Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2007, Article ID 31056, 15 pages doi:10.1155/2007/31056 Research Article Weak and Strong Convergence of Multistep Iteration for Finite Family of Asymptotically Nonexpansive Mappings Balwant Singh Thakur and Jong Soo Jung Received 14 March 2007; Accepted 26 May 2007 Recommended by Nan-Jing Huang Strong and weak convergence theorems for multistep iterative scheme with errors for finite family of asymptotically nonexpansive mappings are established in Banach spaces. Our results extend and improve the corresponding results of Chidume and Ali (2007), Cho et al. (2004), K han and Fukhar-ud-din (2005), Plubtieng et al.(2006), Xu and Noor (2002), and many others. Copyright © 2007 B. S. Thakur and J. S. Jung. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, dis- tribution, and reproduction in a ny medium, provided the original work is properly cited. 1. Introduction and preliminaries Let K beanonemptysubsetofarealnormedspaceE. A self-mapping T : K → K is said to be nonexpansive if Tx− Ty≤x − y for all x, y in K. T is said to be asymptotically nonexpansive if there exists a sequence {r n } in [0,∞)withlim n→∞ r n = 0suchthatT n x − T n y≤(1 + r n )x − y for all x, y in K and n ∈ N. The class of asymptotically nonexpansive mappings which is an important generaliza- tion of that of nonexpansive mappings was introduced by Goebel and Kirk [ 6]. Iteration processes for nonexpansive and asy mptotically nonexpansive mappings in Banach spaces including Mann [7] and Ishikawa [8] iteration processes have been studied extensively by many authors to solve the nonlinear operators as well as variational inequalities; see [1–22, 25]. Noor [13] introduced a three-step iterative scheme and studied the approximate so- lution of variational inclusion in Hilbert spaces by using the techniques of updating the solution and auxiliary principle. Glowinski and Le Tallec [9] used three-step iterative schemes to find the approximate solutions of the elastoviscoplasticity problem, liquid crystal theory, and eigenvalue computation. It has been shown in [9] that the three-step 2 Fixed Point Theory and Applications iterative scheme gives better numerical results than the two-step and one-step approx- imate iterations. Thus, we conclude that the three-step scheme plays an important and significant role in solving various problems which arise in pure and applied sciences. Re- cently, Xu and Noor [5] introduced and studied a three-step scheme to approximate fixed points of asymptotically nonexpansive mappings in Banach space. Cho et al. [2]extended the work of Xu and Noor [5] to the three-step iterative scheme with errors in a Banach space and gave weak and strong convergence theorems for asymptotically nonexpansive mappings in a Banach space. Moreover, Suantai [20] gave weak and strong convergence theorems for a new three-step iterative scheme of asymptotically nonexpansive mappings. More recently, Plubtieng et al. [4] introduced a three-step iterative scheme with errors for three asymptotically nonexpansive mappings and established strong convergence of this scheme to common fixed point of three asymptotically nonexpansive mappings. Very recently, Chidume and Ali [1] considered multistep scheme for finite family of asymptot- ically nonexpansive mappings and gave weak convergence theorems for this scheme in a uniformly convex Banach space whose the