Open Access

Stability of the Ishikawa iteration scheme with errors for two strictly hemicontractive operators in Banach spaces

Fixed Point Theory and Applications20122012:160

https://doi.org/10.1186/1687-1812-2012-160

Received: 3 March 2012

Accepted: 7 September 2012

Published: 25 September 2012

Abstract

The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu (J. Math. Anal. Appl. 224:91-101, 1998) for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces.

Keywords

Ishikawa iteration scheme with errors strictly hemicontractive operators Lipschitz operators Banach space

1 Preliminaries

Let K be a nonempty subset of an arbitrary Banach space E and E 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 respectively (for a single-valued map T : X X , x X is called a fixed point of T iff T ( x ) = x ). We denote by J the normalized duality mapping from E to 2 E defined by
J ( x ) = { f E : x , f = x 2 = f 2 } .

Let T be a self-mapping of K.

Definition 1 Then T is called Lipshitzian if there exists L > 0 such that
T x T y L x y
(1.1)

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[2, 3]
  1. 1.
    The mapping T is said to be pseudocontractive if the inequality
    x y x y + t ( ( I T ) x ( I T ) y )
    (1.2)
     
holds for each x , y K and for all t > 0 . As a consequence of a result of Kato [4], it follows from the inequality (1.2) that T is pseudocontractive if and only if there exists j ( x y ) J ( x y ) such that
T x T y , j ( x y ) x y 2
(1.3)
for all x , y K .
  1. 2.
    T is said to be strongly pseudocontractive if there exists a t > 1 such that
    x y ( 1 + r ) ( x y ) r t ( T x T y )
    (1.4)
     
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 a t x > 1 such that
    x y ( 1 + r ) ( x y ) r t x ( T x T y )
    (1.5)
     
for all y D ( T ) and r > 0 .
  1. 4.
    T is said to be strictly hemicontractive if F ( T ) φ and if there exists a t > 1 such that
    x q ( 1 + r ) ( x q ) r t ( T x q )
    (1.6)
     

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

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. Osilike [5] proved that an iteration scheme which is almost T-stable on X may fail to be T-stable on X.

Clearly, each strongly pseudocontractive operator is local strongly pseudocontractive.

Chidume [6] 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. Afterwards, several authors generalized this result of Chidume in various directions. Chidume [7] proved a similar result by removing the restriction lim n α n = 0 . Tan and Xu [8] extended that result of Chidume to the Ishikawa iteration scheme in a p-uniformly smooth Banach space. Chidume and Osilike [2] improved the result of Chidume [6] to strictly hemicontractive mappings defined on a real uniformly smooth Banach space.

Recently, some researchers have generalized the results to real smooth Banach spaces, real uniformly smooth Banach spaces, real Banach spaces; or to the Mann iteration method, the Ishikawa iteration method; or to strongly pseudocontractive operators, local strongly pseudocontractive operators, strictly hemicontractive operators [919].

The main purpose of this paper is to establish the convergence, almost common-stability and common-stability of the Ishikawa iteration scheme with error terms in the sense of Xu [1] for two Lipschitz strictly hemicontractive operators in arbitrary Banach spaces. Our results extend, improve and unify the corresponding results in [2, 3, 10, 11, 1518, 2025].

2 Main results

We need the following results.

Lemma 3[26]

Let { α n } n = 0 , { β n } n = 0 , { γ n } n = 0 and { ω n } n = 0 be nonnegative real sequences such that
α n + 1 ( 1 ω n ) α n + ω n β n + γ n , n 0 ,

with { ω n } n = 0 [ 0 , 1 ] , n = 0 ω n = , n = 0 γ n < and lim n β n = 0 . Then lim n α n = 0 .

Lemma 4[27]

Let { a n } n = 0 , { b n } n = 0 be sequences of nonnegative real numbers and 0 θ < 1 , so that
a n + 1 θ a n + b n , for all n 0 .
  1. (i)

    If lim n b n = 0 , then lim n a n = 0 .

     
  2. (ii)

    If n = 0 b n < , then n = 0 a n < .

