Báo cáo hóa học: "WEAK AND STRONG CONVERGENCE OF FINITE FAMILY WITH ERRORS OF NONEXPANSIVE NONSELF-MAPPINGS" docx

WEAK AND STRONG CONVERGENCE OF FINITE FAMILY WITH ERRORS OF NONEXPANSIVE NONSELF-MAPPINGS S. PLUBTIENG AND K. UNGCHITTRAKOOL Received 27 September 2005; Revised 5 May 2006; Accepted 8 May 2006 We are concerned with the study of a multistep iterative scheme with errors involving a finite family of nonexpansive nonself-mappings. We approximate the common fixed points of a finite family of nonexpansive nonself-mappings by weak and strong conver- gence of the scheme in a unifor mly convex Banach space. Our results extend and improve some recent results, Shahzad (2005) and many others. Copyright © 2006 S. Plubtieng and K. Ungchittrakool. This is an open access article dis- tributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is prop- erly cited. 1. Introduction Let K be a subset of a real normed linear space E and let T be a self-mapping on K. T is said to be nonexpansive provided Tx− Ty  x − y for all x, y ∈ K. Fixed-point iteration process for nonexpansive mappings in Banach spaces includ- ing Mann and Ishikawa iteration processes has been studied extensively by many au- thors to solve the nonlinear operator equations in Hilbert spaces and Banach spaces; see [3, 7, 10, 11, 15, 16]. Tan and Xu [15] introduced and studied a modified Ishikawa process to approximate fixed points of nonexpansive mappings defined on nonempty closed convex bounded subsets of a uniformly convex Banach space E.Fiveyearslater, Xu [18] introduced iterative schemes known as Mann iterative scheme with er rors and Ishikawa iterative scheme with errors. Takahashi and Tamura [14] introduced and stud- ied a generalization of Ishikawa iterative schemes for a pair of nonexpansive mappings in Banach spaces. Recently, Khan and Fukhar-ud-din [6] extended their scheme to the modified Ishikawa iterative schemes with errors for two mappings and gave weak and strong convergence theorems. On the other hand, iterative techniques for approximat- ing fixed p oints of nonexpansive nonself-mappings have been studied by various au- thors; see [4, 8, 13, 19]. Shahzad [12] introduced and studied an iteration scheme for Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2006, Article ID 81493, Pages 1–12 DOI 10.1155/FPTA/2006/81493 2 Weak and strong convergence approximating a fixed point of nonexpansive nonself-mappings (when such a fixed point exists) and gave some strong and weak convergence theorems for such mappings. Inspired and motivated by these facts, we introduce and study a multistep iter ative scheme with errors for a finite family of nonexpansive nonself-mappings. Our schemes can be viewed as an extension for two-step iterative schemes of Shahzad [12]. The scheme is defined as follows. Let K be a nonempty closed convex subset of a uniformly convex Banach space E, which is also a nonexpansive retract of E.AndletT 1 ,T 2 , , T N : K → E be nonexpansive mappings, the following iteration scheme is studied: x 1 n = P  α 1 n T 1 x n + β 1 n x n + γ 1 n u 1 n  , x 2 n = P  α 2 n T 2 x 1 n + β 2 n x n + γ 2 n u 2 n  , . . . . . . x n+1 = x N n = P  α N n T N x N n − 1+β N n x n + γ N n u N n  (1.1) with x 1 ∈ K, n  1, where P is a nonexpansive retraction with respect to K and {α 1 n }, {α 2 n }, , {α N n }, {β 1 n },{β 2 n }, , {β N n }, {γ 1 n },{γ 2 n }, , {γ N n } are sequences in [0,1] with α i n + β i n + γ i n = 1foralli = 1,2, ,N,and{u 1 n },{u 2 n }, , {u N n } are bounded sequences in K. For N = 2, T 1 = T 2 ≡ T, β n = α 1 n , α n = α 2 n ,andγ 1 n = γ 2 n ≡ 0, then (1.1) reduces to the scheme for a mapping defined by Shahzad [12]: x 1 = x ∈ K, x n+1 = P  1 − α n  x n + α n TP  1 − β n  x n + β n Tx n  , (1.2) where {α n }, {β n } are sequences in [0,1]. For N = 2, T 1 ,T 2 : K → K, T 1 = T, T 2 = S,andy n = x 1 n ,then(1.1) reduces to the scheme with errors for two mappings defined by x 1 = x ∈ K, y n = α 1 n Tx n + β 1 n x n + γ 1 n u 1 n , x n+1 = x 2 n = α 2 n Sy n + β 2 n x n + γ 2 n u 2 n , (1.3) where {α 1 n }, {α 2 n }, {β 1 n }, {β 2 n }, {γ 1 n }, {γ 2 n } are s equences in [0,1] with α 1 n + β 1 n + γ 1 n =1 = α 2 n + β 2 n + γ 2 n and {u 1 n }, {u 2 n } are bounded sequences in K. It is our purpose in this paper to establish several weak and strong convergence theorems of the multistep iterative scheme with errors for a finite family of nonexpansive nonself-mappings. More precisely, we prove weak convergence of these implicit iteration S. Plubtieng and K. Ungchittrakool 3 processes in a uniformly convex Banach space which has the Kadec-Klee property. The results presented in this paper extend and improve the corresponding ones announced by Shahzad [12], and many others. 2. Preliminaries In this section, we recall the well-known concepts and results. Let E be a real Banach space. A subset K of E is said to be a retract of E if there exists acontinuousmapP : E → K such that Px = x for all x ∈ K.AmapP : E → E is said to be a retraction if P 2 = P. It follows that if a map P is a retraction, then Py= y for all y in the range of P.AmappingT : K → E is called demiclosed with respect to y ∈ E if for each sequence {x n } in K and each x ∈ E, x n  x and Tx n → y imply that x ∈ K and Tx = y.A Banach space E is said to satisfy Opial’s condition [9] if for any sequence {x n } in E, x n  x implies that limsup n→∞   x n − x   < limsup n→∞   x n − y   (2.1) for all y ∈ E with x = y.ABanachspaceE is said to have the Kadec-Klee property if for ever y sequence {x n } in E, x n  x and x n →x together imply x n − x→0. A family {T i : i = 1,2, ,N} of N nonself-mappings of K (i.e., T i : K → E)withF =  N i =1 F(T i ) = ∅ is said to satisfy condition (B)onK if there is a nondecreasing function f :[0,∞) → [0,∞) with f (0) = 0and f (r) > 0forallr ∈ (0,∞) such that for all x ∈ K, max 1iN    x − T i x     f  d(x, F)  . (2.2) The family {T i : i = 1,2, ,N} is said to satisfy condition (A N )if(2.2)isreplacedby 1/N  N i =1 x − T i x  f (d(x,F)) for all x ∈ K. Note that condition (B) reduces to condi- tion (A N )whenx − T 1 x=x − T 2 x = ··· = x − T N x. AmappingT : K → E is called semicompact if any sequence {x n } in K satisfying x n − Tx n →0asn →∞has a convergent subsequence. Lemma 2.1 (Tan and Xu [15]). Let {s n }, {t n } be two nonnegative sequences satisfying s n+1  s n + t n , ∀n  1. (2.3) If  ∞ n=1 t n < ∞, then lim n→∞ s n exists. Moreover, if there ex ists a subsequence {s n j } of {s n } such that s n j → 0 as j →∞, then s n → 0 as n →∞. Lemma 2.2 (Xu [17]). Let p>1 and R>0 be two fixed numbers and E a Banach space. Then E is uniformly convex if and only if there exists a continuous, st rictly increasing, and convex function g :[0, ∞) → [0,∞) with g(0) = 0 such that λx +(1− λ)y p  λx p + (1 − λ)y p − W p (λ)g(x − y) for all x, y ∈ B R (0) ={x ∈ E : x  R},andλ ∈ [0,1], where W p (λ) = λ(1 − λ) p + λ p (1 − λ). Lemma 2.3 (Kaczor [5]). 