dual space satisfies the Kadec-Klee property. They also proved a strong convergence theorem under some appropriate conditions on finite family of asymptotically nonexpansive mappings. Inspired by the above facts, in this paper, a new multistep iteration scheme with errors for finite family of asymptotically nonexpansive mappings is introduced and strong and weak convergence theorems of this scheme to common fixed point of asymptotically non- expansive mappings are proved. In particular, our weak convergence theorem is proved in a uniformly convex Banach space whose the dual has a Kadec-Klee property. It is worth mentioning that there are uniformly convex Banach spaces, which have neither a Fr ´ echet differentiable norm nor Opial property; however, their dual does have the Kadec-Klee property. This means that our weak convergence result can apply not only to L p -spaces with 1 <p< ∞ but also to other spaces which do not satisfy Opial’s condition or have a Fr ´ echet differentiable norm. Our theorems improve and generalize some previous results in [1–5, 15, 17–19]. Our iterative scheme is defined as below. Let K beanonemptyclosedsubsetofanormedspaceE,andlet {T 1 ,T 2 , ,T N } : K → K be N asymptotically nonexpansive mappings. For a given x 1 ∈ K and a fixed N ∈ N (N denote the set of all positive integers), compute the sequence {x n } by x n+1 = x (N) n = α (N) n T n N x (N−1) n + β (N) n x n + γ (N) n u (N) n , x (N−1) n = α (N−1) n T n N −1 x (N−2) n + β (N−1) n x n + γ (N−1) n u (N−1) n , . . . x (3) n = α (3) n T n 3 x (2) n + β (3) n x n + γ (3) n u (3) n , x (2) n = α (2) n T n 2 x (1) n + β (2) n x n + γ (2) n u (2) n , x (1) n = α (1) n T n 1 x n + β (1) n x n + γ (1) n u (1) n , (1.1) where, {u (1) n },{u (2) n }, ,{u (N) n } are bounded sequences in K and {α (i) n }, {β (i) n }, {γ (i) n } are appropriate real sequences in [0,1] such that α (i) n + β (i) n + γ (i) n = 1foreachi ∈{1,2, ,N}. We now give some preliminaries and results which will be used in the rest of this paper. B. S. Thakur and J. S. Jung 3 ABanachspaceE is said to satisfy Opial’s condition if for each sequence x n in E,the condition, that the sequence x n → x weakly, implies limsup n→∞   x n − x   < limsup n→∞   x n − y   (1.2) for all y ∈ E with y = x. ABanachspaceE is said to have Kadec-Klee property if for every sequence {x n } in E, x n → x weakly and x n →x strongly together imply that x n − x→0. We will make use of the following lemmas. Lemma 1.1 [2]. Let E be a uniformly convex Banach space, let K be a nonempty closed convex subset of E, and let T : K → K be an asymptotically nonexpansive mapping. Then, I − T is demiclosed at zero, that is, for each sequence {x n } in K,if{x n } converges weakly to q ∈ K and {(I − T)x n } converges strongly to 0, then (I − T)q = 0. Lemma 1.2 [16]. Let {a n }, {b n }, and {c n } be sequences of nonnegative real numbers satis- fy ing the inequality a n+1 ≤  1+δ n  a n + b n , n ≥ 1. (1.3) If  ∞ n=1 δ n < ∞ and  ∞ n=1 b n < ∞, then lim n→∞ a n exists. If, in addition, {a n } has a subse- quence which converges strongly to zero, then lim n→∞ a n = 0. Lemma 1.3 [19]. Let E be a uniformly convex Banach space and let b, c be two constants with 0 <b<c<1.Supposethat {t n } is a real sequence in [b,c] and {x n }, {y n } are two sequences in E such that limsup n→∞   x n   ≤ a, limsup n→∞   y n   ≤ a, lim n→∞   t n x n +  1 − t n  y n   = a. (1.4) Then, lim