Open Access

Convergence theorems for mixed type asymptotically nonexpansive mappings

Fixed Point Theory and Applications20122012:224

DOI: 10.1186/1687-1812-2012-224

Received: 27 April 2012

Accepted: 16 November 2012

Published: 11 December 2012

Abstract

In this paper, we introduce a new two-step iterative scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and prove strong and weak convergence theorems for the new two-step iterative scheme in uniformly convex Banach spaces.

Keywords

mixed type asymptotically nonexpansive mapping strong and weak convergence common fixed point uniformly convex Banach space

1 Introduction

Let K be a nonempty subset of a real normed linear space E. A mapping T : K K is said to be asymptotically nonexpansive if there exists a sequence { k n } [ 1 , ) with lim n k n = 1 such that
T n x T n y k n x y
(1.1)

for all x , y K and n 1 .

In 1972, Goebel and Kirk [1] introduced the class of asymptotically nonexpansive self-mappings, which is an important generalization of the class of nonexpansive self-mappings, and proved that if K is a nonempty closed convex subset of a real uniformly convex Banach space E and T is an asymptotically nonexpansive self-mapping of K, then T has a fixed point.

Since then, some authors proved weak and strong convergence theorems for asymptotically nonexpansive self-mappings in Banach spaces (see [216]), which extend and improve the result of Goebel and Kirk in several ways.

Recently, Chidume et al. [10] introduced the concept of asymptotically nonexpansive nonself-mappings, which is a generalization of an asymptotically nonexpansive self-mapping, as follows.

Definition 1.1 [10]

Let K be a nonempty subset of a real normed linear space E. Let P : E K be a nonexpansive retraction of E onto K. A nonself-mapping T : K E is said to be asymptotically nonexpansive if there exists a sequence { k n } [ 1 , ) with k n 1 as n such that
T ( P T ) n 1 x T ( P T ) n 1 y k n x y
(1.2)

for all x , y K and n 1 .

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.

In 2003, also, Chidume et al. [10] studied the following iteration scheme:
{ x 1 K , x n + 1 = P ( ( 1 α n ) x n + α n T 1 ( P T 1 ) n 1 x n )
(1.3)

for each n 1 , where { α n } is a sequence in ( 0 , 1 ) and P is a nonexpansive retraction of E onto K, and proved some strong and weak convergence theorems for an asymptotically nonexpansive nonself-mapping.

In 2006, Wang [11] generalized the iteration process (1.3) as follows:
{ x 1 K , x n + 1 = P ( ( 1 α n ) x n + α n T 1 ( P T 1 ) n 1 y n ) , y n = P ( ( 1 β n ) x n + β n T 2 ( P T 2 ) n 1 x n )
(1.4)

for each n 1 , where T 1 , T 2 : K E are two asymptotically nonexpansive nonself-mappings and { α n } , { β n } are real sequences in [ 0 , 1 ) , and proved some strong and weak convergence theorems for two asymptotically nonexpansive nonself-mappings. Recently, Guo and Guo [12] proved some new weak convergence theorems for the iteration process (1.4).

The purpose of this paper is to construct a new iteration scheme of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappings and to prove some strong and weak convergence theorems for the new iteration scheme in uniformly convex Banach spaces.

2 Preliminaries

Let E be a real Banach space, K be a nonempty closed convex subset of E and P : E K be a nonexpansive retraction of E onto K. Let S 1 , S 2 : K K be two asymptotically nonexpansive self-mappings and T 1 , T 2 : K E be two asymptotically nonexpansive nonself-mappings. Then we define the new iteration scheme of mixed type as follows:
{ x 1 K , x n + 1 = P ( ( 1 α n ) S 1 n x n + α n T 1 ( P T 1 ) n 1 y n ) , y n = P ( ( 1 β n ) S 2 n x n + β n T 2 ( P T 2 ) n 1 x n )
(2.1)

for each n 1 , where { α n } , { β n } are two sequences in [ 0 , 1 ) .

If S 1 and S 2 are the identity mappings, then the iterative scheme (2.1) reduces to the sequence (1.4).

We denote the set of common fixed points of S 1 , S 2 , T 1 and T 2 by F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) and denote the distance between a point z and a set A in E by d ( z , A ) = inf x A z x .

Now, we recall some well-known concepts and results.