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 x ∗ , y ∗ ∈ ω w (x n );hereω w (x n ) 4 Weak and strong convergence denote the set of all weak subsequential limits of {x n }.Supposelim n→∞ tx n +(1−t)x ∗ − y ∗  exists for all t ∈ [0,1]. Then x ∗ = y ∗ . Lemma 2.4 (Browder [1]). Let E be a uniformly convex Banach space, K anonemptyclosed convex subset of E,andT : K → E a nonexpansive mapping. Then I − T is demiclosed at zero. 3. Main results In this section, we prove weak and strong convergence theorems of the iterative scheme given in (1.1) for a finite family of nonexpansive mappings in a Banach space. In order to prove our main results, the following lemmas are needed. Lemma 3.1. Let E be a uniformly convex Banach space and K a nonempty closed con- vex subset of E which is also a nonexpansive retract of E.LetT 1 ,T 2 , , T N : K → E be nonexpansive mappings. Let {x n } bethesequencedefinedby(1.1)with  ∞ n=1 γ i n < ∞ for each i = 1,2, ,N.If  N i =1 F(T i ) = ∅, then lim n→∞ x n − x ∗  exists for all x ∗ ∈  N i =1 F(T i ). Proof. For each n  1, we note that   x 1 n − x ∗    α 1 n   T 1 x n − x ∗   + β 1 n   x n − x ∗   + γ 1 n   u 1 n − x ∗    α 1 n   x n − x ∗   + β 1 n   x n − x ∗   + γ 1 n   u 1 n − x ∗      x n − x ∗   + d 0 n , (3.1) where d 0 n = γ 1 n u 1 n − x ∗ .Since  ∞ n=1 γ 1 n < ∞,  ∞ n=1 d 0 n < ∞. Next, we note that   x 2 n − x ∗    α 2 n   T 2 x 1 n − x ∗   + β 2 n   x n − x ∗   + γ 2 n   u 2 n − x ∗    α 2 n   x 1 n − x ∗   + β 2 n   x n − x ∗   + γ 2 n   u 2 n − x ∗   =  α 2 n + β 2 n    x n − x ∗   + α 2 n d 0 n + γ 2 n   u 2 n − x ∗      x n − x ∗   + d 1 n , (3.2) where d 1 n = α 2 n d 0 n + γ 2 n u 2 n − x ∗ .Since  ∞ n=1 d 0 n < ∞ and  ∞ n=1 γ 2 n < ∞,  ∞ n=1 d 1 n < ∞. Simi- larly, we have   x 3 n − x ∗    α 3 n   x 2 n − x ∗   + β 3 n   x n − x ∗   + γ 3 n   u 3 n − x ∗    α 3 n    x n − x ∗   + d 1 n  + β 3 n   x n − x ∗   + γ 3 n   u 3 n − x ∗      x n − x ∗   + α 3 n d 1 n + γ 3 n   u 3 n − x ∗   =   x n − x ∗   + d 2 n , (3.3) where d 2 n = α 3 n d 1 n + γ 3 n u 3 n − x ∗ ,so  ∞ n=1 d 2 n < ∞. S. Plubtieng and K. Ungchittrakool 5 By continuing the above method, there exists a nonnegative real sequence {d i−1 n } such that  ∞ n=1 d i−1 n < ∞ and   x i n − x ∗      x n − x ∗   + d i−1 n , ∀n  1, ∀i = 1,2, , N. (3.4) Thus x n+1 − x ∗ =x N n − x ∗   x n − x ∗  + d N−1 n for all n ∈ N.Hence,byLemma 2.1, lim n→∞ x n − x ∗  exists. This completes the proof.  Lemma 3.2. Let E be a uniformly convex Banach space and K anonemptyclosedconvex subset of E whichisalsoanonexpansiveretractofE.LetT 1 ,T 2 , , T N : K → E be nonex- pansive mappings. Let {x n } bethesequencedefinedby(1.1)with  ∞ n=1 γ i n < ∞ and {α i n }⊆ [ε,1 − ε] for all i = 1,2, ,N,forsomeε ∈ (0,1).If  N i =1 F(T i ) = ∅, then lim n→∞ x n − T i x n =0 for all i = 1, 2, ,N. Proof. Let x ∗ ∈  N i =1 F(T i ). Then, by Lemma 3.1,lim n→∞ x n − x ∗  exists. Let lim n→∞  x n − x ∗ =r.Ifr = 0, then by the continuity of each T i the conclusion follows. Sup- pose that r>0. Firstly, we are n ow to show that lim n→∞ T N x n − x n =0. Since {x n } and {u i n } are bounded for all i = 1,2, ,N, there