n→∞ x n − y n =0,wherea ≥ 0 is some constant. Lemma 1.4 [12]. Let E be a real reflexive Banach space such that its dual E ∗ has the Kadec- Klee property . Let {x n } be a bounded sequence in E and p,q ∈ ω w (x n ),whereω w (x n ) denotes the weak w-limit set of {x n }.Supposethatlim n→∞ tx n +(1− t)p − q exists for all t ∈ [0,1]. Then p = q. 2. Main results In this section, we prove strong and weak convergence theorems for multistep iteration with errors in Banach spaces. In order to prove our main results, we need the following lemmas. Lemma 2.1. Let E be a real normed space and let K beanonemptyclosedconvexsubsetof E.Let {T 1 ,T 2 , ,T N } : K → K be N asymptotically nonexpansive mappings with sequences {r (i) n } such that  ∞ n=1 r (i) n < ∞, 1 ≤ i ≤ N.Let{x n } be the sequence defined by (1.1)with  ∞ n=1 γ (i) n < ∞, 1 ≤ i ≤ N.IfF =  N i =1 F(T i ) =∅, then lim n→∞ x n − p exists for all p ∈ F. 4 Fixed Point Theory and Applications Proof. For any p ∈ F, we note that   x (1) n − p   ≤ α (1) n   T n 1 x n − p   + β (1) n   x n − p   + γ (1) n   u (1) n − p   ≤ α (1) n  1+r n    x n − p   + β (1) n   x n − p   + γ (1) n   u (1) n − p   ≤  1+r n    x n − p   + t (1) n , (2.1) where t (1) n = γ (1) n u (1) n − p.Since{u (1) n } is bounded and  ∞ n=1 γ (1) n < ∞,wecanseethat  ∞ n=1 t (1) n < ∞.Itfollowsfrom(2.1)that   x (2) n − p   ≤ α (2) n   T n 2 x (1) n − p   + β (2) n   x n − p   + γ (2) n   u (2) n − p   ≤ α (2) n  1+r n    x (1) n − p   + β (2) n   x n − p   + γ (2) n   u (2) n − p   ≤ α (2) n  1+r n  1+r n    x n − p   + t (1) n  + β (2) n   x n − p   + γ (2) n   u (2) n − p   ≤ α (2) n  1+r n  2   x n − p   + α (2) n t (1) n  1+r n  + β (2) n   x n − p   + γ (2) n   u (2) n − p   ≤ α (2) n  1+r n  2   x n − p   + α (2) n t (1) n  1+r n  + β (2) n  1+r n  2   x n − p   + γ (2) n   u (2) n − p   ≤  α (2) n + β (2) n  1+r n  2   x n − p   + α (2) n t (1) n  1+r n  + γ (2) n   u (2) n − p   ≤  1+r n  2   x n − p   + α (2) n t (1) n  1+r n  + γ (2) n   u (2) n − p   ≤  1+r n  2   x n − p   + t (2) n , (2.2) where t (2) n = α (2) n t (1) n (1 + r n )+γ (2) n u (2) n − p.Since{u (2) n } is bounded and  ∞ n=1 t (1) n < ∞, we can see that  ∞ n=1 t (2) n < ∞. Similarly, we see that   x (3) n − p   ≤ α (3) n  1+r n  1+r n  2   x n − p   + t (2) n  + β (3) n   x n − p   + γ (3) n   u (3) n − p   ≤  α (3) n + β (3) n  1+r n  3   x n − p   + α (3) n t (2) n  1+r n  + γ (3) n   u (3) n − p   ≤  1+r n  3   x n − p   + α (3) n t (2) n  1+r n  + γ (3) n   u (3) n − p   ≤  1+r n  3   x n − p   + t (3) n , (2.3) where t (3) n = α (3) n t (2) n (1 + r n )+γ (3) n u (3) n − p.Since{u (3) n } is bounded and  ∞ n=1 t (2) n < ∞, we can see that  ∞ n=1 t (3) n < ∞. Continuing the above process, we get   x n+1 − p   =   x (N) n − p   ≤  1+r n  N   x n − p   + t (N) n , (2.4) where {t (N) n } is nonnegative real sequence such that  ∞ n=1 t (N) n < ∞.ByLemma 1.2, lim n→∞ x n − p exists. This completes the proof.  