Open Access

Almost stability of the Mann type iteration method with error term involving strictly hemicontractive mappings in smooth Banach spaces

  • Nawab Hussain1,
  • Arif Rafiq2,
  • Ljubomir B Ciric3 and
  • Saleh Al-Mezel1Email author
Journal of Inequalities and Applications20122012:207

https://doi.org/10.1186/1029-242X-2012-207

Received: 9 March 2012

Accepted: 6 September 2012

Published: 24 September 2012

Abstract

Let K be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and T : K K be a continuous strictly hemicontractive mapping. Under some conditions, we obtain that the Mann iteration method with error term converges strongly to a unique fixed point of T and is almost T-stable on K. As an application of our results, we establish strong convergence of a multi-step iteration process.

Keywords

Mann iteration method with error termstrictly hemicontractive operatorsstrongly pseudocontractive operatorslocal strongly pseudocontractive operatorscontinuous mappingsLipschitz mappingssmooth Banach spaces

1 Introduction

Chidume [1] established that the Mann iteration sequence converges strongly to the unique fixed point of T in case T is a Lipschitz strongly pseudo-contractive mapping from a bounded closed convex subset of L p (or l p ) into itself. Schu [2] generalized the result in [1] to both uniformly continuous strongly pseudo-contractive mappings and real smooth Banach spaces. Park [3] extended the result in [1] to both strongly pseudocontractive mappings and certain smooth Banach spaces. Rhoades [4] proved that the Mann and Ishikawa iteration methods may exhibit different behavior for different classes of nonlinear mappings. Harder and Hicks [5, 6] revealed the importance of investigating the stability of various iteration procedures for various classes of nonlinear mappings. Harder [7] established applications of stability results to first-order differential equations. Afterwords, several generalizations have been made in various directions (see, for example, [2, 4, 821].

Let K be a nonempty closed bounded convex subset of an arbitrary smooth Banach space X and T : K K be a continuous strictly hemicontractive mapping. Under some conditions, we obtain that the Mann iteration method with error term converges strongly to a unique fixed point of T and is almost T-stable on K. As an application, we shall also establish strong convergence of a multi-step iteration process. The results presented here generalize the corresponding results in [24, 10, 11, 22].

2 Preliminaries

Let K be a nonempty subset of an arbitrary Banach space X and X be its dual space. The symbols D ( T ) , R ( T ) and F ( T ) stand for the domain, the range and the set of fixed points of T : X X respectively (x is called a fixed point of T iff T ( x ) = x ). We denote by J the normalized duality mapping from X to 2 X defined by
J ( x ) = { f X : x , f = x 2 = f 2 } .

Let T be a self-mapping of K.

Definition 1 The mapping T is called Lipshitzian if there exists L > 0 such that
T x T y L x y

for all x , y K . If L = 1 , then T is called non-expansive and if 0 L < 1 , T is called contraction.

Definition 2 [10, 22]

  1. 1.
    The mapping T is said to be pseudocontractive if the inequality
    x y x y + t [ ( I T ) x ( I T ) y ]
    (2.1)
     
holds for each x , y K and for all t > 0 .
  1. 2.
    T is said to be strongly pseudocontractive if there exists t > 1 such that
    x y ( 1 + r ) ( x y ) r t ( T x T y )
    (2.2)
     
for all x , y D ( T ) and r > 0 .
  1. 3.
    T is said to be local strongly pseudocontractive if for each x D ( T ) , there exists t x > 1 such that
    x y ( 1 + r ) ( x y ) r t x ( T x T y )
    (2.3)
     
for all y D ( T ) and r > 0 .
  1. 4.
    T is said to be strictly hemicontractive if F ( T ) and if there exists t > 1 such that
    x q ( 1 + r ) ( x q ) r t ( T x q )
    (2.4)
     

for all x D ( T ) , q F ( T ) and r > 0 .

Clearly, each strongly pseudocontractive operator is local strongly pseudocontractive.

