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Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2008, Article ID 745010, 7 pages doi:10.1155/2008/745010 Research Article Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals Satit Saejung Department of Mathematics, Faculty of Science, Khon Kaen University, Khon Kaen 40002, Thailand Correspondence should be addressed to Satit Saejung, saejung@kku.ac.th Received 28 November 2007; Revised 15 January 2008; Accepted 30 January 2008 Recommended by William A. Kirk We prove a convergence theorem by the new iterative method introduced by Takahashi et al. 2007. Our result does not use Bochner integrals so it is different from that by Takahashi et al. We also cor- rect the strong convergence theorem recently proved by He and Chen 2007. Copyright q 2008 Satit Saejung. 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 H be a real Hilbert space with the inner product ·, · and the norm ·.Let{Tt : t ≥ 0} be a family of mappings from a subset C of H into itself. We call it a nonexpansive semigroup on C if the following conditions are satisfied: 1 T0x x for all x ∈ C; 2 Ts tTsTt for all s, t ≥ 0; 3 for each x ∈ C the mapping t → Ttx is continuous; 4 Ttx − Tty≤x − y for all x, y ∈ C and t ≥ 0. Motivated by Suzuki’s result 1 and Nakajo-Takahashi’s results 2,HeandChen3 recently proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hy- brid method in the mathematical programming. However, their proof of the main result 3, Theorem 2.3 is very questionable. Indeed, the existence of the subsequence {s j } such that 2.16 of 3 are satisfied, that is, s j −→ 0, x j − T s j x j s j −→ 0, 1.1 needs to be proved precisely. So, the aim of this short paper is to correct He-Chen’s result and also to give a new result by using the method recently introduced by Takahashi et al. 2 Fixed Point Theory and Applications We need the following lemma proved by Suzuki 4, Lemma 1. Lemma 1.1. Let {t n } be a real sequence and let τ be a real number such that lim inf n t n ≤ τ ≤ lim sup n t n . Suppose that either of the following holds: i lim sup n t n1 − t n ≤ 0,or ii lim inf n t n1 − t n ≥ 0. Then τ is a cluster point of {t n }. Moreover, for ε>0, k, m ∈ N,thereexistsm 0 ≥ m such that |t j −τ| <ε for every integer j with m 0 ≤ j ≤ m 0 k. 2. Results 2.1. The shrinking projection method The following method is introduced by Takahashi et al. in 5. We use this method to approx- imate a common fixed point of a nonexpansive semigroup without Bochner integrals as was the case in 5, Theorem 4.4. Theorem 2.1. Let C be a closed convex subset of a real Hilbert space H.Let{Tt : t ≥ 0} be a nonexpansive semigroup on C with a nonempty common fixed point F,thatis,F ∩ t≥0 FTt / ∅. Suppose that {x n } is a sequence iteratively generated by the following scheme: x 0 ∈ H taken arbitrary, C 1 C, x 1 P C 1 x 0 , y n α n x n 1 − α n T t n x n , C n1 z ∈ C n : y n − z ≤ x n − z , x n1 P C n1 x 0 . 2.1 where {α n }⊂0,a ⊂ 0, 1, lim inf n t n 0, lim sup n t n > 0,andlim n t n1 − t n 0.Thenx n → P F x 0 . Proof. It is well known that F is closed and convex. We first show that the iterative scheme is well defined. To see that each C n is nonempty, it suffices to show that F ⊂ C n . The proof is by induction. Clearly, F ⊂ C 1 . Suppose that F ⊂ C k . Then, for z ∈ F ⊂ C k , y k − z ≤ α k x k − z 1 − α k T t k x k − z ≤ α k x k − z 1 − α k x k − z x k − z. 2.2 That is, z ∈ C k1 as required. Notice that C n : z ∈ H : y n − z ≤ x n − z 2.3 Satit Saejung 3 is convex since y n − z ≤ x n − z ⇐⇒ 2 x n − y n ,z ≤ x n 2 − y n 2 . 2.4 This implies that each subset C n C ∩ C 1 ∩···∩ C n−1 is convex. It is also clear that C n is closed. Hence the first claim is proved. Next, we prove that {x n } is bounded. As x n P C n x 0 , x n − x 0 ≤ z − x 0 ∀z ∈ C n . 