RESEARC H Open Access An implicit iterative algorithm with errors for two families of generalized asymptotically nonexpansive mappings Ravi P Agarwal 1 , Xiaolong Qin 2 and Shin Min Kang 3* * Correspondence: smkang@gnu. ac.kr 3 Department of Mathematics and RINS, Gyeongsang National University, Jinju 660-701, Korea Full list of author information is available at the end of the article Abstract In this paper, an implicit iterative algorithm with errors is considered for two families of generalized asymptotically nonex pansive 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 subpr oblems of a more general problem. These constraints can be given by fixed-point problems, see, for example, [1-3]. Study of fixed points of non- linear mappings and its approximation algorithms constitutes a topic of intensive research efforts. Many well-known problems arising in various branches of scie nce 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 nonexpan- sive mappings, see, for example, [3-5]. 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 algori thm can converge to the unique fixed point. However, for more general nonexpansive mappings, Picard iterative algorithm fails to convergen ce 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, [6-24]. A classical convergence theorem of nonexpansive mappings has been established by Xu a nd Ori [23]. In 2006, Chang et al. [6] considered an implicit iterative algorithm with error s for asymptotic ally nonexpansive mappings in a Banach space. Strong and weak convergence theorems are Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 © 2011 Agarwal et al; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creati vecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. established. Recently, Cianciaruso et al. [9] considered an Ishikaw a-type iterative algo- rithm for the class of asymptotically nonexpansive mappings. Strong and weak conver- gence theorems are also established. In this pape r, based on the c lass of general ized asymptotically nonexpansive mappings, an Ishikawa-type implicit iterative algorithm with errors for two families o f mappings is considered. Strong and weak convergence theorems of common fixed points are established. The results presented in this paper mainly improve t he corresponding results announced in Chang et a l. [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 Su antai [22], Xu and Ori [23], Zhou and Chang [24]. 2 Preliminaries Let C be a nonempty closed convex subset of a Banach space E.LetT : C ® C 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 Tx − 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 ® 1asn ® ∞ 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 n onempt y bounded closed convex subset of a uniformly convex Banach space E,thenevery 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 continu- ous 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, sup x,yÎC (||T n x - T n y|| - ||x - y||)}, we see that ξ n ® 0asn ® ∞. 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 con- vex and bounded subset of a real Hilbert space, then every asymptotically nonexpan- sive 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 ® 1asn ® ∞ such that Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 Page 2 of 17 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,sup x,yÎC (||T n x - T n y|| - h n ||x - y||)}, we see that ξ n ® 0asn ® ∞. 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 c lass 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 {T 1 , T 2 , , T N }with{a n }areal sequence in (0, 1), {u n } a bounded sequence in C and an initial point x 0 Î 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 n 1 x N+1 + u N+1 , ··· x 2N = α 2N x 2N−1 +(1− α 2N )T 2 N x 2N + u 2N , x 2N+1 = α 2N+1 x 2N +(1− α 2N+1 )T 3 1 x 2N+1 + u 2N+1 , ··· . The above table can be rewritten in the following compact form: x n = α n x n−1 +(1− α n )T j(n) i ( n ) x n + u n , ∀n ≥ 1 , where for each n ≥ 1fixed,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 {T 1 , T 2 , , 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 j(n) i(n) x n + δ n v n , x n =(1− α n − γ n )x n−1 + α n T j(n) i ( n ) y n + γ n u n , ∀n ≥ 1 , (2:3) where {a n }, {b n }, {g n }, and {δ n } are rea l 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 asymptoti- cally nonexpansive mappings {T 1 , T 2 , , T N } and {S 1 , S 2 , , S N }. Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 Page 3 of 17 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 2N = α 2N x 2N−1 + β 2N T 2N (α 2N x 2N + β 2N S 2N x 2N + γ 2N v 2N ) + γ 2N u 2N , x 2N+1 = α 2N+1 x 2N + β 2N+1 T 2N+1 (α 2N+1 x 2N+1 + β 2N+1 S 2N+1 x 2N+ 1 + γ 2N+1 v 2N+1 )+γ 2N+1 u 2N+1 , ··· , where {a n }, {b n }, {g 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 j(n) i ( n ) (α n x n + β n S j(n) i ( n ) x n + γ n v n )+γ n u n , n ≥ 1 , where for each n ≥ 1fixed,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 algo- rithm: y n = α n x n + β n S j(n) i(n) x n + γ n v n , x n = α n x n−1 + β n T j(n) i ( 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 i t -Lipschitz generalized asymptotically nonexpansive mapping with a sequence {h i n }⊂[1, ∞ ) such that h i n → 1 as n ® ∞ and S i be a L i s -Lipschitz generalized asymptotically nonexpansive mapping with sequences {k i n }⊂[1, ∞ ) such that k i n → 1 as n ® ∞ for each 1 ≤ i ≤ N. Define a mapping W n : C ® C by W n (x)=α n x n−1 + β n T j(n) i ( n ) (α n x + β n S j(n) i ( n ) x + γ n v n )+γ n u n , ∀n ≥ 1 . It follows that W n (x) − W n (y) ≤ β n T j(n) i(n) (α n x + β n S j(n) i(n) x + γ n v n ) − T j(n) i(n) (α n y + β n S j(n) i(n) y + γ n v n ) ≤ β n L(α n x − y +β n S j(n) i(n) x − S j(n) i(n) y ) ≤ β n L ( α n + β n L ) x − y , ∀x, y ∈ C, Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 Page 4 of 17 where L =max L 1 t , , L N t , L 1 s , L N s . (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 Î C such that x n = W n (x n )=α n x n−1 + β n T j ( n ) i(n) (α n x + β n S j ( n ) i(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 Î E :||x|| = 1}. E is said to be uniformly convex if for any ε Î (0, 2] there exists δ > 0 such that for any x, y Î 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 }inE, 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 Î E and y ≠ x. It is well known [30] that each l p (1 ≤ p < ∞) and Hilbert spaces satisfy Opial ’ s conditi on. 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 : C ® C is said to be demiclosed at the origin if for each sequence {x n }inC, the condition x n ® x 0 weakly and Tx n ® 0 strongly implies Tx 0 = 0. T is said to be semicompact if any bounded sequence {x n }inC satisfying lim n®∞ ||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 n 0 is some nonnegative integer. If ∞ n = 0 c n < ∞ and ∞ n = 0 b n < ∞ , then lim n®∞ a n exists. Lemma 2.2. [17]Let E be a uniformly convex Banach space and 0<l ≤ t n ≤ h <1for 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 lim n®∞ ||x n - y n || = 0. Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 Page 5 of 17 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 : C ® C 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 : C ® C be a uniformly L i t -Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {h i n }⊂[1, ∞ ) , where h i n → 1 as n ® ∞ and S i : C ® C be a uniformly L i s -Lipschitz and generalized asymptotically nonex- pansive mapping with a sequence {k i n }⊂[1, ∞ ) , where k i n → 1 as n ® ∞ for each 1 ≤ i ≤ N. Assume that F = N i =1 F( T i ) N i =1 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 i n :1≤ i ≤ N } and k n =sup{k i n :1≤ i ≤ N } . Let {a n }, {b n }, {g 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 i n =max{0, sup x, y ∈C ( T n i x − T n i y −h i n x − y ) } and ν i n =max{0, sup x, y ∈C ( S n i x − S n i y −k i n x − y ) } . Let ξ n =max{μ n , ν n }, where ν n =max{ν i n :1≤ i ≤ N } and ν n =max{ν i n :1≤ i ≤ N } . Assume that the following restrictions are satisfied: (a) ∞ n =1 γ n < ∞ and ∞ n =1 γ n < ∞ ; (b) ∞ n =1 (e n − 1) < ∞ and ∞ n =1 ξ n < ∞ ; (c) β n L(α n + β n L) < 1 , where L is defined in (2.5); (d) there exist constants l, h Î (0, 1) such that l ≤ a 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 j(n) i(n) x n + γ n v n − f ≤ α n x n − f +β n S j(n) i(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 j(n) i(n) y n + γ n u n − f ≤ α n x n−1 − f +β n T j(n) i(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) Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 Page 6 of 17 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 2 