + u n + 1 q 2 ] .
(2.1)
Observe that
u n + 1 u n = a n ( T n u n u n ) a n T n u n T n q + q u n a n ( 1 + L ) u n q .
(2.2)
Substituting (2.2) into (2.1), we obtain
u n + 1 q 2 ( 1 a n ) 2 u n q 2 + a n 2 L ( 1 + L ) ( u n q 2 + u n + 1 q 2 ) + 2 a n [ k n u n + 1 q 2 Φ ( u n + 1 q ) 1 + Φ ( u n + 1 q ) + u n + 1 q 2 ] .
(2.3)
Since a n 0 and k n 1 as n , without loss of generality, we assume that
1 2 < 1 a 2 L ( 1 + L ) 2 a n k n < 1 , n 1 .
Then (2.3) implies that
(2.4)

Since n = 1 { 4 a n ( k n 1 ) + 2 a n 2 [ 1 + 2 L ( 1 + L ) ] } < , by Lemma 1.9, lim n u n q exists. Denote M = sup n 1 { u n q } .

On the other hand, from (2.4), we have
(2.5)
Let inf n 1 Φ ( u n + 1 q ) 1 + Φ ( u n + 1 q ) + u n + 1 q 2 = δ . Then δ = 0 . Assume δ > 0 . Then we have
Φ ( u n + 1 q ) 1 + Φ ( u n + 1 q ) + u n + 1 q 2 δ , n 1 .
It follows from (2.5) that
2 a n δ u n q 2 u n + 1 q 2 + [ 4 a n ( k n 1 ) + 2 a n 2 ( 1 + 2 L ( 1 + L ) ) ] M 2 ,
(2.6)
which implies that
2 δ n = 1 a n u 1 q 2 + n = 1 [ 4 a n ( k n 1 ) + 2 a n 2 ( 1 + 2 L ( 1 + L ) ) ] M 2 < ,
(2.7)
which is a contradiction, and so δ = 0 . Thus, there exists a subsequence
{ Φ ( u n i + 1 q ) 1 + Φ ( u n i + 1 q ) + u n i + 1 q 2 }
of
{ Φ ( u n + 1 q ) 1 + Φ ( u n + 1 q ) + u n + 1 q 2 }
such that
lim i Φ ( u n i + 1 q ) 1 + Φ ( u n i + 1 q ) + u n i + 1 q 2 = 0 .
Since 0 u n q M , it follows that
0 Φ ( u n i + 1 q ) 1 + Φ ( M ) + M 2 Φ ( u n i + 1 q ) 1 + Φ ( u n i + 1 q ) + u n i + 1 q 2 .

Thus, lim i Φ ( u n i + 1 q ) = 0 . By the strictly increasing continuous function Φ, we obtain that lim i u n i + 1 q = 0 and so lim n u n q = 0 . This completes the proof. □

Theorem 2.2 Let E be a real Banach space and D be a nonempty closed convex subset of E. Let T : D D be a uniformly L-Lipschitz asymptotically pseudocontractive mapping with a sequence { k n } [ 1 , + ) such that lim n k n = 1 . Suppose that { a n } and { b n } are two real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (a)

    a n 0 as n and n = 1 a n = ;

     
  2. (b)

    n = 1 a n 2 < and n = 1 a n ( k n 1 ) < ;

     
  3. (c)

    n = 1 a n b n < .

     
For any u 1 , x 1 D , let { u n } and { x n } be the modified Mann and Ishikawa iterations defined by (1.5) and (1.6), respectively. If F ( T ) , q F ( T ) and there exists a strictly increasing continuous function Φ : [ 0 , + ) [ 0 , + ) with Φ ( 0 ) = 0 such that
where j ( x n + 1 u n + 1 ) J ( x n + 1 u n + 1 ) . Then the following two assertions are equivalent:
  1. (1)

    { u n } converges strongly to the fixed point q of T.

     
  2. (2)

    { x n } converges strongly to the fixed point q of T.

     

Proof If the iteration (1.6) converges to a fixed point q, then, by putting b n = 0 , we can get the convergence of the iteration (1.5).

Conversely, we only need to prove that the iteration (1.5) the iteration (1.6), i.e., u n q 0 as n x n q 0 as n . Here, without loss of generality, let u n q 1 . Then T n u n u n ( 1 + L ) .

