Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 948529, 8 pages
doi:10.1155/2010/948529
Research Article

Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions

Ci-Shui Ge,1 Jin Liang,2 and Ti-Jun Xiao3

1Department of Mathematics and Physics, Anhui University of Architecture, Hefei, Anhui 230022, China
2Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China
3School of Mathematical Sciences, Fudan University, Shanghai 200433, China

Received 8 October 2009; Revised 29 November 2009; Accepted 22 January 2010

Academic Editor: Anthony To Ming Lau

Copyright © 2010 Ci-Shui Ge et al. 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.

Abstract

We introduce and study some new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. General results for asymptotically strict pseudocontractions are established. The main result extends the previous results.

1. Introduction

Let 𝐻 be a real Hilbert space, 𝐶 a nonempty closed convex subset of 𝐻 , 𝑇 ∶ 𝐶 → 𝐶 a self-mapping of 𝐶 and F i x ( 𝑇 ) ∶ = { 𝑥 ∈ 𝐶 ∶ 𝑇 𝑥 = 𝑥 } .

Recall that a mapping 𝑇 ∶ 𝐶 → 𝐶 is called to be nonexpansive if

‖ 𝑇 𝑥 − 𝑇 𝑦 ‖ ≤ ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐶 . ( 1 . 1 ) 𝑇 is called to be asymptotically nonexpansive [1] if there exists a sequence { 𝑘 𝑛 } with 𝑘 𝑛 ≥ 1 and l i m 𝑛 → ∞ 𝑘 𝑛 = 1 such that

‖ 𝑇 𝑛 𝑥 − 𝑇 𝑛 𝑦 ‖ ≤ 𝑘 𝑛 ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐶 , a n d a l l i n t e g e r s 𝑛 ≥ 1 . ( 1 . 2 ) 𝑇 is called to be an asymptotically 𝜅 -strict pseudocontraction, if there exist 0 ≤ 𝜅 < 1 a n d 0 ≤ 𝛾 𝑛 → 0 ( 𝑛 → ∞ ) such that

‖ 𝑇 𝑛 𝑥 − 𝑇 𝑛 𝑦 ‖ 2 ≤ 1 + 𝛾 𝑛 ‖ 𝑥 − 𝑦 ‖ 2 + 𝜅 ‖ ( 𝐼 − 𝑇 𝑛 ) 𝑥 − ( 𝐼 − 𝑇 𝑛 ‖ ) 𝑦 2 ( 1 . 3 ) for all 𝑥 , 𝑦 ∈ 𝐶 and all integers 𝑛 ≥ 1 .

As 𝜅 = 0 , asymptotically 𝜅 -strict pseudocontraction 𝑇 is asymptotically nonexpansive.

In [2], Nakajo and Takahashi studied the iterative approximation of fixed points of nonexpansive mappings and proved the following strong convergence theorem.

Theorem 1 A. Let 𝐶 be a nonempty closed convex subset of a Hilbert space 𝐻 and let 𝑇 be a nonexpansive mapping of 𝐶 into itself such that F i x ( 𝑇 ) ≠ ∅ . Suppose { 𝑥 𝑛 } is given by 𝑥 0 𝑦 ∈ 𝐶 c h o s e n a r b i t r a r i l y , 𝑛 = 𝛼 𝑛 𝑥 𝑛 + 1 − 𝛼 𝑛 𝑇 𝑥 𝑛 , 𝐶 𝑛 = ‖ ‖ 𝑦 𝑧 ∈ 𝐶 ∶ 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 − 𝑧 𝑛 ‖ ‖ , 𝑄 − 𝑧 𝑛 = 𝑧 ∈ 𝐶 ∶ ⟨ 𝑥 𝑛 − 𝑧 , 𝑥 0 − 𝑥 𝑛 , 𝑥 ⟩ ≥ 0 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝑄 𝑛 𝑥 0 , 𝑛 ∈ ℕ , ( 1 . 4 ) where 𝑃 𝐶 𝑛 ∩ 𝑄 𝑛 is the metric projection from C onto 𝐶 𝑛 ∩ 𝑄 𝑛 and 𝛼 𝑛 is chosen so that 0 ≤ 𝛼 𝑛 ≤ 𝑎 < 1 . Then, { 𝑥 𝑛 } converges strongly to 𝑃 F i x ( 𝑇 ) 𝑥 0 , where 𝑃 F i x ( 𝑇 ) is the metric projection from C onto F i x ( 𝑇 ) .

