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Convergence of perturbed composite implicit iteration process for a finite family of asymptotically nonexpansive mappings

Fixed Point Theory and Applications20132013:97

https://doi.org/10.1186/1687-1812-2013-97

  • Received: 4 July 2012
  • Accepted: 26 March 2013
  • Published:

Abstract

In this paper, we introduce a perturbed composite implicit iterative process with errors for a finite family of asymptotically nonexpansive mappings. Under Opial’s condition, semicompact and lim inf n d ( x n , F ( T ) ) = 0 conditions, respectively, we prove that this iterative scheme converges weakly or strongly to a common fixed point of a finite family of asymptotically nonexpansive mappings in uniformly convex Banach spaces. The results presented in this paper generalize and improve the corresponding results of Sun (J. Math. Anal. Appl. 286:351-358, 2003), Chang (J. Math. Anal. Appl. 313:273-283, 2006), Gu (J. Math. Anal. Appl. 329:766-776, 2007), Thakur (Appl. Math. Comput. 190:965-973, 007), Rafiq (Rostock. Math. Kolloqu. 62:21-39, 2007) and some others.

MSC:47H9, 47H10.

Keywords

  • asymptotically nonexpansive mapping
  • uniformly convex Banach space
  • perturbation
  • composite implicit iterative process
  • weak and strong convergence
  • common fixed point

1 Introduction

Let E be a real Banach space, K be a nonempty convex subset of E. Let { T 1 , T 2 , , T N } be a finite family of mappings from K into itself, and F ( T i ) be the set of fixed points of T i ( i I = { 1 , 2 , , N } ). F ( T ) denotes the set of common fixed points of { T 1 , T 2 , , T N } .

Recently, Xu and Ori [1] have introduced an implicit iteration process for a finite family of nonexpansive mappings as follows:
x n = α n x n 1 + ( 1 α n ) T n x n , n 1 ,
(1)

where T n = T n ( mod N ) (here the modN function takes values in I), { α n } be a real sequence in [ 0 , 1 ] , x 0 be an initial point in K.

Sun [2] have extended this iterative process defined by Xu and Ori to a new iterative process for a finite family of asymptotically nonexpansive mappings, which is defined as follows:
x n = α n x n 1 + ( 1 α n ) T i k x n , n 1 ,
(2)

where n = ( k 1 ) N + i , i I .

Chang [3] have discussed the convergence of the implicit iteration process with errors for a finite family of asymptotically nonexpansive mappings as follows:
x n = α n x n 1 + ( 1 α n ) T i ( n ) k ( n ) x n + u n , n 1 ,
(3)

where n = ( k ( n ) 1 ) N + i ( n ) , i ( n ) I , and k ( n ) 1 with k ( n ) as n . Under the hypotheses n = 1 u n < and some appropriate conditions, they proved some results of weak and strong convergence for { x n } defined by (3). However, the condition n = 1 u n < is not too reasonable, because this implies that { u n } are very small for n sufficiently big.

Gu [4] has extended the above implicit iteration processes. A composite implicit iteration process with random errors was introduced as follows:
{ x n = ( 1 α n γ n ) x n 1 + α n T n y n + γ n u n , n 1 ; y n = ( 1 β n δ n ) x n + β n T n x n + δ n v n , n 1 ,
(4)

where { α n } , { β n } , { γ n } , { δ n } are four real sequences in [ 0 , 1 ] satisfying α n + γ n 1 and β n + δ n 1 for all n 1 , { u n } , { v n } are two sequences in K and x 0 is an initial point. Some theorems were established on the strong convergence of the composite implicit iteration process defined by (4) for a finite family of mappings in real Banach spaces.

Thakur [5] has improved the composite implicit iteration process defined by (4) as follows:
{ x n = ( 1 α n ) x n 1 + α n T i ( n ) k ( n ) y n , n 1 ; y n = ( 1 β n ) x n + β n T i ( n ) k ( n ) x n , n 1 .
(5)

Some theorems were proved on the weak and strong convergence of the composite implicit iteration process defined by (5) for a finite family of mappings in real uniformly convex Banach spaces.