     

Lemma 5[4]

Let x , y X . Then x x + r y for every r > 0 if and only if there is f J ( x ) such that Re y , f 0 .

Lemma 6[2]

Let T : D ( T ) X X be an operator with F ( T ) . 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 J ( x q ) satisfying
Re x T x , j ( x q ) ( 1 1 t ) x q 2 .

Lemma 7[24]

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

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

     
  2. (ii)

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

     

In the sequel, let k = t 1 t ( 0 , 1 ) , where t is the constant appearing in (1.6). Further L denotes the common Lipschitz constant of T and S, and I denotes the identity mapping on an arbitrary Banach space X.

Definition 8 Let K be a nonempty convex subset of X and T , S : K K be two operators. Assume that x o K and x n + 1 = f ( T , S , 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 ) F ( S ) φ . Let { y n } n = 0 be any bounded sequence in K and put ε n = y n + 1 f ( T , S , y n ) .
  1. (i)

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

     
  2. (ii)

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

     

We now establish our main results.

Theorem 9 Let K be a nonempty closed convex subset of an arbitrary Banach space X and T , S : K K be two Lipschitz strictly hemicontractive operators. Suppose that { u n } n = 0 , { v n } n = 0 are arbitrary bounded sequences in K, and { a n } n = 0 , { b n } n = 0 , { c n } n = 0 , { a n } n = 0 , { b n } n = 0 and { c n } n = 0 are any sequences in [ 0 , 1 ] satisfying
  1. (i)

    a n + b n + c n = 1 = a n + b n + c n ,

     
  2. (ii)

    c n = o ( b n ) ,

     
  3. (iii)

    lim n c n = 0 ,

     
  4. (iv)

    n = 0 b n = ,

     
  5. (v)

    L [ ( 1 + L ) 2 b n + c n + ( 1 + L ) ( b n + c n ) ] + c n b n k ( k s ) , n 0 ,

     
where s is a constant in ( 0 , 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 z n + c n v n , z n = a n x n + b n S x n + c n u n , n 0 .
(2.1)
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 w n + c n v n , w n = a n y n + b n S y n + c n u n , n 0 .
(2.2)
Then
  1. (a)
    the sequence { x n } n = 0 converges strongly to the common fixed point q of T and S. Also,
    x n + 1 q ( 1 s b n ) x n q + L ( 1 + L ) k 1 b n c n u n q + ( 1 + L ) k 1 c n v n q , n 0 ,
    (b)
    y n + 1 q ( 1 s b n ) y n q + L ( 1 + L ) k 1 b n c n u n q + ( 1 + L ) k 1 c n v n q + ε n , n 0 ,
     
  2. (c)

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

     
  3. (d)