Let E be a real Banach space, E be the dual space of E and J : E 2 E be the normalized duality mapping defined by
J ( x ) = { f E : x , f = x f , f = x }

for all x E , where , denotes duality pairing between E and E . A single-valued normalized duality mapping is denoted by j.

A subset K of a real Banach space E is called a retract of E [10] if there exists a continuous mapping P : E K such that P x = x for all x K . Every closed convex subset of a uniformly convex Banach space is a retract. A mapping P : E E is called a retraction if P 2 = P . It follows that if a mapping P is a retraction, then P y = y for all y in the range of P.

A Banach space E is said to satisfy Opial’s condition [17] if, for any sequence { x n } of E, x n x weakly as n implies that
lim sup n x n x < lim sup n x n y

for all y E with y x .

A Banach space E is said to have a Fréchet differentiable norm [18] if, for all x U = { x E : x = 1 } ,
lim t 0 x + t y x t

exists and is attained uniformly in y U .

A Banach space E is said to have the Kadec-Klee property [19] if for every sequence { x n } in E, x n x weakly and x n x , it follows that x n x strongly.

Let K be a nonempty closed subset of a real Banach space E. A nonself-mapping T : K E is said to be semi-compact [11] if, for any sequence { x n } in K such that x n T x n 0 as n , there exists a subsequence { x n j } of { x n } such that { x n j } converges strongly to some x K .

Lemma 2.1 [15]

Let { a n } , { b n } and { c n } be three nonnegative sequences satisfying the following condition:
a n + 1 ( 1 + b n ) a n + c n

for each n n 0 , where n 0 is some nonnegative integer, n = n 0 b n < and n = n 0 c n < . Then lim n a n exists.

Lemma 2.2 [8]

Let E be a real uniformly convex Banach space and 0 < p t n q < 1 for each n 1 . Also, suppose that { x n } and { y n } are two sequences of E such that
lim sup n x n r , lim sup n y n r , lim n t n x n + ( 1 t n ) y n = r

hold for some r 0 . Then lim n x n y n = 0 .

Lemma 2.3 [10]

Let E be a real uniformly convex Banach space, K be a nonempty closed convex subset of E and T : K E be an asymptotically nonexpansive mapping with a sequence { k n } [ 1 , ) and k n 1 as n . Then I T is demiclosed at zero, i.e., if x n x weakly and x n T x n 0 strongly, then x F ( T ) , where F ( T ) is the set of fixed points of T.

Lemma 2.4 [16]

Let X be a uniformly convex Banach space and C be a convex subset of X. Then there exists a strictly increasing continuous convex function γ : [ 0 , ) [ 0 , ) with γ ( 0 ) = 0 such that, for each mapping S : C C with a Lipschitz constant L > 0 ,
α S x + ( 1 α ) S y S [ α x + ( 1 α ) y ] L γ 1 ( x y 1 L S x S y )

for all x , y C and 0 < α < 1 .

Lemma 2.5 [16]

Let X be a uniformly convex Banach space such that its dual space X has the Kadec-Klee property. Suppose { x n } is a bounded sequence and f 1 , f 2 W w ( { x n } ) such that
lim n α x n + ( 1 α ) f 1 f 2

exists for all α [ 0 , 1 ] , where W w ( { x n } ) denotes the set of all weak subsequential limits of  { x n } . Then f 1 = f 2 .

3 Strong convergence theorems

In this section, we prove strong convergence theorems for the iterative scheme given in (2.1) in uniformly convex Banach spaces.

Lemma 3.1 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let S 1 , S 2 : K K be two asymptotically nonexpansive self-mappings with { k n ( 1 ) } , { k n ( 2 ) } [ 1 , ) and T 1 , T 2 : K E be two asymptotically nonexpansive nonself-mappings with { l n ( 1 ) } , { l n ( 2 ) } [ 1 , ) such that n = 1 ( k n ( i ) 1 ) < and n = 1 ( l n ( i ) 1 ) < for i = 1 , 2 , respectively, and F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) . Let { x n } be the sequence defined by (2.1), where { α n } and { β n } are two real sequences in [ 0 , 1 ) . Then
  1. (1)

    lim n x n q exists for any q F ;

     
  2. (2)

    lim n d ( x n , F ) exists.

