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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 428241, 11 pages doi:10.1155/2008/428241 ResearchArticleConvergenceTheoremsforCommonFixedPointsofNonselfAsymptoticallyQuasi-Non-Expansive Mappings Chao Wang and Jinghao Zhu Department of Applied Mathematics, Tongji University, Shanghai 200092, China Correspondence should be addressed to Chao Wang, wangchaoxj20002000@yahoo.com.cn Received 1 April 2008; Revised 12 June 2008; Accepted 19 July 2008 Recommended by Simeon Reich We introduce a new three-step iterative scheme with errors. Several convergencetheoremsof this scheme are established forcommon fixed pointsofnonselfasymptoticallyquasi-non-expansive mappings in real uniformly convex Banach spaces. Our theorems improve and generalize recent known results in the literature. Copyright q 2008 C. Wang and J. Zhu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction Let K be a nonempty closed convex subset of real normed linear space E. Recall that a mapping T : K → K is called asymptotically nonexpansive if there exists a sequence{r n }⊂ 0, ∞, with lim n→∞ r n 0 such that T n x − T n y≤1 r n x − y, for all x, y ∈ K and n ≥ 1. Moreover, it is uniformly L-Lipschitzian if there exists a constant L>0 such that T n x − T n y≤Lx − y, for all x, y ∈ K and each n ≥ 1. Denote and define by FT{x ∈ K : Tx x} the set of fixed pointsof T. Suppose FT / ∅. A mapping T is called asymptoticallyquasi-non-expansive if there exists a sequence {r n }⊂0, ∞,with lim n→∞ r n 0 such that T n x − p≤1 r n x − p, for all x, y ∈ K, p ∈ FT,andn ≥ 1. It is clear from the above definitions that an asymptotically nonexpansive mapping must be uniformly L-Lipschitzian as well as asymptotically quasi-non-expansive, but the converse does not hold. Iterative technique forasymptotically nonexpansive self-mapping in Hilbert spaces and Banach spaces including Mann-type and Ishikawa-type iteration processes has been studied extensively by many authors; see, for example, 1–6. Recently, Chidume et al. 7 have introduced the concept ofnonselfasymptotically nonexpansive mappings, which is the generalization ofasymptotically nonexpansive mappings. Similarly, the concept ofnonselfasymptoticallyquasi-non-expansive mappings 2 Fixed Point Theory and Applications can also be defined as the generalization ofasymptoticallyquasi-non-expansive mappings and nonselfasymptotically nonexpansive mappings. These mappings are defined as follows. Definition 1.1. Let K be a nonempty closed convex subset of real normed linear space E,let P : E → K be the nonexpansive retraction of E onto K,andletT : K → E be a nonself mapping. i T is said to be a nonselfasymptotically nonexpansive mapping if there exists a sequence {r n }⊂0, ∞, with lim n→∞ r n 0 such that TPT n−1 x − TPT n−1 y ≤ 1 r n x − y, 1.1 for all x, y ∈ K and