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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 962303, 7 pages doi:10.1155/2009/962303 Research Article Strong Convergence Theorems for Infinitely Nonexpansive Mappings in Hilbert Space Yi-An Chen College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China Correspondence should be addressed to Yi-An Chen, chenyian1969@sohu.com Received 23 June 2009; Accepted 12 October 2009 Recommended by Anthony To Ming Lau We introduce a modified Ishikawa iterative process for approximating a fixed point of two infinitely nonexpansive self-mappings by using the hybrid method in a Hilbert space and prove that the modified Ishikawa iterative sequence converges strongly to a common fixed point of two infinitely nonexpansive self-mappings. Copyright q 2009 Yi-An Chen. 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 C be a nonempty closed convex subset of a Hilbert space H, T a self-mapping of C. Recall that T is said to be nonexpansive if Tx − Ty≤x − y, for all x, y ∈ C. Construction of fixed points of nonexpansive mappings via Mann’s iteration 1 has extensively been investigated in literature see, e.g., 2–5 and reference therein.Butthe convergence about Mann’s iteration and Ishikawa’s iteration is in general not strong see the counterexample in 6. In order to get strong convergence, one must modify them. In 2003, Nakajo and Takahashi 7 proposed such a modification for a nonexpansive mapping T. Consider the algorithm, x 0 ∈ C chosen arbitrarity, y n  α n x n   1 − α n  Tx n , C n   v ∈ C :   y n − v   ≤  x n − v   , Q n  { v ∈ C :  x n − v, x n − x 0  ≤ 0 } , x n1  P C n ∩Q n  x 0  , 1.1 2 Fixed Point Theory and Applications where P C denotes the metric projection from H onto a closed convex subset C of H. They prove the sequence {x n } generated by that algorithm 1.1 converges strongly to a fixed point of T provided that the control sequence {α n } is chosen so that sup n≥0 α n < 1. Let {T n } ∞ n1 be a sequence of nonexpansive self-mappings of C, {λ n } ∞ n1 a sequence of nonnegative numbers in 0, 1. For each n ≥ 1, defined a mapping W n of C into itself as follows: U n,n1  I, U n,n  λ n T n U n,n1   1 − λ n  I, U n,n−1  λ n−1 T n−1 U n,n   1 − λ n−1  I, . . . U n,k  λ k T k U n,k1   1 − λ k  I, U n,k−1  λ k−1 T k−1 U n,k   1 − λ k−1  I, . . . U n,2  λ 2 T 2 U n,3   1 − λ 2  I, W n  U n,1  λ 1 T 1 U n,2   1 − λ 1  I. 1.2 Such a mapping W n is called the W-mapping generated by T n ,T n−1 , ,T 1 and λ n ,λ n−1 , ,λ 1 ; see 8. In this paper, motivated by 9, for any given x i ∈ C i  0, 1, ,q,q ∈ N is a fixed number, we will propose the following iterative progress for two infinitely nonexpansive mappings {T 1 n } and {T 2 n } in a Hilbert space H: x 0 ,x 1 , ,x q ∈ C chosen arbitrarity, y n  α n x n   1 − α n  W  1  n z n−q , z n  α n x n   1 − α n  W  2  n x n , C n   v ∈ K :   y n − v   2 ≤  x n − v  2   1 − α n     x n−q − x ∗   2 −  x n − x ∗  2  , Q n   v ∈ K :  x n − v, x n − x q  ≤ 0  , x n1  P C n ∩Q n  x q  ,n≥ q 1.3 and prove, {x n } converges strongly to a fixed point of {T 1 n } and {T 2 n }. We will use the notation:  for weak convergence and → for strong convergence. ω w x n {x : ∃x n j x} denotes the weak ω-limit set of x n . Fixed Point Theory and Applications 3 2. Preliminaries In this paper, we need some facts and tools which are listed as lemmas below. Lemma 2.1 see 10. Let H be a Hilbert space, C a nonempty closed convex subset of H, and T a nonexpansive mapping with FixT /  ∅.If{x n } is a sequence in C weakly converging to x and if {I − Tx n } converges strongly to y,thenI − Tx  y. Lemma 2.2 see 11. Let C be a nonempty bounded closed convex subset of a Hilbert space H. Given also a real number a ∈ R and x, y, z ∈ H. Then the set D : {v ∈ C : y − v 2 ≤x − v 2  z, v  a} is closed and convex. Let {T n } ∞ n1 be a