     
Proof (1) Set h n = max { k n ( 1 ) , k n ( 2 ) , l n ( 1 ) , l n ( 2 ) } . For any q F , it follows from (2.1) that
y n q ( 1 β n ) ( S 2 n x n q ) + β n ( T 2 ( P T 2 ) n 1 x n q ) ( 1 β n ) h n x n q + β n h n x n q = h n x n q
(3.1)
and so
x n + 1 q ( 1 α n ) ( S 1 n x n q ) + α n ( T 1 ( P T 1 ) n 1 y n q ) ( 1 α n ) h n x n q + α n h n y n q ( 1 α n ) h n 2 x n q + α n h n 2 x n q = [ 1 + ( h n 2 1 ) ] x n q .
(3.2)
Since n = 1 ( k n ( i ) 1 ) < and n = 1 ( l n ( i ) 1 ) < for i = 1 , 2 , we have n = 1 ( h n 2 1 ) < . It follows from Lemma 2.1 that lim n x n q exists.
  1. (2)
    Taking the infimum over all q F in (3.2), we have
    d ( x n + 1 , F ) [ 1 + ( h n 2 1 ) ] d ( x n , F )
     

for each n 1 . It follows from n = 1 ( h n 2 1 ) < and Lemma 2.1 that the conclusion (2) holds. This completes the proof. □

Lemma 3.2 Let E be a real uniformly convex Banach space and K be a nonempty closed convex subset of E. Let S 1 , S 2 : K K be two asymptotically nonexpansive self-mappings with { k n ( 1 ) } , { k n ( 2 ) } [ 1 , ) and T 1 , T 2 : K E be two asymptotically nonexpansive nonself-mappings with { l n ( 1 ) } , { l n ( 2 ) } [ 1 , ) such that n = 1 ( k n ( i ) 1 ) < and n = 1 ( l n ( i ) 1 ) < for i = 1 , 2 , respectively, and F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) . Let { x n } be the sequence defined by (2.1) and the following conditions hold:
  1. (a)

    { α n } and { β n } are two real sequences in [ ϵ , 1 ϵ ] for some ϵ ( 0 , 1 ) ;

     
  2. (b)

    x T i y S i x T i y for all x , y K and i = 1 , 2 .

     

Then lim n x n S i x n = lim n x n T i x n = 0 for i = 1 , 2 .