Such algorithm in (1.4) is referred to be the (CQ) algorithm in [3], due to the fact that each iterate 𝑥 𝑛 + 1 is obtained by projecting 𝑥 0 onto the intersection of the suitably constructed closed convex sets 𝐶 𝑛 and 𝑄 𝑛 . It is known that the (CQ) algorithm in (1.4) is of independent interest, and the (CQ) algorithm has been extended to various mappings by many authors (cf., e.g., [3–11]).

Very recently, by extending the (CQ) algorithm, Takahashi et al. [9] studied a family of nonexpansive mappings and gave some good strong convergence theorems. Kim and Xu [5] extended the (CQ) algorithm to study asymptotically 𝜅 -strict pseudocontractions and established the following interesting result with the help of some boundedness conditions.

Theorem 1 B. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and let 𝑇 ∶ 𝐶 → 𝐶 be an asymptotically 𝜅 -strict pseudocontractions for some 0 ≤ 𝜅 < 1 . Assume that the fixed point set F i x ( 𝑇 ) of T is nonempty and bounded. Let { 𝑥 𝑛 } ∞ 𝑛 = 0 be the sequence generated by the following (CQ) algorithm: 𝑥 0 𝑦 ∈ 𝐶 , c h o s e n a r b i t r a r i l y , 𝑛 = 𝛼 𝑛 𝑥 𝑛 + 1 − 𝛼 𝑛 𝑇 𝑛 𝑥 𝑛 , 𝐶 𝑛 = ‖ ‖ 𝑦 𝑧 ∈ 𝐶 ∶ 𝑛 ‖ ‖ − 𝑧 2 ≤ ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑧 2 + 𝜅 − 𝛼 𝑛 1 − 𝛼 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑇 𝑥 𝑛 ‖ ‖ 2 + 𝜃 𝑛 , 𝑄 𝑛 = 𝑧 ∈ 𝐶 ∶ ⟨ 𝑥 𝑛 − 𝑧 , 𝑥 0 − 𝑥 𝑛 , 𝑥 ⟩ ≥ 0 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝑄 𝑛 𝑥 0 , ( 1 . 5 ) where 𝜃 𝑛 = Δ 2 𝑛 1 − 𝛼 𝑛 𝛾 𝑛 ⟶ 0 ( 𝑛 ⟶ ∞ ) , Δ 𝑛 ‖ ‖ 𝑥 = s u p 𝑛 ‖ ‖ − 𝑧 ∶ 𝑧 ∈ F i x ( 𝑇 ) < ∞ . ( 1 . 6 ) Assume that control sequence { 𝛼 𝑛 } ∞ 𝑛 = 0 is chosen so that l i m s u p 𝑛 → ∞ 𝛼 𝑛 < 1 − 𝜅 . Then { 𝑥 𝑛 } converges strongly to 𝑃 F i x ( 𝑇 ) 𝑥 0 .

It is our purpose in this paper to try to obtain some new fixed point theorems for asymptotically strict pseudocontractions without the boundedness conditions as in Theorem B. Motivated by Nakajo and Takahashi [2], Takahashi et al. [9], and Kim and Xu [5], we introduce and study certain new CQ-type iterative algorithms with variable coefficients for asymptotically strict pseudocontractions in real Hilbert spaces. Our results improve essentially the corresponding results of [5].

2. Results and Proofs

Throughout this paper,

(i) 𝑥 𝑛 ⇀ 𝑥 means that { 𝑥 𝑛 } converges weakly to 𝑥 . (ii) 𝑥 𝑛 → 𝑥 means that { 𝑥 𝑛 } converges strongly to 𝑥 . (iii) 𝜔 𝑤 ( 𝑥 𝑛 ) ∶ = { 𝑥 ∶ ∃ 𝑥 𝑛 𝑗 ⇀ 𝑥 } , that is, the weak 𝜔 -limit set of { 𝑥 𝑛 } . (iv) 𝐵 𝑟 ( 𝑥 0 ) ∶ = { 𝑥 ∈ 𝐻 ∶ ‖ 𝑥 − 𝑥 0 ‖ ≤ 𝑟 } .(v) ℕ is the set of nonnegative integers.

