RESEARC H Open Access Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces Lanping Zhu, Qianglian Huang * and Xiaoru Chen * Correspondence: qlhmath@yahoo.com.cn College of Mathematics, Yangzhou University, Yangzhou 225002, China Abstract In this article, we introduce a new three-step iterative scheme for the mappings which are asymptotically nonexpansive in the intermediate sense in Banach spaces. Weak convergence theorem is established for this three-step iterative scheme in a uniformly convex Banach space that satisfies Opial’s condition or whose dual space has the Kadec-Klee property. Furthermore, we give an example of the nonself mapping which is asymptotically nonexpansive in the intermediate sense but not asymptotically nonexpansive. The results obtained in this article extend and improve many recent results in this area. AMS classification: 47H10; 47H09; 46B20. Keywords: asymptotically nonexpansive in the intermediate sense non-self mapping, Kadec-Klee property, Opial?’?s condition, common fixed point 1 Introduction Fixed- point iterations process for nonexpansive and asymptotically nonexpansive map- pings in Banach spaces have been studied extensively by various authors [1-13]. In 1991, Schu [4] considered the following modified Mann iteration process for an asymptotically nonexpansive map T on C and a sequence {a n } in [0, 1]: x 1 ∈ C, x n+1 = α n x n + ( 1 − α n ) T n x n , n ≥ 1 . (1:1) Since then, Schu’s iteration process (1.1) has been widely used to approximate fixed points of asymptotically nonexpansive mappings in Hilbert spaces or Banach spaces [7,8,10-13]. Noor, in 2000, introduced a three-step iterative scheme and studied the approximate solutions of variational inclusion in Hilbert spaces [6]. Later, Xu and Noor [7], Cho et al. [8], Suantai [9], Plubtieng et al. [12] studied t he convergence of the three-step iterations for asymptotically nonexpansive mappings in a uniformly con- vex Banach space which satisfies Opial’ s condition or whose norm is Fréchet differentiable. In most of these articles, the operator T remains a self-mapping of a nonempty closed convex subset C of a uniformly convex Banach space X. If, however, the domain of T, D(T), is a proper subset of X (and this is the case in several applications), and T maps D(T)intoX, then the iterative sequence {x n } may fail to be well defined. One method that has been used to overcome this is to introduce a retraction P. A subset C Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 © 2011 Zhu et al; licensee Springer. This is a n Open Access article distributed under the terms of t he 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 . of X is said to be retract if there exists continuous mapping P : X ® C such that Px = x for all x Î C and P is said to be a retraction. Recent results on approximation of fixed points of nonexp ansive or asymptotically nonexpansive nonself mappings can be found in [14-19] and the references cited therein. For example, in 2003, Chidume et al. [16] introduced the following modified Mann iteration process and