Definition 3 [57]

Let K be a nonempty convex subset of X and T : K K be an operator. Assume that x o K and x n + 1 = f ( T , x n ) defines an iteration scheme which produces a sequence { x n } n = 0 K . Suppose, furthermore, that { x n } n = 0 converges strongly to q F ( T ) . Let { y n } n = 0 be any bounded sequence in K and put ε n = y n + 1 f ( T , y n ) .
  1. (1)

    The iteration scheme { x n } n = 0 defined by x n + 1 = f ( T , x n ) is said to be T-stable on K if lim n ε n = 0 implies that lim n y n = q .

     
  2. (2)

    The iteration scheme { x n } n = 0 defined by x n + 1 = f ( T , x n ) is said to be almost T-stable on K if n = 0 ε n < implies that lim n y n = q .

     

It is easy to verify that an iteration scheme { x n } n = 0 which is T-stable on K is almost T-stable on K.

Lemma 4 [3]

Let X be a smooth Banach space. Suppose one of the following holds:
  1. (1)

    J is uniformly continuous on any bounded subsets of X,

     
  2. (2)

    x y , j ( x ) j ( y ) x y 2 for all x, y in X,

     
  3. (3)
    for any bounded subset D of X, there is a c : [ 0 , ) [ 0 , ) such that
    Re x y , j ( x ) j ( y ) c ( x y ) ,
     
for all x , y D , where c satisfies
lim t 0 + c ( t ) t = 0 .
(2.5)
Then for any ϵ > 0 and any bounded subset K, there exists δ > 0 such that
s x + ( 1 s ) y 2 ( 1 2 s ) y 2 + 2 s Re x , j ( y ) + 2 s ϵ
(2.6)

for all x , y K and s [ 0 , δ ] .

Lemma 5 [10]

Let T : D ( T ) X X be an operator with F ( T ) φ . Then T is strictly hemicontractive if and only if there exists t > 1 such that for all x D ( T ) and q F ( T ) , there exists j ( x q ) J ( x q ) satisfying
Re x T x , j ( x q ) ( 1 1 t ) x q 2 .
(2.7)

Lemma 6 [4]

Let X be an arbitrary normed linear space and T : D ( T ) X X be an operator.
  1. (1)

    If T is a local strongly pseudocontractive operator and F ( T ) , then F ( T ) is a singleton and T is strictly hemicontractive.

     
  2. (2)

    If T is strictly hemicontractive, then F ( T ) is a singleton.

     

3 Main results

We now prove our main results.

Lemma 7 Let { α n } n = 0 , { β n } n = 0 and { γ n } n = 0 be nonnegative real sequences, and let ϵ > 0 be a constant satisfying
β n + 1 ( 1 α n l ) β n + ϵ α n + γ n ; l 1 , n 0 ,

where n = 0 α n l = , α n 1 for all n 0 and n = 0 γ n < . Then, lim n sup β n ϵ .

Proof By a straightforward argument, for n k 0 ,
β n + 1 β k j = k n ( 1 α j l ) + ϵ j = k n α j i = j + 1 n ( 1 α i l ) + j = k n γ j i = j + 1 n ( 1 α i l ) ,
(3.1)
where we put i = n + 1 n ( 1 α i l ) = 1 . Note that j = k n α j i = j + 1 n ( 1 α i l ) 1 . It follows from (3.1) that
β n + 1 exp ( j = k n α j l ) β k + ϵ + j = k n γ j .
(3.2)
For a given δ > 0 , there exists a positive integer k such that j = k γ j < δ . Thus (3.2) ensures that
lim n sup β n ϵ + δ .

Letting δ 0 + yields lim n sup β n ϵ . □

Remark 8
  1. (i)

    If γ n = 0 for each n 0 , then Lemma 7 reduces to Lemma 1 of Park [3].

     
  2. (ii)

    If l = 1 , then Lemma 7 reduces to Lemma 2.1 of Liu et al. [4].