The following lemmas are basic (cf., e.g., [6] for Lemma 2.1, and [5] for Lemmas 2.2-2.3).

Lemma 2.1. Let 𝐾 be a closed convex subset of a real Hilbert space 𝐻 . Given 𝑥 ∈ 𝐻 , 𝑧 ∈ 𝐾 . Then 𝑧 = 𝑃 𝐾 𝑥 if and only if ⟨ 𝑥 − 𝑧 , 𝑦 − 𝑧 ⟩ ≤ 0 , ∀ 𝑦 ∈ 𝐾 , ( 2 . 1 ) where 𝑃 𝐾 𝑥 is the unique point in 𝐾 with the property ‖ ‖ 𝑥 − 𝑃 𝐾 𝑥 ‖ ‖ ≤ ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑦 ∈ 𝐾 . ( 2 . 2 )

Lemma 2.2. Let 𝐾 be a closed convex subset of a real Hilbert space 𝐻 , { 𝑥 𝑛 } ⊂ 𝐻 , 𝑢 ∈ 𝐻 , and 𝑞 = 𝑃 𝐾 𝑢 . Suppose that { 𝑥 𝑛 } satisfies ‖ ‖ 𝑥 𝑛 ‖ ‖ ≤ − 𝑢 ‖ 𝑢 − 𝑞 ‖ , ∀ 𝑛 ∈ ℕ , ( 2 . 3 ) and 𝜔 𝑤 ( 𝑥 𝑛 ) ⊂ 𝐾 . Then 𝑥 𝑛 → 𝑞 .

Lemma 2.3. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 and 𝑇 ∶ 𝐶 → 𝐶 an asymptotically 𝜅 -strict pseudocontraction. Then
( I ) for each 𝑛 ≥ 1 , 𝑇 𝑛 satisfies the Lipschitz condition: ‖ 𝑇 𝑛 𝑥 − 𝑇 𝑛 𝑦 ‖ ≤ 𝐿 𝑛 ‖ 𝑥 − 𝑦 ‖ , ∀ 𝑥 , 𝑦 ∈ 𝐶 , ( 2 . 4 ) where 𝐿 𝑛 = √ 𝜅 + 1 + 𝛾 𝑛 ( 1 − 𝜅 ) , 𝛾 1 − 𝜅 𝑛 i s a s i n ( 1 . 3 ) ; ( 2 . 5 )
( I I ) if { 𝑥 𝑛 } is a sequence in 𝐶 such that 𝑥 𝑛 ⇀ ̃ 𝑥 and l i m s u p 𝑚 → ∞ l i m s u p 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑇 𝑚 𝑥 𝑛 ‖ ‖ = 0 , ( 2 . 6 ) then ( 𝐼 − 𝑇 ) 𝑥 𝑛 ⟶ 0 ⟹ ( 𝐼 − 𝑇 ) ̃ 𝑥 = 0 . ( 2 . 7 ) In particular, 𝑥 𝑛 ⇀ ̃ 𝑥 , ( 𝐼 − 𝑇 ) 𝑥 𝑛 ⟶ 0 ⟹ ( 𝐼 − 𝑇 ) ̃ 𝑥 = 0 . ( 2 . 8 )
( I I I ) F i x ( 𝑇 ) is closed and convex so that the projection 𝑃 F i x ( 𝑇 ) is well defined.