sequence of nonexpansive self-mappings on C,whereC is a nonempty closed convex subset of a strictly convex Banach space E. Given a sequence {λ n } ∞ n1 in 0, 1, one defines a sequence {W n } ∞ n1 of self-mappings on C by 1.2. Then one has the following results. Lemma 2.3 see 8. Let C be a nonempty closed convex subset of a strictly convex Banach space E, {T n } ∞ n1 a sequence of nonexpansive self-mappings on C such that  ∞ n1 FT n  /  ∅ and let {λ n } be a sequence in 0,b for some b ∈ 0, 1. Then, for every x ∈ C and k ≥ 1 the limit lim n →∞ U n,k x exists. Remark 2.4. It can be known from Lemma 2.3 that if D is a nonempty bounded subset of C, then for ε>0 there exists n 0 ≥ k such that sup x∈D U n,k x − U k x≤ε for all n>n 0 . Remark 2.5. Using Lemma 2.3, we can define a mapping W : C → C as follows: Wx  lim n →∞ W n x  lim n →∞ U n,1 x 2.1 for all x ∈ C. Such a W is called the W-mapping generated by T 1 ,T 2 , and λ 1 ,λ 2 , Since W n is nonexpansive mapping, W : C → C is also nonexpansive. Indeed, observe that for each x, y ∈ C,   Wx − Wy    lim n →∞   W n x − W n y   ≤   x − y   . 2.2 If {x n } is a bounded sequence in C, then we put D  {x n : n ≥ 0}. Hence, it is clear from Remark 2.4 that for ε>0 there exists N 0 ≥ 1 such that for all n>N 0 , W n x n − Wx n   U n,1 x n − U 1 x n ≤sup x∈D U n,1 x − U 1 x≤ε. This implies that lim n →∞  W n x n − Wx n   0. 2.3 Lemma 2.6 see 8. Let C be a nonempty closed convex subset of a strictly convex Banach space E. Let {T n } ∞ n1 be a sequence of nonexpansive self-mappings on C such that  ∞ n1 FT n  /  ∅ and let {λ n } be a sequence in 0,b for some b ∈ 0, 1. Then, FW  ∞ n1 FT n . 4 Fixed Point Theory and Applications 3. Strong Convergence Theorem Theorem 3.1. Let C be a closed convex subset of a Hilbert space H and let {W 1 n } and {W 2 n } be defined as 1.2. Assume that α n ≤ a for all n and for some 0 <a<1, and {α n }∈b, c for all n and 0 <b<c<1.IfF   ∞ n1 FT 1 n   FT 2 n  /  ∅, then {x n } generated by 1.3 converges strongly to P F x q . Proof. Firstly, we observe that C n is convex by Lemma 2.2. Next, we show that F ⊂ C n for all n. Indeed, for all x ∗ ∈ F,   y n − x ∗   2 ≤ α n  x n − x ∗  2   1 − α n    z n−q − x ∗   2   x n − x ∗  2   1 − α n     z n−q − x ∗   2 −  x n − x ∗  2  ,   z n−q − x ∗   2     α n−q x n−q   1 − α n−q  W 2 n−q x n−q − x ∗     α n−q   x n−q − x ∗   2   1 − α n−q     W 2 n−q x n−q − x ∗    2 − α n−q  1 − α n−q     W 2 n−q x n−q − x n−q    2 ≤ α n−q   x n−q − x ∗   2   1 − α n−q    x n−q − x ∗   2 − α n−q  1 − α n−q     W 2 n−q x n−q − x n−q    2    x n−q − x ∗   2 − α n−q  1 − α n−q     W 2 n−q x n−q − x n−q    2 ≤   x n−q − x ∗   2 . 3.1 Therefore,   y n − x ∗   2 ≤  x n − x ∗  2   1 − α n     x n−q − x ∗   2 −  x n − x ∗  2  . 3.2 That is x ∗ ∈ C n for all n ≥ q. Next we show that F ⊂ Q n for all n ≥ q. We prove this by induction. For n  q, we have F ⊂ C  Q q . Assume that F ⊂ Q n for all n ≥ q  1, since x n1 is the projection of x q onto C n  Q n , so x n1 − z, x q − x n1 ≥0, ∀z ∈ C n  Q n . 3.3 As F ⊂ C n  Q n by the induction assumption, the last inequality holds, in particular, for all x ∗ ∈ F. This together with definition of Q n1 implies that F ⊂ Q n1 . Hence F ⊂ C n  Q n for all n ≥ q. Notice that the definition of Q n implies x n  P Q n x q . This together with the fact F ⊂ Q n further implies x n − x q ≤x ∗ − x q  for all x ∗ ∈ F. Fixed Point Theory and Applications 5 The fact x n1 ∈ Q n asserts that x n1 − x n ,x n − x q ≥0 implies  x n1 − x n  2    x n1 − x q  − x n − x q    2    x n1 − x q   2 −   x n − x q   2 − 2x n1 − x n ,x n − x q  ≤   x n1 − x q   2 −   x n − x q   2 −→ 0  n −→ ∞  . 3.4 We now claim that W 1 x n − x n →0andW 2 x n − x n →0. Indeed,    x n − W 1 n z n−q       x n − y n   1 − α n ≤  x n − x n1     x n1 − y n   1 − α n , 3.5 since x n1 ∈ C n , we have   y n − x n1   2 ≤  x n − x n1  2   1 − α n     x n−q − x ∗   2 −  x n − x ∗  2  −→ 0. 