Applying the iterations (1.5), (1.6) and Lemma 1.8, we have
x n + 1 u n + 1 2 = ( 1 a n ) ( x n u n ) + a n ( T n y n T n u n ) 2 ( 1 a n ) 2 x n u n 2 + 2 a n T n y n T n u n , j ( x n + 1 u n + 1 ) = ( 1 a n ) 2 x n u n 2 + 2 a n T n y n T n x n + 1 , j ( x n + 1 u n + 1 ) + 2 a n T n x n + 1 T n u n + 1 , j ( x n + 1 u n + 1 ) + 2 a n T n u n + 1 T n u n , j ( x n + 1 u n + 1 ) ( 1 a n ) 2 x n u n 2 + 2 a n L y n x n + 1 x n + 1 u n + 1 + 2 a n [ k n x n + 1 u n + 1 2 Φ ( x n + 1 u n + 1 ) 1 + Φ ( x n + 1 u n + 1 ) + x n + 1 u n + 1 2 ] + 2 a n L u n + 1 u n x n + 1 u n + 1 .
(2.8)
Observe that
(2.9)
(2.10)
(2.11)
where A n = b n ( L + 1 ) + a n [ 1 + L ( 1 + b n L ) ] 0 and B n = b n ( L + 1 ) + a n ( L + 1 ) ( 1 + L b n ) 0 as n . Substituting (2.9), (2.11) into (2.8), we obtain
(2.12)
Since a n , b n , A n , B n , k n 1 0 as n , without loss of generality, we assume that
1 2 < 1 a n L A n a n L B n a n 2 L ( 1 + L ) 2 a n k n < 1 , n 1 .
Then (2.12) implies that
(2.13)
Since
n = 1 [ 4 a n L A n + 2 a n L B n + 2 a n 2 ( 1 + L + L 2 ) + 4 a n ( k n 1 ) ] <

and n = 1 [ 2 a n L B n + 2 a n 2 L ( 1 + L ) ] < , by Lemma 1.9, lim n x n u n exists. Denote M 0 = sup n 1 { x n u n } .

On the other hand, from (2.13), it follows that
(2.14)
Let inf n 1 Φ ( x n + 1 u n + 1 ) 1 + Φ ( x n + 1 u n + 1 ) + x n + 1 u n + 1 2 = δ . Then δ = 0 . Assume δ > 0 . Then we have
Φ ( x n + 1 u n + 1 ) 1 + Φ ( x n + 1 u n + 1 ) + x n + 1 u n + 1 2 δ , n 1 .
It follows from (2.14) that
2 a n δ x n u n 2 x n + 1 u n + 1 2 + [ 4 a n L A n + 2 a n L B n + 2 a n 2 ( 1 + L + L 2 ) + 4 a n ( k n 1 ) ] M 2 + 2 a n L B n + 2 a n 2 L ( 1 + L ) ,
(2.15)
which implies that
2 δ n = 1 a n x 1 u 1 2 + n = 1 [ 4 a n L A n + 2 a n L B n + 2 a n 2 ( 1 + L ) + 4 a n ( k n 1 ) ] M 2 + n = 1 [ 2 a n L B n + 2 a n 2 L ( 1 + L ) ] < ,
(2.16)
which is a contradiction, and so δ = 0 . Thus, there exists a subsequence
{ Φ ( x n i + 1 u n i + 1 ) 1 + Φ ( x n i + 1 u n i + 1 ) + x n i + 1 u n i + 1 2 }
of
{ Φ ( x n + 1 u n i + 1 ) 1 + Φ ( x n + 1 u n i + 1 ) + x n + 1 u n + 1 2 }
such that
lim i Φ ( x n i + 1 u n i + 1 ) 1 + Φ ( x n i + 1 u n i + 1 ) + x n i + 1 u n i + 1 2 = 0 .
Since 0 x n u n M , it follows that
0 Φ ( x n i + 1 u n i + 1 ) 1 + Φ ( M ) + M 2 Φ ( x n i + 1 u n i + 1 ) 1 + Φ ( x n i + 1 u n i + 1 ) + x n i + 1 u n i + 1 2 .

Thus, lim i Φ ( x n i + 1 u n i + 1 ) = 0 . By the strictly increasing continuous function Φ, we obtain that lim i x n i + 1 u n i + 1 = 0 and so lim n x n u n = 0 . Using the inequality x n q x n u n + u n q 0 as n , we know that x n q as n . This completes the proof. □

Theorem 2.3 Let E be a real Banach space and D be a nonempty closed convex subset of E. Let T : D D be a uniformly L-Lipschitz asymptotically pseudocontractive mapping with a sequence { k n } [ 1 , + ) such that lim n k n = 1 . Suppose that { a n } and { b n } are two real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (a)

    a n 0 as n and n = 1 a n = ;

     
  2. (b)

    n = 1 a n 2 < and n = 1 a n ( k n 1 ) < ;

     
  3. (c)

    n = 1 a n b n < .