    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 7 that F ( T ) F ( S ) is a singleton; that is, F ( T ) F ( S ) = { q } for some q K . Set
M = max { sup n 0 { u n q } , sup n 0 { v n q } } .
Since T is strictly hemicontractive, it follows form Lemma 6 that
Re x T x , j ( x q ) k x q 2 , x K ,
which implies that
Re ( I T k I ) x ( I T k I ) q , j ( x q ) 0 , x K .
In view of Lemma 5, we have
x q x q + r [ ( I T k I ) x ( I T k I ) q ] , x K , r > 0 .
(2.3)
Also,
( 1 b n ) x n = ( 1 ( 1 k ) b n ) x n + 1 + b n ( I T k I ) x n + 1 + b n ( T x n + 1 T z n ) c n ( v n x n ) ,
(2.4)
and
( 1 b n ) q = ( 1 ( 1 k ) b n ) q + b n ( I T k I ) q .
(2.5)
From (2.4) and (2.5), we infer that for all n 0 ,
( 1 b n ) x n q ( 1 ( 1 k ) b n ) ( x n + 1 q ) + b n ( I T k I ) ( x n + 1 q ) b n T x n + 1 T z n c n v n x n = ( 1 ( 1 k ) b n ) x n + 1 q + b n 1 ( 1 k ) b n ( I T k I ) ( x n + 1 q ) b n T x n + 1 T z n c n v n x n ( 1 ( 1 k ) b n ) x n + 1 q b n T x n + 1 T z n c n v n x n ,
which implies that for all n 0 ,
(2.6)
(2.7)
(2.8)
Substituting (2.8) in (2.7), we have
x n + 1 z n [ b n + c n + ( 1 + L ) b n + c n ] x n q + L b n [ [ 1 + ( 1 + L ) b n + c n ] x n q + c n u n q ] + c n v n q + c n u n q = [ ( 1 + L ) b n + L ( 1 + L ) b n b n + ( 1 + L ) b n + c n + ( 1 + L b n ) c n ] x n q + c n v n q + ( 1 + L b n ) c n u n q .
(2.9)
Substituting (2.9) in (2.6), we get
x n + 1 q ( 1 k b n + k 1 c n ) x n q + k 1 L b n [ [ ( 1 + L ) b n + L ( 1 + L ) b n b n + ( 1 + L ) b n + c n + ( 1 + L b n ) c n ] x n q + c n v n q + ( 1 + L b n ) c n u n q ] + k 1 c n v n q = [ 1 b n [ k k 1 L ( ( 1 + L ) b n + L ( 1 + L ) b n b n + ( 1 + L ) b n + c n + ( 1 + L b n ) c n ) k 1 t n ] ] x n q + k 1 L b n ( 1 + L b n ) c n u n q + k 1 ( 1 + L b n ) c n v n q [ 1 b n [ k k 1 L ( ( 1 + L ) 2 b n + ( 1 + L ) b n + c n + ( 1 + L ) c n ) k 1 t n ] ] x n q + k 1 L ( 1 + L ) b n c n u n q + k 1 ( 1 + L ) c n v n q ( 1 s b n ) x n q + k 1 L ( 1 + L ) b n c n u n q + k 1 ( 1 + L ) c n v n q ( 1 s b n ) x n q + k 1 L ( 1 + L ) b n c n M + k 1 ( 1 + L ) b n t n M = ( 1 s b n ) x n q + k 1 ( 1 + L ) M b n ( L c n + t n ) .
Put
we have
α n + 1 ( 1 ω n ) α n + ω n β n + γ n , n 0 .

Observe that n = 0 ω n = , ω n [ 0 , 1 ] and lim n β n = 0 . It follows from Lemma 3 that lim n x n q = 0 .

We also have
( 1 b n ) y n = ( 1 ( 1 k ) b n ) p n + b n ( I T k I ) p n + b n ( T p n T w n ) c n ( v n y n ) .
(2.10)
From (2.5) and (2.10), it follows that for all n 0 ,
( 1 b n ) y n q ( 1 ( 1 k ) b n ) ( p n q ) + b n ( I T k I ) ( p n q ) b n T p n T w n c n v n y n = ( 1 ( 1 k ) b n ) p n q + b n 1 ( 1 k ) b n ( I T k I ) ( p n q ) b n T p n T w n c n v n y n ( 1 ( 1 k ) b n ) p n q b n T p n T w n c n v n y n ,
which implies that for all n 0 ,
(2.11)
(2.12)
(2.13)
Substituting (2.13) in (2.12), we have
p n w n [ b n + c n + ( 1 + L ) b n + c n ] y n q + L b n [ [ 1 + ( 1 + L ) b n + c n ] y n q + c n u n q ] + c n v n q + c n u n q = [ ( 1 + L ) b n + L ( 1 + L ) b n b n + ( 1 + L ) b n + c n + ( 1 + L b n ) c n ] y n q + c n v n q + ( 1 + L b n ) c n u n q .
(2.14)
Substituting (2.14) in (2.11), we get
p n q ( 1 k b n + k 1 c n ) y n q + k 1 L b n [ [ ( 1 + L ) b n + L ( 1 + L ) b n b n + ( 1 + L ) b n + c n + ( 1 + L b n ) c n ] y n q + c n v n q + ( 1 + L b n ) c n u n q ] + k 1 c n v n q = [ 1 b n [ k k 1 L ( ( 1 + L ) b n + L ( 1 + L ) b n b n + ( 1 + L ) b n + c n + ( 1 + L b n ) c n ) k 1 t n ] ] y n q + k 1 L b n ( 1 + L b n ) c n u n q + k 1 ( 1 + L b n ) c n v n q [ 1 b n [ k k 1 L ( ( 1 + L ) 2 b n + ( 1 + L ) b n + c n + ( 1 + L ) c n ) k 1 t n ] ] y n q + k 1 L ( 1 + L ) b n c n u n q + k 1 ( 1 + L ) c n v n q ( 1 s b n ) y n q + k 1 L ( 1 + L ) b n c n u n q + k 1 ( 1 + L ) c n v n q
(2.15)
for any n 0 . Thus (2.15) implies that
y n + 1 q y n + 1 p n + p n q ( 1 s b n ) y n q + k 1 L ( 1 + L ) b n c n u n q + k 1 ( 1 + L ) c n v n q + ε n = ( 1 ω n ) y n q + ω n β n + γ n .
(2.16)
With
we have
α n + 1 ( 1 ω n ) α n + ω n β n + γ n , n 0 .