Proof Set h n = max { k n ( 1 ) , k n ( 2 ) , l n ( 1 ) , l n ( 2 ) } . For any given q F , lim n x n q exists by Lemma 3.1. Now, we assume that lim n x n q = c . It follows from (3.2) and n = 1 ( h n 2 1 ) < that
lim n ( 1 α n ) ( S 1 n x n q ) + α n ( T 1 ( P T 1 ) n 1 y n q ) = c
and
lim sup n S 1 n x n q lim sup n k n ( 1 ) x n q = c .
Taking lim sup on both sides in (3.1), we obtain lim sup n y n q c and so
lim sup n T 1 ( P T 1 ) n 1 y n q lim sup n l n ( 1 ) y n q c .
Using Lemma 2.2, we have
lim n S 1 n x n T 1 ( P T 1 ) n 1 y n = 0 .
(3.3)
By the condition (b), it follows that
x n T 1 ( P T 1 ) n 1 y n S 1 n x n T 1 ( P T 1 ) n 1 y n
and so, from (3.3), we have
lim n x n T 1 ( P T 1 ) n 1 y n = 0 .
(3.4)
Since
x n q x n T 1 ( P T 1 ) n 1 y n + T 1 ( P T 1 ) n 1 y n q x n T 1 ( P T 1 ) n 1 y n + l n ( 1 ) y n q .
Taking lim inf on both sides in the inequality above, we have
lim inf n y n q c
by (3.4) and so
lim n y n q = c .
Using (3.1), we have
lim n ( 1 β n ) ( S 2 n x n q ) + β n ( T 2 ( P T 2 ) n 1 x n q ) = c .
In addition, we have
lim sup n S 2 n x n q lim sup n k n ( 2 ) x n q = c
and
lim sup n T 2 ( P T 2 ) n 1 x n q lim sup n l n ( 2 ) x n q = c .
It follows from Lemma 2.2 that
lim n S 2 n x n T 2 ( P T 2 ) n 1 x n = 0 .
(3.5)
Now, we prove that
lim n x n T 1 x n = lim n x n T 2 x n = 0 .
Indeed, since x n T 2 ( P T 2 ) n 1 x n S 2 n x n T 2 ( P T 2 ) n 1 x n by the condition (b). It follows from (3.5) that
lim n x n T 2 ( P T 2 ) n 1 x n = 0 .
(3.6)
Since S 2 n x n = P ( S 2 n x n ) and P : E K is a nonexpansive retraction of E onto K, we have
y n S 2 n x n β n S 2 n x n T 2 ( P T 2 ) n 1 x n
and so
lim n y n S 2 n x n = 0 .
(3.7)
Furthermore, we have
y n x n y n S 2 n x n + S 2 n x n T 2 ( P T 2 ) n 1 x n + T 2 ( P T 2 ) n 1 x n x n .
Thus it follows from (3.5), (3.6) and (3.7) that
lim n x n y n = 0 .
(3.8)
Since x n T 1 ( P T 1 ) n 1 x n S 1 n x n T 1 ( P T 1 ) n 1 x n by the condition (b) and
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equv_HTML.gif
Using (3.3) and (3.8), we have
lim n S 1 n x n T 1 ( P T 1 ) n 1 x n = 0
(3.9)
and
lim n x n T 1 ( P T 1 ) n 1 x n = 0 .
(3.10)
It follows from
x n + 1 S 1 n x n = P [ ( 1 α n ) S 1 n x n + α n T 1 ( P T 1 ) n 1 y n ] P ( S 1 n x n ) α n S 1 n x n T 1 ( P T 1 ) n 1 y n
and (3.3) that
lim n x n + 1 S 1 n x n = 0 .
(3.11)
In addition, we have
x n + 1 T 1 ( P T 1 ) n 1 y n x n + 1 S 1 n x n + S 1 n x n T 1 ( P T 1 ) n 1 y n .
Using (3.3) and (3.11), we obtain that
lim n x n + 1 T 1 ( P T 1 ) n 1 y n = 0 .
(3.12)
Thus, using (3.9), (3.10) and the inequality
S 1 n x n x n S 1 n x n T 1 ( P T 1 ) n 1 x n + T 1 ( P T 1 ) n 1 x n x n ,
we have lim n S 1 n x n x n = 0 . It follows from (3.6) and the inequality
S 1 n x n T 2 ( P T 2 ) n 1 x n S 1 n x n x n + x n T 2 ( P T 2 ) n 1 x n
that
lim n S 1 n x n T 2 ( P T 2 ) n 1 x n = 0 .
(3.13)
Since
x n + 1 T 2 ( P T 2 ) n 1 y n x n + 1 S 1 n x n + S 1 n x n T 2 ( P T 2 ) n 1 x n + l n ( 2 ) x n y n ,
from (3.8), (3.11) and (3.13), it follows that
lim n x n + 1 T 2 ( P T 2 ) n 1 y n = 0 .
(3.14)
Again, since ( P T i ) ( P T i ) n 2 y n 1 , x n K for i = 1 , 2 and T 1 , T 2 are two asymptotically nonexpansive nonself-mappings, we have
https://static-content.springer.com/image/art%3A10.1186%2F1687-1812-2012-224/MediaObjects/13663_2012_Article_329_Equ20_HTML.gif
(3.15)
for i = 1 , 2 . It follows from (3.12), (3.14) and (3.15) that
lim n T i ( P T i ) n 1 y n 1 T i x n = 0
(3.16)
for i = 1 , 2 . Moreover, we have
x n + 1 y n x n + 1 T 1 ( P T 1 ) n 1 y n + T 1 ( P T 1 ) n 1 y n x n + x n y n .
Using (3.4), (3.8) and (3.12), we have
lim n x n + 1 y n = 0 .
(3.17)
In addition, we have
x n T i x n x n T i ( P T i ) n 1 x n + T i ( P T i ) n 1 x n T i ( P T i ) n 1 y n 1 + T i ( P T i ) n 1 y n 1 T i x n x n T i ( P T i ) n 1 x n + max { sup n 1 l n ( 1 ) , sup n 1 l n ( 2 ) } x n y n 1 + T i ( P T i ) n 1 y n 1 T i x n
for i = 1 , 2 . Thus it follows from (3.6), (3.10), (3.16) and (3.17) that
lim n x n T 1 x n = lim n x n T 2 x n = 0 .
Finally, we prove that
lim n x n S 1 x n = lim n x n S 2 x n = 0 .
In fact, by the condition (b), we have
x n S i x n x n T i ( P T i ) n 1 x n + S i x n T i ( P T i ) n 1 x n x n T i ( P T i ) n 1 x n + S i n x n T i ( P T i ) n 1 x n
for i = 1 , 2 . Thus it follows from (3.5), (3.6), (3.9) and (3.10) that
lim n x n S 1 x n = lim n x n S 2 x n = 0 .