n ≥ 1. ii T is said to be a nonself uniformly L-Lipschitzian mapping if there exists a constant L>0 such that TPT n−1 x − TPT n−1 y ≤ Lx − y, 1.2 for all x, y ∈ K and n ≥ 1. iii T is said to be a nonselfasymptoticallyquasi-non-expansive mapping if FT / ∅ and there exists a sequence {r n }⊂0, ∞, with lim n→∞ r n 0 such that TPT n−1 x − p ≤ 1 r n x − p, 1.3 for all x, y ∈ K, p ∈ FT,andn ≥ 1. By studying the following iteration process Mann-type iteration: x 1 ∈ K, x n1 P 1 − α n x n α n TPT n−1 x n , ∀n ≥ 1, 1.4 where {α n }⊂0, 1, Chidume et al. 7 obtained many convergencetheoremsfor the fixed pointsofnonselfasymptotically nonexpansive mapping T. Later on, Wang 8 generalized the iteration process 1.4 as follows Ishikawa-type iteration: x 1 ∈ K, x n1 P 1 − α n x n α n T 1 PT 1 n−1 y n , y n P 1 − β n x n β n T 2 PT 2 n−1 x n , ∀n ≥ 1 1.5 where T 1 ,T 2 : K → E are nonselfasymptotically nonexpansive mappings and {α n }, {β n }⊂ 0, 1. Also, he got several convergencetheoremsof the iterative scheme 1.5 under proper conditions. In 2000, Noor 9 first introduced a three-step iterative sequence and studied the approximate solutions of variational inclusion in Hilbert spaces by using the techniques of updating the solution and the auxiliary principle. Glowinski and Tallec 10 showed that the three-step iterative schemes perform better than the Mann-type and Ishikawa-type iterative schemes. On the other hand, Xu and Noor 11 introduced and studied a three-step scheme to approximate fixed pointsofasymptotically nonexpansive mappings in Banach spaces. Cho et al. 12 and Plubtieng et al. 13 extended the work of Xu and Noor to the three- step iterative scheme with errors, and gave weak and strong convergencetheoremsforasymptotically nonexpansive mappings in Banach spaces. C. Wang and J. Zhu 3 Inspired and motivated by these facts, a new class of three-step iterative schemes with errors, for three nonselfasymptoticallyquasi-non-expansive mappings, is introduced and studied in this paper. This scheme can be viewed as an extension for 1.4, 1.5, and others. This scheme is defined as follows. Let K be a nonempty convex subset of real normed linear space X,letP : E → K be the nonexpansive retraction of E onto K,andletT 1 ,T 2 ,T 3 : K → E be three nonselfasymptoticallyquasi-non-expansive mappings. Compute the sequences{x n }, {y n },and{z n } by x 1 ∈ K, x n1 P α n T 1 PT 1 n−1 y n β n x n γ n w n , y n P α n T 2 PT 2 n−1 z n β n x n γ n v n , z n P α n T 3 PT 3 n−1 x n β n x n γ n u n , ∀n ≥ 1 1.6 where {α n }, {α n }, {α n }, {β n }, {β n }, {β n }, {γ n }, {γ n },and{γ n } are real sequences in 0, 1 with α n β n γ n α n β n γ n α n β n γ n 1, and {u n }, {v n },and{w n } are bounded sequences in K. Remark 1.2. i If T 1 T 2 T 3 : T, γ n γ n γ n 0, and α n α n 0, then scheme 1.6 reduces to t he Mann-type iteration 1.4. ii If T 2 T 3 , γ n γ n γ n 0, and α n 0, then scheme 1.6 reduces to the Ishikawa- type iteration 1.5. iii If T 1 ,T 2 ,andT 3 are three self-asymptotically nonexpansive mappings, then scheme 1.6 reduces to the three-step iteration with errors defined by 12, 13, and others. The purpose of this paper is to study the iterative sequences 1.6 to converge to a common fixed point of three nonselfasymptoticallyquasi-non-expansive mappings in real uniformly convex Banach spaces. Our results extend and improve the corresponding results in 5, 7, 8, 11–13, and many others. 2. Preliminaries and lemmas In this section, we first recall some well-known definitions. A real Banach space E is said to be uniformly convex if the modulus of convexity of E: δ E εinf 1 − x y 2 : x y 1, x − y ε > 0, 2.1 for all 0 <ε≤ 2 i.e., δ E ε is a function 0, 2 → 0, 1. AsubsetK of E is said to be a retract if there exists continuous mapping P : E → K such that Px x, for all x ∈ K, and every closed convex subset of a uniformly convex Banach space is a retract. A mapping P : E → E is said to be a retraction if P 2 P. A mapping T : K → E with FT / ∅ is said to satisfy condition Asee 14 if there exists a nondecreasing function f : 0, ∞ → 0, ∞ with f00, for all r ∈ 0, ∞, such that x − Tx≥f d x, FT , 2.2 for all x ∈ K, where dx, FT inf{x − x ∗ : x ∗ ∈ FT}. We modify this condition for three mappings T 1 ,T 2 ,T 3 : K → E as follows. Three mappings T 1 ,T 2 ,T 3 : K → E, where K is a subset of E, are said to satisfy condition B if there 4 Fixed Point Theory and Applications exist a real number α>0 and a nondecreasing function f : 0, ∞ → 0, ∞ with f00, for all r ∈ 0, ∞, such that x − T 1 x ≥ αf dx, F or x − T 2 x ≥ αf dx, F or x − T 3 x ≥ αf dx, F , 2.3 for all x ∈ K, where F FT 1 ∩FT 2 ∩FT 3 / ∅. Note that condition B reduces to condition A when T 1 T 2 T 3 and α 1. A mapping T : K → E is said to be semicompact if, for any sequence {x n } in K such that x n −Tx n →0 n →∞, there exists subsequence {x n j } of {x n } such that {x n j } converges strongly to x ∗ ∈ K. Next we state the following useful lemmas. Lemma 2.1 see 5. Let {a n }, {b n }, and {c n } be sequences of nonnegative real numbers satisfying the inequality a n1 ≤ 1 c n a n b n , ∀n ≥ 1. 2.4 If ∞ n1 c n < ∞ and ∞ n1 b n < ∞,thenlim n→∞ a n exists. Lemma 2.2 see 15. Let E be a real uniformly convex Banach space and 0 ≤ k ≤ t n ≤ q<1, for all positive integer n ≥ 1. Suppose that {x n } and {y n } are two 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;thenlim n→∞ x n − y n 0. 