Observe that n = 0 ω n = , ω n [ 0 , 1 ] and lim n β n = 0 . It follows from Lemma 3 that lim n y n q = 0 .

Suppose that lim n y n = q . It follows from equation (2.15) that
ε n y n + 1 q + p n q ( 1 s b n ) y n q + k 1 L ( 1 + L ) b n c n u n q + k 1 ( 1 + L ) c n v n q + y n + 1 q 0 ,

as n ; that is, ε n 0 as n . □

Using the techniques in the proof of Theorem 9, we have the following results.

Theorem 10 Let X, K, T, S, s, { u n } n = 0 , { v n } n = 0 , { x n } n = 0 , { z n } n = 0 , { w n } n = 0 , { y n } n = 0 and { p n } n = 0 be as in Theorem  9. Suppose that { a n } n = 0 , { b n } n = 0 , { c n } n = 0 , { a n } n = 0 , { b n } n = 0 and { c n } n = 0 are sequences in [ 0 , 1 ] satisfying conditions (i), (iii)-(v) of Theorem  9 with
n = 0 c n < .

Then the conclusions of Theorem  9 hold.

Theorem 11 Let X, K, T, S, s, { u n } n = 0 , { v n } n = 0 , { x n } n = 0 , { z n } n = 0 , { w n } n = 0 , { y n } n = 0 and { p n } n = 0 be as in Theorem  9. Suppose that { a n } n = 0 , { b n } n = 0 , { c n } n = 0 , { a n } n = 0 , { b n } n = 0 and { c n } n = 0 are sequences in [ 0 , 1 ] satisfying condition (i), (iii) and (v) of Theorem  9 with
where m is a constant. Then
  1. (a)
    the sequence { x n } n = 0 converges strongly to the common fixed point q of T and S. Also,
    x n + 1 q ( 1 s m ) x n q + C , n 0 ,
     
where
C = k 1 ( 1 + L ) [ L sup n 0 { c n u n q } + sup n 0 { c n v n q } ] ,
(b)
y n + 1 q ( 1 s m ) y n q + k 1 L ( 1 + L ) c n u n q + k 1 ( 1 + L ) c n v n q + ε n , n 0 ,
  1. (c)

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

     
Proof As in the proof of Theorem 9, we conclude that F ( T ) F ( S ) = { q } and
x n + 1 q ( 1 s b n ) x n q + k 1 L ( 1 + L ) b n c n u n q + k 1 ( 1 + L ) c n v n q ( 1 s m ) x n q + k 1 L ( 1 + L ) c n u n q + k 1 ( 1 + L ) c n v n q ( 1 s m ) x n q + C , n 0 .
Let

Observe that 0 θ < 1 and lim n b n = 0 . It follows from Lemma 4 that lim n x n q = 0 .

Also, from (2.15), we have
y n + 1 q ( 1 s b n ) y n q + k 1 L ( 1 + L ) b n c n u n q + k 1 ( 1 + L ) c n v n q + ε n ( 1 s m ) y n q + k 1 L ( 1 + L ) c n u n q + k 1 ( 1 + L ) c n v n q + ε n .
Suppose that lim n y n = q . It follows from equation (2.15) that
ε n y n + 1 q + p n q ( 1 s m ) y n q + k 1 L ( 1 + L ) c n u n q + k 1 ( 1 + L ) c n v n q + y n + 1 q 0 ,

as n ; that is, ε n 0 as n .