This completes the proof. □

Now, we find two mappings, S 1 = S 2 = S and T 1 = T 2 = T , satisfying the condition (b) in Lemma 3.2 as follows.

Example 3.1 [20]

Let be the real line with the usual norm | | and let K = [ 1 , 1 ] . Define two mappings S , T : K K by
T x = { 2 sin x 2 , if  x [ 0 , 1 ] , 2 sin x 2 , if  x [ 1 , 0 ) ,
and
S x = { x , if  x [ 0 , 1 ] , x , if  x [ 1 , 0 ) .
Now, we show that T is nonexpansive. In fact, if x , y [ 0 , 1 ] or x , y [ 1 , 0 ) , then we have
| T x T y | = 2 | sin x 2 sin y 2 | | x y | .
If x [ 0 , 1 ] and y [ 1 , 0 ) or x [ 1 , 0 ) and y [ 0 , 1 ] , then we have
| T x T y | = 2 | sin x 2 + sin y 2 | = 4 | sin x + y 4 cos x y 4 | | x + y | | x y | .

This implies that T is nonexpansive and so T is an asymptotically nonexpansive mapping with k n = 1 for each n 1 . Similarly, we can show that S is an asymptotically nonexpansive mapping with l n = 1 for each n 1 .

Next, we show that two mappings S, T satisfy the condition (b) in Lemma 3.2. For this, we consider the following cases:

Case 1. Let x , y [ 0 , 1 ] . Then we have
| x T y | = | x + 2 sin y 2 | = | S x T y | .
Case 2. Let x , y [ 1 , 0 ) . Then we have
| x T y | = | x 2 sin y 2 | | x 2 sin y 2 | = | S x T y | .
Case 3. Let x [ 1 , 0 ) and y [ 0 , 1 ] . Then we have
| x T y | = | x + 2 sin y 2 | | x + 2 sin y 2 | = | S x T y | .
Case 4. Let x [ 0 , 1 ] and y [ 1 , 0 ) . Then we have
| x T y | = | x 2 sin y 2 | = | S x T y | .

Therefore, the condition (b) in Lemma 3.2 is satisfied.

Theorem 3.1 Under the assumptions of Lemma  3.2, if one of S 1 , S 2 , T 1 and T 2 is completely continuous, then the sequence { x n } defined by (2.1) converges strongly to a common fixed point of S 1 , S 2 , T 1 and T 2 .

Proof Without loss of generality, we can assume that S 1 is completely continuous. Since { x n } is bounded by Lemma 3.1, there exists a subsequence { S 1 x n j } of { S 1 x n } such that { S 1 x n j } converges strongly to some q . Moreover, we know that
lim j x n j S 1 x n j = lim j x n j S 2 x n j = 0
and
lim j x n j T 1 x n j = lim j x n j T 2 x n j = 0
by Lemma 3.2, which imply that
x n j q x n j S 1 x n j + S 1 x n j q 0
as j and so x n j q K . Thus, by the continuity of S 1 , S 2 , T 1 and T 2 , we have
q S i q = lim j x n j S i x n j = 0
and
q T i q = lim j x n j T i x n j = 0

for i = 1 , 2 . Thus it follows that q F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) . Furthermore, since lim n x n q exists by Lemma 3.1, we have lim n x n q = 0 . This completes the proof. □

Theorem 3.2 Under the assumptions of Lemma  3.2, if one of S 1 , S 2 , T 1 and T 2 is semi-compact, then the sequence { x n } defined by (2.1) converges strongly to a common fixed point of S 1 , S 2 , T 1 and T 2 .

Proof Since lim n x n S i x n = lim n x n T i x n = 0 for i = 1 , 2 by Lemma 3.2 and one of S 1 , S 2 , T 1 and T 2 is semi-compact, there exists a subsequence { x n j } of { x n } such that { x n j } converges strongly to some q K . Moreover, by the continuity of S 1 , S 2 , T 1 and T 2 , we have q S i q = lim j x n j S i x n j = 0 and q T i q = lim j x n j T i x n j = 0 for i = 1 , 2 . Thus it follows that q F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) . Since lim n x n q exists by Lemma 3.1, we have lim n x n q = 0 . This completes the proof. □

Theorem 3.3 Under the assumptions of Lemma  3.2, 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 ) ) x S 1 x + x S 2 x + x T 1 x + x T 2 x

for all x K , where F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) , then the sequence { x n } defined by (2.1) converges strongly to a common fixed point of S 1 , S 2 , T 1 and T 2 .