     

Theorem 9 Let Xbe a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. Let K be a nonempty closed bounded convex subset of X and T : K K be a continuous strictly hemicontractive mapping. Suppose that { u n } n = 0 is an arbitrary sequence in K and { a n } n = 0 , { b n } n = 0 and { c n } n = 0 are any sequences in [ 0 , 1 ] satisfying conditions (i) a n + b n + c n = 1 , (ii) c n = o ( b n ) , (iii) lim n b n = 0 and (iv) n = 0 b n = .

For a sequence { v n } n = 0 in K, suppose that { x n } n = 0 is the sequence generated from an arbitrary x 0 K by
x n + 1 = a n x n + b n T v n + c n u n , n 0 ,
(3.3)

and satisfying lim n v n x n = 0 .

Let { y n } n = 0 be any sequence in K and define { ε n } n = 0 by
ε n = y n + 1 p n , n 0 ,
(3.4)

where p n = a n y n + b n T v n + c n u n , such that lim n v n y n = 0 .

Then
  1. (a)

    the sequence { x n } n = 0 converges strongly to a unique fixed point q of T,

     
  2. (b)

    n = 0 ε n < implies that lim n y n = q , so that { x n } n = 0 is almost T-stable on K,

     
  3. (c)

    lim n y n = q implies that lim n ε n = 0 .

     

Proof From (ii), we have c n = t n b n , where t n 0 as n .

It follows from Lemma 6 that F ( T ) is a singleton. That is, F ( T ) = { q } for some q K .

Set M = 1 + diam K . For all n 0 , it is easy to verify that
M = sup n 0 x n q + sup n 0 T v n q + sup n 0 u n q + sup n 0 { p n q } + sup n 0 y n q .
(3.5)
For given any ϵ > 0 and the bounded subset K, there exists a δ > 0 satisfying (2.6). Note that (ii), (iii), lim n v n x n = 0 and the continuity of T ensure that there exists an N such that
b n < min { δ , 1 2 ( 1 k ) } , t n ϵ 16 M 2 , T v n T x n ϵ 4 M , n N ,
(3.6)
where k = 1 t and t satisfies (2.7). Using (3.3) and Lemma 4, we infer that
x n + 1 q 2 = ( 1 b n ) ( x n q ) + b n ( T v n q ) + c n ( u n x n ) 2 ( ( 1 b n ) ( x n q ) + b n ( T v n q ) + 2 M c n ) 2 ( 1 b n ) ( x n q ) + b n ( T v n q ) 2 + 8 M 2 c n ( 1 2 b n ) x n q 2 + 2 b n Re ( T v n q , j ( x n q ) ) + 2 ϵ b n + 8 M 2 c n = ( 1 2 b n ) x n q 2 + 2 b n Re ( T x n q , j ( x n q ) ) + 2 b n Re ( T v n T x n , j ( x n q ) ) + 2 ϵ b n + 8 M 2 c n ( 1 2 b n ) x n q 2 + 2 k b n x n q 2 + 2 b n T v n T x n x n q + 2 ϵ b n + 8 M 2 c n ( 1 2 ( 1 k ) b n ) x n q 2 + 2 M b n T v n T x n + 2 ϵ b n + 8 M 2 c n ( 1 2 ( 1 k ) b n ) x n q 2 + 3 ϵ b n ,
(3.7)

for all n N .

Put
we have from (3.7)
β n + 1 ( 1 α n ) β n + ϵ α n + γ n , n 0 .
Observe that n = 0 α n = , α n < 1 for all n 0 . It follows from Lemma 7 that
lim n sup x n q 2 ϵ .

Letting ϵ 0 + , we obtain that lim n sup x n q 2 = 0 , which implies that x n q as n .