Theorem 2.4. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 , 𝑇 ∶ 𝐶 → 𝐶 an asymptotically 𝜅 -strict pseudocontraction for some 0 ≤ 𝜅 < 1 , and F i x ( 𝑇 ) ≠ ∅ . Let { 𝑥 𝑛 } be the sequence generated by the following CQ-type algorithm with variable coefficients: 𝑥 0 𝑦 ∈ 𝐶 c h o s e n a r b i t r a r i l y , 𝑛 = ̂ 𝛽 1 − 𝑛 𝑥 𝑛 + ̂ 𝛽 𝑛 𝑇 𝑛 𝑥 𝑛 , 𝐶 𝑛 = ‖ ‖ 𝑦 𝑧 ∈ 𝐶 ∶ 𝑛 ‖ ‖ − 𝑧 2 ≤ ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑧 2 + ̂ 𝛽 𝑛 ̂ 𝛽 𝜅 + 𝑛 ‖ ‖ 𝑥 − 1 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 + 𝜃 𝑛 , 𝑄 𝑛 = 𝑧 ∈ 𝐶 ∶ ⟨ 𝑥 𝑛 − 𝑧 , 𝑥 0 − 𝑥 𝑛 , 𝑥 ⟩ ≥ 0 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝑄 𝑛 𝑥 0 , 𝑛 ∈ ℕ , ( 2 . 9 ) where ̂ 𝛽 𝑛 = 𝛽 𝑛 ‖ ‖ 𝑥 1 + 𝑛 − 𝑥 0 ‖ ‖ 2 , 𝛽 𝑛 ∈ 1 2 , 1 , 𝜃 𝑛 = 2 1 + 𝑟 2 0 𝛽 𝑛 𝛾 𝑛 , ( 2 . 1 0 ) the sequence { 𝛽 𝑛 } is chosen so that 𝛽 𝑛 → 1 ( 𝑛 → ∞ ) , the positive real number 𝑟 0 is chosen so that 𝐵 𝑟 0 ( 𝑥 0 ) ∩ F i x ( 𝑇 ) ≠ ∅ , and { 𝛾 𝑛 } is as in (1.3). Then { 𝑥 𝑛 } converges strongly to 𝑃 F i x ( 𝑇 ) 𝑥 0 .

