Hindawi Publishing 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 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 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 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≤Lx − y, for all x, y ∈ K and each n ≥ 1. Denote and define by FT{x ∈ K : Tx  x} the set of fixed points of T. Suppose FT /  ∅. A mapping T is called asymptotically quasi-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 ∈ FT,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 for asymptotically 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 of nonself asymptotically nonexpansive mappings, which is the generalization of asymptotically nonexpansive mappings. Similarly, the concept of nonself asymptotically quasi-non-expansive mappings 2 Fixed Point Theory and Applications can also be defined as the generalization of asymptotically quasi-non-expansive mappings and nonself asymptotically 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 nonself asymptotically nonexpansive mapping if there exists a sequence {r n }⊂0, ∞, with lim n→∞ r n  0 such that   TPT n−1 x − TPT 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   TPT n−1 x − TPT n−1 y   ≤ Lx − y, 1.2 for all x, y ∈ K and n ≥ 1. iii T is said to be a nonself asymptotically quasi-non-expansive mapping if FT /  ∅ and there exists a sequence {r n }⊂0, ∞, with lim n→∞ r n  0 such that   TPT n−1 x − p   ≤  1  r n  x − p, 1.3 for all x, y ∈ K, p ∈ FT,andn ≥ 1. By studying the following iteration process Mann-type iteration: x 1 ∈ K, x n1  P  1 − α n  x n  α n TPT n−1 x n  , ∀n ≥ 1, 1.4 where {α n }⊂0, 1, Chidume et al. 7 obtained many convergence theorems for the fixed points of nonself asymptotically nonexpansive mapping T. Later on, Wang 8 generalized the iteration process 1.4 as follows Ishikawa-type iteration: x 1 ∈ K, x n1  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 nonself asymptotically nonexpansive mappings and {α n }, {β n }⊂ 0, 1. Also, he got several convergence theorems of 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 points of asymptotically 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 convergence theorems for asymptotically 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 nonself asymptotically quasi-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 nonself asymptotically quasi-non-expansive mappings. Compute the sequences{x n }, {y n },and{z n } by x 1 ∈ K, x n1  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 nonself asymptotically quasi-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 εinf  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 FT /  ∅ is said to satisfy condition Asee 14 if there exists a nondecreasing function f : 0, ∞ → 0, ∞ with f00, for all r ∈ 0, ∞, such that x − Tx≥f  d  x, FT  , 2.2 for all x ∈ K, where dx, FT  inf{x − x ∗  : x ∗ ∈ FT}. 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 f00, for all r ∈ 0, ∞, such that   x − T 1 x   ≥ αf  dx, F  or   x − T 2 x   ≥ αf  dx, F  or   x − T 3 x   ≥ αf  dx, F  , 2.3 for all x ∈ K, where F  FT 1 ∩FT 2 ∩FT 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 n1 ≤  1  c n  a n  b n , ∀n ≥ 1. 2.4 If  ∞ n1 c n < ∞ and  ∞ n1 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 convergence of the iteration scheme 1.6 to a common fixed point of nonself asymptotically quasi-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 nonself asymptotically quasi-non-expansive mappings with sequences {r i n } such that  ∞ n1 r i n < ∞, for all i  1, 2, 3. Suppose that {x n } is defined by 1.6 with  ∞ n1 γ n < ∞,  ∞ n1 γ  n < ∞, and  ∞ n1 γ  n < ∞.IfF  FT 1  ∩ FT 2  ∩ FT 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  ∞ n1 r n < ∞ and  ∞ n1 k n < ∞.By 1.6, we have   x n1 − p      P  α n T 1  PT n−1 1  y n  β n x n  γ n w n  − Pp   ≤   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  − Pp   ≤   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 2r n M. Since  ∞ n1 r n < ∞ and  ∞ n1 k n < ∞, then  ∞ n1 m n < ∞. Substituting 3.5 into 3.2, we have   x n1 − 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  ∞ n1 r n < ∞,  ∞ n1 k n < ∞, and  ∞ n1 m n < ∞, then  ∞ n1 c n < ∞ and  ∞ n1 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 nonself asymptotically quasi-non-expansive mappings with sequences {r i n } such that  ∞ n1 r i n < ∞, for all i  1, 2, 3. Suppose that {x n } is defined by 1.6 with  ∞ n1 γ n < ∞,  ∞ n1 γ  n < ∞, and  ∞ n1 γ  n < ∞,whereα n ,α  n , and α  n are three sequences in ε, 1 − ε,forsomeε>0.IfF  FT 1  ∩ FT 2  ∩ FT 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,  ∞ n1 r n < ∞ and  ∞ n1 k n < ∞.From3.5, 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  ∞ n1 r n < ∞ and  ∞ n1 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,and3.15, 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,and3.25 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 n1 − 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 n1 −x n      x n1 −T 1  PT 1  n x n1      T 1  PT 1  n x n1 −T 1  PT 1  n x n      T 1  PT 1  n x n −T 1 x n   ≤   x n1 − x n      x n1 − T 1  PT 1  n x n1    L   x n1 − x n    L   T 1  PT 1  n−1 x n − x n   . 