exists R>0suchthatx n − x ∗ + γ i n (u i n − x n ), T i x i−1 n − x ∗ + γ i n (u i n − x n ) ∈ B R (0) for all n  1andforalli = 1,2, ,N. Using Lemma 2.2, we have   x N n − x ∗   2    α N n T N x N−1 n + β N n x n + γ N n u N n − x ∗   2 =   α N n  T N x N−1 n − x ∗ + γ N n  u N n − x n  +  1 − α N n  x n − x ∗ + γ N n  u N n − x n    2  α N n   T N x N−1 n − x ∗ + γ N n  u N n − x n    2 +  1 − α N n    x n − x ∗ + γ N n  u N n − x n    2 − W 2  α N n  g    T N x N−1 n − x n     α N n    x N−1 n − x ∗   + γ N n   u N n − x n    2 +  1 − α N n    x n − x ∗   + γ N n   u N n − x n    2 − W 2  α N n  g    T N x N−1 n − x n     α N n    x n − x ∗   + d N−2 n + γ N n   u N n − x n    2 +  1 − α N n    x n − x ∗   + d N−2 n + γ N n   u N n − x n    2 − W 2  α N n  g    T N x N−1 n − x n    =    x n − x ∗   + λ N−2 n  2 − W 2  α N n  g    T N x N−1 n − x n    , (3.5) where λ N−2 n := d N−2 n + γ N n u N n − x ∗ .Observethatε 3  W 2 (α N n )now(3.5) implies that ε 3 g(T N x N−1 n − x n )  x n − x ∗  2 −x n+1 − x ∗  2 + ρ N−2 n ,whereρ N−2 n := 2λ N−2 n x n − x ∗  2 +(λ N−2 n ) 2 .Since  ∞ n=1 d N−2 n < ∞ and  ∞ n=1 γ N−2 n < ∞,weget  ∞ n=1 ρ N−2 n < ∞. This implies that lim n→∞ g(T N x N−1 n − x n ) = 0. Since g is strictly increasing and continuous 6 Weak and strong convergence at 0, it follows that lim n→∞ T N x N−1 n − x n =0. Note that   x n − x ∗      x n − T N x N−1 n   +   T N x N−1 n − x ∗      x n − T N x N−1 n   +   x N−1 n − x ∗   , (3.6) for all n  1. Thus r = lim n→∞ x n − x ∗   liminf n→∞ x N−1 n − x ∗   limsup n→∞ x N−1 n − x ∗   r and therefore lim n→∞ x N−1 n − x ∗ =r. Using the same argument in the proof above, we have   x N−1 n − x ∗   2  α N−1 n    x N−2 n − x ∗   + γ N−1 n   u N−1 n − x ∗    2 +  1 − α N−1 n    x n − x ∗   + γ N−1 n   u N−1 n − x ∗    2 − W 2  α N−1 n  g    T N−1 x N−2 n − x n     α N−1 n    x n − x ∗   + d N−3 n + γ N−1 n   u N−1 n − x ∗    2 +  1 − α N−1 n    x n − x ∗   + d N−3 n + γ N−1 n   u N−1 n − x ∗    2 − W 2  α N−1 n  g    T N−1 x N−2 n − x n       x n − x ∗   2 + ρ N−3 n − W 2  α N−1 n  g    T N−1 x N−2 n − x n    . (3.7) This implies that ε 3 g(T N−1 x N−2 n − x n )  x n − x ∗  2 −x N−1 n − x ∗  2 + ρ N−3 n and there- fore lim n→∞ T N−1 x N−2 n − x n =0. Thus, we have   x n − T N x n      x n − T N x N−1 n   +   T N x N−1 n − T N x n      x n − T N x N−1 n   +   x N−1 n − x n   =   x n − T N x N−1 n   +   P  α N−1 n T N−1 x N−2 n + β N−1 n x n + γ N−1 n u N−1 n  − Px n      x n − T N x N−1 n   + α N−1 n   x n − T N−1 x N−2 n   + γ N−1 n   u N−1 n − x n   . (3.8) Since lim n→∞ x n − T N x N−1 n =0, lim n→∞ x n − T N−1 x N−2 n =0, and  ∞ n=1 γ N−1 n < ∞,it follows that lim n→∞ x n − T N x n =0. Similarly, by using the same argument as in the proof above, we have lim n→∞ x n − T N−2 x N−3 n =lim n→∞ x n − T N−3 x N−4 n =, ,= lim n→∞ x n −T 2 x 1 n =0. This implies that lim n→∞ x n −T N−1 x n =lim n→∞ x n − T N−2 x n = , , = lim n→∞ x n − T 3 x n =0. It remains to show that lim n→∞   x n − T 1 x n   = 0, lim n→∞   x n − T 2 x n   = 0. (3.9) S. Plubtieng and K. Ungchittrakool 7 Note that   x 1 n − x ∗   2  α 1 n    x n − x ∗   + γ 1 n   u 1 n − x ∗    2 +  1 − α 1 n    x n − x ∗   + γ 1 n   u 1 n − x ∗    2 − W 2  α 1 n  g    T 1 x n − x n    =    x n − x ∗   + γ 1 n   u 1 n − x ∗    2 − W 2  α 1 n  g    T 1 x n − x n    . (3.10) Thus, we have ε 3 g(T 1 x n − x n )  (x n − x ∗  + γ 1 n u 1 n − x ∗ ) 2 −x 1 n − x ∗  2 and there- fore lim n→∞ T 1 x n − x n =0. Since x n − T 2 x n   x n − T 2 x 1 n  + α 1 n T 1 x n − x n  + γ 1 n u 1 n − x n , it implies that lim n→∞ T 2 x n − x n =0. Therefore lim n→∞ T i x n − x n =0forall i = 1,2, ,N.  