j(n) 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 n 0 such that (1 − α n )e 2 j ( n ) ≤ 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 2 j(n) − 1) 1 − (1 − α n )e 2 j(n) 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 2 j(n) ≤ 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 M 1 = sup n≥1 {e n }, M 2 = sup n≥1 {||v n - f||}, and M 3 = sup n≥1 {||u n - f ||}. In view of Lemma 2.1, we see that lim n®∞ ||x n - f|| exists for each f ∈ F . This implies that the sequence {x n } is bounded. Next, we assume that lim n®∞ ||x n - f|| = d > 0. From (3.1), we see that T j ( n ) i(n) y n − f + γ n (u n − T j ( n ) i(n) y n ) ≤ T j(n) i(n) y n − f +γ n u n − T j(n) i(n) y n ≤ e j(n) y n − f +ξ j(n) + γ n u n − T j(n) i(n) y n ≤ e 2 j(n) 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 j(n) i ( n ) y n . This implies from the restrictions (a) and (b) that lim sup n →∞ T j(n) i(n) y n − f + γ n (u n − T j(n) i(n) y n ) ≤ d . Notice that x n−1 − f + γ n (u n − T j(n) i ( n ) y n ) ≤ x n−1 − f +γ n u n − T j(n) i ( n ) y n . This shows from the restriction (a) that lim sup n →∞ x n−1 − f + γ n (u n − T j(n) i(n) y n ) ≤ d . Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 Page 7 of 17 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 j(n) i(n) y n )) +(1− α n )(T j(n) i ( n ) y n − f + γ n (u n − T j(n) i ( n ) y n )) . It follows from Lemma 2.2 that lim n→∞ T j ( n ) i(n) y n − x n−1 =0 . (3:4) Notice that x n − x n−1 ≤ β n T j ( n ) i ( 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 j(n) i ( n ) x n ) ≤ x n − f +γ n v n − S j(n) i ( n ) x n and S j(n) i(n) x n − f + γ n (v n − S j(n) i(n) x n ) ≤ S j(n) i(n) x n − f +γ n v n − S j(n) i(n) x n ≤ e j(n) x n − f +ξ j(n) + γ n v n − S j(n) i ( n ) x n . which in turn imply that lim sup n →∞ x n − f + γ n (v n − S j(n) i(n) x n ) ≤ d and lim sup n →∞ S j(n) i(n) x n − f + γ n (v n − S j(n) i(n) x n ) ≤ d . On the other hand, we have x n − f = α n x n−1 + β n T j(n) i(n) y n + γ n u n − f ≤ α n x n−1 − f +β n T j(n) i(n) y n − f +γ n u n − f ≤ α n x n−1 − T k(n) i(n) y n + T j(n) i(n) y n − f +γ n u n − f ≤ α n x n−1 − T k(n) i ( n ) y n +e j(n) y n − f +ξ j(n) + γ n u n − f , Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 Page 8 of 17 from which it follows that lim inf n®∞ ||y n - f|| ≥ d. In view of (3.1), we see that lim sup n®∞ ||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 j(n) i(n) x n )) +(1− α n )(S j(n) i ( n ) x n − f + γ n (v n − S j(n) i ( n ) x n )) . This implies from Lemma 2.2 that lim n →∞ S j(n) i(n) x n − x n =0 . (3:7) On the other hand, we have T j(n) i(n) x n − x n ≤ T j(n) i(n) x n − T j(n) i(n) y n + T j(n) i(n) y n − x n−1 + x n−1 − x n ≤ T j(n) i(n) x n − T j(n) i(n) y n + T j(n) i(n) y n − x n−1 + x n−1 − x n ≤ L x n − y n + T j(n) i(n) y n − x n−1 + x n−1 − x n ≤ Lβ n S j(n) i(n) x n − x n +Lγ n v n − x n + T j(n) i(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 j(n) i(n) x n − x n =0 . (3:8) Since n =(j(n)-1)N + i(n), where i(n) Î {1, 2, , N}, we see that x n − S i(n) x n ≤x n − S j(n) i(n) x n + S j(n) i(n) x n − S i(n) x n ≤ x n − S j(n) i(n) x n +L S j(n)−1 i(n) x n − x n ≤ x n − S j(n) i(n) x n +L( S j(n)−1 i(n) x n − S j(n)−1 i(n−N) x n−N + S j(n)−1 i ( n−N ) x n−N − x n−N + x n−N − x n ). (3:9) Notice that j ( n − N ) = j ( n ) − 1andi ( n − N ) = i ( n ). This in turn implies that S j ( n ) −1 i(n) x n − S j ( n ) −1 i(n−N) x n−N = S j ( n ) −1 i(n) x n − S j ( n ) −1 i(n) x n−N ≤ L x n − x n−N (3:10) and S j(n)−1 i ( n−N ) x n−N − x n−N = S j(n−N) i ( n−N ) x n−N − x n−N . (3:11) Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 Page 9 of 17 Substituting (3.10) and (3.11) into (3.9) yields that x n − S i(n) x n ≤x n − S j(n) i(n) x n +L(L x n − x n−N + S j(n−N) i ( 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 jN+1 − S 1 x jN+1 =0, lim j→∞ x jN+2 − S 2 x jN+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 jN+s − S r x jN+s ≤ x jN+s − x jN+r + x jN+r − S r x jN+r + S r x jN+r − S r x jN+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 jN+s − S r x jN+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 j ( n ) i(n) x n + T j ( n ) i(n) x n − T i(n) x n ≤ x n − T j(n) i(n) x n +L T j(n)−1 i(n) x n − x n ≤ x n − T j(n) i(n) x n +L( T j(n)−1 i(n) x n − T j(n)−1 i(n−N) x n−N + T j(n)−1 i ( n−N ) x n−N − x n−N + x n−N − x n ). (3:14) On the other hand, we have T j(n)−1 i(n) x n − T j(n)−1 i(n−N) x n−N = T j(n)−1 i(n) x n − T j(n)−1 i(n) x n−N ≤ L x n − x n−N (3:15) and T j(n)−1 i ( n−N ) x n−N − x n−N = T j(n−N) i ( 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 k(n) i(n) x n +L(L x n − x n−N + T k(n−N) i ( 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) Agarwal et al. Fixed Point Theory and Applications 2011, 2011:58 http://www.fixedpointtheoryandapplications.com/content/2011/1/58 Page 10 of 17 [...]