Open Access

An implicit iterative algorithm with errors for two families of generalized asymptotically nonexpansive mappings

Fixed Point Theory and Applications20112011:58

https://doi.org/10.1186/1687-1812-2011-58

Received: 13 April 2011

Accepted: 27 September 2011

Published: 27 September 2011

Abstract

In this paper, an implicit iterative algorithm with errors is considered for two families of generalized asymptotically nonexpansive mappings. Strong and weak convergence theorems of common fixed points are established based on the implicit iterative algorithm.

Mathematics Subject Classification (2000) 47H09 · 47H10 · 47J25

Keywords

Asymptotically nonexpansive mapping common fixed point implicit iterative algorithm generalized asymptotically nonexpansive mapping

1 Introduction

In nonlinear analysis theory, due to applications to complex real-world problems, a growing number of mathematical models are built up by introducing constraints which can be expressed as subproblems of a more general problem. These constraints can be given by fixed-point problems, see, for example, [13]. Study of fixed points of nonlinear mappings and its approximation algorithms constitutes a topic of intensive research efforts. Many well-known problems arising in various branches of science can be studied by using algorithms which are iterative in their nature. The well-known convex feasibility problem which captures applications in various disciplines such as image restoration, computer tomography, and radiation therapy treatment planning is to find a point in the intersection of common fixed point sets of a family of nonexpansive mappings, see, for example, [35].

For iterative algorithms, the most oldest and simple one is Picard iterative algorithm. It is known that T enjoys a unique fixed point, and the sequence generated in Picard iterative algorithm can converge to the unique fixed point. However, for more general nonexpansive mappings, Picard iterative algorithm fails to convergence to fixed points of nonexpansive even that it enjoys a fixed point.

Recently, Mann-type iterative algorithm and Ishikawa-type iterative algorithm (implicit and explicit) have been considered for the approximation of common fixed points of nonlinear mappings by many authors, see, for example, [624]. A classical convergence theorem of nonexpansive mappings has been established by Xu and Ori [23]. In 2006, Chang et al. [6] considered an implicit iterative algorithm with errors for asymptotically nonexpansive mappings in a Banach space. Strong and weak convergence theorems are established. Recently, Cianciaruso et al. [9] considered an Ishikawa-type iterative algorithm for the class of asymptotically nonexpansive mappings. Strong and weak convergence theorems are also established. In this paper, based on the class of generalized asymptotically nonexpansive mappings, an Ishikawa-type implicit iterative algorithm with errors for two families of mappings is considered. Strong and weak convergence theorems of common fixed points are established. The results presented in this paper mainly improve the corresponding results announced in Chang et al. [6], Chidume and Shahzad [7], Cianciaruso et al. [9], Guo and Cho [10], Khan et al. [12], Plubtieng et al. [14], Qin et al. [15], Shzhzad and Zegeye [18], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], Zhou and Chang [24].

2 Preliminaries

Let C be a nonempty closed convex subset of a Banach space E. Let T : CC be a mapping. Throughout this paper, we use F(T) to denote the fixed point set of T.

Recall the following definitions.

T is said to be nonexpansive if
T x - T y x - y , x , y C .
T is said to be asymptotically nonexpansive if there exists a positive sequence {h n } [1, ∞) with h n → 1 as n → ∞ such that
T n x - T n y h n x - y , x , y C , n 1 .

It is easy to see that every nonexpansive mapping is asymptotically nonexpansive with the asymptotical sequence {1}. The class of asymptotically nonexpansive mappings was introduced by Goebel and Kirk [25] in 1972. It is known that if C is a nonempty bounded closed convex subset of a uniformly convex Banach space E, then every asymptotically nonexpansive mapping on C has a fixed point. Further, the set F(T) of fixed points of T is closed and convex. Since 1972, a host of authors have studied weak and strong convergence problems of implicit iterative processes for such a class of mappings.

T is said to be asymptotically nonexpansive in the intermediate sense if it is continuous and the following inequality holds:
lim sup n sup x , y C ( T n x - T n y - x - y ) 0 .
(2.1)
Putting ξ n = max{0, supx,yC(||T n x - T n y|| - ||x - y||)}, we see that ξ n → 0 as n → ∞. Then, (2.1) is reduced to the following:
T n x - T n y x - y + ξ n , x , y C , n 1 .

The class of asymptotically nonexpansive mappings in the intermediate sense was introduced by Bruck et al. [26] (see also Kirk [27]) as a generalization of the class of asymptotically nonexpansive mappings. It is known that if C is a nonempty closed convex and bounded subset of a real Hilbert space, then every asymptotically nonexpansive self mapping in the intermediate sense has a fixed point; see [28] more details.

T is said to be generalized asymptotically nonexpansive if it is continuous and there exists a positive sequence {h n } [1, ∞) with h n → 1 as n → ∞ such that
lim sup n sup x , y C ( T n x - T n y - h n x - y ) 0 .
(2.2)
Putting ξ n = max{0, supx,yC(||T n x - T n y|| - h n ||x - y||)}, we see that ξ n → 0 as n → ∞. Then, (2.2) is reduced to the following:
T n x - T n y h n x - y + ξ n , x , y C , n 1 .

We remark that if h n ≡ 1, then the class of generalized asymptotically nonexpansive mappings is reduced to the class of asymptotically nonexpansive mappings in the intermediate.