Theorem 3.3. Let E be a uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E.LetT 1 ,T 2 , , T N : K → E be nonexpansive mappings which are satisfying condition (B).Let {x n } bethesequencedefined by (1.1)with  ∞ n=1 γ i n < ∞ and {α i n }⊆[ε,1 − ε] for all i = 1,2, ,N for some ε ∈ (0,1).If F : =  N i =1 F(T i ) = ∅, then {x n } converges strongly to a common fixed point in F. Proof. By Lemma 3.2,lim n→∞ T i x n − x n =0foralli = 1,2, , N. Now by condition (B), f (d(x n ,F))  M n := max 1iN {T i x n − x n } for all n ∈ N.Hencelim n→∞ f (d(x n ,F)) = 0. Since f is a nondecreasing function and f (0) = 0, therefore lim n→∞ d(x n ,F) = 0. Now we can choose a subsequence {x n j } of {x n } and a sequence {y j }∈F such that x n j − y j  < 2 − j . By the following method of the proof of Tan and Xu [15], we get that {y j } is a Cauchy sequence in F and so it converges. Let y j → y.SinceF is closed, therefore y ∈ F and then x n j → y.ByLemma 3.1,lim n→∞ x n − x ∗  exists for all x ∗ ∈ F, x n → y ∈ F.  For N = 2, T 1 = T 2 ≡ T, β n = α 1 n , α n = α 2 n ,andγ 1 n = γ 2 n ≡ 0inTheorem 3.3,weobtain the following results. Corollary 3.4 (see [12, Theorem 3.6]). Let E be a real uniformly convex Banach space and K a nonempty closed convex subset of E whichisalsoanonexpansiveretractofE.Let T : K → E be a nonexpansive mapping with F(T) = ∅.Let{α n } and {β n } be sequences in [ε,1 − ε] for some ε ∈ (0,1).Fromarbitraryx 1 ∈ K, de fine the sequence {x n } by the recursion (1.2). Suppose T satisfies condition (A 1 ). Then {x n } converges strongly to some fixed point of T. When N = 2, S = T 1 , T = T 2 : C → C,andy n = x 1 n in Theorem 3.3,weobtainstrong convergence theorem as follows. Corollary 3.5. Let E be a uniformly convex Banach space and let C be a none mpty closed convex subset of E whichisalsoanonexpansiveretractofE.LetS, T be nonexpan- sive mappings of C into itself satisfying condition (A 2 ),andlet{x n } be sequence defined by (1.3)with  ∞ n=1 γ 1 n < ∞,  ∞ n=1 γ 2 n < ∞ and 0 <δ α 1 n , α 2 n  1 − δ<1 for all n ∈ N.If F : = F(S) ∩ F(T) = ∅, then {x n } converges strongly to a common fixed point of S and T. 8 Weak and strong convergence Theorem 3.6. Let E be a uniformly convex Banach space and let K be a nonempty closed convex subset of E which is also a nonexpansive retract of E.LetT 1 ,T 2 , , T N : K → E be nonexpansive mappings. Suppose that one of the mappings in {T i : i = 1,2, ,N} is semi- compact. Let {x n } bethesequencedefinedby(1.1)with  ∞ n=1 γ i n < ∞ and {α i n }⊆[ε,1− ε] for all i = 1,2, ,N for some ε ∈ (0,1).IfF :=  N i =1 F(T i ) = ∅, then {x n } converges strongly to a common fixed point in F. Proof. Suppose that T i 0 is semicompact for some i 0 = 1,2, ,N.ByLemma 3.1,wehave lim n→∞ x n − T i 0 x n =0. So there exists a subsequence {x n j } of {x n } such that x n j → x ∗ ∈ K as j →∞.NowLemma 3.2 guarantees that lim j→∞ x n j − T l x n j =0foralll = 1,2, , N and so x ∗ − T l x ∗ =0foralll = 1,2, , N. This implies that x ∗ ∈ F.By Lemma 3.1,lim n→∞ x n − x ∗  exists and then lim n→∞ x n − x ∗ =lim j→∞ x n j − x ∗ =0. This completes the proof.  