3.6 Thus    x n − W 1 n z n−q    −→ 0. 3.7 We now show lim n →∞ W 2 n x n − x n   0. Let {W 2 n k x n k − x n k } be any subsequence of {W 2 n x n − x n }. Since C is a bounded subset of H, there exists a subsequence {x n k j } of {x n k } such that lim j →∞    x n k j − x ∗     lim sup k →∞  x n k − x ∗  : r. 3.8 Since    x n k j − x ∗    ≤    x n k j − W 1 n k j z n k j −q        W 1 n k j z n k j −q − x ∗    ≤    x n k j − W 1 n k j z n k j −q        z n k j −q − x ∗    , 3.9 it follows that r  lim j →∞ x n k j − x ∗ ≤lim inf j →∞ z n k j − x ∗ .By3.1, we have    z n k j − x ∗    ≤    x n k j − x ∗    2 . 3.10 Hence lim sup j →∞    z n k j − x ∗    ≤ lim j →∞    x n k j − x ∗     r. 3.11 6 Fixed Point Theory and Applications Thus, lim j →∞    z n k j − x ∗     r  lim j →∞    x n k j − x ∗    . 3.12 Using 3.1 again, we obtain that α n k j −q  1 − α n k j −q     W 2 n k j −q x n k j −q − x n k j −q    2 ≤    x n k j −q − x ∗    2 −    z n k j −q − x ∗    2 −→ 0. 3.13 This imply that lim j →∞ W 2 n k j x n k j − x n k j   0. For the arbitrariness of {x n k }⊂{x n }, we have lim n →∞ W 2 n x n − x n   0and  z n − x n    1 − α n     W 2 n x n − x n    −→ 0. 3.14 Thus, by 3.4, 3.7 and 3.14, we have    W 1 n x n − x n    ≤    W 1 n x n − W 1 n z n−q        W 1 n z n−q − x n    ≤   z n−q − x n       W 1 n z n−q − x n    ≤    W 1 n z n−q − x n       z n−q − x n−q      x n−q − x n−q1      x n−q1 − x n−q2    ···  x n−1 − x n  −→ 0. 3.15 Since lim n →∞ W 1 n x n − W 1 x n   0 and lim n →∞ W 2 n x n − W 2 x n   0, we have lim n →∞    W 1 x n − x n     0, lim n →∞    W 2 x n − x n     0. 3.16 Thus, using 3.16, Lemma 2.1, and the boundedness of {x n },wegetthat∅ /  ω w x n  ⊂ F. Since x n  P Q n x q  and F ⊂ Q n , we have x n −x q ≤x ∗ −x q  where x ∗ : P F x q . By the weak lower semicontinuity of the norm, we have w −x q ≤x ∗ − x q  for all w ∈ ω w x n . However, since ω w x n  ⊂ F, we must have w  x ∗ for all w ∈ ω w x n . Hence x n x ∗  P F x q  and  x n − x ∗  2    x n − x q   2  2x n − x q ,x q − x ∗     x q − x ∗   2 ≤ 2    x ∗ − x q   2   x n − x q ,x q − x ∗   −→ 0. 3.17 That is, {x n } converges to P F x q . This completes the proof. Fixed Point Theory and Applications 7 Acknowledgment This work is supported by Grant KJ080725 of the Chongqing Municipal Education Commission. References 1 W. R. Mann, “Mean value methods in iteration,” Proceedings of the American Mathematical Society, vol. 4, pp. 506–510, 1953. 2 S. Ishikawa, “Fixed points by a new iteration method,” Proceedings of the American Mathematical Society, vol. 44, pp. 147–150, 1974. 3 L. Deng, “On Chidume’s open questions,” Journal of Mathematical Analysis and Applications, vol. 174, no. 2, pp. 441–449, 1993. 4 C. E. 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Liu, “Iterative scheme for nonself generalized asymptotically quasi-nonexpansive mappings,” Applied Mathematics and Computation, vol. 205, no. 1, pp. 317–324, 2008. 10 K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. 11 C. Martinez-Yanes and H K. Xu, “Strong convergence of the CQ method for fixed point iteration processes,” Nonlinear Analysis: Theory, Methods & Applications, vol. 64, no. 11, pp. 2400–2411, 2006. . Anthony To Ming Lau We introduce a modified Ishikawa iterative process for approximating a fixed point of two in nitely nonexpansive self -mappings by using the hybrid method in a Hilbert space and. Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009, Article ID 962303, 7 pages doi:10.1155/2009/962303 Research Article Strong Convergence Theorems for In nitely Nonexpansive. In nitely Nonexpansive Mappings in Hilbert Space Yi-An Chen College of Mathematics and Statistics, Chongqing Technology and Business University, Chongqing 400067, China Correspondence should be addressed to Yi-An