got the conver- gence theorems for asymptotically nonexpansive nonself-mapping: x 1 ∈ C, x n+1 = P[α n x n + ( 1 − α n ) T ( PT ) n−1 x n ], n ≥ 1 . (1:2) Recently, Thianwan [18] generalized the iteration process (1.2) as follows: x 1 Î C, x n+1 = P[α n y n +(1− α n )T 1 (PT 1 ) n−1 y n ]; y n = P[β n x n + ( 1 − β n ) T 2 ( PT 2 ) n−1 x n ] . (1:3) Obviously, if b n =1foralln ≥ 1, then (1.3) reduces to (1.2). Thianwan [18] proved the weak convergence theorem of the iteration process (1.3) in uniformly convex Banach spaces that satisfy Opial’s condition. The concept of asymptotical ly nonexpansive in the intermediate sense nonself map- pings was introduced by Chidume et al. [20] as an important generalization of asymp- totically nonexpansive in the intermediate sense self-mappings. Definition 1.1 Let C be a nonempty subset of a Banach space X. Let P : X ® Cbea nonexpansive retraction of X onto C. A nonself mapping T : C ® X is called asymptoti- cally nonexpansive in the intermediate sense if T is continuous and the following inequality holds: lim sup n→+∞ sup x, y ∈C ( T(PT) n−1 x − T(PT) n−1 y −x − y ) ≤ 0 . It should be noted that in [20- 22], the asymptotically nonexpansive in the intermedi- ate sense mapping is required to be uniformly continuous. In this article, we assume the continuity of T instead of un iform continuity. Chidume et al. [ 20] gave the w eak convergence theorem for u niforml y continuous nonself mapping which is asymptoti- cally nonexpansive in the intermediate sense in uniformly convex Banach space whose dual space has the Kadec-Klee property. Inspired and motivated by [16,18,20], we investigate the weak convergence theo rem of three-step iteration process for co ntinuous nonself mappings which are asymptoti- cally nonexpansive in the intermediate sense in this article. Since the asymptotically nonexpansive in the intermediate sense mappings are non-Lipschitzian and Bruck ’s Lemma [23] do not extend beyond Lipschitzian mappi ngs, new tec hniques are needed for this more general case. Utilizing the technique of the modulus of convexity and a new demiclosed principle for nonself-maps of Kazor [24], we establish the we ak con- vergence theorem of the three-step iterative scheme in a uniformly convex Banach space that satisfies Opial’s condition or whose dual space has the Kadec-Klee property, which extends an d improves the recently announced ones in [4,16,18-20]. It should be noted that our theorems are new even in the case that the space has a Fréchet differ- entiable norm. In the end, to illustrate our theorem, we give a nonself mapping which is asymptotically nonexpansive in the intermediate sense but not asymptotical ly nonexpansive. Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 2 of 13 2 Preliminaries Let X be a Banach space and X* be its dual, then the value of x* Î X*atx Î X will be denoted by 〈x, x*〉 and we associate the set J( x ) = {x ∗ ∈ X ∗ : x, x ∗  = ||x|| 2 = ||x ∗ || 2 } . It follows from the Hahn-Bana ch theorem that J(x) ≠ ∅ for any x Î X.Thenthe multi-valued operator J : X ↦ X* is called the normalized duality mapping of X.Recall that a Banach space X is said to be uniformly convex if for each ε Î [0, 2], the modu- lus of convexity of X defined by δ(ε)=inf{1 − 1 2  x + y : x ≤ 1,  y ≤ 1,  x − y ≥ ε} , satisfies the inequality δ(ε) >0 for all ε >0. Note that every closed convex subset of a uni- formly convex Banach space is a retract. We say that X has the Kadec-Klee property if for every sequence {x n } ⊂ X, whenever x n ⇀ x with ||x n || ® ||x||, it follows that x n ® x.We would li ke to remark that a reflexive Banach space X with a Fréchet differentiable norm implies that its dual X* has Kadec-Klee property, while the converse implication fails [25]. Recall that a Banach space X is said to satisfies Opial’s condition if x n ⇀ x and x ≠ y implies that lim sup n →+∞  x n − x < lim sup n →+∞  x n − y  . The following lemmas are needed to prove our main results in next section. Lemma 2.1 [5]Let the nonnegative number sequences {c n } and {w n } satisfy c n +1 ≤ c n + w n , n ∈ N If +∞  n=1 w n < + ∞ , then lim n →+∞ c n exists. Lemma 2.2 [4] Suppose that X is a uniformly convex Banach space and for al l posi- tive integers n,0<p≤ t n ≤ q<1. If {x n } and {y n } are two sequences of X such that lim sup n →+∞  y 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 . Lemma 2.3 [3]Let X be a uniformly convex Banach space. If ||x|| ≤ 1, ||y|| ≤ 1 and ||x - y|| ≥ ε >0, then for all l Î [0, 1],  λx + ( 1 − λ ) y ≤ 1 − 2λ ( 1 − λ ) δ ( ε ). Lemma 2.4 [26]Let X be a Banach space and J be the normalized duali ty mapping. Then for given x, y Î X and j(x + y) Î J(x + y), we have  x + y 2 ≤ x 2 +2y, j ( x + y )  . Lemma 2.5 (De miclosed principle for nonself-map [ 24]) Let C be a nonempty closed convex subset of a uniformly convex Banach space X and T : C ® X be a nonself map- ping which is continuous and asymptotically nonexpansive in the interme diate sense. If {x n } is a sequence in C converging weakly to x and Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 3 of 13 lim k→+∞ lim sup n →+∞  x n − T(PT) k−1 x n =0 , then x Î F(T), i.e., Tx = x. 3 Main results Let C be a nonempty closed convex subset of a uniformly convex Banach space X and P : X ® C beanonexpansiveretractionfromX onto C.LetT 1 , T 2 , T 3 : C ® X be three continuous nonself mappings which are asymptotically nonexpansive in the inter- mediate sense. Suppose that r n =max{0, sup x, y ∈C;i=1,2,3.  T i (PT i ) n−1 x − T i (PT i ) n−1 y −x − y } , then r n ≥ 0, lim n →+∞ r n = 0 and for all x, y Î C and n Î N,  T i ( PT i ) n−1 x − T i ( PT i ) n−1 y −x − y ≤ r n , i =1,2,3 . For a given x 1 Î C, we define the sequence {x n } ⊂ C by x n+1 = P[α (1) n z n +(1− α (1) n )T 1 (PT 1 ) n−1 z n ]; z n = P[α (2) n y n +(1− α (2) n )T 2 (PT 2 ) n−1 y n ]; y n = P[α (3) n x n + ( 1 − α (3) n ) T 3 ( PT 3 ) n−1 x n ] . (3:1) where { α ( i ) n } is in 0[1] with 0 < p ≤ α (i) n ≤ q < 1 , i =1,2,3. We also assume that the sequence {r n } satisfies +∞  n =1 r n < + ∞ andthesetofcommon fixed points of {T i } 3 i = 1 is nonempty, i.e., F = ∩ 3 i =1 F( T