Theorem 3.7. Let E be a uniformly convex Banach space satisfying the Opial’s condition and K a nonempty closed convex subset of E whichisalsoanonexpansiveretractofE.Let T 1 ,T 2 , , T N : K → E be nonexpansive mappings and let {x n } be a sequence defined by (1.1) with  ∞ n=1 γ i n < ∞ and {α i n }⊆[ε,1 − ε] for all i = 1,2, ,N for some ε ∈ (0,1).IfF :=  N i =1 F(T i ) = ∅, then {x n } converges weakly to a common fixed point in F. Proof. Let x ∗ ∈ F.ThenasprovedinLemma 3.1,lim x→∞ x n − x ∗  exists. Now we prove that {x n } has a unique weak subsequential limit in F. To prove this, let x n i  z 1 and x n j  z 2 for some subsequences {x n i }, {x n j } of {x n }.ByLemma 3.2, lim i→∞   x n i − T k x n i   = 0 = lim j→∞   x n j − T k x n j   (3.11) for all k = 1,2, ,N and by Lemma 2.4 insures that I − T k aredemiclosedatzerofor all k = 1,2, ,N. T herefore we o btain T k z 1 = z 1 and T k z 2 = z 2 for all k = 1,2, ,N. Then z 1 ,z 2 ∈ F. Next, we prove the uniqueness. Suppose that z 1 = z 2 , then by the Opial’s condition lim n→∞ x n − z 1 =lim i→∞ x n i − z 1  < lim i→∞ x n i − z 2 =lim j→∞ x n j − z 2  < lim j→∞ x n j − z 1 =lim n→∞ x n − z 1 . This is a contradiction. Hence {x n } converges weakly to a point in F.  Lemma 3.8. Let E be a real uniformly c onvex Banach space and K a nonempty closed convex subset of E whichisalsoanonexpansiveretractofE.LetT 1 ,T 2 , , T N : K → E be nonexpan- sive mappings. From arbitrary x 1 ∈ K, define the sequence {x n } by the recursion (1.1)with for each i = 1,2, ,N,  ∞ n=1 γ i n < ∞.IfF :=  N i =1 F(T i ) = ∅,thenforallu,v ∈ F, the limit lim n→∞   tx n +(1− t)u − v   (3.12) exists for all t ∈ [0,1]. Proof. By Lemma 3.1,wehavelim n→∞ x n − x ∗  exists for all x ∗ ∈ F. This implies that {x n } is bounded. Observe that there exists R>0suchthat{x n }⊂C := B R (0) ∩ K,and hence C is a nonempty closed convex bounded subset of E.Leta n (t):=tx n +(1−t)u−v. Then lim n→∞ a n (0) =u − v,andfromLemma 3.1,lim n→∞ a n (1) = lim n→∞ x n − v ex- ists. Without loss of generality, we may assume that lim n→∞ x n − u=r>0and S. Plubtieng and K. Ungchittrakool 9 t ∈ (0,1). For any n  1andforalli = 1,2, ,N,wedefineA i n : C → C by A 1 n := P  α 1 n T 1 + β 1 n I + γ 1 n u 1 n  , A 2 n := P  α 2 n T 2 A 1 n + β 2 n I + γ 2 n u 2 n  , . . . . . . . . . A N n := P  α N n T N A N−1 n + β N n I + γ N n u N n  . (3.13) Thus, for all x, y ∈ K,wehaveA i n x − A i n y  α i n A i−1 n x − A i−1 n y + β i n x − y for all i = 2, ,N,andA 1 n x − A 1 n y  α 1 n x − y + β 1 n x − y. This implies that   A N n x − A N n y    x − y. (3.14) Set S n,m := A N n+m −1 A N n+m −2 ···A N n , m  1, and b n,m :=S n,m (tx n +(1− t)u) − (tS n,m x n + (1 −t)S n,m u). It easy to see that A N n x n =x n+1 , S n,m x n =x n+m ,andS n,m x−S n,m y x− y. We show first that, for any x ∗ ∈ F, S n,m x ∗ − x ∗ →0 uniformly for all m  1asn → ∞ . Indeed, for any x ∗ ∈ F,wehave   A i n x ∗ − x ∗    α i n   A i−1 n x ∗ − x ∗   + γ i n   u i n − x ∗   (3.15) for all i = 2, ,N,andA 1 n x ∗ − x ∗   γ 1 n u 1 n − x ∗ . Therefore   A N n x ∗ − x ∗    σ 2 n γ 1 n   u 1 n − x ∗   + σ 3 n γ 2 n   u 2 n − x ∗   + ···+ σ N n γ N−1 n   u N−1 n − x ∗   + γ N n   u N n − x ∗    M N  i=1 γ i n , (3.16) for all n  1, where M = max{sup n1 {u 1 n − x ∗ }, ,sup n1 {u N n − x ∗ }} and σ k n =  N i =k