Rafiq [6] have improved the implicit iterative process. The Mann type implicit iteration process was introduced in Hilbert spaces as follows:
x n = α n x n 1 + ( 1 α n ) T v n , n 1 ,
(6)

where v n is a perturbation of x n , and satisfy n 1 x n v n < . Moreover, Ciric [7] also did some work in this respect.

Inspired and motivated by the above works, in this paper we will extend and improve the above iterative process to a perturbed composite implicit iterative process for a finite family of asymptotically nonexpansive mappings as follows:
{ x n = ( 1 α n γ n ) x n 1 + α n T i ( n ) k ( n ) y n + γ n u n , n 1 ; y n = ( 1 β n δ n ) x n 1 + β n T i ( n ) k ( n ) x ˜ n + δ n v n , n 1 ,
(7)

where n = ( k ( n ) 1 ) N + i ( n ) , i ( n ) I , T n = T n ( mod N ) , { α n } , { β n } , { γ n } , { δ n } are four real sequences in [ 0 , 1 ] satisfying α n + γ n 1 and β n + δ n 1 for all n 1 , { u n } , { v n } are two sequences in K and x 0 is an initial point. { x ˜ n } be a sequence in K satisfying n 1 x n x ˜ n < , which implies that x n x ˜ n 0 ( n ). Therefore, x ˜ n is known as the perturbation of x n , and { x ˜ n } is known as the perturbed sequence of { x n } . This sequence { x n } defined by (7) is said to be the perturbed composite implicit iterative sequence with random errors.

Especially, (I) in the iterative process defined by (7), when β n = 0 , δ n = 0 for all n 1 , we have
x n = ( 1 α n γ n ) x n 1 + α n T i ( n ) k ( n ) x n 1 + γ n u n , n 1 .
(8)

At this time, the perturbed composite implicit iterative sequence generated by (7) becomes a Mann-type iterative sequence with random errors.

(II) In the iterative process defined by (7), when β n = 1 , δ n = 0 for all n 1 , we have
x n = ( 1 α n γ n ) x n 1 + α n T i ( n ) 2 k ( n ) x ˜ n + γ n u n , n 1 .
(9)

At this time, the perturbed composite implicit iterative sequence generated by (7) becomes a perturbed implicit iterative sequence with random errors.

(III) In the iterative process defined by (7), when x n 1 = x ˜ n for all n 1 , we have
{ x n = ( 1 α n γ n ) x n 1 + α n T i ( n ) k ( n ) y n + γ n u n , n 1 ; y n = ( 1 β n δ n ) x n 1 + β n T i ( n ) k ( n ) x n 1 + δ n v n , n 1 .
(10)

At this time, the perturbed composite implicit iterative sequence generated by (7) becomes an Ishikawa-type iterative sequence with random errors for a finite family of asymptotically nonexpansive mappings { T i , i I } .

From the above iterative processes defined by (1)-(6) and (8)-(10), we know that the iterative process (7) improves and extends some iterative process introduced by the recent literature. Moreover, we point out that the iterative process, defined by (7), in which it is not necessary to compute the value of the given operator at x n , but compute an approximate point of x n , are particularly useful in the numerical analysis. Therefore, the iterative sequence generated by (7) is better than some implicit iterative sequences at the existent aspect.

The main purpose of this paper is to study the convergence of the perturbed composite implicit iterative sequence { x n } defined by (7) for a finite family of asymptotically nonexpansive mappings under Opial’s condition, semicompact and lim inf n d ( x n , F ( T ) ) = 0 conditions, respectively. The results presented in this paper generalized and improve the corresponding results of Sun [2], Chang [3], Gu [4], Thakur [5], Rafiq [6], and some others [1, 715].

2 Preliminaries

For the sake of convenience, we first recall some definitions and conclusions.

Definition 2.1 Let K be a closed subset of the real Banach space E and T : K K be a mapping.
  1. 1.

    T is said to be semicompact, if for any bounded sequence { x n } in K such that T x n x n 0 ( n ), then there exists a subsequence { x n i } of { x n } such that x n i x E ;

     
  2. 2.