Proof Since lim n x n S i x n = lim n x n T i x n = 0 for i = 1 , 2 by Lemma 3.2, we have lim n f ( d ( x n , F ) ) = 0 . Since f : [ 0 , ) [ 0 , ) is a nondecreasing function satisfying f ( 0 ) = 0 , f ( r ) > 0 for all r ( 0 , ) and lim n d ( x n , F ) exists by Lemma 3.1, we have lim n d ( x n , F ) = 0 .

Now, we show that { x n } is a Cauchy sequence in K. In fact, from (3.2), we have
x n + 1 q [ 1 + ( h n 2 1 ) ] x n q
for each n 1 , where h n = max { k n ( 1 ) , k n ( 2 ) , l n ( 1 ) , l n ( 2 ) } and q F . For any m, n, m > n 1 , we have
x m q [ 1 + ( h m 1 2 1 ) ] x m 1 q e h m 1 2 1 x m 1 q e h m 1 2 1 e h m 2 2 1 x m 2 q e i = n m 1 ( h i 2 1 ) x n q M x n q ,
where M = e i = 1 ( h i 2 1 ) . Thus, for any q F , we have
x n x m x n q + x m q ( 1 + M ) x n q .
Taking the infimum over all q F , we obtain
x n x m ( 1 + M ) d ( x n , F ) .

Thus it follows from lim n d ( x n , F ) = 0 that { x n } is a Cauchy sequence. Since K is a closed subset of E, the sequence { x n } converges strongly to some q K . It is easy to prove that F ( S 1 ) , F ( S 2 ) , F ( T 1 ) and F ( T 2 ) are all closed and so F is a closed subset of K. Since lim n d ( x n , F ) = 0 , q F , the sequence { x n } converges strongly to a common fixed point of S 1 , S 2 , T 1 and T 2 . This completes the proof. □

4 Weak convergence theorems

In this section, we prove weak convergence theorems for the iterative scheme defined by (2.1) in uniformly convex Banach spaces.

Lemma 4.1 Under the assumptions of Lemma  3.1, for all q 1 , q 2 F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) , the limit
lim n t x n + ( 1 t ) q 1 q 2

exists for all t [ 0 , 1 ] , where { x n } is the sequence defined by (2.1).

Proof Set a n ( t ) = t x n + ( 1 t ) q 1 q 2 . Then lim n a n ( 0 ) = q 1 q 2 and, from Lemma 3.1, lim n a n ( 1 ) = lim n x n q 2 exists. Thus it remains to prove Lemma 4.1 for any t ( 0 , 1 ) .

Define the mapping G n : K K by
G n x = P [ ( 1 α n ) S 1 n x + α n T 1 ( P T 1 ) n 1 P ( ( 1 β n ) S 2 n x + β n T 2 ( P T 2 ) n 1 x ) ]
for all x K . It is easy to prove that
G n x G n y h n 4 x y
(4.1)
for all x , y K , where h n = max { k n ( 1 ) , k n ( 2 ) , l n ( 1 ) , l n ( 2 ) } . Letting h n = 1 + v n , it follows from 1 j = n h j 4 e 4 j = n v j and n = 1 v n < that lim n j = n h j 4 = 1 . Setting
S n , m = G n + m 1 G n + m 2 G n
(4.2)
for each m 1 , from (4.1) and (4.2), it follows that
S n , m x S n , m y ( j = n n + m 1 h j 4 ) x y
for all x , y K and S n , m x n = x n + m , S n , m q = q for any q F . Let
b n , m = t S n , m x n + ( 1 t ) S n , m q 1 S n , m ( t x n + ( 1 t ) q 1 ) .
(4.3)
Then, using (4.3) and Lemma 2.4, we have
b n , m ( j = n n + m 1 h j 4 ) γ 1 ( x n q 1 ( j = n n + m 1 h j 4 ) 1 S n , m x n S n , m q 1 ) ( j = n h j 4 ) γ 1 ( x n q 1 ( j = n h j 4 ) 1 x n + m q 1 ) .
It follows from Lemma 3.1 and lim n j = n h j 4 = 1 that lim n b n , m = 0 uniformly for all m. Observe that
a n + m ( t ) S n , m ( t x n + ( 1 t ) q 1 ) q 2 + b n , m = S n , m ( t x n + ( 1 t ) q 1 ) S n , m q 2 + b n , m ( j = n n + m 1 h j 4 ) t x n + ( 1 t ) q 1 q 2 + b n , m ( j = n h j 4 ) a n ( t ) + b n , m .

