Strong Convergence Theorems for Nonexpansive Semigroups without Bochner Integrals
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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, [email protected] 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 correct 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 {T t : 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 T 0x x for all x ∈ C; 2 T s t T sT t for all s, t ≥ 0; 3 for each x ∈ C the mapping t → T tx is continuous; 4 T tx − T ty ≤ x − y for all x, y ∈ C and t ≥ 0. Motivated by Suzuki’s result 1 and Nakajo-Takahashi’s results 2, He and Chen 3 recently proved a strong convergence theorem for nonexpansive semigroups in Hilbert spaces by hybrid 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 {sj } such that 2.16 of 3 are satisfied, that is, x j − T s j xj sj −→ 0, −→ 0, 1.1 sj 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.
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Fixed Point Theory and Applications We need the following lemma proved by Suzuki 4, Lemma 1.
Lemma 1.1. Let {tn } be a real sequence and let τ be a real number such that lim infn tn ≤ τ ≤ lim supn tn . Suppose that either of the following holds: i lim supn tn1 − tn ≤ 0, or ii lim infn tn1 − tn ≥ 0. Then τ is a cluster point of {tn }. Moreover, for ε > 0, k, m ∈ N, there exists m0 ≥ m such that |tj −τ| < ε for every integer j with m0 ≤ j ≤ m0 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 approximate 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 {T t : t ≥ 0} be a nonexpansive semigroup on C with a nonempty common fixed point F, that is, F ∩t≥0 FT t / ∅. Suppose that {xn } is a sequence iteratively generated by the following scheme: x0 ∈ H taken arbitrary, C1 C, x1 PC1 x0 , yn αn xn
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