    T is said to be demiclosed at the origin, if for each sequence { x n } in K, the conditions x n x 0 weakly and T x n 0 strongly imply T x 0 = 0 ;

     
  3. 3.
    T is said to be asymptotically nonexpansive, if there exists a sequence h n [ 1 , + ) with lim n h n = 1 such that
    T n x T n y h n x y , x , y K , n 1 .
    (11)
     
  4. 4.
    Let T is said to be uniformly L-Lipschitizian if there exists a constant L > 0 such that
    T n x T n y L x y , x , y E , n 1 .
     

Definition 2.2 [16]

A Banach space X is said to satisfy Opial’s condition if x n x weakly as n and x y imply that lim sup n x n x < lim sup n x n y .

Lemma 2.1 Let K be a nonempty subset of E, T 1 , T 2 , , T N : K K be N asymptotically nonexpansive mappings. Then
  1. (i)
    there exists a sequence { h n } [ 1 , + ) with lim n h n = 1 such that
    T i n x T i n y h n x y , x , y K , i I , n 1 ;
    (12)
     
  2. (ii)
    { T 1 , T 2 , , T N } is uniformly Lipschitzian, i.e., there exists a constant L such that
    T i n x T i n y L x y , x , y K , i I , n 1 .
    (13)
     
Proof Since T 1 , T 2 , , T N : K K are N asymptotically nonexpansive mappings, then for every i I and n N , there exists h n ( i ) [ 1 , + ) with lim n h n ( i ) = 1 such that
T i n x T i n y h n ( i ) x y , x , y E .

Taking h n = max { h n ( 1 ) , h n ( 2 ) , , h n ( N ) } , then h n [ 1 , + ) , lim n h n = 1 and (12) holds.

An asymptotically nonexpansive mapping must is a uniformly Lipschitzian mapping. Hence, for every i I and n N , there exists L i such that
T i n x T i n y L i x y , x , y E .

Taking L = max { L 1 , L 2 , , L N } , it is obvious that (13) holds. □

Lemma 2.2 [17]

Let E be a uniformly convex Banach space, K be a nonempty, closed and convex subset of E and T : K K be an asymptotically nonexpansive mapping. Then I T is demi-closed at zero, i.e., for each sequence { x n } in K, if { x n } convergence weakly to q E and { ( I T ) x n } converges strongly to 0, then ( I T ) q = 0 .

Lemma 2.3 [18]

Let E be a Banach space satisfying Opial’s condition, { x n } be a sequence in E. Let u , v E be such that lim n x n u and lim n x n v exist. If { x n k } and { x n l } are two subsequences of { x n } which converge weakly to u and v, respectively, then u = v .

Lemma 2.4 [19]

Let E be a uniformly convex Banach space, b, c be two constants with 0 < b < c < 1 . Suppose that { t n } is a sequence in [ b , c ] and { x n } , { y n } are two sequences in E. Then the conditions lim n t n x n + ( 1 t n ) y n = d , lim sup n x n d , lim sup n y n d imply that lim n x n y n = 0 , where d is a nonnegative constant.

Lemma 2.5 [20]

Let { a n } , { b n } , { δ n } are three sequences of nonnegative real numbers, if there exists n 0 such that
a n + 1 ( 1 + δ n ) a n + b n , n > n 0 ,
where n = 1 δ n < and n = 1 b n < . Then
  1. (i)

    lim n a n exists;

     
  2. (ii)

    lim n a n = 0 whenever lim inf n a n = 0 .

     
Lemma 2.6 Let E be a real Banach space and K be a nonempty closed convex subset of E. Let T 1 , T 2 , , T N : K K be N asymptotically nonexpansive mappings with F ( T ) = i = 1 N F ( T i ) . Let { u n } and { v n } are two bounded sequences in K. If { α n } , { β n } , { γ n } , { δ n } be four real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (i)

    α n + γ n 1 and β n + δ n 1 for all n 1 ;

     
  2. (ii)

    lim sup n α n = α < 1 or lim sup n β n = β < 1 ;

     
  3. (iii)

    n = 1 γ n < , n = 1 δ n < , n = 1 ( h n 1 ) < ;

     
  4. (iv)

    n = 1 x ˜ n x n < .