Lemma 2.2. Let E be a real uniformly convex Banach space and let K be a nonempty closed convex subset of E.Let {T 1 ,T 2 , ,T N } : K → K be N asymptotically nonexpansive mapping s B. S. Thakur and J. S. Jung 5 w ith sequences {r (i) n } such that  ∞ n=1 r (i) n < ∞, 1 ≤ i ≤ N and let F =  N i =1 F(T i ) =∅.Let {x n } bethesequencedefinedby(1.1)andsomeα,β ∈ (0,1) with the following restrictions: (i) 0 <α ≤ α (i) n ≤ β<1, 1 ≤ i ≤ N,foralln ≥ n 0 for some n 0 ∈ N; (ii)  ∞ n=1 γ i n < ∞, 1 ≤ i ≤ N. Then, lim n→∞ x n − T i x n =0. Proof. For any p ∈ F(T), it follows from Lemma 2.1 that lim n→∞ x n − p exists. Let lim n→∞ x n − p=a for some a ≥ 0. We note that   x N−1 n − p   ≤  1+r n  N−1   x n − p   + t (N−1) n , ∀n ≥ 1, (2.5) where {t (N−1) n } is nonnegative real sequence such that  ∞ n=1 t (N−1) n < ∞. It follows that limsup n→∞   x (N−1) n − p   ≤ limsup n→∞  1+r n  N−1   x n − p   + t N−1 n  = lim n→∞   x n − p   = a (2.6) and so limsup n→∞   T n N x (N−1) n − p   ≤ limsup n→∞  1+r n    x (N−1) n − p   = limsup n→∞   x (N−1) n − p   ≤ a. (2.7) Next, consider   T n N x (N−1) n − p + γ (N) n  u (N) n − x n    ≤   T n N x (N−1) n − p   + γ (N) n   u (N) n − x n   . (2.8) Thus, limsup n→∞   T n N x (N−1) n − p + γ (N) n  u (N) n − x n    ≤ a. (2.9) Also,   x n − p + γ (N) n  u (N) n − x n    ≤   x n − p   + γ (N) n   u (N) n − x n   (2.10) gives that limsup n→∞   x n − p + γ (N) n  u (N) n − x n    ≤ a, (2.11) and we observe that x (N) n − p = α (N) n T n N x (N−1) n − α (N) n p + α (N) n γ (N) n u (N) n − α (N) n γ (N) n x n +  1 − α (N) n  x n −  1 − α (N) n  p − γ (N) n x n + γ (N) n u (N) n − α (N) n γ (N) n u (N) n + α (N) n γ (N) n x n = α (N) n  T n N x (N−1) n − p + γ (N) n  u (N) n − x n  +  1 − α (N) n  x n − p  −  1 − α (N) n  γ (N) n x n +  1 − α (N) n  γ (N) n u (N) n = α (N) n  T n N x (N−1) n − p + γ (N) n  u (N) n − x n  +  1 − α (N) n  x n − p + γ (N) n  u (N) n − x n  . (2.12) 6 Fixed Point Theory and Applications Therefore, a = lim n→∞   x (N) n − p   = lim n→∞   α (N) n  T n N x (N−1) n − p + γ (N) n  u (N) n − x n  +  1 − α (N) n  x n − p + γ (N) n  u (N) n − x n    . (2.13) By (2.9), (2.14), and Lemma 1.3,wehave lim n→∞   T n N x (N−1) n − x n   = 0. (2.14) Now, we will show that lim n→∞ T n N −1 x (N−2) n − x n =0. For each n ≥ 1,   x n − p   ≤   T n N x (N−1) n − x n   +   T n N x (N−1) n − p   ≤   T n N x (N−1) n − x n   +  1+r n    x (N−1) n − p   . (2.15) Using (2.14), we have a = lim n→∞   x n − p   ≤ liminf n→∞   x (N−1) n − p   . (2.16) It follows that a ≤ liminf n→∞   x (N−1) n − p   ≤ limsup n→∞   x (N−1) n − p   ≤ a. (2.17) This implies that lim n→∞   x (N−1) n − p   = a. (2.18) On the other hand, we have   x (N−2) n − p   ≤  1+r n  N−2   x n − p   + t (N−2) n , ∀n ≥ 1, (2.19) where  ∞ n=1 t (N−2) n < ∞.Therefore, limsup n→∞   x (N−2) n − p   ≤ limsup n→∞  1+r n  N−2   x n − p   + t (N−2) n = a, (2.20) and hence, limsup n→∞   T n N −1 x (N−2) n − p   ≤ limsup n→∞  1+r n    x (N−2) n − p   ≤ a. (2.21) Next, consider   T n N −1 x (N−2) n − p + γ (N−1) n  u (N−1) n − x n    ≤   T n N −1 x (N−2) n − p   + γ (N−1) n   u (N−1) n − x n   . (2.22) Thus, limsup n→∞   T n N −1 x (N−2) n − p + γ (N−1) n  u (N−1) n − x n    ≤ a. (2.23) B. S. Thakur and J. S. Jung 7 Also,   x n − p + γ (N−1) n  u (N−1) n − x n    ≤   x n − p   + γ (N−1) n   u (N−1) n − x n   (2.24) gives that limsup n→∞   x n − p + γ (N−1) n  u (N−1) n − x n    ≤ a, (2.25) and we observe that x (N−1) n − p = α (N−1) n T n N −1 x (N−2) n +  1 − α (N−1) n  x n − γ (N−1) n x n + γ (N−1) n u (N−1) n −  1 − α (N−1) n  p − α (N−1) n p = α (N−1) n  T n N −1 x (N−2) n − p + γ (N−1) n  u (N−1) n − x n  +  1 − α (N−1) n  x n − p + γ (N−1) n  u (N−1) n − x n  , (2.26) and hence a = lim n→∞   x (N−1) n − p   = lim n→∞   α (N−1) n  T n N −1 x (N−2) n − p + γ (N−1) n  u (N−1) n − x n  +  1 − α (N−1) n  x n − p + γ (N−1) n  u (N−1) n − x n    . (2.27) By (2.23), (2.25), and Lemma 1.3,wehave lim n→∞   T n N −1 x (N−2) n − x n   = 0. (2.28) Similarly, by using the same argument as in the proof above, we have lim n→∞   T n N −1 x (N−2) n − x n   = 0. (2.29) Continuing similar process, we have lim n→∞   T N−i x (N−i−1) n − x n   = 0, 0 ≤ i ≤ (N − 2). (2.30) Now,   T n 1 x n − p + γ (1) n  u (1) n − x n    ≤   T n 1 x n − p   + γ (1) n   u (1) n − x n   . (2.31) Thus, limsup n→∞   T n 1 x n − p + γ (1) n  u (1) n − x n    ≤ a. (2.32) Also,   x n − p + γ (1) n  u (1) n − x n    ≤   x n − p   + γ (1) n   u (1) n − x n   (2.33) 8 Fixed Point Theory and Applications gives that limsup n→∞   x n − p + γ (1) n  u (1) n − x n    ≤ a, (2.34) and hence, a = lim n→∞   x (1) n − p   = lim n→∞   α (1) n  T n 1 x n − p + γ (1) n  u (1) n − x n  +  1 − α (1) n  x n − p + γ (1) n  u (1) n − x n    . (2.35) By (2.32), (2.34), and Lemma 1.3,wehave lim n→∞   T n 1 x n − x n   = 0, (2.36) and this implies that   x n+1 − x n   =   α (N) n T n N x (N−1) n +  1 − α (N) n − γ (N) n  x n + γ (N) n u (N) n − x n   ≤ α (N) n   T n N x (N−1) n − x n   + γ (N) n   u (N) n − x n   −→ ∞ ,asn −→ ∞ . (2.37) Thus, we have   T n N x n − x n   ≤   T n N x n − T n N x (N−1) n   +   T n N x (N−1) n − x n   ≤  1+r n    x n − x (N−1) n   +   T n N x (N−1) n − x n   =  1+r n    x n − α (N−1) n T n N −1 x (N−2) n +  1 − α (N−1) n − γ (N−1) n  x n + γ (N−1) n u (N−1) n   +   T n N x (N−1) n − x n   ≤  1+r n  α (N−1) n   x n − T n N −1 x (N−2) n   + γ (N−1) n   u (N−1) n − x n    +   T n N x (N−1) n − x n   −→ ∞ ,asn −→ ∞ , (2.38) and we have   T N x n − x n   ≤   x n+1 − x n   +   x n+1 − T n+1 N x n+1   +   T n+1 N x n+1 − T n+1 N x n   +   T n+1 N x n − T N x n   ≤   x n+1 − x n   +   x n+1 − T n+1 N x n+1   +  1+r n+1    x n+1 − x n   +  1+r 1    T n N x n − x n   . (2.39) It follows from (2.37), (2.38), and (2.39)that lim n→∞   T N x n − x n   = 0. (2.40) B. S. Thakur and J. S. Jung 9 Next, we consider   T n N −1 x n − x n   ≤   T n N −1 x n − T n N −1 x (N−2) n   +   T n N −1 x (N−2) n − x n   ≤  1+r n    x n − x (N−2) n   +   T n N −1 x (N−2) n − x n   ≤  1+r n  α (N−2) n   x n − T n N −2 x (N−3) n   + γ (N−2) n   u (N−2) n − x n    +   T n N −1 x (N−2) n − x n   −→ ∞ ,asn −→ ∞ , (2.41)   T N−1 x n − x n   ≤   x n+1 − x n   +   x n+1 − T n+1 N −1 x n+1   +   T n+1 N −1 x n+1 − T n+1 N −1 x n   +   T n+1 N −1 x n − T N−1 x n   ≤   x n+1 − x n   +   x n+1 − T n+1 N −1 x n+1   +  1+r n+1    x n+1 − x n   +  1+r 1    T n N −1 x n − x n   . (2.42) It follows from (2.37), (2.41) and the above inequality that lim n→∞   T N−1 x n − x n   = 0. (2.43) Continuing similar process, we have lim n→∞   T N−i x n − x n   = 0, 0 ≤ i ≤ (N − 2). (2.44) Now,   T 1 x n − x n   ≤   x n+1 − x n   +   x n+1 − T n+1 1 x n+1   +   T n+1 1 x n+1 − T n+1 1 x n   +   T n+1 1 x n − T 1 x n   ≤   x n+1 − x n   +   x n+1 − T n+1 1 x n+1   +  1+r n+1    x n+1 − x n   +  1+r 1    T n 1 x n − x n   . (2.45) It follows from (2.36), (2.37) and the above inequality that lim n→∞   T 1 x n − x n   = 0, (2.46) and hence, lim n→∞   T N−i x n − x n   = 0, 0 ≤ i ≤ (N − 1). (2.47) This completes the proof.  