Thus we have lim sup n a n ( t ) lim inf n a n ( t ) , that is, lim n t x n + ( 1 t ) q 1 q 2 exists for all t ( 0 , 1 ) . This completes the proof. □

Lemma 4.2 Under the assumptions of Lemma  3.1, if E has a Fréchet differentiable norm, then, for all q 1 , q 2 F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) , the limit
lim n x n , j ( q 1 q 2 )

exists, where { x n } is the sequence defined by (2.1). Furthermore, if W w ( { x n } ) denotes the set of all weak subsequential limits of { x n } , then x y , j ( q 1 q 2 ) = 0 for all q 1 , q 2 F and x , y W w ( { x n } ) .

Proof This follows basically as in the proof of Lemma 3.2 of [12] using Lemma 4.1 instead of Lemma 3.1 of [12]. □

Theorem 4.1 Under the assumptions of Lemma  3.2, if E has a Fréchet differentiable norm, then the sequence { x n } defined by (2.1) converges weakly to a common fixed point of S 1 , S 2 , T 1 and T 2 .

Proof Since E is a uniformly convex Banach space and the sequence { x n } is bounded by Lemma 3.1, there exists a subsequence { x n k } of { x n } which converges weakly to some q K . By Lemma 3.2, we have
lim k x n k S i x n k = lim k x n k T i x n k = 0

for i = 1 , 2 . It follows from Lemma 2.3 that q F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) .

Now, we prove that the sequence { x n } converges weakly to q. Suppose that there exists a subsequence { x m j } of { x n } such that { x m j } converges weakly to some q 1 K . Then, by the same method given above, we can also prove that q 1 F . So, q , q 1 F W w ( { x n } ) . It follows from Lemma 4.2 that
q q 1 2 = q q 1 , j ( q q 1 ) = 0 .

Therefore, q 1 = q , which shows that the sequence { x n } converges weakly to q. This completes the proof. □

Theorem 4.2 Under the assumptions of Lemma  3.2, if the dual space E of E has the Kadec-Klee property, then the sequence { x n } defined by (2.1) converges weakly to a common fixed point of S 1 , S 2 , T 1 and T 2 .

Proof Using the same method given in Theorem 4.1, we can prove that there exists a subsequence { x n k } of { x n } which converges weakly to some q F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) .

Now, we prove that the sequence { x n } converges weakly to q. Suppose that there exists a subsequence { x m j } of { x n } such that { x m j } converges weakly to some q K . Then, as for q, we have q F . It follows from Lemma 4.1 that the limit
lim n t x n + ( 1 t ) q q

exists for all t [ 0 , 1 ] . Again, since q , q W w ( { x n } ) , q = q by Lemma 2.5. This shows that the sequence { x n } converges weakly to q. This completes the proof. □

Theorem 4.3 Under the assumptions of Lemma  3.2, if E satisfies Opial’s condition, then the sequence { x n } defined by (2.1) converges weakly to a common fixed point of S 1 , S 2 , T 1 and T 2 .

Proof Using the same method as given in Theorem 4.1, we can prove that there exists a subsequence { x n k } of { x n } which converges weakly to some q F = F ( S 1 ) F ( S 2 ) F ( T 1 ) F ( T 2 ) .

Now, we prove that the sequence { x n } converges weakly to q. Suppose that there exists a subsequence { x m j } of { x n } such that { x m j } converges weakly to some q ¯ K and q ¯ q . Then, as for q, we have q ¯ F . Using Lemma 3.1, we have the following two limits exist:
lim n x n q = c , lim n x n q ¯ = c 1 .
Thus, by Opial’s condition, we have
c = lim sup k x n k q < lim sup k x n k q ¯ = lim sup j x m j q ¯ < lim sup j x m j q = c ,

which is a contradiction and so q = q ¯ . This shows that the sequence { x n } converges weakly to q. This completes the proof. □

Declarations

Acknowledgements

The project was supported by the National Natural Science Foundation of China (Grant Number: 11271282) and the second author was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant Number: 2012-0008170).