     

Let { x n } be the perturbed composite implicit iterative sequence defined by (7), then lim n x n p exists for all p F ( T ) .

Proof Take p F ( T ) , it follows from (7) and Lemma 2.1 that
x n p ( 1 α n γ n ) x n 1 + α n T i ( n ) k ( n ) y n + γ n u n p ( 1 α n γ n ) x n 1 p + α n h n y n p + γ n u n p
(14)
and
y n p ( 1 β n δ n ) x n 1 + β n T i ( n ) k ( n ) x ˜ n + δ n v n p ( 1 β n δ n ) x n 1 p + β n h n x ˜ n p + δ n v n p ( 1 β n δ n ) x n 1 p + β n h n x ˜ n x n + β n h n x n p + δ n v n p .
(15)
Substituting (15) into (14) and simplifying, we obtain
( 1 α n β n h n 2 ) x n p [ 1 α n γ n + α n h n ( 1 β n δ n ) ] x n 1 p + α n β n h n 2 x ˜ n x n + α n δ n h n v n p + γ n u n p .
(16)
We notice the hypotheses on { α n } , { β n } and { h n } , by lim sup n α n = α < 1 , there exists n 0 N such that
1 α n β n h n 2 1 α n h n 2 1 2 ( 1 α ) > 0 , n n 0 .
It follows from (16) that for n n 0
x n p 1 α n γ n + α n h n ( 1 β n δ n ) 1 α n β n h n 2 x n 1 p + 1 1 α n β n h n 2 ( α n β n h n 2 x ˜ n x n + α n δ n h n v n p + γ n u n p ) [ 1 + α n β n h n 2 α n + α n h n ( 1 β n ) 1 α n β n h n 2 ] x n 1 p + 2 1 α ( α n β n h n 2 x ˜ n x n + α n δ n h n v n p + γ n u n p ) { 1 + 2 1 α [ α n β n h n ( h n 1 ) + α n ( h n 1 ) ] } x n 1 p + 2 1 α ( α n β n h n 2 x ˜ n x n + α n δ n h n v n p + γ n u n p ) .
Hence, we have
x n p ( 1 + θ n ) x n 1 p + η n , n n 0 ,
(17)
where
θ n = 2 1 α [ α n β n h n ( h n 1 ) + α n ( h n 1 ) ] , n n 0
and
η n = 2 1 α ( α n β n h n 2 x ˜ n x n + α n δ n h n v n p + γ n u n p ) , n n 0 .

From condition (iii), it is obvious that n = 1 θ n < . In addition, since { u n } , { v n } are all bounded, we deduce that n = 1 η n < form (iii)-(iv). By virtue of (17) and Lemma 2.5, we obtain that lim n x n p exists. This completes the proof of Lemma 2.6. □

3 Main results and proofs

Theorem 3.1 Let E be a real Banach space and K be a nonempty, closed and convex subset of E. Let T 1 , T 2 , , T N : K K be N asymptotically nonexpansive mappings with F ( T ) = i = 1 N F ( T i ) . Let { u n } and { v n } are two bounded sequences in K. If { α n } , { β n } , { γ n } , { δ n } be four real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (i)

    α n + γ n 1 and β n + δ n 1 for all n 1 ;

     
  2. (ii)

    lim sup n α n < 1 or lim sup n β n < 1 ;

     
  3. (iii)

    n = 1 γ n < , n = 1 δ n < , n = 1 ( h n 1 ) < ;

     
  4. (iv)

    n = 1 x ˜ n x n < .

     

Then the perturbed composite implicit iterative sequence { x n } defined by (7) converges strongly to a common fixed point of { T 1 , T 2 , , T N } if and only if lim inf n d ( x n , F ( T ) ) = 0 .

Proof The necessity of Theorem 3.1 is obvious. Now we prove the sufficiency of Theorem 3.1.