In 2006, Chang et al. [6] considered the following implicit iterative algorithms for a finite family of asymptotically nonexpansive mappings {T1, T2,..., T N } with {α n } a real sequence in (0, 1), {u n } a bounded sequence in C and an initial point x0 C:
x 1 = α 1 x 0 + ( 1 - α 1 ) T 1 x 1 + u 1 , x 2 = α 2 x 1 + ( 1 - α 2 ) T 2 x 2 + u 2 , x N = α N x N - 1 + ( 1 - α N ) T N x N + u N , x N + 1 = α N + 1 x N + ( 1 - α N + 1 ) T 1 n x N + 1 + u N + 1 , x 2 N = α 2 N x 2 N - 1 + ( 1 - α 2 N ) T N 2 x 2 N + u 2 N , x 2 N + 1 = α 2 N + 1 x 2 N + ( 1 - α 2 N + 1 ) T 1 3 x 2 N + 1 + u 2 N + 1 , .
The above table can be rewritten in the following compact form:
x n = α n x n - 1 + ( 1 - α n ) T i ( n ) j ( n ) x n + u n , n 1 ,

where for each n ≥ 1 fixed, j(n) - 1 denotes the quotient of the division of n by N and i(n) the rest, i.e., n = (j(n) - 1)N + i(n).

Based on the implicit iterative algorithm, they obtained, under the assumption that C + C C, weak and strong convergence theorems of common fixed points for a finite family of asymptotically nonexpansive mappings {T1, T2,..., T N }; see [6] for more details.

Recently, Cianciaruso et al. [9] considered a Ishikawa-like iterative algorithm for the class of asymptotically nonexpansive mappings in a Banach space. To be more precise, they introduced and studied the following implicit iterative algorithm with errors.
y n = ( 1 - β n - δ n ) x n + β n T i ( n ) j ( n ) x n + δ n v n , x n = ( 1 - α n - γ n ) x n - 1 + α n T i ( n ) j ( n ) y n + γ n u n , n 1 ,
(2.3)

where {α n }, {β n }, {γ n }, and {δ n } are real number sequences in [0,1], {u n } and {v n } are bounded sequence in C. Weak and strong convergence theorems are established in a uniformly convex Banach space; see [29] for more details.

In this paper, motivated and inspired by the results announced in Chang et al. [6], Chidume and Shahzad [7], Cianciaruso et al. [9], Guo and Cho [10], Plubtieng et al. [14], Qin et al. [15], Shzhzad and Zegeye [18], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], Zhou and Chang [24], we consider the following Ishikawa-like implicit iteration algorithm with errors for two finite families of generalized asymptotically nonexpansive mappings {T1, T2,..., T N } and {S1, S2,..., S N }.
x 0 C , x 1 = α 1 x 0 + β 1 T 1 ( α 1 x 1 + β 1 S 1 x 1 + γ 1 v 1 ) + γ 1 u 1 , x 2 = α 2 x 1 + β 2 T 2 ( α 2 x 2 + β 2 S 2 x 2 + γ 2 v 2 ) + γ 2 u 2 , x N = α N x N 1 + β N T N ( α N x N + β N S N x N + γ N v N ) + γ N u N , x N + 1 = α N + 1 x N + β N + 1 T N + 1 ( α N + 1 x N + 1 + β N + 1 S N + 1 x N + 1 + γ N + 1 v N + 1 ) + γ N + 1 u N + 1 , x 2 N = α 2 N x 2 N 1 + β 2 N T 2 N ( α 2 N x 2 N + β 2 N S 2 N x 2 N + γ 2 N v 2 N ) + γ 2 N u 2 N , x 2 N + 1 = α 2 N + 1 x 2 N + β 2 N + 1 T 2 N + 1 ( α 2 N + 1 x 2 N + 1 + β 2 N + 1 S 2 N + 1 x 2 N + 1 + γ 2 N + 1 v 2 N + 1 ) + γ 2 N + 1 u 2 N + 1 , ,
where {α n }, {β n }, {γ n }, { α n } , { β n } , and { γ n } are sequences in [0,1] such that α n + β n + γ n = α n + β n + γ n = 1 for each n ≥ 1. We have rewritten the above table in the following compact form:
x n = α n x n - 1 + β n T i ( n ) j ( n ) ( α n x n + β n S i ( n ) j ( n ) x n + γ n v n ) + γ n u n , n 1 ,

where for each n ≥ 1 fixed, j(n) - 1 denotes the quotient of the division of n by N and i(n) the rest, i.e., n = (j(n) - 1)N + i(n).

Putting y n = α n x n + β n S n x n + γ n v n , we have the following composite iterative algorithm:
y n = α n x n + β n S i ( n ) j ( n ) x n + γ n v n , x n = α n x n - 1 + β n T i ( n ) j ( n ) y n + γ n u n , n 1 .
(2.4)

We remark that the implicit iterative algorithm (2.4) is general which includes (2.3) as a special case.

Now, we show that (2.4) can be employed to approximate fixed points of generalized asymptotically nonexpansive mappings which is assumed to be Lipschitz continuous. Let T i be a L t i -Lipschitz generalized asymptotically nonexpansive mapping with a sequence { h n i } [ 1 , ) such that h n i 1 as n → ∞ and S i be a L s i -Lipschitz generalized asymptotically nonexpansive mapping with sequences { k n i } [ 1 , ) such that k n i 1 as n → ∞ for each 1 ≤ iN. Define a mapping W n : CC by
W n ( x ) = α n x n - 1 + β n T i ( n ) j ( n ) ( α n x + β n S i ( n ) j ( n ) x + γ n v n ) + γ n u n , n 1 .
It follows that
W n ( x ) - W n ( y ) β n T i ( n ) j ( n ) ( α n x + β n S i ( n ) j ( n ) x + γ n v n ) - T i ( n ) j ( n ) ( α n y + β n S i ( n ) j ( n ) y + γ n v n ) β n L ( α n x - y + β n S i ( n ) j ( n ) x - S i ( n ) j ( n ) y ) β n L ( α n + β n L ) x - y , x , y C ,
where
L = max L t 1 , , L t N , L s 1 , L s N .
(2.5)
If β n L ( α n + β n L ) < 1 for all n ≥ 1, then W n is a contraction. By Banach contraction mapping principal, we see that there exists a unique fixed point x n C such that
x n = W n ( x n ) = α n x n - 1 + β n T i ( n ) j ( n ) ( α n x + β n S i ( n ) j ( n ) x + γ n v n ) + γ n u n , n 1 .