We recall the following definitions: (i) A mapping T : K → K with F(T) =∅is said to satisfy condition (A) [21]onK if there exists a nondecreasing function f :[0, ∞) → [0, ∞)with f (0) = 0and f (r) >r for all r ∈ (0,∞)suchthatforallx ∈ K x − Tx≥ f (d(x, F)), where d(x,F(T)) = inf{x − p : p ∈ F(T)}. (ii) A finite family {T 1 , ,T N } of N self mappings of K with F =  i=1 N F(T i ) =∅ is said to satisfy condition (B) on K [1] if there exist f and d as in (i) such that max 1≤i≤N x − T i x≥ f (d(x,F)) for all x ∈ K. 10 Fixed Point Theory and Applications (iii) A finite family {T 1 , ,T N } of N self mappings of K with F =  i=1 N F(T i ) =∅ is said to satisfy condit ion (C) on K [1] if there exist f and d as in (i) such that (1/N)  N i =1 x − T i x≥ f (d(x,F)) for all x ∈ K. Note that condition (B) reduces to condition (A) when T i = T,foralli = 1,2, ,N. It is well known that every continuous and demicompact mapping must satisfy condi- tion (A) (see [21]). Since every completely continuous mapping T : K → K is continuous and demicompact, it satisfies condition (A). Therefore, to study strong convergence of {x n } defined by (1.1), we use condition (B) instead of the complete continuity of map- pings T 1 ,T 2 , ,T N . Theorem 2.3. Let E be a real uniformly convex Banach space and K let be a nonempt y closed convex subse t of E.Let {T 1 , ,T N } : K → K be N asymptotically nonexpansive mappings w ith sequences {r (i) n } such that  ∞ n=1 r (i) n < ∞ for all 1 ≤ i ≤ N and F =  N i =1 F(T i ) =∅. Suppose that {T 1 ,T 2 , ,T N } satisfies condition (B). Let {x n } be the sequence defined by (1.1)andsomeα,β ∈ (0,1) with the following restrictions: (i) 0 <α ≤ α (i) n ≤ β<1, 1 ≤ i ≤ N ∀ n ≥ n 0 for some n 0 ∈ N; (ii)  ∞ n=1 γ i n < ∞, 1 ≤ i ≤ N. Then, {x n } converges strongly to a common fixed point of the mappings {T 1 , ,T N }. Proof. By Lemma 2.1,weseethatlim n→∞ x n − p exists for all p ∈ F.Letlim n→∞ x n − p=a for some a ≥ 0. Without loss of generality, if a = 0, there is nothing to prove. Assume that a>0, as proved in Lemma 2.1,wehave   x n+1 − p   =   x (N) n − p   ≤  1+r n  N   x n − p   + t (N) n , (2.48) where {t (N) n } is nonnegative real sequence such that  ∞ n=1 t (N) n < ∞. This gives that d  x n+1 ,F  ≤  1+r n  N d  x n ,F  + t (N) n . (2.49) Applying Lemma 1.2 to the above inequality, we obtain that lim n→∞ d(x n ,F) exists. Also, by Lemma 2.2,lim n→∞ x n − T i x n =0, for all i = 1,2, ,N.Since{T 1 ,T 2 , ,T N } satis- fies condition (B), we conclude that lim n→∞ d(x n ,F) = 0. Next, we show that {x n } is a Cauchy sequence. Since lim n→∞ d(x n ,F) = 0, given any ε>0, there exists a natural number n 0 such that d(x n ,F) <ε/3foralln ≥ n 0 .So,wecan find p ∗ ∈ F such that x n 0 − p ∗  <ε/2. For all n ≥ n 0 and m ≥ 1, we have   x n+m − x n   ≤   x n+m − p ∗   +   x n − p ∗   ≤   x n 0 − p ∗   +   x n 0 − p ∗   < ε 2 + ε 2 = ε. (2.50) This shows that {x n } is a Cauchy sequence and so is convergent since E is complete. Let lim n→∞ x n = q ∗ .Thenq ∗ ∈ K. It remains to show that q ∗ ∈ F.Letε 1 > 0begiven. 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Theory and Applications Volume 2007, Article ID 31056, 15 pages doi:10.1155/2007/31056 Research Article Weak and Strong Convergence of Multistep Iteration for Finite Family of Asymptotically Nonexpansive. Banach space and gave weak and strong convergence theorems for asymptotically nonexpansive mappings in a Banach space. Moreover, Suantai [20] gave weak and strong convergence theorems for a new. established strong convergence of this scheme to common fixed point of three asymptotically nonexpansive mappings. Very recently, Chidume and Ali [1] considered multistep scheme for finite family of asymptot- ically