3. Main results In this section, we will prove the strong convergenceof the iteration scheme 1.6 to a common fixed point ofnonselfasymptoticallyquasi-non-expansive mappings T 1 ,T 2 ,andT 3 . We first prove the following lemmas. Lemma 3.1. Let K be a nonempty closed convex subset of a real normed linear space E.LetT 1 ,T 2 ,T 3 : K → E be nonselfasymptoticallyquasi-non-expansive mappings with sequences {r i n } such that ∞ n1 r i n < ∞, for all i 1, 2, 3. Suppose that {x n } is defined by 1.6 with ∞ n1 γ n < ∞, ∞ n1 γ n < ∞, and ∞ n1 γ n < ∞.IfF FT 1 ∩ FT 2 ∩ FT 3 / ∅,thenlim n→∞ x n − p exists, for all p ∈ F. Proof. Let p ∈ F. Since {u n }, {v n },and{w n } are bounded sequences in K, therefore there exists M>0 such that M max sup n≥1 u n − p , sup n≥1 v n − p , sup n≥1 w n − p . 3.1 Let r n max{r 1 n ,r 2 n ,r 3 n } and k n max{γ n ,γ n ,γ n }. Then ∞ n1 r n < ∞ and ∞ n1 k n < ∞.By 1.6, we have x n1 − p P α n T 1 PT n−1 1 y n β n x n γ n w n − Pp ≤ α n T 1 PT n−1 1 y n β n x n γ n w n − α n β n γ n p ≤ α n T 1 PT n−1 1 y n − p β n x n − p γ n w n − p ≤ α n 1 r n y n − p β n x n − p k n w n − p , 3.2 y n − p P α n T 2 PT n−1 2 z n β n x n γ n v n − Pp ≤ α n T 2 PT n−1 2 z n β n x n γ n v n − α n β n γ n p ≤ α n 1 r n z n − p β n x n − p k n v n − p , 3.3 C. Wang and J. Zhu 5 and similarly, we also have z n − p ≤ α n 1 r n x n − p β n x n − p k n u n − p . 3.4 Substituting 3.4 into 3.3,weobtain y n − p ≤ α n 1 r n α n 1 r n x n − p β n x n − p k n u n − p β n x n − p k n v n − p ≤ α n α n 1 r n 2 x n − p α n β n 1 r n x n − p β n x n − p α n k n 1 r n u n − p k n v n − p ≤ 1 − β n − γ n α n 1 r n 2 x n − p 1 − β n − γ n β n 1 r n x n − p β n x n − p k n 1 r n u n − p k n v n − p ≤ 1 − β n − γ n α n β n 1 r n 2 x n − p β n x n − p m n ≤ 1 − β n 1 r n 2 x n − p β n 1 r n 2 x n − p m n ≤ 1 r n 2 x n − p m n , 3.5 where m n k n 2r n M. Since ∞ n1 r n < ∞ and ∞ n1 k n < ∞, then ∞ n1 m n < ∞. Substituting 3.5 into 3.2, we have x n1 − p ≤ α n 1 r n 1 r 2 n x n − p m n β n x n − p γ n w n − p ≤ α n 1 r n 3 β n x n − p α n 1 r n m n γ n w n − p ≤ α n β n 1 r n 3 x n − p 1 r n m n k n w n − p ≤ 1 r n 3 x n − p 1 r n m n k n M ≤ 1 c n x n − p b n , 3.6 where c n 1 r n 3 − 1andb n 1 r n m n k n M. Since ∞ n1 r n < ∞, ∞ n1 k n < ∞, and ∞ n1 m n < ∞, then ∞ n1 c n < ∞ and ∞ n1 b n < ∞. It follows from Lemma 2.1 that lim n→∞ x n − p exists. This completes the proof. Lemma 3.2. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.Let T 1 ,T 2 ,T 3 : K → E be uniformly L-Lipschitzian nonselfasymptoticallyquasi-non-expansive mappings with sequences {r i n } such that ∞ n1 r i n < ∞, for all i 1, 2, 3. Suppose that {x n } is defined by 1.6 with ∞ n1 γ n < ∞, ∞ n1 γ n < ∞, and ∞ n1 γ n < ∞,whereα n ,α n , and α n are three sequences in ε, 1 − ε,forsomeε>0.IfF FT 1 ∩ FT 2 ∩ FT 3 / ∅,then lim n→∞ x n − T 1 x n lim n→∞ x n − T 2 x n lim n→∞ x n − T 3 x n 0. 3.7 Proof. For any p ∈ F,byLemma 3.1, we see that lim n→∞ x n − p exists. Assume lim n→∞ x n − p a, for some a ≥ 0. For all n ≥ 1, let r n max{r 1 n ,r 2 n ,r 3 n } and k n max{γ n ,γ n ,γ n }. 6 Fixed Point Theory and Applications Then, ∞ n1 r n < ∞ and ∞ n1 k n < ∞.From3.5, we have y n − p ≤ 1 r n 2 x n − p m n . 