3.32 It follows from 3.30, 3.31,and3.32 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 n1 −x n      x n1 −T 2  PT 2  n x n1      T 2  PT 2  n x n1 −T 2  PT 2  n x n      T 2  PT 2  n x n −T 2 x n   ≤   x n1 − x n      x n1 − T 2  PT 2  n x n1    L   x n1 − 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,and3.35 that lim n→∞   x n − T 2 x n    0. 3.36 Finally, we consider   x n − T 3 x n   ≤   x n1 −x n      x n1 −T 3  PT 3  n x n1      T 3  PT 3  n x n1 −T 3  PT 3  n x n      T 3  PT 3  n x n −T 3 x n   ≤   x n1 − x n      x n1 − T 3  PT 3  n x n1    L   x n1 − x n    L   T 3  PT 3  n−1 x n − x n   . 3.37 It follows from 3.29, 3.30,and3.37 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 theorems of 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 nonself asymptotically quasi-non-expansive mappings with sequences {r i n } such that  ∞ n1 r i n < ∞, for all i  1, 2, 3, satisfying condition (B). Suppose that {x n } is defined by 1.6 with  ∞ n1 γ n < ∞,  ∞ n1 γ  n < ∞, and  ∞ n1 γ  n < ∞,where α n ,α  n , and α  n are three sequences in ε, 1 − ε,forsomeε>0.IfF  FT 1  ∩ FT 2  ∩ FT 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→∞ dx n ,F0. 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 nonself asymptotically nonexpansive mappings with sequences {r i n } such that  ∞ n1 r i n < ∞, for all i  1, 2, 3, satisfying condition (B). Suppose that {x n } is defined by 1.6 with  ∞ n1 γ n < ∞,  ∞ n1 γ  n < ∞, and  ∞ n1 γ  n < ∞,whereα n ,α  n , and α  n are three sequences in ε, 1 − ε, for some ε>0.IfF  FT 1  ∩ FT 2  ∩ FT 3  /  ∅,then{x n } converges strongly to a common fixed point of T 1 ,T 2 , and T 3 . Proof. Since every nonself asymptotically nonexpansive mapping is uniformly L-Lipschitzian and nonself asymptotically 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 nonself asymptotically quasi-non-expansive mappings with sequences {r i n } such that  ∞ n1 r i n < ∞, for all i  1, 2, 3. Suppose that {x n } is defined by 1.6 with  ∞ n1 γ n < ∞,  ∞ n1 γ  n < ∞, and  ∞ n1 γ  n < ∞,whereα n ,α  n , and α  n are three sequences in ε, 1 − ε, for some ε>0.IfF  FT 1  ∩ FT 2  ∩ FT 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 nonself asymptotically nonexpansive mappings with sequences {r i n } such that  ∞ n1 r i n < ∞, for all i  1, 2, 3. Suppose that {x n } is defined by 1.6 with  ∞ n1 γ n < ∞,  ∞ n1 γ  n < ∞, and  ∞ n1 γ  n < ∞,whereα n ,α  n , and α  n are three sequences in ε, 1 − ε, for some ε>0.IfF  FT 1  ∩ FT 2  ∩ FT 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. Ishikawa, “Fixed points and iteration of a nonexpansive mapping in a Banach space,” Proceedings of the American Mathematical Society, vol. 59, no. 1, pp. 65–71, 1967. 3 K. K. Tan and H. K. Xu, “Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process,” Journal of Mathematical Analysis and Applications, vol. 178, no. 2, pp. 301–308, 1993. 4 J. Schu, “Weak and strong convergence to fixed points of asymptotically nonexpansive mappings,” Bulletin of the Australian Mathematical Society, vol. 43, no. 1, pp. 153–159, 1991. 5 Q. H. Liu, “Iterative sequences for asymptotically quasi-nonexpansive mappings with error member,” Journal of Mathematical Analysis and Applications, vol. 259, no. 1, pp. 18–24, 2001. 6 N. Shahzad and A. Udomene, “Approximating common fixed points of two asymptotically quasi- nonexpansive mappings in Banach spaces,” Fixed Point Theory and Applications, vol. 2006, Article ID 18909, 10 pages, 2006. 7 C. E. Chidume, E. U. Ofoedu, and H. Zegeye, “Strong and weak convergence theorems for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol. 280, no. 2, pp. 364–374, 2003. [...]... the convergence of modified Noor iterations with errors for asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol 322, no 2, pp 1018–1029, 2006 14 H F Senter and W G Dotson Jr., “Approximating fixed points of nonexpansive mappings,” Proceedings of the American Mathematical Society, vol 44, no 2, pp 375–380, 1974 15 J Schu, “Iterative construction of fixed points of. .. Mechanics, vol 9 of SIAM Studies in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1989 11 B Xu and M A Noor, Fixed- point iterations for asymptotically nonexpansive mappings in Banach spaces,” Journal of Mathematical Analysis and Applications, vol 267, no 2, pp 444–453, 2002 12 Y J Cho, H Zhou, and G Guo, “Weak and strong convergence theorems for three-step iterations with errors for asymptotically. ..C Wang and J Zhu 11 8 L Wang, “Strong and weak convergence theorems for common fixed point of nonself asymptotically nonexpansive mappings,” Journal of Mathematical Analysis and Applications, vol 323, no 1, pp 550–557, 2006 9 M A Noor, “New approximation schemes for general variational inequalities,” Journal of Mathematical Analysis and Applications, vol 251, no 1, pp 217–229, . 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