Proof. We divide the proof into five steps.Step 1. We prove that 𝐶 𝑛 ∩ 𝑄 𝑛 is nonempty, convex and closed.
Clearly, both 𝑄 𝑛 and 𝐶 𝑛 are convex and closed, so is 𝐶 𝑛 ∩ 𝑄 𝑛 . Since 𝑇 ∶ 𝐶 → 𝐶 is an asymptotically 𝜅 -strict pseudocontraction, we have by (1.3), ‖ 𝑇 𝑛 𝑥 − 𝑝 ‖ 2 ≤ 1 + 𝛾 𝑛 ‖ 𝑥 − 𝑝 ‖ 2 + 𝜅 ‖ ( 𝐼 − 𝑇 𝑛 ) 𝑥 − ( 𝐼 − 𝑇 𝑛 ) 𝑝 ‖ 2 ≤ 1 + 𝛾 𝑛 ‖ 𝑥 − 𝑝 ‖ 2 + 𝜅 ‖ 𝑥 − 𝑇 𝑛 𝑥 ‖ 2 , ( 2 . 1 1 ) for all 𝑥 ∈ 𝐶 , 𝑝 ∈ F i x ( 𝑇 ) , and all integers 𝑛 ≥ 1 .
By (2.9) and (2.11), we deduce that for each 𝑝 ∈ 𝐵 𝑟 0 ( 𝑥 0 ) ∩ F i x ( 𝑇 ) , 𝑛 ∈ ℕ , ‖ ‖ 𝑦 𝑛 ‖ ‖ − 𝑝 2 = ‖ ‖ ̂ 𝛽 1 − 𝑛 𝑥 𝑛 + ̂ 𝛽 − 𝑝 𝑛 𝑇 𝑛 𝑥 𝑛 ‖ ‖ − 𝑝 2 = ̂ 𝛽 1 − 𝑛 ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑝 2 + ̂ 𝛽 𝑛 ‖ ‖ 𝑇 𝑛 𝑥 𝑛 ‖ ‖ − 𝑝 2 − ̂ 𝛽 𝑛 ̂ 𝛽 1 − 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 = ̂ 𝛽 1 − 𝑛 ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑝 2 + ̂ 𝛽 𝑛 1 + 𝛾 𝑛 ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑝 2 ‖ ‖ 𝑥 + 𝜅 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 − ̂ 𝛽 𝑛 ̂ 𝛽 1 − 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 ≤ ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑝 2 + ̂ 𝛽 𝑛 ̂ 𝛽 𝜅 + 𝑛 ‖ ‖ 𝑥 − 1 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 + 𝛽 𝑛 𝛾 𝑛 2 ‖ ‖ 𝑥 𝑛 − 𝑥 0 ‖ ‖ 2 + ‖ ‖ 𝑥 0 ‖ ‖ − 𝑝 2 ‖ ‖ 𝑥 1 + 𝑛 − 𝑥 0 ‖ ‖ 2 ≤ ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑝 2 + ̂ 𝛽 𝑛 ̂ 𝛽 𝜅 + 𝑛 ‖ ‖ 𝑥 − 1 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 + 2 1 + 𝑟 2 0 𝛽 𝑛 𝛾 𝑛 = ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑝 2 + ̂ 𝛽 𝑛 ̂ 𝛽 𝜅 + 𝑛 ‖ ‖ 𝑥 − 1 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 + 𝜃 𝑛 . ( 2 . 1 2 ) Therefore, 𝐵 𝑟 𝑥 0 ∩ F i x ( 𝑇 ) ⊂ 𝐶 𝑛 , ∀ 𝑛 ∈ ℕ . ( 2 . 1 3 ) Next, we prove by induction that 𝐵 𝑟 0 𝑥 0 ∩ F i x ( 𝑇 ) ⊂ 𝑄 𝑛 , ∀ 𝑛 ∈ ℕ . ( 2 . 1 4 ) Obviously, 𝐵 𝑟 0 ( 𝑥 0 ) ∩ F i x ( 𝑇 ) ⊂ 𝐶 = 𝑄 0 , that is, (2.14) holds for 𝑛 = 0 . Assume that 𝐵 𝑟 0 ( 𝑥 0 ) ∩ F i x ( 𝑇 ) ⊂ 𝑄 𝑛 for some 𝑛 ∈ ℕ . Then, (2.13) implies that 𝐵 𝑟 0 ( 𝑥 0 ) ∩ F i x ( 𝑇 ) ⊂ 𝐶 𝑛 ∩ 𝑄 𝑛 ≠ ∅ and 𝑥 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝑄 𝑛 𝑥 0 is well defined.
By Lemma 2.1, we get ⟨ 𝑥 𝑛 + 1 − 𝑧 , 𝑥 0 − 𝑥 𝑛 + 1 ⟩ ≥ 0 , ∀ 𝑧 ∈ 𝐶 𝑛 ∩ 𝑄 𝑛 . In particular, for each 𝑧 ∈ 𝐵 𝑟 0 ( 𝑥 0 ) ∩ F i x ( 𝑇 ) , we have ⟨ 𝑥 𝑛 + 1 − 𝑧 , 𝑥 0 − 𝑥 𝑛 + 1 ⟩ ≥ 0 . This together with the definition of 𝑄 𝑛 + 1 , the inequality (2.14) holds for 𝑛 + 1 . So (2.14) is true. Step 2. We prove that l i m 𝑛 → ∞ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ = 0 .
By the definition of 𝑄 𝑛 and Lemma 2.1, we get 𝑥 𝑛 = 𝑃 𝑄 𝑛 𝑥 0 . Hence, ‖ ‖ 𝑥 𝑛 − 𝑥 0 ‖ ‖ ≤ ‖ ‖ 𝑝 − 𝑥 0 ‖ ‖ , ∀ 𝑝 ∈ 𝐵 𝑟 0 𝑥 0 ∩ F i x ( 𝑇 ) . ( 2 . 1 5 ) Denoting 𝑀 ∶ = ‖ 𝑥 0 ‖ + ‖ 𝑝 − 𝑥 0 ‖ , we have ‖ 𝑥 𝑛 ‖ ≤ 𝑀 , for all 𝑛 ∈ ℕ , and ‖ ‖ 𝑥 𝑛 − 𝑥 0 ‖ ‖ ≤ ‖ ‖ 𝑞 − 𝑥 0 ‖ ‖ , ∀ 𝑛 ∈ ℕ , ( 2 . 1 6 ) where 𝑞 = 𝑃 F i x ( 𝑇 ) 𝑥 0 ⊂ 𝐵 𝑟 0 ( 𝑥 0 ) ∩ F i x ( 𝑇 ) . The definition of 𝑥 𝑛 + 1 shows that 𝑥 𝑛 + 1 ∈ 𝑄 𝑛 , that is, ⟨ 𝑥 𝑛 + 1 − 𝑥 𝑛 , 𝑥 𝑛 − 𝑥 0 ⟩ ≥ 0 . This implies that ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ 2 = ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 0 ‖ ‖ 2 − ‖ ‖ 𝑥 𝑛 − 𝑥 0 ‖ ‖ 2 𝑥 − 2 𝑛 + 1 − 𝑥 𝑛 , 𝑥 𝑛 − 𝑥 0 ≤ ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 0 ‖ ‖ 2 − ‖ ‖ 𝑥 𝑛 − 𝑥 0 ‖ ‖ 2 . ( 2 . 1 7 ) Thus { ‖ 𝑥 𝑛 − 𝑥 0 ‖ } is increasing. Since { 𝑥 𝑛 } is bounded, l i m 𝑛 → ∞ ‖ 𝑥 𝑛 − 𝑥 0 ‖ exists and l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ = 0 . ( 2 . 1 8 ) Step 3. We prove that l i m 𝑛 → ∞ ‖ 𝑥 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ = 0 .