On the same lines, we obtain
p n q 2 = ( 1 b n ) ( y n q ) + b n ( T v n q ) + c n ( u n y n ) 2 ( ( 1 b n ) ( y n q ) + b n ( T v n q ) + 2 M c n ) 2 ( 1 b n ) ( y n q ) + b n ( T v n q ) 2 + 8 M 2 c n ( 1 2 b n ) y n q 2 + 2 b n Re ( T v n q , j ( y n q ) ) + 2 ϵ b n + 8 M 2 c n = ( 1 2 b n ) y n q 2 + 2 b n Re ( T y n q , j ( y n q ) ) + 2 b n Re ( T v n T y n , j ( y n q ) ) + 2 ϵ b n + 8 M 2 c n ( 1 2 b n ) y n q 2 + 2 k b n y n q 2 + 2 b n T v n T y n y n q + 2 ϵ b n + 8 M 2 c n ( 1 2 ( 1 k ) b n ) y n q 2 + 2 M b n T v n T y n + 2 ϵ b n + 8 M 2 c n ( 1 2 ( 1 k ) b n ) y n q 2 + 3 ϵ b n ,
(3.8)

for all n N .

Suppose that n = 0 ε n < . In view of (3.4) and (3.8), we infer that
y n + 1 q 2 ( y n + 1 p n + p n q ) 2 p n q 2 + 3 M ε n [ 1 2 b n ( 1 k ) ] y n q 2 + 3 ϵ b n + 3 M ε n ,
(3.9)

for all n N .

Now, put
and we have from (3.9)
β n + 1 ( 1 α n ) β n + ϵ α n + γ n , n 0 .
Observe that n = 0 α n = , α n < 1 and n = 0 γ n < for all n 0 . It follows from Lemma 7 that
lim n sup y n q 2 ϵ .

Letting ϵ 0 + , we obtain that lim n sup y n q 2 = 0 , which implies that y n q as n .

Conversely, suppose that lim n y n = q , then (iii) and (3.8) imply that
ε n y n + 1 q + p n q y n + 1 q + [ [ 1 2 ( 1 k ) b n ] y n q 2 + 3 ϵ b n ] 1 2 0 ,

as n , that is, ε n 0 as n . □

Using the methods of the proof of Theorem 9, we can easily prove the following.

Theorem 10 Let X, K, T and { u n } n = 0 , be as in Theorem 9. Suppose that { a n } n = 0 , { b n } n = 0 and { c n } n = 0 are sequences in [ 0 , 1 ] satisfying conditions (i), (iii)-(iv) and
n = 0 c n < .

If { x n } n = 0 , { v n } n = 0 , { y n } n = 0 , { p n } n = 0 and { ε n } n = 0 are as in Theorem 9, then the conclusions of Theorem 9 hold.

Corollary 11 Let X be a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. Let K be a nonempty closed bounded convex subset of X and T : K K be a Lipschitz strictly hemicontractive mapping. Suppose that { u n } n = 0 is an arbitrary sequence in K and { a n } n = 0 , { b n } n = 0 and { c n } n = 0 are any sequences in [ 0 , 1 ] satisfying conditions (i) a n + b n + c n = 1 , (ii) c n = o ( b n ) , (iii) lim n b n = 0 and (iv) n = 0 b n = .

For a sequence { v n } n = 0 in K, suppose that { x n } n = 0 is the sequence generated from an arbitrary x 0 K by
x n + 1 = a n x n + b n T v n + c n u n , n 0 ,

and satisfying lim n v n x n = 0 .

Let { y n } n = 0 be any sequence in K and define { ε n } n = 0 by
ε n = y n + 1 p n , n 0 ,

where p n = a n y n + b n T v n + c n u n , such that lim n v n y n = 0 .

Then
  1. (a)

    the sequence { x n } n = 0 converges strongly to a unique fixed point q of T,

     
  2. (b)

    n = 0 ε n < implies that lim n y n = q , so that { x n } n = 0 is almost T-stable on K,

     
  3. (c)

    lim n y n = q implies that lim n ε n = 0 .

     
Corollary 12 Let X, K, T and { u n } n = 0 be as in Corollary 11. Suppose that { a n } n = 0 , { b n } n = 0 and { c n } n = 0 are sequences in [ 0 , 1 ] satisfying conditions (i), (iii)-(iv) and
n = 0 c n < .

If { x n } n = 0 , { v n } n = 0 , { y n } n = 0 , { p n } n = 0 and { ε n } n = 0 are as in Corollary 11, then the conclusions of Corollary 11 hold.