That is, the implicit iterative algorithm (2.4) is well defined.

The purpose of this paper is to establish strong and weak convergence theorem of fixed points of generalized asymptotically nonexpansive mappings based on (2.4).

Next, we recall some well-known concepts.

Let E be a real Banach space and U E = {x E : ||x|| = 1}. E is said to be uniformly convex if for any ε (0, 2] there exists δ > 0 such that for any x, y U E ,
x - y ε implies x + y 2 1 - δ .

It is known that a uniformly convex Banach space is reflexive and strictly convex.

Recall that E is said to satisfy Opial's condition[30] if for each sequence {x n } in E, the condition that the sequence x n x weakly implies that
lim inf n x n - x < lim inf n x n - y

for all y E and yx. It is well known [30] that each l p (1 ≤ p < ∞) and Hilbert spaces satisfy Opial's condition. It is also known [29] that any separable Banach space can be equivalently renormed to that it satisfies Opial's condition.

Recall that a mapping T : CC is said to be demiclosed at the origin if for each sequence {x n } in C, the condition x n x0 weakly and Tx n → 0 strongly implies Tx0 = 0. T is said to be semicompact if any bounded sequence {x n } in C satisfying limn→∞||x n - Tx n || = 0 has a convergent subsequence.

In order to prove our main results, we also need the following lemmas.

Lemma 2.1. [20]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 , n n 0 ,

where n0is some nonnegative integer. If n = 0 c n < and n = 0 b n < , then limn→∞a n exists.

Lemma 2.2. [17]Let E be a uniformly convex Banach space and 0 < λt n η < 1 for all n ≥ 1. Suppose that {x n } and {y n } are sequences of E such that
lim sup n x n r , lim sup n y n r
and
lim n t n x n + ( 1 - t n ) y n = r

hold for some r ≥ 0. Then limn→∞||x n - y n || = 0.

The following lemma can be obtained from Qin et al. [31] or Sahu et al. [32] immediately.

Lemma 2.3. Let C be a nonempty closed convex subset of a real Hilbert space H. Let T : CC be a Lipschitz generalized asymptotically nonexpansive mapping. Then I - T is demiclosed at origin.

3 Main results

Now, we are ready to give our main results in this paper.

Theorem 3.1. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T i : CC be a uniformly L t i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { h n i } [ 1 , ) , where h n i 1 as n → ∞ and S i : CC be a uniformly L s i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { k n i } [ 1 , ) , where k n i 1 as n → ∞ for each 1 ≤ iN. Assume that F = i = 1 N F ( T i ) i = 1 N F ( S i ) . Let {u n }, {v n } be bounded sequences in C and e n = max{h n , k n }, where h n = sup { h n i : 1 i N } and k n = sup { k n i : 1 i N } . Let {α n }, {β n }, {γ n }, { α n } , { β n } and { γ n } be sequences in [0,1] such that α n + β n + γ n = α n + β n + γ n = 1 for each n ≥ 1. Let {x n } be a sequence generated in (2.4). Put μ n i = max { 0 , sup x , y C ( T i n x - T i n y - h n i x - y ) } and ν n i = max { 0 , sup x , y C ( S i n x - S i n y - k n i x - y ) } . Let ξ n = max{μ n , ν n }, where μ n = max { μ n i : 1 i N } and ν n = max { ν n i : 1 i N } . Assume that the following restrictions are satisfied:
  1. (a)

    n = 1 γ n < and n = 1 γ n < ;

     
  2. (b)

    n = 1 ( e n - 1 ) < and n = 1 ξ n < ;

     
  3. (c)

    β n L ( α n + β n L ) < 1 , where L is defined in (2.5);

     
  4. (d)

    there exist constants λ, η (0, 1) such that λα n , α n η .