For arbitrary p F ( T ) , it follows from (17) in Lemma 2.6 that
x n p ( 1 + θ n ) x n 1 p + η n , n n 0 ,
where n = 1 θ n < and n = 1 η n < . Hence, we have
d ( x n , F ( T ) ) ( 1 + θ n ) d ( x n 1 , F ( T ) ) + η n , n n 0 .
(18)
It follows from (18) and Lemma 2.5 that limit lim n d ( x n , F ( T ) ) exists. By the assumption, we have lim n d ( x n , F ( T ) ) = 0 . Consequently, for any given ε > 0 , there exists a positive integer N 1 ( N 1 > n 0 ) such that
d ( x n , F ( T ) ) < ε 8 , k = n η k < ε 8 , k = n θ k < 1 , n N 1 ,
and there exists p 1 F ( T ) such that x n p 1 < ε / 8 , n N 1 . By (18) and the inequality 1 + x e x ( x 0 ), for any n N 1 and all m 1 , we have
x n + m x n exp { θ n + m 1 } x n + m 1 p 1 + η n + m 1 + x n p 1 exp { θ n + m 1 + θ n + m 2 } x n + m 2 p 1 + exp { θ n + m 1 } η n + m 2 + η n + m 1 + x n p 1 [ exp { k = n n + m 1 θ k } + 1 ] x n p 1 + exp { k = n n + m 1 θ k } k = n n + m 1 η k < ε .
Hence, { x n } is a Cauchy sequence in E. By the completeness of E, we can assume that x n x K . Next we prove that F ( T ) is a close subset of K. Let { p n } is a sequence in F ( T ) which converges strongly to some p, then we have for any i I
p T i p p p n + p n T i p ( 1 + L ) p p n 0 ( n ) .

Thus, p F ( T ) , and F ( T ) is closed. Since lim n d ( x n , F ( T ) ) = 0 , then x F ( T ) . Consequently, { x n } defined by (7) converges strongly to a common fixed point of { T 1 , T 2 , , T N } in K. This completes the proof of Theorem 3.1. □

Theorem 3.2 Let E be a real uniformly convex Banach space satisfying Opial’s condition and K be a nonempty closed convex subset of E. Let T 1 , T 2 , , T N : K K be N asymptotically nonexpansive mappings with F ( T ) = i = 1 N F ( T i ) . Let { u n } and { v n } are two bounded sequences in K. If { α n } , { β n } , { γ n } , { δ n } be four real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (i)

    α n + γ n 1 and β n + δ n 1 for all n 1 ;

     
  2. (ii)

    0 < lim inf n α n lim sup n α n < 1 , lim sup n β n < 1 ;

     
  3. (iii)

    n = 1 γ n < , n = 1 δ n < , n = 1 ( h n 1 ) < ;

     
  4. (iv)

    n = 1 x ˜ n x n < .

     

Then the perturbed composite implicit iterative sequence { x n } defined by (7) converges weakly to a common fixed point of { T 1 , T 2 , , T N } in K.

Proof First, we prove that lim n x n T j x n = 0 for all j I .