Authors’ Affiliations

(1)
School of Mathematics and Physics, Suzhou University of Science and Technology
(2)
Department of Mathematics Education and the RINS College of Education, Gyeongsang National University
(3)
Department of Aerospace Engineering and Mechanics, University of Minnesota

References

  1. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mapping. Proc. Am. Math. Soc. 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView Article
  2. Chang SS, Cho YJ, Zhou HY: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. J. Korean Math. Soc. 2001, 38: 1245–1260.MathSciNet
  3. Chang SS, Tan KK, Lee HWJ, Chan CK: On the convergence of implicit iteration process with error for a finite family of asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2006, 313: 273–283. 10.1016/j.jmaa.2005.05.075MathSciNetView Article
  4. Cho YJ, Kang JI, Zhou HY: Approximating common fixed points of asymptotically nonexpansive mappings. Bull. Korean Math. Soc. 2005, 42: 661–670.MathSciNetView Article
  5. Cho YJ, Zhou HY, Guo GT: Weak and strong convergence theorems for three-step iterations with errors for asymptotically nonexpansive mappings. Comput. Math. Appl. 2004, 47: 707–717. 10.1016/S0898-1221(04)90058-2MathSciNetView Article
  6. Chen W, Guo W: Convergence theorems for two finite families of asymptotically nonexpansive mappings. Math. Comput. Model. 2011, 54: 1311–1319. 10.1016/j.mcm.2011.04.002View Article
  7. Guo W, Cho YJ: On strong convergence of the implicit iterative processes with errors for a finite family of asymptotically nonexpansive mappings. Appl. Math. Lett. 2008, 21: 1046–1052. 10.1016/j.aml.2007.07.034MathSciNetView Article
  8. Schu J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull. Aust. Math. Soc. 1991, 43: 153–159. 10.1017/S0004972700028884MathSciNetView Article
  9. Zhou YY, Chang SS: Convergence of implicit iterative process for a finite family of asymptotically nonexpansive mappings in Banach spaces. Numer. Funct. Anal. Optim. 2002, 23: 911–921. 10.1081/NFA-120016276MathSciNetView Article
  10. Chidume CE, Ofoedu EU, Zegeye H: Strong and weak convergence theorems for asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2003, 280: 364–374. 10.1016/S0022-247X(03)00061-1MathSciNetView Article
  11. Wang L: Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings. J. Math. Anal. Appl. 2006, 323: 550–557. 10.1016/j.jmaa.2005.10.062MathSciNetView Article
  12. Guo W, Guo W: Weak convergence theorems for asymptotically nonexpansive nonself-mappings. Appl. Math. Lett. 2011, 24: 2181–2185. 10.1016/j.aml.2011.06.022MathSciNetView Article
  13. Pathak HK, Cho YJ, Kang SM: Strong and weak convergence theorems for nonself-asymptotically perturbed nonexpansive mappings. Nonlinear Anal. 2009, 70: 1929–1938. 10.1016/j.na.2008.02.092MathSciNetView Article
  14. Zhou HY, Cho YJ, Kang SM: A new iterative algorithm for approximating common fixed points for asymptotically nonexpansive mappings. Fixed Point Theory Appl. 2007., 2007: Article ID 64974
  15. Sun ZH: Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings. J. Math. Anal. Appl. 2003, 286: 351–358. 10.1016/S0022-247X(03)00537-7MathSciNetView Article
  16. Falset JG, Kaczor W, Kuczumow T, Reich S: Weak convergence theorems for asymptotically nonexpansive mappings and semigroups. Nonlinear Anal. 2001, 43: 377–401. 10.1016/S0362-546X(99)00200-XMathSciNetView Article
  17. Opial Z: Weak convergence of successive approximations for nonexpansive mappings. Bull. Am. Math. Soc. 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView Article
  18. Osilike MO, Udomene A: Demiclosedness principle and convergence theorems for strictly pseudocontractive mappings of Browder-Petryshyn type. J. Math. Anal. Appl. 2001, 256: 431–445. 10.1006/jmaa.2000.7257MathSciNetView Article
  19. Goebel K, Kirk WA Cambridge Studies in Advanced Mathematics 28. In Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge; 1990.View Article
  20. Liu Z, Feng C, Ume JS, Kang SM: Weak and strong convergence for common fixed points of a pair of nonexpansive and asymptotically nonexpansive mappings. Taiwan. J. Math. 2007, 11: 27–42.MathSciNet

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