α i n .Hence   S n,m x ∗ − x ∗      A N n+m −1 A N n+m −2 ···A N n x ∗ − A N n+m −1 A N n+m −2 ···A N n+1 x ∗   +   A N n+m −1 A N n+m −2 ···A N n+1 x ∗ − A N n+m −1 A N n+m −2 ···A N n+2 x ∗   . . . . . . +   A N n+m −1 x ∗ − x ∗      A N n x ∗ − x ∗   +   A N n+1 x ∗ − x ∗   + ···+   A N n+m −1 x ∗ − x ∗    M N  i=1  γ i n + γ i n+1 + ···+ γ i n+m −1   M N  i=1 ∞  k=n γ i k := δ x ∗ n . (3.17) 10 Weak and strong convergence Since  ∞ n=1 γ i n <∞,foralli=1,2, , N,wehaveδ x ∗ n →0asn→∞ and hence S n,m x ∗ −x ∗ →0 as n →∞.Observethat a n+m (t) =   tS n,m x n +(1− t)u − v      tS n,m x n +(1− t)u − S n,m  tx n +(1− t)u    +   S n,m  tx n +(1− t)u  − v   =   tS n,m x n +(1− t)S n,m u − S n,m  tx n +(1− t)u  +(1− t)  u − S n,m u    +   S n,m  tx n +(1− t)u  − v    b n,m +   S n,m  tx n +(1− t)u  − v   +(1− t)   u − S n,m u    b n,m +   S n,m  tx n +(1− t)u  − S n,m v   +   S n,m v − v   +(1− t)   u − S n,m u    b n,m + a n (t)+   S n,m v − v   +(1− t)   u − S n,m u    b n,m + a n (t)+δ v n +(1− t)δ u n . (3.18) By using [2, Theorem 2.3], we have b n,m  ϕ −1    x n − u   −   S n,m x n − S n,m u    = ϕ −1    x n − u   −   x n+m − u + u − S n,m u     ϕ −1    x n − u   −    x n+m − u   −   S n,m u − u    , (3.19) and so the sequence {b n,m } convergesuniformlyto0asn →∞for all m  1. Thus, fixing n and letting m →∞in (3.19), we have limsup m→∞ a n+m (t)  ϕ −1    x n − u   −  lim m→∞   x m − u   − δ u n  + a n (t)+δ v n +(1− t)δ u n (3.20) and again letting n →∞, limsup n→∞ a n (t)  ϕ −1 (0) + liminf n→∞ a n (t)+0+0= liminf n→∞ a n (t). (3.21) This completes the proof.  Theorem 3.9. Let E be a real uniformly convex Banach space such that its dual E ∗ has the Kaded-Klee property and K anonemptyclosedconvexsubsetofE which is also a nonexpan- sive ret ract of E.LetT 1 ,T 2 , , T N : K → E be nonexpansive mappings with F :=  N i =1 F(T i )= ∅.Fromarbitraryx 1 ∈ K, define the sequence {x n } by the recursion (1.1) with for each i = 1,2, ,N,  ∞ n=1 γ i n < ∞ and α i n ∈ [ε,1 − ε] for some ε ∈ (0,1). Then {x n } converges weakly to some fixed point of T. Proof. Lemma 3.1 guarantees that {x n } is bounded. Since E is reflexive, there exists a subsequence {x n j } of {x n } converging w eakly to some x ∗ ∈ K.ByLemma 3.2,wehave [...]... Khan and H Fukhar-ud-din, Weak and strong convergence of a scheme with errors for two nonexpansive mappings, Nonlinear Analysis 61 (2005), no 8, 1295–1301 [7] W R Mann, Mean value methods in iteration, Proceedings of the American Mathematical Society 4 (1953), no 3, 506–510 [8] S.-Y Matsushita and D Kuroiwa, Strong convergence of averaging iterations of nonexpansive nonself-mappings, Journal of Mathematical... Iterative construction of fixed points of asymptotically nonexpansive mappings, Journal of Mathematical Analysis and Applications 158 (1991), no 2, 407–413 [12] N Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Analysis 61 (2005), no 6, 1031–1039 [13] W Takahashi and G.-E Kim, Strong convergence of approximants to fixed points of nonexpansive nonself-mappings... Takahashi and T Tamura, Convergence theorems for a pair of nonexpansive mappings, Journal of Convex Analysis 5 (1998), no 1, 45–56 [15] K.