... of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform opial property Colloq Math 65, 169–179 (1993) 27 Kirk, WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type Israel J Math 17, 339–346 (1974) doi:10.1007/BF02757136 28 Xu, HK: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type Nonlinear Anal... RG: An implicit iteration process for nonexpansive mappings Numer Funct Anal Optim 22, 767–773 (2001) doi:10.1081/NFA-100105317 24 Zhou, YY, Chang, SS: Convergence of implicit iterative process for a finite of asymptotically nonexpansive mappings in Banach spaces Numer Funct Anal Optim 23, 911–921 (2002) doi:10.1081/NFA-120016276 25 Goebel, K, Kirk, WA: A fixed point theorem for asymptotically nonexpansive. .. Florida, USA (1987) 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 313, 273–283 (2006) doi:10.1016/j.jmaa.2005.05.075 7 Chidume, CE, Shahzad, N: Strong convergence of an implicit iteration process for a finite family of nonexpansive mappings Nonlinear Anal 62, 1149–1156 (2005)... Convergence of an implicit algorithm for two families of nonexpansive mappings Comput Math Appl 59, 3084–3091 (2010) doi:10.1016/j.camwa.2010.02.029 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 71, e2839–e2848 (2009) doi:10.1016/j.na.2009.06.090 14 Plubtieng, S, Ungchittrakool, K, Wangkeeree,... Ungchittrakool, K, Wangkeeree, R: Implicit iterations of two finite families for nonexpansive mappings in Banach spaces Numer Funct Anal Optim 28, 737–749 (2007) doi:10.1080/01630560701348525 15 Qin, X, Cho, YJ, Shang, M: Convergence analysis of implicit iterative algorithms for asymptotically nonexpansive mappings Appl Math Comput 210, 542–550 (2009) doi:10.1016/j.amc.2009.01.018 16 Qin, X, Kang, SM, Agarwal, RP:... the class of asymptotically nonexpansive mappings to the class of generalized asymptotically nonexpansive mappings If S r = I for each r Î {1, 2, , N} and γn = 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 Ti : C ® C be a uniformly Lit-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence... be a real uniformly convex Banach space and C be a nonempty closed convex subset of E Let Ti : C ® C be a uniformly Lit-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {hin } ⊂ [1, ∞), where hin → 1as n ® ∞ and Si : C ® C be a uniformly Lis-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {kin } ⊂ [1, ∞), where kin → 1as n ® ∞ for each 1 ≤... convergence of an implicit iterative process for generalized asymptotically quasinonexpansive mappings Fixed Point Theory Appl 2010, 19 (2010) Article ID 714860 17 Schu, J: Weak and strong convergence of fixed points of asymptotically nonexpansive mappings Bull Austral Math Soc 43, 153–159 (1991) doi:10.1017/S0004972700028884 18 Shzhzad, N, Zegeye, H: Strong convergence of an implicit iteration process for. .. 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 Ti : C ® C be a uniformly Lit-Lipschitz and generalized asymptotically nonexpansive mapping with a sequence {hin } ⊂ [1, ∞), where hin → 1as n ® ∞ and Si : C ® C be a uniformly Lis-Lipschitz and generalized asymptotically nonexpansive mapping with. .. 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) Extend the mappings from one family of mappings to two families of mappings; (2) Extend . Access An implicit iterative algorithm with errors for two families of generalized asymptotically nonexpansive mappings Ravi P Agarwal 1 , Xiaolong Qin 2 and Shin Min Kang 3* * Correspondence: smkang@gnu. ac.kr 3 Department. 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. one family of mappings to two families of mappings; (2) Extend the mappings from the class of asymptotically nonexpansive mappings to the class of generalized asymptotically nonexpansive mappings. If