     
Then
lim n x n - T r x n = lim n x n - S r x n = 0 , r { 1 , 2 , , N } .
Proof. Fixing f F , we see that
y n - f = α n x n + β n S i ( n ) j ( n ) x n + γ n v n - f α n x n - f + β n S i ( n ) j ( n ) x n - f + γ n v n - f α n x n - f + β n e j ( n ) x n - f + β n ξ j ( n ) + γ n v n - f e j ( n ) x n - f + β n ξ j ( n ) + γ n v n - f
(3.1)
and
x n - f = α n x n - 1 + β n T i ( n ) j ( n ) y n + γ n u n - f α n x n - 1 - f + β n T i ( n ) j ( n ) y n - f + γ n u n - f α n x n - 1 - f + β n e j ( n ) y n - f + β n ξ j ( n ) + γ n u n - f . α n x n - 1 - f + ( 1 - α n ) e j ( n ) y n - f + β n ξ j ( n ) + γ n u n - f .
(3.2)
Substituting (3.1) into (3.2), we see that
x n - f α n x n - 1 - f + ( 1 - α n ) e j ( n ) ( e j ( n ) x n - f + β n ξ j ( n ) + γ n v n - f ) + β n ξ j ( n ) + γ n u n - f . α n x n - 1 - f + ( 1 - α n ) e j ( n ) 2 x n - f + ( 1 + e j ( n ) ) ξ j ( n ) + e j ( n ) γ n v n - f + γ n u n - f .
Notice that n = 1 ( e n - 1 ) < . We see from the restrictions (b) and (d) that there exists a positive integer n0 such that
( 1 - α n ) e j ( n ) 2 R < 1 , n n 0 ,
where R = ( 1 - λ ) ( 1 + λ 2 - 2 λ ) . It follows that
x n - f 1 + ( 1 - α n ) ( e j ( n ) 2 - 1 ) 1 - ( 1 - α n ) e j ( n ) 2 x n - 1 - f + ( 1 + e j ( n ) ) ξ j ( n ) + e j ( n ) γ n v n - f + γ n u n - f 1 - ( 1 - α n ) e j ( n ) 2 1 + ( 1 + M 1 ) ( e j ( n ) - 1 ) 1 - R x n - 1 - f + ( 1 + M 1 ) ξ j ( n ) + M 1 M 2 γ n + M 3 γ n 1 - R ,
(3.3)
where M1 = supn≥1{e n }, M2 = supn≥1{||v n - f||}, and M3 = supn≥1{||u n - f||}. In view of Lemma 2.1, we see that limn→∞||x n - f|| exists for each f F . This implies that the sequence {x n } is bounded. Next, we assume that limn→∞||x n - f|| = d > 0. From (3.1), we see that
T i ( n ) j ( n ) y n - f + γ n ( u n - T i ( n ) j ( n ) y n ) T i ( n ) j ( n ) y n - f + γ n u n - T i ( n ) j ( n ) y n e j ( n ) y n - f + ξ j ( n ) + γ n u n - T i ( n ) j ( n ) y n e j ( n ) 2 x n - f + e j ( n ) ξ j ( n ) + e j ( n ) γ n v n - f + ξ j ( n ) + γ n u n - T i ( n ) j ( n ) y n .
This implies from the restrictions (a) and (b) that
lim sup n T i ( n ) j ( n ) y n - f + γ n ( u n - T i ( n ) j ( n ) y n ) d .
Notice that
x n - 1 - f + γ n ( u n - T i ( n ) j ( n ) y n ) x n - 1 - f + γ n u n - T i ( n ) j ( n ) y n .
This shows from the restriction (a) that
lim sup n x n - 1 - f + γ n ( u n - T i ( n ) j ( n ) y n ) d .
On the other hand, we have
d = lim n x n - f = lim n α n ( x n - 1 - f + γ n ( u n - T i ( n ) j ( n ) y n ) ) + ( 1 - α n ) ( T i ( n ) j ( n ) y n - f + γ n ( u n - T i ( n ) j ( n ) y n ) ) .
It follows from Lemma 2.2 that
lim n T i ( n ) j ( n ) y n - x n - 1 = 0 .
(3.4)
Notice that
x n - x n - 1 β n T i ( n ) j ( n ) y n - x n - 1 + γ n u n - x n - 1 .
It follows from (3.4) and the restriction (a) that
lim n x n - x n - 1 = 0 .
(3.5)
This implies that
lim n x n - x n + l = 0 , l = 1 , 2 , , N .
(3.6)
Notice that
x n - f + γ n ( v n - S i ( n ) j ( n ) x n ) x n - f + γ n v n - S i ( n ) j ( n ) x n
and
S i ( n ) j ( n ) x n - f + γ n ( v n - S i ( n ) j ( n ) x n ) S i ( n ) j ( n ) x n - f + γ n v n - S i ( n ) j ( n ) x n e j ( n ) x n - f + ξ j ( n ) + γ n v n - S i ( n ) j ( n ) x n .
which in turn imply that
lim sup n x n - f + γ n ( v n - S i ( n ) j ( n ) x n ) d
and
lim sup n S i ( n ) j ( n ) x n - f + γ n ( v n - S i ( n ) j ( n ) x n ) d .
On the other hand, we have
x n - f = α n x n - 1 + β n T i ( n ) j ( n ) y n + γ n u n - f α n x n - 1 - f + β n T i ( n ) j ( n ) y n - f + γ n u n - f α n x n - 1 - T i ( n ) k ( n ) y n + T i ( n ) j ( n ) y n - f + γ n u n - f α n x n - 1 - T i ( n ) k ( n ) y n + e j ( n ) y n - f + ξ j ( n ) + γ n u n - f ,
from which it follows that lim infn→∞||y n - f|| ≥ d. In view of (3.1), we see that lim supn→∞||y n - f|| ≤ d. This proves that
lim n y n - f = d .
Notice that
lim n y n - f = lim n α n ( x n - f + γ n ( v n - S i ( n ) j ( n ) x n ) ) + ( 1 - α n ) ( S i ( n ) j ( n ) x n - f + γ n ( v n - S i ( n ) j ( n ) x n ) ) .
This implies from Lemma 2.2 that
lim n S i ( n ) j ( n ) x n - x n = 0 .
(3.7)
On the other hand, we have
T i ( n ) j ( n ) x n - x n T i ( n ) j ( n ) x n - T i ( n ) j ( n ) y n + T i ( n ) j ( n ) y n - x n - 1 + x n - 1 - x n T i ( n ) j ( n ) x n - T i ( n ) j ( n ) y n + T i ( n ) j ( n ) y n - x n - 1 + x n - 1 - x n L x n - y n + T i ( n ) j ( n ) y n - x n - 1 + x n - 1 - x n L β n S i ( n ) j ( n ) x n - x n + L γ n v n - x n + T i ( n ) j ( n ) y n - x n - 1 + x n - 1 - x n .
This combines with (3.4), (3.5), and (3.7) gives that
lim n T i ( n ) j ( n ) x n - x n = 0 .
(3.8)
Since n = (j(n) - 1)N + i(n), where i(n) {1, 2,..., N}, we see that
x n S i ( n ) x n x n S i ( n ) j ( n ) x n + S i ( n ) j ( n ) x n S i ( n ) x n x n S i ( n ) j ( n ) x n + L S i ( n ) j ( n ) 1 x n x n x n S i ( n ) j ( n ) x n + L ( S i ( n ) j ( n ) 1 x n S i ( n N ) j ( n ) 1 x n N + S i ( n N ) j ( n ) 1 x n N x n N + x n N x n ).
(3.9)
Notice that
j ( n - N ) = j ( n ) - 1 a n d i ( n - N ) = i ( n ) .
This in turn implies that
S i ( n ) j ( n ) - 1 x n - S i ( n - N ) j ( n ) - 1 x n - N = S i ( n ) j ( n ) - 1 x n - S i ( n ) j ( n ) - 1 x n - N L x n - x n - N
(3.10)
and
S i ( n - N ) j ( n ) - 1 x n - N - x n - N = S i ( n - N ) j ( n - N ) x n - N - x n - N .
(3.11)
Substituting (3.10) and (3.11) into (3.9) yields that
x n S i ( n ) x n x n S i ( n ) j ( n ) x n + L ( L x n x n N + S i ( n N ) j ( n N ) x n N x n N ).
It follows from (3.6) and (3.7) that
lim n x n - S i ( n ) x n = 0 .
(3.12)
In particular, we see that
lim j x j N + 1 - S 1 x j N + 1 = 0 , lim j x j N + 2 - S 2 x j N + 2 = 0 , lim j x j N + N - S N x j N + N = 0 .
For any r, s = 1, 2,..., N, we obtain that
x j N + s - S r x j N + s x j N + s - x j N + r + x j N + r - S r x j N + r + S r x j N + r - S r x j N + s ( 1 + L ) x j N + s - x j N + r + x j N + r - S r x j N + r .
Letting j → ∞, we arrive at
lim j x j N + s - S r x j N + s = 0 ,
which is equivalent to
lim n x n - S r x n = 0 .
(3.13)
Notice that
x n T i ( n ) x n x n T i ( n ) j ( n ) x n + T i ( n ) j ( n ) x n T i ( n ) x n x n T i ( n ) j ( n ) x n + L T i ( n ) j ( n ) 1 x n x n x n T i ( n ) j ( n ) x n + L ( T i ( n ) j ( n ) 1 x n T i ( n N ) j ( n ) 1 x n N + T i ( n N ) j ( n ) 1 x n N x n N + x n N x n ).
(3.14)
On the other hand, we have
T i ( n ) j ( n ) - 1 x n - T i ( n - N ) j ( n ) - 1 x n - N = T i ( n ) j ( n ) - 1 x n - T i ( n ) j ( n ) - 1 x n - N L x n - x n - N
(3.15)
and
T i ( n - N ) j ( n ) - 1 x n - N - x n - N = T i ( n - N ) j ( n - N ) x n - N - x n - N .
(3.16)
Substituting (3.15) and (3.16) into (3.14) yields that
x n T i ( n ) x n x n T i ( n ) k ( n ) x n + L ( L x n x n N + T i ( n N ) k ( n N ) x n N x n N ).
It follows from (3.6) and (3.8) that
lim n x n - T i ( n ) x n = 0 .
(3.17)
In particular, we see that
lim k x j N + 1 - T 1 x j N + 1 = 0 , lim k x j N + 2 - T 2 x j N + 2 = 0 , lim k x j N + N - T N x j N + N = 0 .
For any r, s = 1, 2,..., N, we obtain that
x j N + s - T r x j N + s x j N + s - x j N + r + x j N + r - T r x j N + r + T r x j N + r - T r x j N + s ( 1 + L ) x j N + s - x j N + r + x j N + r - T r x j N + r .
Letting j → ∞, we arrive
lim j x j N + s - T r x j N + s = 0 ,
which is equivalent to
lim n x n - T r x n = 0 .
(3.18)