Conversely, suppose that lim n ε n = 0 . Put

Observe that 0 θ < 1 and lim n b n = 0 . It follows from Lemma 4 that lim n y n q = 0 . □

As an immediate consequence of Theorems 9 and 11, we have the following:

Corollary 12 Let K be a nonempty closed convex subset of an arbitrary Banach space X and T , S : K K be two Lipschitz strictly hemicontractive operators. Suppose that { α n } n = 0 , { β n } n = 0 are any sequences in [ 0 , 1 ] satisfying
  1. (vi)

    n = 0 α n = ,

     
  2. (vii)

    L [ ( 1 + L ) 2 α n + ( 1 + L ) β n ] k ( k s ) , n 0 ,

     
where s is a constant in ( 0 , k ) . Suppose that { x n } n = 0 is the sequence generated from an arbitrary x 0 K by
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 = ( 1 α n ) y n + α n T w n ,
and
w n = ( 1 β n ) y n + β n S y n , n 0 .
Then
  1. (a)

    the sequence { x n } n = 0 converges strongly to the common fixed point q of T and S,

     
  2. (b)

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

     
  3. (c)

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

     
Corollary 13 Let X, K, T, S, s, { x n } n = 0 , { z n } n = 0 , { w n } n = 0 , { y n } n = 0 and { p n } n = 0 be as in Theorem  9. Suppose that { α n } n = 0 , { β n } n = 0 are sequences in [ 0 , 1 ] satisfying conditions (vi)-(vii) and (iii) of Theorem  9 with
α n m > 0 , n 0 ,
where m is a constant. Then
  1. (a)
    the sequence { x n } n = 0 converges strongly to the common fixed point q of T and S. Also,
    x n + 1 q ( 1 s m ) x n q , n 0 ,
    (b)
    y n + 1 q ( 1 s m ) y n q + ε n , n 0 ,
     
  2. (c)

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

     
Example 14 Let R denote the set of real numbers with the usual norm, K = R , and define T , S : R R by
T x = 2 5 sin 2 x , and S x = 4 5 x .
Set L = 4 5 , t = 5 4 , s = 1 400 . Clearly, F ( T ) F ( S ) = { 0 } and
| T x T y | 2 5 | sin x sin y | | sin x + sin y | L | x y | , x , y R .

Clearly both T and S are Lipschitz operators on R .

Also, it follows from (1.1) that
| ( 1 + r ) ( x y ) r t ( T x T y ) | ( 1 + r ) | x y | r t | T x T y | = | x y | + r ( | x y | t | T x T y | ) | x y |
for any x , y R and r > 0 . Thus T is strongly pseudocontractive and Lemma 7 ensures that T is strictly hemicontractive. Put
then it can be easily seen that
L [ ( 1 + L ) 2 b n + c n + ( 1 + L ) ( b n + c n ) ] + c n b n 0.456 0.049375 , n 0 .

It follows from Theorem 9 that the sequence { x n } n = 0 defined by (2.1) converges strongly to the common fixed point 0 of T and S in K and the iterative scheme defined by (2.1) is T-stable.

Declarations

Acknowledgements

The first author gratefully acknowledges the support from the Deanship of Scientific Research (DSR) at King Abdulaziz University (KAU) during this research.