Corollary 13 Let X be a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. Let K be a nonempty closed bounded convex subset of X and T : K K be a continuous strictly hemicontractive mapping. Suppose that { α n } n = 0 is a sequence in [ 0 , 1 ] satisfying conditions (i) lim n α n = 0 and (ii) n = 0 α n = .

For a sequence { v n } n = 0 in K, suppose that { x n } n = 0 is the sequence generated from an arbitrary x 0 K by
x n + 1 = α n x n + ( 1 α n ) T v n , n 0 ,

and satisfying lim n v n x n = 0 .

Let { y n } n = 0 be any sequence in K and define { ε n } n = 0 by
ε n = y n + 1 p n , n 0 ,

where p n = α n y n + ( 1 α n ) T v n , such that lim n v n y n = 0 .

Then
  1. (a)

    the sequence { x n } n = 0 converges strongly to a unique fixed point q of T,

     
  2. (b)

    n = 0 ε n < implies that lim n y n = q , so that { x n } n = 0 is almost T-stable on K,

     
  3. (c)

    lim n y n = q implies that lim n ε n = 0 .

     

Corollary 14 Let X be a smooth Banach space satisfying any one of the Axioms (1)-(3) of Lemma 4. Let K be a nonempty closed bounded convex subset of X and T : K K be a Lipschitz strictly hemicontractive mapping. Suppose that { α n } n = 0 is a sequence in [ 0 , 1 ] satisfying conditions (i) lim n α n = 0 and (ii) n = 0 α n = .

For a sequence { v n } n = 0 in K, suppose that { x n } n = 0 is the sequence generated from an arbitrary x 0 K by
x n + 1 = α n x n + ( 1 α n ) T v n , n 0 ,

and satisfying lim n v n x n = 0 .

Let { y n } n = 0 be any sequence in K and define { ε n } n = 0 by
ε n = y n + 1 p n , n 0 ,

where p n = α n y n + ( 1 α n ) T v n , such that lim n v n y n = 0 .

Then
  1. (a)

    the sequence { x n } n = 0 converges strongly to a unique fixed point q of T,

     
  2. (b)

    n = 0 ε n < implies that lim n y n = q , so that { x n } n = 0 is almost T-stable on K,

     
  3. (c)

    lim n y n = q implies that lim n ε n = 0 .

     

4 Applications to a multi-step iteration process

Khan et al. [23] have introduced and studied a multi-step iteration process for a finite family of selfmappings. We now introduce a modified multi-step process as follows:

Let K be a nonempty closed convex subset of a real normed space E and T 1 , T 2 , , T p : K K ( p 2 ) be a family of selfmappings.

Algorithm 1 For a given x 0 K , compute the sequence { x n } n 0 by the iteration process of arbitrary fixed order p 2 ,
(4.1)

which is called the modified multi-step iteration process, where { α n } n 0 , { β n i } n 0 [ 0 , 1 ] , i = 1 , 2 , , p 1 .

For p = 3 , we obtain the following three-step iteration process:

Algorithm 2 For a given x 0 K , compute the sequence { x n } n 0 by the iteration process:
(4.2)

where { α n } n 0 , { β n 1 } n 0 and { β n 2 } n 0 are three real sequences in [ 0 , 1 ] .

For p = 2 , we obtain the Ishikawa [24] iteration process:

Algorithm 3 For a given x 0 K , compute the sequence { x n } n 0 by the iteration process
x n + 1 = ( 1 α n ) x n + α n T 1 y n 1 , y n 1 = ( 1 β n 1 ) x n + β n 1 T 2 x n , n 0 ,
(4.3)

where { α n } n 0 and { β n 1 } n 0 are two real sequences in [ 0 , 1 ] .

If T 1 = T , T 2 = I , β n 1 = 0 in (4.3), we obtain the Mann iteration process [14]:

Algorithm 4 For any given x 0 K , compute the sequence { x n } n 0 by the iteration process
x n + 1 = ( 1 α n ) x n + α n T x n , n 0 ,
(4.4)

where { α n } is a real sequence in [ 0 , 1 ] .