This completes the proof. □

Now, we are in a position to give weak convergence theorems.

Theorem 3.2. Let E be a real Hilbert space and C be a nonempty closed convex subset of E. Let T i : CC be a uniformly L t i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { h n i } [ 1 , ) , where h n i 1 as n → ∞ and S i : CC be a uniformly L s i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { k n i } [ 1 , ) , where k n i 1 as n → ∞ for each 1 ≤ iN. Assume that F = i = 1 N F ( T i ) i = 1 N F ( S i ) . Let {u n }, {v n } be bounded sequences in C and e n = max{h n , k n }, where h n = sup { h n i : 1 i N } and k n = sup { k n i : 1 i N } . Let {α n }, {β n }, {γ n }, { α n } , { β n } and { γ n } be sequences in [0,1] such that α n + β n + γ n = α n + β n + γ n = 1 for each n ≥ 1. Let {x n } be a sequence generated in (2.4). Put μ n i = max { 0 , sup x , y C ( T i n x - T i n y - h n i x - y ) } and ν n i = max { 0 , sup x , y C ( S i n x - S i n y - k n i x - y ) } . Let ξ n = max{μ n , ν n }, where μ n = max { μ n i : 1 i N } and ν n = max { ν n i : 1 i N } . Assume that the following restrictions are satisfied:
  1. (a)

    n = 1 γ n < and n = 1 γ n < ;

     
  2. (b)

    n = 1 ( e n - 1 ) < and n = 1 ξ n < ;

     
  3. (c)

    β n L ( α n + β n L ) < 1 , where L is defined in (2.5);

     
  4. (d)

    there exist constants λ, η (0, 1) such that λα n , α n η .

     

Then the sequence {x n } converges weakly to some point in F .

Proof. Since E is a Hilbert space and {x n } is bounded, we can obtain that there exists a subsequence { x n p } of {x n } converges weakly to x* C. It follows from (3.13) and (3.18) that
lim p x n p - T r x n p = lim p x n p - S r x n p = 0 .
Since I - S r and I - T r are demiclosed at origin by Lemma 2.3, we see that x * F . Next, we show that the whole sequence {x n } converges weakly to x*. Suppose the contrary. Then there exists some subsequence { x n q } of {x n } such that { x n q } converges weakly to x** C. In the same way, we can show that x * * F . Notice that we have proved that limn→∞||x n - f|| exists for each f F . By virtue of Opial's condition of E, we have
lim inf p x n p - x * < lim inf p x n p - x * * = lim inf q x n q - x * * < lim inf q x n q - x * .