2.5 In particular, for z ∈ F ⊂ C n for all n ∈ N, the sequence {x n − x 0 } is bounded and hence so is {x n }. Next, we show that {x n } is a Cauchy sequence. As x n1 ∈ C n1 ⊂ C n and x n P C n x 0 , x n − x 0 ≤ x n1 − x 0 ∀n. 2.6 Moreover, since the sequence {x n } is bounded, lim n→∞ x n − x 0 exists. 2.7 Note that x 0 − x n ,x n − v ≥ 0 ∀v ∈ C n . 2.8 In particular, since x nk ∈ C nk ⊂ C n for all k ∈ N, x nk − x n 2 x nk − x 0 2 − x n − x 0 2 − 2 x nk − x n ,x n − x 0 ≤ x nk − x 0 2 − x n − x 0 2 . 2.9 It then follows from the existence of lim n x n − x 0 2 that {x n } is a Cauchy sequence. In fact, for ε>0, there exists a natural number N such that, for all n ≥ N, x n − x 0 2 − a < ε 2 , 2.10 where a lim n x n − x 0 2 . In particular, if n ≥ N and k ∈ N,then x nk − x n 2 ≤ x nk − x 0 2 − x n − x 0 2 ≤ a ε 2 − a − ε 2 ε. 2.11 Moreover, x n1 − x n −→ 0. 2.12 4 Fixed Point Theory and Applications We now assume that x n → p for some p ∈ C. Now since α n ≤ a<1 for all n ∈ N and x n1 ∈ C n , x n − T t n x n 1 1 − α n y n − x n ≤ 1 1 − a y n − x n1 x n1 − x n ≤ 2 1 − a x n1 − x n −→ 0. 2.13 The last convergence follows from 2.12. We choose a sequence {t n k } of positive real number such that t n k −→ 0, 1 t n k x n k − T t n k x n k −→ 0. 2.14 We now show that h ow such a special subsequence can be constructed. First we fix δ>0 such that lim inf n t n 0 <δ<lim sup n t n . 2.15 From 2.13, there exists m 1 ∈ N such that Tt n x n − x n < 1/3 2 for all n ≥ m 1 .ByLemma 1.1, δ/2 is a cluster point of {t n }. In particular, there exists n 1 >m 1 such that δ/3 <t n 1 <δ.Next, we choose m 2 >n 1 such that Tt n x n − x n < 1/4 2 for all n ≥ m 2 . Again, by Lemma 1.1, δ/3 is a cluster point of {t n } and this implies that there exists n 2 >m 2 such that δ/4 <t n 2 <δ/2. Continuing in this way, we obtain a subsequence {n k } of {n} satisfying T t n k x n k − x n k < 1 k 2 2 , δ k 2 <t n k < δ k ∀k ∈ N. 2.16 Consequently, 2.14 is satisfied. We next show that p ∈ F.Toseethis,wefixt>0, x n k − Ttp ≤ t/t n k −1 j0 T jt n k x n k − T j 1t n k x n k T t t n k t n k x n k − T t t n k t n k p T t t n k t n k p − Ttp ≤ t t n k x n k − Tt n k x n k x n k − p T t − t t n k t n k p − p ≤ t t n k x n k − T t n k x n k x n k − p sup Tsp − p :0≤ s ≤ t n k . 2.17 As x n k → p and 2.14,wehavex n k → Ttp and so Ttp p. Finally, we show that p P F x 0 . Since F ⊂ C n1 and x n1 P C n1 x 0 , x n1 − x 0 ≤ q − x 0 ∀n ∈ N,q∈ F. 2.18 Satit Saejung 5 But x n → p;wehave p − x 0 ≤ q − x 0 ∀q ∈ F. 2.19 Hence p P F x 0 as required. This completes the proof. 2.2. The hybrid method We consider the iterative scheme computing by the hybrid method some authors call the CQ- method. The following result is proved by He and Chen 3. However, the important part of the proof seems to be overlooked. Here we present the correction under some additional restriction on the parameter {t n }. Theorem 2.2. Let C be a closed convex subset of a real Hilbert space H.Let{Tt : t ≥ 0} be a nonexpansive semigroup on C with a nonempty common fixed point F,thatis,F ∩ t≥0 FTt / ∅. Suppose that {x n } is a sequence iteratively generated by the following scheme: x 0 ∈ C taken arbitrary, y n α n x n 1 − α n T t n x n , C n z ∈ C : y n − z ≤ x n − z , Q n z ∈ C : x n − x 0 ,z− x n ≥ 0 , x n1 P C n ∩Q n x 0 , 2.20 where {α n }⊂0,a ⊂ 0, 1, lim inf n t n 0, lim sup n t n > 0,andlim n t n1 − t n 0.Thenx n → P F x 0 . Proof. For the sake of clarity, we give the whole sketch proof even though some parts of the proof are the same as 3. To see that the scheme is well defined, it suffices to show that both C n and Q n are closed and convex, and C n ∩ Q n / ∅ for all n ∈ N. It follows easily from the definition that C n and Q n are just the intersection of C and the half-spaces, respectively, C n : z ∈ H :2 x n − y n ,z ≤ x n 2 − y n 2 , Q n : z ∈ H : x n − x 0 ,z− x n ≥ 0 . 