The definition of 𝑥 𝑛 + 1 shows that 𝑥 𝑛 + 1 ∈ 𝐶 𝑛 , that is, ‖ ‖ 𝑦 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ 2 ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ 2 + ̂ 𝛽 𝑛 ̂ 𝛽 𝜅 + 𝑛 ‖ ‖ 𝑥 − 1 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 + 𝜃 𝑛 . ( 2 . 1 9 ) By (2.19) and the definition of 𝑦 𝑛 in (2.9), we deduce that ̂ 𝛽 2 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 = ‖ ‖ 𝑦 𝑛 − 𝑥 𝑛 ‖ ‖ 2 ≤ ‖ ‖ 𝑦 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ 2 + ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ 2 ‖ ‖ 𝑦 + 2 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ ⋅ ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ ≤ ̂ 𝛽 𝑛 ̂ 𝛽 𝜅 + 𝑛 ‖ ‖ 𝑥 − 1 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 + 𝜃 𝑛 ‖ ‖ 𝑥 + 2 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ 2 ‖ ‖ 𝑦 + 2 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ ⋅ ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ . ( 2 . 2 0 ) Further, we have ̂ 𝛽 ( 1 − 𝜅 ) 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 ‖ ‖ 𝑥 ≤ 2 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ 2 ‖ ‖ 𝑥 + 2 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ ⋅ ‖ ‖ 𝑦 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + 𝜃 𝑛 . ( 2 . 2 1 ) Thus, (2.19) and (2.21) imply that ( ̂ 𝛽 1 − 𝜅 ) 𝑛 ‖ ‖ 𝑥 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 ‖ ‖ 𝑥 ≤ 4 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ 2 ‖ ‖ 𝑥 + 2 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ ⋅ ‖ ‖ 𝑥 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ ̂ 𝛽 𝑛 | | ̂ 𝛽 𝜅 + 𝑛 | | ‖ ‖ 𝑥 − 1 + 2 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ √ 𝜃 𝑛 + 𝜃 𝑛 . ( 2 . 2 2 ) Noticing ‖ 𝑥 𝑛 ‖ ≤ 𝑀 , 𝛽 𝑛 ∈ [ 1 / 2 , 1 ] , we get ̂ 𝛽 𝑛 = 𝛽 𝑛 ‖ ‖ 𝑥 1 + 𝑛 ‖ ‖ 2 ≥ 1 2 1 + 𝑀 2 > 0 . ( 2 . 2 3 ) From l i m 𝑛 → ∞ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ = 0 , l i m 𝑛 → ∞ 𝜃 𝑛 = 0 , and (2.22), it follows that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ = 0 . ( 2 . 2 4 ) Step 4. We prove that l i m 𝑛 → ∞ ‖ ‖ 𝑥 𝑛 − 𝑇 𝑥 𝑛 ‖ ‖ = 0 . ( 2 . 2 5 ) By Lemma 2.3 and the definition of 𝑇 , we obtain ‖ ‖ 𝑥 𝑛 − 𝑇 𝑥 𝑛 ‖ ‖ ≤ ‖ ‖ 𝑥 𝑛 − 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝑇 𝑛 + 1 𝑥 𝑛 + 1 ‖ ‖ + ‖ ‖ 𝑇 𝑛 + 1 𝑥 𝑛 + 1 − 𝑇 𝑛 + 1 𝑥 𝑛 ‖ ‖ + ‖ ‖ 𝑇 𝑛 + 1 𝑥 𝑛 − 𝑇 𝑥 𝑛 ‖ ‖ ≤ 1 + 𝐿 𝑛 + 1 ‖ ‖ 𝑥 𝑛 + 1 − 𝑥 𝑛 ‖ ‖ + ‖ ‖ 𝑥 𝑛 + 1 − 𝑇 𝑛 + 1 𝑥 𝑛 + 1 ‖ ‖ + 𝐿 1 ‖ ‖ 𝑥 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ , ( 2 . 2 6 ) where 𝐿 𝑛 = √ 𝜅 + 1 + 𝛾 𝑛 ( 1 − 𝜅 ) , 𝛾 1 − 𝜅 𝑛 i s a s i n ( 1 . 3 ) . ( 2 . 2 7 ) By (2.18), (2.24), and (2.26), we know that (2.25) holds.Step 5. Finally, by Lemma 2.3 and (2.25), we have 𝜔 𝑤 ( 𝑥 𝑛 ) ⊂ F i x ( 𝑇 ) . Furthermore, it follows from (2.16) and Lemma 2.2 that the sequence { 𝑥 𝑛 } converges strongly to 𝑞 = 𝑃 F i x ( 𝑇 ) 𝑥 0 .