This is a contradiction. Hence, x* = x**. This completes the proof. □

Remark 3.3. Theorem 3.2 which includes the corresponding results announced in Chang et al. [6], Chidume and Shahzad [7], Guo and Cho [10], Plubtieng et al. [14], Qin et al. [15], Thakur [21], Thianwan and Suantai [22], Xu and Ori [23], and Zhou and Chang [24] as special cases mainly improves the results of Cianciaruso et al. [9] in the following aspects.
  1. (1)

    Extend the mappings from one family of mappings to two families of mappings;

     
  2. (2)

    Extend the mappings from the class of asymptotically nonexpansive mappings to the class of generalized asymptotically nonexpansive mappings.

     

If S r = I for each r {1, 2,..., N} and γ n = 0 , then Theorem 3.2 is reduced to the following.

Corollary 3.4. Let E be a real Hilbert space and C be a nonempty closed convex subset of E. Let T i : CC be a uniformly L t i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { h n i } [ 1 , ) , where h n i 1 as n → ∞ for each 1 ≤ iN. Assume that F = i = 1 N F ( T i ) . Let {u n } be a bounded sequence in C and h n = sup { h n i : 1 i N } . Let {α n }, {β n } and {γ n } be sequences in [0,1] such that α n + β n + γ n = 1 for each n ≥ 1. Let {x n } be a sequence generated in the following process:
x 0 C , x n = α n x n - 1 + β n T i ( n ) j ( n ) x n + γ n u n , n 1 .
(3.19)
Put μ n i = max { 0 , sup x , y C ( T i n x - T i n y - h n i x - y ) } . Let μ n = max { μ n i : 1 i N } . Assume that the following restrictions are satisfied:
  1. (a)

    n = 1 γ n < ;

     
  2. (b)

    n = 1 ( h n - 1 ) < and n = 1 μ n < ;

     
  3. (c)

    β n L < 1, where L = max { L t i : 1 i N } ;

     
  4. (d)

    there exist constants λ, η (0, 1) such that λα n , α n η .

     

Then the sequence {x n } converges weakly to some point in F .

Next, we are in a position to state strong convergence theorems in a Banach space.

Theorem 3.5. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T i : CC be a uniformly L t i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { h n i } [ 1 , ) , where h n i 1 as n → ∞ and S i : CC be a uniformly L s i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { k n i } [ 1 , ) , where k n i 1 as n → ∞ for each 1 ≤ iN. Assume that F = i = 1 N F ( T i ) i = 1 N F ( S i ) . Let {u n }, {v n } be bounded sequences in C and e n = max{h n , k n }, where h n = sup { h n i : 1 i N } and k n = sup { k n i : 1 i N } . Let {α n }, {β n }, {γ n }, { α n } , { β n } and { γ n } be sequences in [0,1] such that α n + β n + γ n = α n + β n + γ n = 1 for each n ≥ 1. Let {x n } be a sequence generated in (2.4). Put μ n i = max { 0 , sup x , y C ( T i n x - T i n y - h n i x - y ) } and ν n i = max { 0 , sup x , y C ( S i n x - S i n y - k n i x - y ) } . Let ξ n = max{μ n , ν n }, where μ n = max { μ n i : 1 i N } and ν n = max { ν n i : 1 i N } . Assume that the following restrictions are satisfied:
  1. (a)

    n = 1 γ n < and n = 1 γ n < ;

     
  2. (b)

    n = 1 ( e n - 1 ) < and n = 1 ξ n < ;

     
  3. (c)

    β n L ( α n + β n L ) < 1 , where L is defined in (2.5);

     
  4. (d)

    there exist constants λ, η (0, 1) such that λ ≥ αn, α n η .

     

If one of mappings in {T1, T2,..., T N } or one of mappings in {S1, S2,..., S N } are semicompact, then the sequence {x n } converges strongly to some point in F .

Proof. Without loss of generality, we may assume that S1 are semicompact. It follows from (3.13) that
lim n x n - S 1 x n = 0 .
By the semicompactness of S1, we see that there exists a subsequence { x n p } of {x n } such that x n p w C strongly. From (3.13) and (3.18), we have
w - S r w w - x n p + x n p - S r x n p + S r x n p - S r w
and
w - T r w w - x n p + x n p - T r x n p + T r x n p - T r w
Since S r and T r are Lipshcitz continuous, we obtain that w F . From Theorem 3.1, we know that limn→∞||x n - f|| exists for each f F . That is, limn→∞||x n - w|| exists. From x n p w , we have
lim n x n - w = 0 .

This completes the proof of Theorem 3.5. □

If S r = I for each r {1, 2,..., N} and γ n = 0 , then Theorem 3.5 is reduced to the following.

Corollary 3.6. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T i : CC be a uniformly L t i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { h n i } [ 1 , ) , where h n i 1 as n → ∞ for each 1 ≤ iN. Assume that F = i = 1 N F ( T i ) . Let {u n } be a bounded sequence in C and h n = sup { h n i : 1 i N } . Let {α n }, {β n } and {γ n } be sequences in [0,1] such that α n + β n + γ n = 1 for each n ≥ 1. Let {x n } be a sequence generated in (3.19). Put μ n i = max { 0 , sup x , y C ( T i n x - T i n y - h n i x - y ) } . Let μ n = max { μ n i : 1 i N } . Assume that the following restrictions are satisfied:
  1. (a)

    n = 1 γ n < ;

     
  2. (b)

    n = 1 ( h n - 1 ) < and n = 1 μ n < ;

     
  3. (c)

    β n L < 1, where L = max { L t i : 1 i N } ;

     
  4. (d)

    there exist constants λ, η (0, 1) such that λ ≥ αn, α n η .

     

If one of mappings in {T1, T2,..., T N } is semicompact, then the sequence {x n } converges strongly to some point in F .