3.8 Taking lim sup n→∞ on both sides in 3.8,since ∞ n1 r n < ∞ and ∞ n1 m n < ∞,weobtain lim sup n→∞ y n − p ≤ lim sup n→∞ x n − p lim n→∞ x n − p a 3.9 so that lim sup n→∞ T 1 PT 1 n−1 y n − p ≤ lim sup n→∞ 1 r n y n − p lim sup n→∞ y n − p ≤ a. 3.10 Next consider T 1 PT 1 n−1 y n − p γ n w n − x n ≤ T 1 PT 1 n−1 y n − p k n w n − x n . 3.11 Since lim n→∞ k n 0, we have lim sup n→∞ T 1 PT 1 n−1 y n − p γ n w n − x n ≤ a. 3.12 In addition, x n − p γ n w n − x n ≤ x n − p k n w n − x n . 3.13 This implies that lim sup n→∞ x n − p γ n w n − x n ≤ a. 3.14 Further, observe that a lim n→∞ x n − p lim n→∞ α n T 1 PT 1 n−1 y n β n x n γ n w n − p lim n→∞ α n T 1 PT 1 n−1 y n 1 − α n x n − γ n x n γ n w n − 1 − α n p − α n p lim n→∞ α n T 1 PT 1 n−1 y n − α n p α n γ n w n − α n γ n x n 1 − α n x n − 1 − α n p − γ n x n γ n w n − α n γ n w n α n γ n x n lim n→∞ α n T 1 PT 1 n−1 y n − p γ n w n − x n 1 − α n x n − p γ n w n − x n . 3.15 By Lemma 2.2, 3.12 , 3.14,and3.15, we have lim n→∞ T 1 PT 1 n−1 y n − x n 0. 3.16 C. Wang and J. Zhu 7 Next we will prove that lim n→∞ T 2 PT 2 n−1 z n − x n 0. Since x n − p ≤ T 1 PT 1 n−1 y n − x n T 1 PT 1 n−1 y n − p ≤ T 1 PT 1 n−1 y n − x n 1 r n y n − p 3.17 and lim n→∞ T 1 PT 1 n−1 y n − x n 0 lim n→∞ r n ,weobtain a lim n→∞ x n − p ≤ lim inf n→∞ y n − p . 3.18 Thus, it follows from 3.10 and 3.18 that lim n→∞ y n − p a. 3.19 On the other hand, from 3.4, we have z n − p ≤ α n 1 r n β n x n − p k n u n − p ≤ 1 r n x n − p k n u n − p . 3.20 By boundedness of t he sequence {u n } and by lim n→∞ r n lim n→∞ k n 0, we have lim sup n→∞ z n − p ≤ lim sup n→∞ x n − p a 3.21 so that lim sup n→∞ T 2 PT 2 n−1 z n − p ≤ lim sup n→∞ 1 r n z n − p ≤ a. 3.22 Next consider T 2 PT 2 n−1 z n − p γ n v n − x n ≤ T 2 PT 2 n−1 z n − p k n v n − x n . 3.23 Thus, we have lim sup n→∞ T 2 PT 2 n−1 z n − p γ n v n − x n ≤ a, x n − p γ n v n − x n ≤ x n − p k n v n − x n . 3.24 This implies that lim sup n→∞ x n − p γ n v n − x n ≤ a. 3.25 Note that a lim n→∞ y n − p lim n→∞ α n T 2 PT 2 n−1 z n β n x n γ n v n − p lim n→∞ α n T 2 PT 2 n−1 z n − p γ n v n − x n 1 − α n x n − p γ n v n − x n . 3.26 8 Fixed Point Theory and Applications It follows from Lemma 2.2, 3.24,and3.25 that lim n→∞ T 2 PT 2 n−1 z n − x n 0. 3.27 Similarly, by using the same argument as in the proof above, we obtain lim n→∞ T 3 PT 3 n−1 x n − x n 0. 3.28 Hence, lim n→∞ T 1 PT 1 n−1 y n − x n lim n→∞ T 2 PT 2 n−1 z n − x n lim n→∞ T 3 PT 3 n−1 x n − x n 0, 3.29 and this implies that x n1 − x n ≤ α n T 1 PT 1 n−1 y n − x n k n w n − x n −→ 0asn −→ ∞ . 3.30 Since T 1 is uniformly L-Lipschitzian mapping, then we have T 1 PT 1 n−1 x n − x n ≤ T 1 PT 1 n−1 x n − T 1 PT 1 n−1 y n T 1 PT 1 n−1 y n − x n ≤ L x n − y n T 1 PT 1 n−1 y n − x n ≤ L x n − α n T 2 PT 2 n−1 z n − β n x n − γ n v n T 1 PT 1 n−1 y n − x n ≤ Lα n T 2 PT 2 n−1 z n − x n Lk n v n − x n T 1 PT 1 n−1 y n − x n −→ 0asn −→ ∞ , 3.31 x n − T 1 x n ≤ x n1 −x n x n1 −T 1 PT 1 n x n1 T 1 PT 1 n x n1 −T 1 PT 1 n x n T 1 PT 1 n x n −T 1 x n ≤ x n1 − x n x n1 − T 1 PT 1 n x n1 L x n1 − x n L T 1 PT 1 n−1 x n − x n . 