-K Tan and H K Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, Journal of Mathematical Analysis and Applications 178 (1993), no 2, 301–308 , Fixed point iteration processes for asymptotically nonexpansive mappings, Proceedings of. .. iteration method, Proceedings of the American Mathematical Society 44 (1974), no 1, 147–150 [4] J S Jung and S S Kim, Strong convergence theorems for nonexpansive nonself-mappings in Banach spaces, Nonlinear Analysis 33 (1998), no 3, 321–329 [5] W Kaczor, Weak convergence of almost orbits of asymptotically nonexpansive commutative semigroups, Journal of Mathematical Analysis and Applications 272 (2002),... Mathematical Analysis and Applications 294 (2004), no 1, 206– 214 [9] Z Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bulletin of the American Mathematical Society 73 (1967), 591–597 [10] B E Rhoades, Fixed point iterations for certain nonlinear mappings, Journal of Mathematical Analysis and Applications 183 (1994), no 1, 118–120 12 Weak and strong convergence. .. 733–739 [17] H K Xu, Inequalities in Banach spaces with applications, Nonlinear Analysis 16 (1991), no 12, 1127–1138 [18] Y Xu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive operator equations, Journal of Mathematical Analysis and Applications 224 (1998), no 1, 91–101 [19] H K Xu and X M Yin, Strong convergence theorems for nonexpansive non-self-mappings, Nonlinear Analysis... result, ωw (xn ) ∩ F is a singleton, and so {xn } converges weakly to some fixed point of T Corollary 3.10 (see [12, Theorem 3.5]) Let E be a real uniformly convex Banach space such that its dual E∗ has the Kadec-Klee property and K a nonempty closed convex subset of E which is also a nonexpansive retract of E Let T : K → E be a nonexpansive mapping with F(T) = ∅ Let {αn } and {βn } be sequences in [ε,1 −... Golden Jubilee Program under Grant PHD/0086/2547, Thailand References [1] F E Browder, Semicontractive and semiaccretive nonlinear mappings in Banach spaces, Bulletin of the American Mathematical Society 74 (1968), 660–665 [2] J Garc´a Falset, W Kaczor, T Kuczumow, and S Reich, Weak convergence theorems for asymptotı ically nonexpansive mappings and semigroups, Nonlinear Analysis 43 (2001), no 3, 377–401... non-self-mappings, Nonlinear Analysis 24 (1995), no 2, 223–228 S Plubtieng: Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand E-mail address: somyotp@nu.ac.th K Ungchittrakool: Department of Mathematics, Faculty of Science, Naresuan University, Phitsanulok 65000, Thailand E-mail address: g47060127@nu.ac.th ... recursion (1.2) Then {xn } converges weakly to some fixed point of T Acknowledgments The authors would like to thank the Faculty of Science, Naresuan University, and the Thailand Research Fund for financial support Moreover, the authors thank the referees for their valuable suggestions to improve this manuscript This work is supported by Thailand Research Fund K Ungchittrakool is also supported by the Royal . WEAK AND STRONG CONVERGENCE OF FINITE FAMILY WITH ERRORS OF NONEXPANSIVE NONSELF-MAPPINGS S. PLUBTIENG AND K. UNGCHITTRAKOOL Received 27 September 2005;. concerned with the study of a multistep iterative scheme with errors involving a finite family of nonexpansive nonself-mappings. We approximate the common fixed points of a finite family of nonexpansive. Ishikawa iterative schemes with errors for two mappings and gave weak and strong convergence theorems. On the other hand, iterative techniques for approximat- ing fixed p oints of nonexpansive nonself-mappings