Theorem 15 Let K be a nonempty closed bounded convex subset of a smooth Banach space X and T 1 , T 2 , , T p ( p 2 ) be selfmappings of K. Let T 1 be a continuous strictly hemicontractive mapping. Let { α n } n 0 , { β n i } n 0 [ 0 , 1 ] , i = 1 , 2 , , p 1 be real sequences in [ 0 , 1 ] satisfying n 0 α n = , lim n α n = 0 and lim n β n 1 = 0 . For arbitrary x 0 K , define the sequence { x n } n 0 by (4.1). Then { x n } n 0 converges strongly to a point in i = 1 p F ( T i ) .

Proof By applying Corollary 13 under assumption that T 1 is continuous strictly hemicontractive mapping, we obtain Theorem 15 which proves strong convergence of the iteration process defined by (4.1). We will check only the condition lim n v n x n = 0 by taking T 1 = T and v n = y n 1 ,
v n x n = y n 1 x n = ( 1 β n 1 ) x n + β n 1 T 2 y n 2 x n = β n 1 T 2 y n 2 x n 2 M β n 1 .

Now, from the condition lim n β n 1 = 0 , it can be easily seen that lim n v n x n = 0 . □

Corollary 16 Let K be a nonempty closed bounded convex subset of a smooth Banach space X and T 1 , T 2 , , T p ( p 2 ) be selfmappings of K. Let T 1 be a Lipschitz strictly hemicontractive mapping. Let { α n } n 0 , { β n i } n 0 [ 0 , 1 ] , i = 1 , 2 , , p 1 be real sequences in [ 0 , 1 ] satisfying n 0 α n = , lim n α n = 0 and lim n β n 1 = 0 . For arbitrary x 0 K , define the sequence { x n } n 0 by (4.1). Then { x n } n 0 converges strongly to a point in i = 1 p F ( T i ) .

Remark 17 Similar results can be found for the iteration processes with error terms, we omit the details.

Declarations

Acknowledgements

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research. The third author gratefully acknowledges the support from the Ministry of Education and Science of Republic Serbia. The fourth author gratefully acknowledges the financial support provided by the University of Tabuk through the project of international cooperation with the University of Texas at El Paso. We are also thankful to the editor and the referees for their suggestions for the improvement of the manuscript.

Authors’ Affiliations

(1)
Department of Mathematics, King Abdulaziz University
(2)
Hajvery University
(3)
Faculty of Mechanical Engineering, University of Belgrade