3.32 It follows from 3.30, 3.31,and3.32 that lim n→∞ x n − T 1 x n 0. 3.33 Next consider T 2 PT 2 n−1 x n − x n ≤ T 2 PT 2 n−1 x n − T 2 PT 2 n−1 z n T 2 PT 2 n−1 z n − x n ≤ L x n − z n T 2 PT 2 n−1 z n − x n ≤ Lα n T 3 PT 3 n−1 x n − x n Lk n u n − x n T 2 PT 2 n−1 z n − x n −→ 0asn −→ ∞ , 3.34 x n − T 2 x n ≤ x n1 −x n x n1 −T 2 PT 2 n x n1 T 2 PT 2 n x n1 −T 2 PT 2 n x n T 2 PT 2 n x n −T 2 x n ≤ x n1 − x n x n1 − T 2 PT 2 n x n1 L x n1 − x n L T 2 PT 2 n−1 x n − x n . 3.35 C. Wang and J. Zhu 9 It follows from 3.30, 3.34,and3.35 that lim n→∞ x n − T 2 x n 0. 3.36 Finally, we consider x n − T 3 x n ≤ x n1 −x n x n1 −T 3 PT 3 n x n1 T 3 PT 3 n x n1 −T 3 PT 3 n x n T 3 PT 3 n x n −T 3 x n ≤ x n1 − x n x n1 − T 3 PT 3 n x n1 L x n1 − x n L T 3 PT 3 n−1 x n − x n . 3.37 It follows from 3.29, 3.30,and3.37 that lim n→∞ x n − T 3 x n 0. 3.38 Therefore, lim n→∞ x n − T 1 x n lim n→∞ x n − T 2 x n lim n→∞ x n − T 3 x n 0. 3.39 This completes the proof. Now, we give our main theoremsof this paper. Theorem 3.3. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T 1 ,T 2 ,T 3 : K → E be uniformly L-Lipschitzian and nonselfasymptoticallyquasi-non-expansive mappings with sequences {r i n } such that ∞ n1 r i n < ∞, for all i 1, 2, 3, satisfying condition (B). Suppose that {x n } is defined by 1.6 with ∞ n1 γ n < ∞, ∞ n1 γ n < ∞, and ∞ n1 γ n < ∞,where α n ,α n , and α n are three sequences in ε, 1 − ε,forsomeε>0.IfF FT 1 ∩ FT 2 ∩ FT 3 / ∅, then {x n } converges strongly to a common fixed point of T 1 ,T 2 , and T 3 . Proof. It follows from Lemma 3.2 that lim n→∞ x n − T 1 x n lim n→∞ x n − T 2 x n lim n→∞ x n − T 3 x n 0. Since T 1 ,T 2 ,andT 3 satisfy condition B, we have lim n→∞ dx n ,F0. From Lemma 3.1 and the proof of Qihou 5, we can obtain that {x n } is a Cauchy sequence in K. Assume that lim n→∞ x n p ∈ K. Since lim n→∞ x n − T 1 x n lim n→∞ x n − T 2 x n lim n→∞ x n − T 3 x n 0, by the continuity of T 1 ,T 2 ,andT 3 , we have p ∈ F,thatis,p is a common fixed point of T 1 ,T 2 ,andT 3 . This completes the proof. Corollary 3.4. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.LetT 1 ,T 2 ,T 3 : K → E be nonselfasymptotically nonexpansive mappings with sequences {r i n } such that ∞ n1 r i n < ∞, for all i 1, 2, 3, satisfying condition (B). Suppose that {x n } is defined by 1.6 with ∞ n1 γ n < ∞, ∞ n1 γ n < ∞, and ∞ n1 γ n < ∞,whereα n ,α n , and α n are three sequences in ε, 1 − ε, for some ε>0.IfF FT 1 ∩ FT 2 ∩ FT 3 / ∅,then{x n } converges strongly to a common fixed point of T 1 ,T 2 , and T 3 . Proof. Since every nonselfasymptotically nonexpansive mapping is uniformly L-Lipschitzian and nonselfasymptotically quasi-non-expansive, the result can be deduced immediately from Theorem 3.3. This completes the proof. 