2.21 As in the proof of the p receding theorem, we have F ⊂ C n for all n ∈ N. Clearly, F ⊂ C Q 1 . Suppose that F ⊂ Q k for some k ∈ N,wehavep ∈ C k ∩Q k . In particular, x k1 −x 0 ,p−x k1 ≥0, that is, p ∈ Q k1 . It follows from the induction that F ⊂ Q n for all n ∈ N. This proves the claim. We next show that x n − Tt n x n → 0. To see this, we first prove that x n1 − x n −→ 0. 2.22 As x n1 ∈ Q n and x n P Q n x 0 , x n − x 0 ≤ x n1 − x 0 ∀n ∈ N. 2.23 6 Fixed Point Theory and Applications For fixed z ∈ F. It follows from F ⊂ Q n for all n ∈ N that x n − x 0 ≤z − x 0 ∀n ∈ N. 2.24 This implies that sequence {x n } is bounded and lim n→∞ x n − x 0 exists. 2.25 Notice that x n1 − x n ,x n − x 0 ≥ 0. 2.26 This implies that x n1 − x n 2 x n1 − x 0 2 − x n − x 0 2 − 2 x n1 − x n ,x n − x 0 ≤ x n1 − x 0 2 − x n − x 0 2 −→ 0. 2.27 It then follows from x n1 ∈ C n that y n − x n1 ≤x n − x n1 and hence T t n x n − x n 1 α n y n − x n ≤ 1 α n y n − x n1 x n1 − x n −→ 0. 2.28 As in Theorem 2.1, we can choose a subsequence {n k } of {n} such that x n k w −−−→ p ∈ C, t n k −→ 0, 1 t n k x n k − T t n k x n k −→ 0. 2.29 Consequently, for any t>0, x n k − Ttp ≤ t t n k x n k − T t n k x n k x n k − p sup Tsp − p :0≤ s ≤ t n k . 2.30 This implies that lim sup k→∞ x n k − Ttp ≤ lim sup k→∞ x n k − p . 2.31 In virtue of Opial’s condition of H,wehavep Ttp for all t>0, that is, p ∈ F.Next,we observe that x 0 − P F x 0 ≤ x 0 − p ≤ lim inf k→∞ x 0 − x n k ≤ lim sup k→∞ x 0 − x n k ≤ x 0 − P F x 0 . 2.32 This implies that lim k→∞ x 0 − x n k x 0 − P F x 0 x 0 − p . 2.33 Consequently, x n k −→ P F x 0 p. 2.34 Hence the whole sequence must converge to P F x 0 p, as required. Satit Saejung 7 Acknowledgments The author would like to thank the referees for his comments and suggestions on the manuscript. This work is supported by the Commission on Higher Education and the Thai- land Research Fund Grant MRG4980022. References 1 T. Suzuki, “On strong convergence to common fixed points of nonexpansive semigroups in Hilbert spaces,” Proceedings of the American Mathematical Society, vol. 131, no. 7, pp. 2133–2136, 2003. 2 K. Nakajo and W. Takahashi, “Strong convergence theorems for nonexpansive mappings and non- expansive semigroups,” Journal of Mathematical Analysis and Applications, vol. 279, no. 2, pp. 372–379, 2003. 3 H. He and R. Chen, “Strong convergence theorems of the CQ method for nonexpansive semigroups,” Fixed Point Theory and Applications, vol. 2007, Article ID 59735, 8 pages, 2007. 4 T. Suzuki, “Strong convergence of Krasnoselskii and Mann’s type sequences for one-parameter non- expansive semigroups without Bochner integrals,” Journal of Mathematical Analysis and Applications, vol. 305, no. 1, pp. 227–239, 2005. 5 W. Takahashi, Y. Takeuchi, and R. Kubota, “Strong convergence theorems by hybrid methods for fam- ilies of nonexpansive mappings in Hilbert spaces,” Journal of Mathematical Analysis and Applications, vol. 341, no. 1, pp. 276–286, 2007. . Applications Volume 2008, Article ID 745010, 7 pages doi:10.1155/2008/745010 Research Article Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals Satit Saejung Department. Chen, Strong convergence theorems of the CQ method for nonexpansive semigroups, ” Fixed Point Theory and Applications, vol. 2007, Article ID 59735, 8 pages, 2007. 4 T. Suzuki, Strong convergence. no. 7, pp. 2133–2136, 2003. 2 K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and non- expansive semigroups, ” Journal of Mathematical Analysis and Applications,
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