References

  1. Chidume CE: Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings. Proc. Am. Math. Soc. 1987, 99(2):283–288.MathSciNetGoogle Scholar
  2. Schu J: Iterative construction of fixed points of strictly pseudocontractive mappings. Appl. Anal. 1991, 40: 67–72. 10.1080/00036819108839994MathSciNetView ArticleGoogle Scholar
  3. Park JA: Mann iteration process for the fixed point of strictly pseudocontractive mapping in some Banach spaces. J. Korean Math. Soc. 1994, 31: 333–337.MathSciNetGoogle Scholar
  4. Liu Z, Kang SM, Shim SH: Almost stability of the Mann iteration method with errors for strictly hemicontractive operators in smooth Banach spaces. J. Korean Math. Soc. 2003, 40(1):29–40.MathSciNetGoogle Scholar
  5. Harder AM, Hicks TL: A stable iteration procedure for nonexpansive mappings. Math. Jpn. 1988, 33: 687–692.MathSciNetGoogle Scholar
  6. Harder AM, Hicks TL: Stability results for fixed point iteration procedures. Math. Jpn. 1988, 33: 693–706.MathSciNetGoogle Scholar
  7. Harder, AM: Fixed point theory and stability results for fixed point iteration procedures. Ph. D. Thesis, University of Missouri-Rolla (1987)Google Scholar
  8. Chang S: Some problems and results in the study of nonlinear analysis. Nonlinear Anal. TMA 1997, 30(7):4197–4208. 10.1016/S0362-546X(97)00388-XView ArticleGoogle Scholar
  9. Chang SS, Cho YJ, Lee BS, Kang SM: Iterative approximations of fixed points and solutions for strongly accretive and strongly pseudocontractive mappings in Banach spaces. J. Math. Anal. Appl. 1998, 224: 149–165. 10.1006/jmaa.1998.5993MathSciNetView ArticleGoogle Scholar
  10. Chidume CE, Osilike MO: Fixed point iterations for strictly hemicontractive maps in uniformly smooth Banach spaces. Numer. Funct. Anal. Optim. 1994, 15: 779–790. 10.1080/01630569408816593MathSciNetView ArticleGoogle Scholar
  11. Ishikawa S: Fixed points by a new iteration method. Proc. Am. Math. Soc. 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5MathSciNetView ArticleGoogle Scholar
  12. Liu Z, Kang SM: Iterative approximation of fixed points for ϕ -hemicontractive operators in arbitrary Banach spaces. Acta Sci. Math. 2001, 67: 821–831.MathSciNetGoogle Scholar
  13. Liu Z, Kang SM: Stability of Ishikawa iteration methods with errors for strong pseudocontractions and nonlinear equations involving accretive operators in arbitrary real Banach spaces. Math. Comput. Model. 2001, 34: 319–330. 10.1016/S0895-7177(01)00064-4MathSciNetView ArticleGoogle Scholar
  14. Liu Z, Kang SM: Convergence theorems for ϕ -strongly accretive and ϕ -hemicontractive operators. J. Math. Anal. Appl. 2001, 253: 35–49. 10.1006/jmaa.2000.6973MathSciNetView ArticleGoogle Scholar
  15. Liu Z, Kang SM: Convergence and stability of the Ishikawa iteration procedures with errors for nonlinear equations of the ϕ -strongly accretive type. Neural Parallel Sci. Comput. 2001, 9: 103–118.MathSciNetGoogle Scholar
  16. Mann WR: Mean value methods in iteration. Proc. Am. Math. Soc. 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3View ArticleGoogle Scholar
  17. Agarwal RP, Cho YJ, Li J, Huang NJ: Stability of iterative procedures with errors approximating common fixed points for a couple of quasi-contractive mappings in q -uniformly smooth Banach spaces. J. Math. Anal. Appl. 2002, 272: 435–447. 10.1016/S0022-247X(02)00150-6MathSciNetView ArticleGoogle Scholar
  18. Hussain N, Rafiq A, Damjanovic B, Lazovic R: On rate of convergence of various iterative schemes. Fixed Point Theory Appl. 2011., 2011: Article ID 45Google Scholar
  19. Rhoades BE: Comments on two fixed point iteration methods. J. Math. Anal. Appl. 1976, 56: 741–750. 10.1016/0022-247X(76)90038-XMathSciNetView ArticleGoogle Scholar
  20. Weng X: Fixed point iteration for local strictly pseudo-contractive mapping. Proc. Am. Math. Soc. 1991, 113(3):727–731. 10.1090/S0002-9939-1991-1086345-8View ArticleGoogle Scholar
  21. Xu Y: Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations. J. Math. Anal. Appl. 1998, 224: 91–101. 10.1006/jmaa.1998.5987MathSciNetView ArticleGoogle Scholar
  22. Shahzad N, Zegeye H: On stability results for ϕ -strongly pseudocontractive mappings. Nonlinear Anal. TMA 2006, 64(12):2619–2630. 10.1016/j.na.2005.09.007MathSciNetView ArticleGoogle Scholar
  23. Khan AR, Domlo AA, Fukhar-ud-din H: Common fixed points of Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 2008, 341: 1–11. 10.1016/j.jmaa.2007.06.051MathSciNetView ArticleGoogle Scholar
  24. Hussain N, et al.: On the rate of convergence of Kirk type iterative schemes. J. Appl. Math. 2012., 2012: Article ID 526503Google Scholar

Copyright

© Hussain et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.