10 Fixed Point Theory and Applications Theorem 3.5. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E. Let T 1 ,T 2 ,T 3 : K → E be uniformly L-Lipschitzian and nonselfasymptoticallyquasi-non-expansive mappings with sequences {r i n } such that ∞ n1 r i n < ∞, for all i 1, 2, 3. Suppose that {x n } is defined by 1.6 with ∞ n1 γ n < ∞, ∞ n1 γ n < ∞, and ∞ n1 γ n < ∞,whereα n ,α n , and α n are three sequences in ε, 1 − ε, for some ε>0.IfF FT 1 ∩ FT 2 ∩ FT 3 / ∅ and one of T 1 ,T 2 , and T 3 is demicompact, then {x n } converges strongly to a common fixed point of T 1 ,T 2 , and T 3 . Proof. Without loss of generality, we may assume that T 1 is demicompact. Since lim n→∞ x n − T 1 x n 0, there exists a subsequence {x n j }⊂{x n } such that x n j → x ∗ ∈ K. Hence, from 3.39, we have x ∗ − T i x ∗ lim n→∞ x n j − T i x n j 0,i 1, 2, 3. 3.40 This implies that x ∗ ∈ F. By the arbitrariness of p ∈ F,fromLemma 3.1, and taking p x ∗ , similarly we can prove that lim n→∞ x n − x ∗ d, 3.41 where d ≥ 0 is some nonnegative number. From x n j → x ∗ , we know that d 0, that is, x n → x ∗ . This completes the proof. Corollary 3.6. Let K be a nonempty closed convex subset of a real uniformly convex Banach space E.LetT 1 ,T 2 ,T 3 : K → E be nonselfasymptotically nonexpansive mappings with sequences {r i n } such that ∞ n1 r i n < ∞, for all i 1, 2, 3. Suppose that {x n } is defined by 1.6 with ∞ n1 γ n < ∞, ∞ n1 γ n < ∞, and ∞ n1 γ n < ∞,whereα n ,α n , and α n are three sequences in ε, 1 − ε, for some ε>0.IfF FT 1 ∩ FT 2 ∩ FT 3 / ∅ and one of T 1 ,T 2 , and T 3 is demicompact, then {x n } converges strongly to a common fixed point of T 1 ,T 2 , and T 3 . Acknowledgments The authors would like to thank the referee and t he editor for their careful reading of the manuscript and their many valuable comments and suggestions. This paper was supported by the National Natural Science Foundation of China Grant no. 10671145. References 1 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, no. 3, pp. 506–510, 1953. 2 S. 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Corporation Fixed Point Theory and Applications Volume 2008, Article ID 428241, 11 pages doi:10.1155/2008/428241 Research Article Convergence Theorems for Common Fixed Points of Nonself Asymptotically Quasi-Non-Expansive. Several convergence theorems of this scheme are established for common fixed points of nonself asymptotically quasi-non-expansive mappings in real uniformly convex Banach spaces. Our theorems. the concept of nonself asymptotically nonexpansive mappings, which is the generalization of asymptotically nonexpansive mappings. Similarly, the concept of nonself asymptotically quasi-non-expansive