For any p F ( T ) , it follows from Lemma 2.6 that lim n x n p exists. Suppose that lim n x n p = d , we have from (7)
lim n x n p = lim n ( 1 α n ) [ x n 1 p + γ n ( u n x n 1 ) ] + α n [ T i ( n ) k ( n ) y n p + γ n ( u n x n 1 ) ] = d .
(19)
Since lim n x n p = d , then { x n } be a bounded sequence. By virtue of the condition (iii) and the boundedness of sequences { x n } and { u n } , we have
(20)
It follows from n = 1 x ˜ n x n < that lim n x ˜ n p = lim n x n p = d . We have
(21)
Therefore, by (19), (20), (21), (ii) and Lemma 2.4, we obtain that
lim n T i ( n ) k ( n ) y n x n 1 = 0 .
Hence,
lim n x n x n 1 lim n [ α n T i ( n ) k ( n ) y n x n 1 + γ n u n x n 1 ] = 0 ,
(22)
which implies that lim n x n x n + j = 0 for all j I . On the other hand, we also have
lim n T i ( n ) k ( n ) x n x n lim n [ x n x n 1 + x n 1 T i ( n ) k ( n ) y n + T i ( n ) k ( n ) y n T i ( n ) k ( n ) x n ] lim n h n y n x n lim n y n x n 1 + lim n x n x n 1 lim n [ β n T i ( n ) k ( n ) x ˜ n x n 1 + δ n v n x n 1 ] lim n [ β n T i ( n ) k ( n ) x ˜ n T i ( n ) k ( n ) x n + β n T i ( n ) k ( n ) x n x n 1 ] lim n [ β n h n x ˜ n x n + β n T i ( n ) k ( n ) x n x n + β n x n 1 x n ] .
(23)
It follows from (22), (23), conditions (ii) and (iv) that
lim n T i ( n ) k ( n ) x n x n = 0 .
(24)
Since for each n > N , n = ( n N ) ( mod N ) , n = ( k ( n ) 1 ) N + i ( n ) , hence n N = [ ( k ( n ) 1 ) 1 ] N + i ( n N ) , i.e. k ( n N ) = k ( n ) 1 and i ( n N ) = i ( n ) . Therefore, we have
T n k ( n ) 1 x n T n N k ( n ) 1 x n N = T n k ( n ) 1 x n T n k ( n ) 1 x n N L x n x n N
(25)
and
T n N k ( n ) 1 x n N x n N = T n N k ( n N ) x n N x n N .
(26)
In view of (25) and (26), we have
x n 1 T n x n x n 1 T n k ( n ) x n + T n x n T n k ( n ) x n x n x n 1 + x n T n k ( n ) x n + L x n T n k ( n ) 1 x n x n x n 1 + x n T n k ( n ) x n + L ( T n k ( n ) 1 x n T n N k ( n ) 1 x n N + T n N k ( n ) 1 x n N x n ) x n x n 1 + x n T n k ( n ) x n + ( L 2 + L ) x n x n N + L T n N k ( n N ) x n N x n N .
(27)
From (24) and (27), it is obviously that lim n x n 1 T n x n = 0 , which implies that
lim n x n T n x n lim n ( x n 1 T n x n + x n x n 1 ) = 0 .
Consequently, we obtain that for all i I
x n T n + i x n x n x n + i + x n + i T n + i x n + i + T n + i x n + i T n + i x n ( 1 + L ) x n x n + i + x n + i T n + i x n + i 0 ( n ) .
(28)

By virtue of (28), we have lim n x n T i x n = 0 for all i I .

Since E is uniformly convex, every bounded subset of E is weakly compact. Again since { x n } is a bounded subset in K, there exists a subsequence { x n k } of { x n } such that { x n k } converges weakly to q in K, and lim n k x n k T i x n k = 0 for all i I . By Lemma 2.2, we have that ( I T i ) q = 0 . Hence, q F ( T i ) for all i I . Therefore, q F ( T ) .

Next, we prove that { x n } converges weakly to q. Suppose that contrary, then there exists a subsequence { x n j } of { x n } such that { x n j } converges weakly to q 1 K and q q 1 . Using the same method, we can prove that q 1 F ( T ) and limit lim n x n q 1 exists. Without loss generality, we assume that lim n x n q = d 1 , lim n x n q 1 = d 2 , where d 1 , d 2 are two nonnegative constants. By virtue of the Opial’s condition of E, we have
d 1 = lim sup n k x n k q < lim sup n k x n k q 1 = lim sup n x n q 1 = lim sup n j x n j q 1 < lim sup n j x n j q = d 1 .

This is contradictory. Hence, q = q 1 , which implies that { x n } converges weakly to q. The proof of Theorem 3.2 is completed. □

Theorem 3.3 Let E be a real uniformly convex Banach space and K be a nonempty, closed and convex subset of E. Let T 1 , T 2 , , T N : K K be N asymptotically nonexpansive mappings with F ( T ) = i = 1 n F ( T i ) and at least there exists T i ( i I ), it is semicompact. Let { u n } and { v n } are two bounded sequences in K. If { α n } , { β n } , { γ n } , { δ n } be four real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (i)

    α n + γ n 1 and β n + δ n 1 for all n 1 ;

     
  2. (ii)

    0 < lim inf n α n lim sup n α n < 1 , lim sup n β n < 1 ;

     
  3. (iii)

    n = 1 γ n < , n = 1 δ n < , n = 1 ( h n 1 ) < ;

     
  4. (iv)

    n = 1 x ˜ n x n < .