i )={x ∈ C : T 1 x = T 2 x = T 3 x = x} = ∅ . Lemma 3.1 lim n →+∞  x n − f = lim n →+∞  y n − f = lim n →+∞  z n − f = r (3:2) exists for all f Î F. Proof. For all f Î F,  y n − f = P [α (3) n x n +(1− α (3) n )T 3 (PT 3 ) n−1 x n ] − f  ≤ [α (3) n x n +(1− α (3) n )T 3 (PT 3 ) n−1 x n ] − f  ≤ α (3) n  x n − f  +(1 − α (3) n )  T 3 (PT 3 ) n−1 x n − f  = x n − f  +r n . Hence  z n − f = P [α (2) n y n +(1− α (2) n )T 2 (PT 2 ) n−1 y n ] − f  ≤ [α (2) n y n +(1− α (2) n )T 2 (PT 2 ) n−1 y n ] − f  ≤ y n − f  +r n ≤ x n − f  +2r n . (3:3) Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 4 of 13 Thus  x n+1 − f = P [α (1) n z n +(1− α (1) n )T 1 (PT 1 ) n−1 z n ] − f  ≤ [α (1) n z n +(1− α (1) n )T 1 (PT 1 ) n−1 z n ] − f  ≤ z n − f  +r n ≤ x n − f  +3r n . (3:4) Put w n =3r n , then we can obtain +∞  n =1 w n < + ∞ and  x n+1 − f ≤ x n − f  +w n . By Lemma 2.1, we can conclude that lim n →+∞  x n − f = r exists. Combining it with (3.4), we have lim n →+∞  z n − f = r . Hence by (3.3), we get lim n →+∞  y n − f = r . This completes the proof. Lemma 3.2 lim k→+∞ lim sup n →+∞  x n − T i (PT i ) k −1 x n =0, i = 1,2,3 . Proof. By (3.2) and (3.4), we can get r = lim n→+∞  [α (1) n z n +(1− α (1) n )T 1 (PT 1 ) n−1 z n ] − f  = lim n →+ ∞  (1 − α (1) n )[T 1 (PT 1 ) n−1 z n − f]+α (1) n (z n − f)  Then it follows from Lemma 2.2 and lim sup n→+∞  T 1 (PT 1 ) n−1 z n − f ≤ r that lim n →+ ∞  T 1 (PT 1 ) n−1 z n − z n =0 . (3:5) According to (3.3), we have r = lim n→+∞  [α (2) n y n +(1− α (2) n )T 2 (PT 2 ) n−1 y n ] − f  = lim n →+ ∞  (1 − α (2) n )[T 2 (PT 2 ) n−1 y n − f]+α (2) n (y n − f)  Noting lim sup n →+∞  T 2 (PT 2 ) n−1 y n − f ≤ r , by Lemma 2.2 again, we can get lim n →+∞  T 2 (PT 2 ) n−1 y n − y n =0 . (3:6) Similarly, we can obtain lim n →+ ∞  T 3 (PT 3 ) n−1 x n − x n =0 . (3:7) Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 5 of 13 Hence, it follows from  y n − x n  = P[α (3) n x n +(1− α (3) n )T 3 (PT 3 ) n−1 x n ] − x n  ≤ [α (3) n x n +(1− α (3) n )T 3 (PT 3 ) n−1 x n ] − x n  ≤ T 3 ( PT 3 ) n−1 x n − x n  . that lim n →+∞  y n − x n = 0 . Also, we can see  z n − y n  = P[α ( 2 ) n y n +(1− α ( 2 ) n )T 2 (PT 2 ) n−1 y n ] − y n  ≤ [α (2) n y n +(1− α (2) n )T 2 (PT 2 ) n−1 y n ] − y n  ≤ T 2 ( PT 2 ) n−1 y n − y n  and  x n+1 − z n  = P[α (1) n z n +(1− α (1) n )T 1 (PT 1 ) n−1 z n ] − z n  ≤ [α (1) n z n +(1− α (1) n )T 1 (PT 1 ) n−1 z n ] − z n  ≤ T 1 ( PT 1 ) n−1 z n − z n  . It follows from (3.5) and (3.6) that lim n →+ ∞  x n+1 − z n = lim n →+ ∞  z n − y n =0 . Hence lim n →+∞  x n+1 − x n = lim n →+∞  z n − x n = 0 . Thus for any fixed k Î N, lim n →+∞  x n+k − x n =0 . Noting (3.7) and  x n − T 3 (PT 3 ) k−1 x n  ≤ x n − x n+k  +  x n+k − T 3 (PT 3 ) n+k−1 x n+k  +  T 3 (PT 3 ) n+k−1 x n+ k −T 3 (PT 3 ) n+k−1 x n  +  T 3 (PT 3 ) n+k−1 x n − T 3 (PT 3 ) k−1 x n  ≤ x n − x n+k  +  x n+k − T 3 (PT 3 ) n+k−1 x n+k  +  x n+k − x n  +r n+k +  T 3 ( PT 3 ) n−1 x n − x n  +r k we have lim sup n →+∞  x n − T 3 (PT 3 ) k−1 x n ≤ r k , which implies lim k→+∞ lim sup n →+∞  x n − T 3 (PT 3 ) k−1 x n =0 . Combining (3.6) with  T 2 (PT 2 ) n−1 x n − x n  ≤ T 2 (PT 2 ) n−1 x n − T 2 (PT 2 ) n−1 y n  +  T 2 (PT 2 ) n−1 y n − y n  +  y n − x n  ≤ 2  x n − y n  +  T 2 ( PT 2 ) n−1 y n − y n  +r n , we can see lim n →+∞  T 2 (PT 2 ) n− 1 x n − x n = 0 . Thus  x n − T 2 (PT 2 ) k −1 x n  ≤ x n − x n+k  +  x n+k − T 2 (PT 2 ) n+k−1 x n+k  +  T 2 (PT 2 ) n+k−1 x n+ k −T 2 (PT 2 ) n+k−1 x n  +  T 2 (PT 2 ) n+k−1 x n − T 2 (PT 2 ) k−1 x n  ≤ x n − x n+k  +  x n+k − T 2 (PT 2 ) n+k−1 x n+k  +  x n+k − x n  +r n+k +  T 2 ( PT 2 ) n−1 x n − x n  +r k Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 6 of 13 which implies lim k→+∞ lim sup n →+∞  x n − T 2 (PT 2 ) k−1 x n =0 . Combining (3.5) with  T 1 (PT 1 ) n−1 x n − x n  ≤ T 1 (PT 1 ) n−1 x n − T 1 (PT 1 ) n−1 z n  +  T 1 (PT 1 ) n−1 z n − z n  +  x n − x n  ≤ 2  x n − z n  +  T 1 ( PT 1 ) n−1 z n − z n  +r n , we can see lim n →+∞  T 1 (PT 1 ) n−1 x n − x n = 0 . Thus  x n − T 1 (PT 1 ) k−1 x n  ≤ x n − x n+k  +  x n+k − T 1 (PT 1 ) n+k−1 x n+k  +  T 1 (PT 1 ) n+k−1 x n+ k −T 1 (PT 1 ) n+k−1 x n  +  T 1 (PT 1 ) n+k−1 x n − T 1 (PT 1 ) k−1 x n  ≤ x n − x n+k  +  x n+k − T 1 (PT 1 ) n+k−1 x n+k  +  x n+k − x n  +r n+k +  T 1 ( PT 1 ) n−1 x n − x n  +r k which implies lim k→+∞ lim sup n →+∞  x n − T 1 (PT 1 ) k−1 x n =0 . This completes the proof. Define the operator W n : C ® C by W n x = P[α ( 1 ) n x (1) +(1− α ( 1 ) n )T 1 (PT 1 ) n−1 x (1) ] ; x (1) = P[α (2) n x (2) +(1− α (2) n )T 2 (PT 2 ) n−1 x (2) ] ; x (2) = P[α (3) n x + ( 1 − α (3) n ) T 3 ( PT 3 ) n−1 x], where x Î C. Then by (3.1), x n+1 = W n x n and for all x, y Î C, we have  x (2) − y (2) ≤α (3) n  x − y  +(1 − α (3) n )  T 3 (PT 3 ) n−1 x − T 3 (PT 3 ) n−1 y  ≤ x − y  +r n ,  x (1) − y (1) ≤x (2) − y (2)  +r n ≤ x − y  +2r n and  W n x − W n y  ≤  x − y  +3r n =  x − y  +w n . For any f Î F, we get W n f = f. Set S n , m = W n+m−1 W n+m−2 ···W n+1 W n : C → C , then x n+m = S n,m x n and for all f Î F, S n,m f = f. Note that for any x, y Î C,  S n,m x − S n,m y ≤ x − y  + ( w n + ···+ w n+m−1 ). (3:8) Lemma 3.3 Let f, g Î F and l Î [0, 1], then h(λ) = lim n →+∞  λx n +(1− λ)f − g  exists. Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 7 of 13 Proof. It follows from Lemma 3.1 that lim n →+∞  x n − f  = r exists. If l =0,1orr =0, then the c onclusion holds. Assume that r>0andl Î (0, 1), then for any ε >0, there exists d>0(d<ε) such that (r + d)[1 − 2λ(1 − λ)δ( ε r + d )] < r − d , (3:9) where δ(·) is the modulus of convexity of the norm. Hence there exists a positive integer n 0 such that for all n>n 0 , r − d 4 ≤ x n − f ≤ r + d 4 (3:10) and +∞  i = n w i ≤ λ(1 − λ) d 4 < ε 4 (3:11) Now we claim that for all n>n 0 ,  S n,m [λx n + ( 1 − λ ) f ] − [λS n,m x n + ( 1 − λ ) f ] ≤ ε. ∀m ∈ N Otherwise, we can suppose that there are some n>n 0 and some m Î N such that  S n,m [λx n + ( 1 − λ ) f ] − [λS n,m x n + ( 1 − λ ) f ] ≥ ε . Put z = lx n +(1-l)f, x =(1-l)(S n,m z - f ), and y = l(S n,m x n - S n,m z), then by (3.8), (3.10), and (3.11), we have  x  =(1− λ)  S n,m z − f  ≤ (1 − λ)[ z − f  +(w n+m−1 + ···+ w n+1 + w n )] ≤ λ(1 − λ)( x n − f  + d 4 ) ≤ λ(1 − λ)(r + d),  y  = λ  S n,m x n − S n,m z  ≤ λ[ x n − z  +(w n+m−1 + ···+ w n+1 + w n )] ≤ λ(1 − λ)( x n − f  + d 4 ) ≤ λ(1 − λ)(r + d),  x − y  = S n,m [λx n + ( 1 − λ ) f ] − [λS n,m x n + ( 1 − λ ) f ] ≥ ε and λx + ( 1 − λ ) y = λ ( 1 − λ )( S n,m x n − f ). So by Lemma 2.3, we get λ(1 − λ)  S n,m x n − f  = λx +(1− λ)y  ≤ λ(1 − λ)(r + d)[1 − 2λ(1 − λ)δ( ε λ(1 − λ)(r + d) ) ] ≤ λ(1 − λ)(r + d)[1 − 2λ(1 − λ)δ( ε r + d )] Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 8 of 13 and then by (3.10), r − d ≤ x n+m − f = S n,m x n − f  ≤ ( r + d)[1 − 2λ(1 − λ)δ( ε r + d )] , which contradicts (3.9). Thus we can conclude that for all n>n 0 ,  S n,m [λx n + ( 1 − λ ) f ] − [λS n,m x n + ( 1 − λ ) f ] ≤ ε, ∀m ∈ N . Hence by (3.11), for all n>n 0 ,  λx n+m +(1− λ)f − g  = λS n,m x n +(1− λ)f − g  ≤ [λS n,m x n +(1− λ)f ] − S n,m [λx n +(1− λ)f ]  +  S n,m [λx n +(1− λ)f ] − g  ≤ ε+  λx n +(1− λ)f − g  +(w n+m−1 + ···+ w n+1 + w n ) ≤ 2ε+  λx n + ( 1 − λ ) f − g  . For any fixed n>n 0 , we can take the limsup for m and obtain lim sup m →+∞  λx m +(1− λ)f − g ≤ λx n +(1− λ)f − g  +2ε . Hence lim sup m →+∞  λx m +(1− λ)f − g ≤ lim inf n→+∞  λx n +(1− λ)f − g  +2ε . Since ε >0 is arbitrary, this implies that h(λ) = lim n →+∞  λx n +(1− λ)f − g  exists. This completes the proof. Remark 3.1 If the mappings are asymptotically nonexpansive, we can use Bruck’s Lemma [23]to prove Lemma 3.3. While Bruck’s Lemma is not valid for non-Lipschitzian mappings, we must introduce new technique to establish a similar inequality. In [20], Chidume et al. also proved that lim n →+∞  λx n +(1− λ)f − g  exists (Lemma 3.12 in [20]). As we have seen, our proof is completely different from theirs in [20]. Lemma 3.4 If f Î ω ω ({x n }) and lim n →+∞  λx n +(1− λ)f − g  exists, then h(λ) = lim n →+∞  λx n +(1− λ)f − g ≤ f − g  . Proof. For any ε >0, there exists n 0 such that for all n ≥ n 0 ,  λx n + ( 1 − λ ) f − g ≤ h ( λ ) + ε . Then for all n ≥ n 0 , λx n + ( 1 − λ ) f − g, J ( f − g ) ≤f − g  ( h ( λ ) + ε ). Since f Î ω ω ({x n }), there exists a subsequence {x n i }⊂{x n } with x n i  f .Hence f ∈ ¯ co{x n i , i ≥ n 0 } and {λf + ( 1 − λ ) f − g, J ( f − g ) }≤f − g  ( h ( λ ) + ε ), i.e., ||f - g|| 2 ≤ ||f - g||(h(l)+ε). Therefore ||f - g|| ≤ h(l). This completes the proof. Now we can prove the weak convergence theorem of the iterative sequence (3.1). Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 9 of 13 Theorem 3.1 Let C be a nonempty close d conve x subset of uniformly conve x Banach space X which satisfies the Opial’s condition or whose dual X* has the Kadec-Klee prop- erty. Let P : X ® C be a nonexpansive retraction from X onto C. Let T 1 , T 2 , T 3 : C ® X be three asymptotically nonexpansive in the intermediate sense nonself mappings with F ≠ ∅ and the nonnegative sequence {r n } satisfy +∞  n=1 r n < + ∞ . Let {x n } be defined by: x 1 Î C and x n+1 = P[α (1) n z n +(1− α (1) n )T 1 (PT 1 ) n−1 z n ] ; z n = P[α (2) n y n +(1− α (2) n )T 2 (PT 2 ) n−1 y n ]; y n = P[α (3) n x n + ( 1 − α (3) n ) T 3 ( PT 3 ) n−1 x n ]. where { α (i) n } is in [0, 1] with 0 < p ≤ α ( i ) n ≤ q < 1 , i =1,2,3.Then {x n }, {y n }, and {z n } converge weakly to a common fixed point of {T i } 3 i = 1 . Proof. It suffices to show that { x n } converges weakly to a common fixed point of {T i } 3 i = 1 .Tothisaim,weonlyneedtoprovethatthesetω ω ({x n }) is singleton. Since X is refl exive and C is bounded, we obtain ω ω ({x n }) ≠ ∅ . Assume that f, g Î ω ω ({x n }), then there exist two subsequences { x n i } and {x n j } in {x n }suchthat x n i  f and x n j  g .Inthe following, we shall show f = g. By Lemmas 2.5 and 3.2, f, g Î F. On one hand, if X satisfies the Opial’s condition and f ≠ g, then by the Lemma 3.1, we get r = lim n→+∞  x n − f = lim i→+∞  x n i − f  < lim i→+∞  x n i − g = lim n→+∞  x n − g = lim j→+∞  x n j − g  < lim j →+∞  x n j − f = lim n→+∞  x n − f = r. This contraction implies f = g.Ontheotherhand,ifX* has Kadec- Klee property, then from Lemmas 2.4, 3.3, and 3.4, we have  λx n +(1− λ)f − g 2 ≤f − g 2 +2λx n − f,J ( λx n + ( 1 − λ ) f − g ) and for all l Î [0, 1], lim inf n →+∞ x n − f,J(λx n +(1− λ)f − g)≥0 . Hence lim inf j →+∞ x n j − f,J(λx n j +(1− λ)f − g)≥0 . Thus for arbitrary k Î N, there exists j k ≥ k,{j k } ↑, such that x n j k − f,J( 1 k x n j k +(1− 1 k )f − g)≥− 1 k . (3:12) Obviously x n j k  g . Put j k = J( 1 k x n j k +(1− 1 k )f − g) , then we may assume that, without loss of generality, j k is weakly convergent to some Zhu et al. Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 Page 10 of 13 [...]... Shahzad, N: Convergence theorems for a common fixed point of finite family of nonself nonexpansive mappings Fixed Point Theory Appl 2005, 1–9 (2005) Shahzad, N: Approximating fixed points of non-self nonexpansive mappings in Banach spaces Nonlinear Anal: Theory, Methods Appl 61, 1031–1039 (2005) doi:10.1016/j.na.2005.01.092 Chidume, CE, Ofoedu, EU, Zegeye, H: Strong and weak convergence theorems for asymptotically... non-Lipschitzian mappings in Banach spaces Comput Math Appl 42, 1565–1570 (2001) doi:10.1016/S0898-1221(01)00262-0 Plubtieng, S, Wangkeeree, R: Strong convergence theorems for three-step iterations with errors for non-Lipschitzian nonself- mappings in Banach Spaces Comput Math Appl 51, 1093–1102 (2006) doi:10.1016/j.camwa.2005.08.035 Bruck, RE: A simple proof of the mean ergodic theorem for nonlinear contractions in. .. SY: Convergence theorems for three-step iterations with errors of asymptotically nonexpansive nonself mappings in Banach spaces Math Appl 24, 540–547 (2011) Chidume, CE, Shahzad, N, Zegeye, H: Convergence theorems for mappings which are asymptotically nonexpansive in the intermediate sense Numer Funct Anal Optimiz 25, 239–257 (2004) Kim, GE, Kim, TH: Mann and Ishikawa iterations with errors for non-Lipschitzian. .. Since X* has Kadec-Klee property, we have jk ® j Taking the limit in (3.12), we get 〈g - f, j〉 ≥ 0, i.e., ||f - g||2 ≤ 0, which implies f = g This completes the proof Remark 3.2 Theorem 3.1 extends the main results in [4,16,18,20]to the case of