Theorem 3.7. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T i : CC be a uniformly L t i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { h n i } [ 1 , ) , where h n i 1 as n → ∞ and S i : CC be a uniformly L s i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { k n i } [ 1 , ) , where k n i 1 as n → ∞ for each 1 ≤ iN. Assume that F = i = 1 N F ( T i ) i = 1 N F ( S i ) . Let {u n }, {v n } be bounded sequences in C and e n = max{h n , k n }, where h n = sup { h n i : 1 i N } and k n = sup { k n i : 1 i N } . Let {α n }, {β n }, {γ n }, { α n } , { β n } and { γ n } be sequences in [0,1] such that α n + β n + γ n = α n + β n + γ n = 1 for each n ≥ 1. Let {x n } be a sequence generated in (2.4). Put μ n i = max { 0 , sup x , y C ( T i n x - T i n y - h n i x - y ) } and ν n i = max { 0 , sup x , y C ( S i n x - S i n y - k n i x - y ) } . Let ξ n = max{μ n , ν n }, where μ n = max { μ n i : 1 i N } and ν n = max { ν n i : 1 i N } . Assume that the following restrictions are satisfied:
  1. (a)

    n = 1 γ n < and n = 1 γ n < ;

     
  2. (b)

    n = 1 ( e n - 1 ) < and n = 1 ξ n < ;

     
  3. (c)

    β n L ( α n + β n L ) < 1 , where L is defined in (2.5);

     
  4. (d)

    there exist constants λ, η (0, 1) such that λα n ,. α n η .

     
If there exists a nondecreasing function g : [0, ∞) → [0, ∞) with g(0) = 0 and g(m) > 0 for all m (0, ∞) such that
max 1 r N { x - S r x } + max 1 r N { x - T r x } g ( d i s t ( x , F ) ) , x C ,

then the sequence {x n } converges strongly to some point in F .

Proof. In view of (3.13) and (3.18) that g ( d i s t ( x n , F ) ) 0 , which implies d i s t ( x n , F ) 0 . Next, we show that the sequence {x n } is Cauchy. In view of (3.3), we obtain by putting
a n = ( 1 + M 1 ) ( e j ( n ) - 1 ) 1 - R a n d b n = ( 1 + M 1 ) ξ j ( n ) + M 1 M 2 γ n + M 3 γ n 1 - R
that
x n - f ( 1 + a n ) x n - 1 - f + b n .
It follows, for any positive integers m, n, where m > n > n0, that
x m - p B x n - p + B i = n + 1 b i + b m ,
where B = exp { n = 1 a n } . It follows that
x n - x m x n - f + x m - f ( 1 + B ) x n - f + B i = n + 1 b i + b m .
Taking the infimum over all f F , we arrive at
x n - x m ( 1 + B ) d i s t ( x n , F ) + B i = n + 1 b i + b m .

In view of n = 1 b n < and d i s t ( x n , F ) 0 , we see that {x n } is a Cauchy sequence in C and so {x n } converges strongly to some x* C. Since T r and S r are Lipschitz for each r {1, 2,..., N}, we see that F is closed. This in turn implies that x * F . This completes the proof. □

If S r = I for each r {1, 2,..., N} and γ n = 0 , then Theorem 3.7 is reduced to the following.

Corollary 3.8. Let E be a real uniformly convex Banach space and C be a nonempty closed convex subset of E. Let T i : CC be a uniformly L t i -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence { h n i } [ 1 , ) , where h n i 1 as n → ∞ for each 1 ≤ iN. Assume that F = i = 1 N F ( T i ) . Let {u n } be a bounded sequence in C and h n = sup { h n i : 1 i N } . Let {α n }, {β n } and {γ n } be sequences in [0,1] such that α n + β n + γ n = 1 for each n ≥ 1. Let {x n } be a sequence generated in (3.19). Put μ n i = max { 0 , sup x , y C ( T i n x - T i n y - h n i x - y ) } . Let μ n = max { μ n i : 1 i N } . Assume that the following restrictions are satisfied:
  1. (a)

    n = 1 γ n < ;

     
  2. (b)

    n = 1 ( h n - 1 ) < and n = 1 μ n < ;

     
  3. (c)

    β n L < 1, where L = max { L t i : 1 i N } ;

     
  4. (d)

    there exist constants λ, η (0, 1) such that, λα n , α n η .

     
If there exists a nondecreasing function g : [0, ∞) → [0, ∞) with g(0) = 0 and g(m) > 0 for all m (0, ∞) such that
max 1 r N { x - T r x } g ( d i s t ( x , F ) ) , x C ,

then the sequence {x n } converges strongly to some point in F .

Declarations

Acknowledgements

The authors are indebted to the referees for their helpful comments. The work was supported partially by Natural Science Foundation of Zhejiang Province (Y6110270).

Authors’ Affiliations

(1)
Department of Mathematics, Texas A&M University - Kingsville
(2)
School of Mathematics and Information Sciences, North China University of Water Resources and Electric Power
(3)
Department of Mathematics and RINS, Gyeongsang National University