     

Then the perturbed composite implicit iterative sequence { x n } defined by (7) converges strongly to a common fixed point of { T 1 , T 2 , , T N } in K.

Proof Without loss of generality, we assume that T 1 is semicompact. By Theorem 3.2, we have lim n x n T 1 x n = 0 . Hence, there exists a subsequence { x n j } of { x n } such that { x n j } x as j . Therefore, we have for all i I
T i x x T i x T i x n j + T i x n j x n j + x n j x .
(29)

It follows from (29) that T i x x = 0 for all i I . This implies that x F ( T ) . Therefore, x be a common fixed point of { T i , i I } . By virtue of Lemma 2.6, lim n x n x exists. It follows from x n j x E that lim n x n x = 0 . Hence, the perturbed composites implicit iterative sequence { x n } generated by (7) strongly converges to a common fixed point of { T i , i I } . This completes the proof of Theorem 3.3. □

Corollary 3.4 Let E be a real Banach space and K be a nonempty closed convex subset of E. Let T 1 , T 2 , , T N : K K be N asymptotically nonexpansive mappings with F ( T ) = i = 1 N F ( T i ) and let { u n } is a bounded sequence in K. If { α n } , { γ n } be two real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (i)

    α n + γ n 1 for all n 1 ;

     
  2. (ii)

    lim sup n α n < 1 ;

     
  3. (iii)

    n = 1 γ n < , n = 1 ( h n 1 ) < ;

     
  4. (iv)

    n = 1 x ˜ n x n < .

     

Then the perturbed implicit iterative sequence { x n } defined by (9) converges strongly to a common fixed point of { T 1 , T 2 , , T N } if and only if lim inf n d ( x n , F ( T ) ) = 0 .

Proof It is enough to take β n = 1 , δ n = 0 for all n N in Theorem 3.1. □

Corollary 3.5 Let E be a real uniformly convex Banach space satisfying Opial’s condition and K be a nonempty closed convex subset of E. Let T 1 , T 2 , , T N : K K be N asymptotically nonexpansive mappings with F ( T ) = i = 1 N F ( T i ) and let { u n } is a bounded sequence in K. If { α n } , { γ n } be two real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (i)

    α n + γ n 1 for all n 1 ;

     
  2. (ii)

    0 < lim inf n α n lim sup n α n < 1 ;

     
  3. (iii)

    n = 1 γ n < , n = 1 ( h n 1 ) < .

     

Then the Mann type iterative sequence { x n } defined by (8) converges weakly to a common fixed point of { T 1 , T 2 , , T N } in K.

Proof It is sufficient to take β n = δ n = 0 for all n N in Theorem 3.2. □

Corollary 3.6 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let T 1 , T 2 , , T N : K K be N asymptotically nonexpansive mappings with F ( T ) = i = 1 n F ( T i ) and at least there exists T i ( i I ), it is semicompact. Let { u n } is a bounded sequence in K. If { α n } , { γ n } be two real sequences in [ 0 , 1 ] satisfying the following conditions:
  1. (i)

    α n + γ n 1 for all n 1 ;

     
  2. (ii)

    0 < lim inf n α n lim sup n α n < 1 ;

     
  3. (iii)

    n = 1 γ n < , n = 1 ( h n 1 ) < .

     

Then the Mann type iterative sequence { x n } defined by (8) converges strongly to a common fixed point of { T 1 , T 2 , , T N } in K.

Proof It is enough to take β n = δ n = 0 for all n N in Theorem 3.3. □

Declarations

Acknowledgements

The authors are grateful to the anonymous referee for valuable suggestions which helped to improve this manuscript.

Authors’ Affiliations

(1)
School of Mathematics and Information Science, Shandong Institute of Business and Technology, Yantai, Shandong, 264005, China

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© Wang; licensee Springer 2013

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