Authors’ Affiliations

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

References

  1. 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
  2. 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
  3. 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
  4. Kato T: Nonlinear semigroups and evolution equations. J. Math. Soc. Jpn. 1967, 19: 508–520. 10.2969/jmsj/01940508View ArticleGoogle Scholar
  5. Osilike MO: Stable iteration procedures for strong pseudocontractions and nonlinear operator equations of the accretive type. J. Math. Anal. Appl. 1996, 204: 677–692. 10.1006/jmaa.1996.0461MathSciNetView ArticleGoogle Scholar
  6. Chidume CE:An iterative process for nonlinear Lipschitzian strongly accretive mappings in L p spaces. J. Math. Anal. Appl. 1990, 151: 453–461. 10.1016/0022-247X(90)90160-HMathSciNetView ArticleGoogle Scholar
  7. Chidume CE: Iterative solutions of nonlinear equations with strongly accretive operators. J. Math. Anal. Appl. 1995, 192: 502–518. 10.1006/jmaa.1995.1185MathSciNetView ArticleGoogle Scholar
  8. Tan KK, Xu HK: Iterative solutions to nonlinear equations of strongly accretive operators in Banach spaces. J. Math. Anal. Appl. 1993, 178: 9–21. 10.1006/jmaa.1993.1287MathSciNetView ArticleGoogle Scholar
  9. Chang SS: 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
  10. 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
  11. Chidume CE: Iterative approximation of fixed points of Lipschitzian strictly pseudocontractive mappings. Proc. Am. Math. Soc. 1987, 99(2):283–288.MathSciNetGoogle Scholar
  12. Chidume CE: Approximation of fixed points of strongly pseudocontractive mappings. Proc. Am. Math. Soc. 1994, 120: 545–551. 10.1090/S0002-9939-1994-1165050-6MathSciNetView ArticleGoogle Scholar
  13. Chidume CE: Iterative solution of nonlinear equations in smooth Banach spaces. Nonlinear Anal. TMA 1996, 26(11):1823–1834. 10.1016/0362-546X(94)00368-RMathSciNetView ArticleGoogle Scholar
  14. Chidume CE, Osilike MO: Nonlinear accretive and pseudocontractive operator equations in Banach spaces. Nonlinear Anal. 1998, 31: 779–789. 10.1016/S0362-546X(97)00439-2MathSciNetView ArticleGoogle Scholar
  15. Deng L: On Chidume’s open questions. J. Math. Anal. Appl. 1993, 174(2):441–449. 10.1006/jmaa.1993.1129MathSciNetView ArticleGoogle Scholar
  16. Deng L: An iterative process for nonlinear Lipschitz and strongly accretive mappings in uniformly convex and uniformly smooth Banach spaces. Acta Appl. Math. 1993, 32: 183–196. 10.1007/BF00998152MathSciNetView ArticleGoogle Scholar
  17. Deng L:Iteration processes for nonlinear Lipschitz strongly accretive mappings in L p spaces. J. Math. Anal. Appl. 1994, 188(1):128–140. 10.1006/jmaa.1994.1416MathSciNetView ArticleGoogle Scholar
  18. Deng L, Ding XP: Iterative approximation of Lipschitz strictly pseudocontractive mappings in uniformly smooth Banach spaces. Nonlinear Anal., Theory Methods Appl. 1995, 24(7):981–987. 10.1016/0362-546X(94)00115-XMathSciNetView ArticleGoogle Scholar
  19. Zeng LC: Iterative approximation of solutions to nonlinear equations of strongly accretive operators in Banach spaces. Nonlinear Anal. TMA 1998, 31: 589–598. 10.1016/S0362-546X(97)00425-2View ArticleGoogle Scholar
  20. Harder AM, Hicks TL: A stable iteration procedure for nonexpansive mappings. Math. Jpn. 1988, 33: 687–692.MathSciNetGoogle Scholar
  21. Harder AM, Hicks TL: Stability results for fixed point iteration procedures. Math. Jpn. 1988, 33: 693–706.MathSciNetGoogle Scholar
  22. 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
  23. Liu LW: Approximation of fixed points of a strictly pseudocontractive mapping. Proc. Am. Math. Soc. 1997, 125(5):1363–1366. 10.1090/S0002-9939-97-03858-6View ArticleGoogle Scholar
  24. 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
  25. 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
  26. Liu LS: Ishikawa and Mann iteration process with errors for nonlinear strongly accretive mappings in Banach spaces. J. Math. Anal. Appl. 1995, 194(1):114–125. 10.1006/jmaa.1995.1289MathSciNetView ArticleGoogle Scholar
  27. Berinde V: Generalized contractions and applications (Romanian). Editura Cub Press 22, Baia Mare; 1997.Google 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.