asymptotically nonexpansive in the intermediate sense mappings and it seems to be new even in the case that the space has a Fréchet differentiable norm In the. .. problems and results in the study of nonlinear analysis Nonlinear Anal: Theory Methods Appl 30, 4197–4208 (1997) doi:10.1016/S0362-546X(97)00388-X Royden, HL: Real Analysis, 3rd edn.Pearson Education (2004) doi:10.1186/1687-1812-2011-106 Cite this article as: Zhu et al.: Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces Fixed Point Theory and Applications... nonexpansive mappings J Math Anal Appl 280, 364–374 (2003) doi:10.1016/S0022-247X(03)00061-1 Wang, C, Zhu, J: Convergence theorems for common fixed points of nonself asymptotically quasi-non-expansive mappings Fixed Point Theory and Applications 2008, 11 (2008) Article ID 428241 Thianwan, S: Common fixed points of new iterations for two asymptotically nonexpansive nonself- mappings in a Banach space... for asymptotically nonexpansive mappings in Banach spaces Nonlinear Anal: Theory Methods Appl 64(3), 558–567 (2006) doi:10.1016/j.na.2005.03.114 Shahzad, N, Udomene, A: Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces Fixed Point Theory and Applications 2006, 10 (2006) Article ID 18909 Plubtieng, S, Wangkeeree, R, Punpaeng, R: On the convergence of. .. terms are added in our recursion formula and are assumed to be bounded, then the results of this article still hold Thus we can get the main results in [19] Acknowledgements This research is supported by the National Natural Science Foundation of China (10971182), the Natural Science Foundation of Jiangsu Province (BK2009179 and BK2010309), the Tianyuan Youth Foundation (11026115), the Jiangsu Government... Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces J Math Anal Appl 267(2), 444–453 (2002) doi:10.1006/jmaa.2001.7649 Page 12 of 13 Zhu et al Fixed Point Theory and Applications 2011, 2011:106 http://www.fixedpointtheoryandapplications.com/content/2011/1/106 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 Cho, Y, Zhou, H, Guo, G: Weak and strong convergence theorems... theorems for three-step iterations with errors for asymptotically nonexpansive mappings Comput Math Appl 47(4-5), 707–717 (2004) doi:10.1016/S0898-1221(04)90058-2 Suantai, S: Weak and strong convergence criteria of Noor iterations for asymptotically nonexpansive mappings J Math Anal Appl 311, 506–517 (2005) doi:10.1016/j.jmaa.2005.03.002 Shahzad, N, Udomene, A: Fixed point solutions of variational inequalities . (2004) doi:10.1186/1687-1812-2011-106 Cite this article as: Zhu et al.: Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces. Fixed Point Theory and Applications 2011 2011:106. Submit. Shahzad, N: Convergence theorems for a common fixed point of finite family of nonself nonexpansive mappings. Fixed Point Theory Appl. 2005,1–9 (2005) 15. Shahzad, N: Approximating fixed points of non-self. RESEARC H Open Access Weak convergence theorem for the three-step iterations of non-Lipschitzian nonself mappings in Banach spaces Lanping Zhu, Qianglian Huang * and Xiaoru Chen *