References

  1. Izmaelov AF, Solodov MV: An active set Newton method for mathematical program with complementary constraints. SIAM J Optim 2008, 19: 1003–1027. 10.1137/070690882MathSciNetView ArticleGoogle Scholar
  2. Kaufman DE, Smith RL, Wunderlich KE: User-equilibrium properties of fixed points in dynamic traffic assignment. Transportation Res Part C 1998, 6: 1–16. 10.1016/S0968-090X(98)00005-9View ArticleGoogle Scholar
  3. Kotzer T, Cohen N, Shamir J: Image restoration by an ovel method of parallel projectio onto constraint sets. Optim Lett 1995, 20: 1172–1174. 10.1364/OL.20.001172View ArticleGoogle Scholar
  4. Bauschke HH, Borwein JM: On projection algorithms for solving convex feasibility problems. SIAM Rev 1996, 38: 367–426. 10.1137/S0036144593251710MathSciNetView ArticleGoogle Scholar
  5. Youla DC: Mathematical theory of image restoration by the method of convex projections. In Image Recovery: Theory and Applications. Edited by: Stark H. Academic Press, Florida, USA; 1987:29–77.Google Scholar
  6. 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 ArticleGoogle Scholar
  7. Chidume CE, Shahzad N: Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings. Nonlinear Anal 2005, 62: 1149–1156. 10.1016/j.na.2005.05.002MathSciNetView ArticleGoogle Scholar
  8. Cho SY, Kang SM: Approximation of fixed points of pseudocontraction semigroups based on a viscosity iterative process. Appl Math Lett 2011, 24: 224–228. 10.1016/j.aml.2010.09.008MathSciNetView ArticleGoogle Scholar
  9. Cianciaruso F, Marino G, Wang X: Weak and strong convergence of the Ishikawa iterative process for a finite family of asymptotically nonexpansive mappings. Appl Math Comput 2010, 216: 3558–3567. 10.1016/j.amc.2010.05.001MathSciNetView ArticleGoogle Scholar
  10. Guo W, Cho YJ: On the 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 ArticleGoogle Scholar
  11. Hao Y, Cho SY, Qin X: Some weak convergence theorems for a family of asymptotically nonexpansive nonself mappings. Fixed Point Theory Appl 2010, 2010: 11. Article ID 218573MathSciNetGoogle Scholar
  12. Khan SH, Yildirim I, Ozdemir M: Convergence of an implicit algorithm for two families of nonexpansive mappings. Comput Math Appl 2010, 59: 3084–3091. 10.1016/j.camwa.2010.02.029MathSciNetView ArticleGoogle Scholar
  13. Kim JK, Nam YM, Sim JY: Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings. Nonlinear Anal 2009, 71: e2839-e2848. 10.1016/j.na.2009.06.090MathSciNetView ArticleGoogle Scholar
  14. Plubtieng S, Ungchittrakool K, Wangkeeree R: Implicit iterations of two finite families for nonexpansive mappings in Banach spaces. Numer Funct Anal Optim 2007, 28: 737–749. 10.1080/01630560701348525MathSciNetView ArticleGoogle Scholar
  15. Qin X, Cho YJ, Shang M: Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings. Appl Math Comput 2009, 210: 542–550. 10.1016/j.amc.2009.01.018MathSciNetView ArticleGoogle Scholar
  16. Qin X, Kang SM, Agarwal RP: On the convergence of an implicit iterative process for generalized asymptotically quasi-nonexpansive mappings. Fixed Point Theory Appl 2010, 2010: 19. Article ID 714860MathSciNetGoogle Scholar
  17. Schu J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings. Bull Austral Math Soc 1991, 43: 153–159. 10.1017/S0004972700028884MathSciNetView ArticleGoogle Scholar
  18. Shzhzad N, Zegeye H: Strong convergence of an implicit iteration process for a finite family of genrealized asymptotically quasi-nonexpansive maps. Appl Math Comput 2007, 189: 1058–1065. 10.1016/j.amc.2006.11.152MathSciNetView ArticleGoogle Scholar
  19. Su Y, Qin X: General iteration algorithm and convergence rate optimal model for common fixed points of nonexpansive mappings. Appl Math Comput 2007, 186: 271–278. 10.1016/j.amc.2006.07.101MathSciNetView ArticleGoogle Scholar
  20. Tan KK, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iterative process. J Math Anal Appl 1993, 178: 301–308. 10.1006/jmaa.1993.1309MathSciNetView ArticleGoogle Scholar
  21. Thakur BS: Weak and strong convergence of composite implicit iteration process. Appl Math Comput 2007, 190: 965–997. 10.1016/j.amc.2007.01.101MathSciNetView ArticleGoogle Scholar
  22. Thianwan S, Suantai S: Weak and strong convergence of an implicity iteration process for a finite family of nonexpansive mappings. Sci Math Japon 2007, 66: 221–229.MathSciNetGoogle Scholar
  23. Xu HK, Ori RG: An implicit iteration process for nonexpansive mappings. Numer Funct Anal Optim 2001, 22: 767–773. 10.1081/NFA-100105317MathSciNetView ArticleGoogle Scholar
  24. Zhou YY, Chang SS: Convergence of implicit iterative process for a finite of asymptotically nonexpansive mappings in Banach spaces. Numer Funct Anal Optim 2002, 23: 911–921. 10.1081/NFA-120016276MathSciNetView ArticleGoogle Scholar
  25. Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proc Amer Math Soc 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3MathSciNetView ArticleGoogle Scholar
  26. Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform opial property. Colloq Math 1993, 65: 169–179.MathSciNetGoogle Scholar
  27. Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Israel J Math 1974, 17: 339–346. 10.1007/BF02757136MathSciNetView ArticleGoogle Scholar
  28. Xu HK: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Anal 1991, 16: 1139–1146. 10.1016/0362-546X(91)90201-BMathSciNetView ArticleGoogle Scholar
  29. van Dulst D: Equivalent norms and the fixed point property for nonexpansive mapping. J Lond Math Soc 1982, 25: 139–144. 10.1112/jlms/s2-25.1.139MathSciNetView ArticleGoogle Scholar
  30. Opial Z: Weak convergence of the sequence of successive appproximations for nonexpansive mappings. Bull Amer Math Soc 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0MathSciNetView ArticleGoogle Scholar
  31. Qin X, Cho SY, Kim JK: Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense. Fixed Point Theory Appl 2010, 2010: 14. Article ID 186874MathSciNetGoogle Scholar
  32. Sahu DR, Xu HK, Yao JC: Asymptotically strict pseudocontractive mappings in the intermediate sens. Nonlinear Anal 2009, 70: 3502–3511. 10.1016/j.na.2008.07.007MathSciNetView ArticleGoogle Scholar

Copyright

© Agarwal et al; licensee Springer. 2011

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.