     
For any x 1 D , let { x n } be the modified Ishikawa iteration defined in (1.6). If F ( T ) , q F ( T ) and there exists a strictly increasing continuous function Φ : [ 0 , + ) [ 0 , + ) with Φ ( 0 ) = 0 such that
T n x n + 1 q , j ( x n + 1 q ) k n x n + 1 q 2 Φ ( x n + 1 q ) 1 + Φ ( x n + 1 q ) + x n + 1 q 2 , n 1 ,

where j ( x n + 1 q ) J ( x n + 1 q ) . Then { x n } converges strongly to the fixed point q of T.

Proof By Theorem 2.1 and Theorem 2.2, we obtain the proof of Theorem 2.3. □

Remark 2.4 Since the condition T n x q , j ( x q ) k n x q 2 Φ ( x q ) 1 + Φ ( x q ) + x q 2 is weaker than T n x q , j ( x q ) k n x q 2 Φ ( x q ) , Theorem 2.1 and Theorem 2.3 generalize the corresponding results of Zeng [14]. Further, our proof methods are different from those of Zeng [14].

For the sake of convenience, we give the following definitions.

Definition 2.5 A mapping T : D E is said to be weak generalized asymptotically φ-hemi-contractive with a sequence { k n } [ 1 , + ) such that lim n k n = 1 if there exists a strictly increasing continuous function φ : [ 0 , + ) [ 0 , + ) with φ ( 0 ) = 0 such that, for any x D and y F ( T ) , there exists j ( x y ) J ( x y ) such that
T n x T n y , j ( x y ) k n x y 2 φ ( x y ) 1 + φ ( x y ) + x y 2 , n 1 .
(2.17)
If the condition (2.17) is replaced by the following inequality:
T n x T n y , j ( x y ) k n x y 2 φ ( x y ) , n 1 ,
(2.18)

then T is called a generalized asymptotically φ-hemi-contractive mapping. Clearly, if T is a generalized asymptotically asymptotically φ-hemi-contractive, then T must be a weak generalized asymptotically asymptotically φ-hemi-contractive mapping. However, the converse is not true in general. This can be seen from the following examples.

Example 2.6 Let E = R be the set of real numbers with the usual norm | | and D = [ 0 , + ) . Define a mapping T : D D by
T x = 2 x 3 1 + 2 x 2 , x D .
Then T is a monotonically increasing function with a fixed point q = 0 D . Define two functions Φ , φ : [ 0 , + ) [ 0 , + ) by Φ ( t ) = t 2 1 + 2 t 2 and φ ( t ) = t 2 , respectively. Then Φ and φ are two strictly increasing continuous functions with Φ ( 0 ) = φ ( 0 ) = 0 . For all x D and q F ( T ) , let k n = 1 . Then we obtain that
(2.19)
(2.20)

Then T is a generalized asymptotically Φ-hemi-contraction and a weak generalized asymptotically φ-hemi-contraction.

Example 2.7 Let E = R be the set of real numbers with the usual norm and R + = [ 0 , + ) . Define a mapping T : R + R by
T x = x + x 3 + x 5 / 2 x 1 / 2 1 + x 3 / 2 + x 2 , x R + .
Then T has a fixed point q = 0 R + . Define a function φ : [ 0 , + ) [ 0 , + ) by φ ( t ) = t 3 / 2 . Then φ is a strictly increasing continuous function with φ ( 0 ) = 0 . For all x R + and q F ( T ) , let n = 1 and k n = 1 . Then we have
T x T q , j ( x q ) = x + x 3 + x 5 / 2 x 1 / 2 1 + x 3 / 2 + x 2 0 , j ( x 0 ) = x + x 3 + x 5 / 2 x 1 / 2 1 + x 3 / 2 + x 2 , x = x 2 + x 4 + x 7 / 2 x 3 / 2 1 + x 3 / 2 + x 2 = x 2 x 3 / 2 1 + x 3 / 2 + x 2 = | x q | 2 | x q | 3 / 2 1 + | x q | 3 / 2 + | x q | 2 = | x q | 2 φ ( | x q | ) 1 + φ ( | x q | ) + | x q | 2 .
(2.21)

Then T is a weak generalized asymptotically φ-hemi-contraction, but not a generalized asymptotically Φ-hemi-contraction with n = 1 .

Declarations

Acknowledgements

The authors are grateful to Professor Yeol-Je Cho for valuable suggestions which helped to improve the manuscript. This work was supported by Hebei Provincial Natural Science Foundation (Grant No. A2011210033).

Authors’ Affiliations

(1)
Department of Mathematics and Physics, Shijiazhuang Tiedao University

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© Xue and Lv; licensee Springer 2013

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