Remark 2.5. Theorem 2.4 improves [5, Theorem 4 . 1 ] since the condition that 𝜃 𝑛 → 0 is satisfied and the boundedness of F i x ( 𝑇 ) is dropped off.

Theorem 2.6. Let 𝐶 be a closed convex subset of a Hilbert space 𝐻 , 𝑇 ∶ 𝐶 → 𝐶 an asymptotically 𝜅 -strict pseudocontraction for some 0 ≤ 𝜅 < 1 , and F i x ( 𝑇 ) be nonempty and bounded. Let { 𝑥 𝑛 } the sequence generated by the following CQ-type algorithm with variable coefficients: 𝑥 0 𝑦 ∈ 𝐶 c h o s e n a r b i t r a r i l y , 𝑛 = ̂ 𝛽 1 − 𝑛 𝑥 𝑛 + ̂ 𝛽 𝑛 𝑇 𝑛 𝑥 𝑛 , 𝐶 𝑛 = ‖ ‖ 𝑦 𝑧 ∈ 𝐶 ∶ 𝑛 ‖ ‖ − 𝑧 2 ≤ ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑧 2 + ̂ 𝛽 𝑛 ̂ 𝛽 𝜅 + 𝑛 ‖ ‖ 𝑥 − 1 𝑛 − 𝑇 𝑛 𝑥 𝑛 ‖ ‖ 2 + 𝜃 𝑛 , 𝑄 𝑛 = 𝑧 ∈ 𝐶 ∶ ⟨ 𝑥 𝑛 − 𝑧 , 𝑥 0 − 𝑥 𝑛 , 𝑥 ⟩ ≥ 0 𝑛 + 1 = 𝑃 𝐶 𝑛 ∩ 𝑄 𝑛 𝑥 0 , 𝑛 ∈ ℕ , ( 2 . 2 8 ) where ̂ 𝛽 𝑛 = 𝛽 𝑛 ‖ ‖ 𝑥 1 + 𝑛 − 𝑥 0 ‖ ‖ 2 , 𝛽 𝑛 ∈ 1 2 , 1 , 𝜃 𝑛 = s u p 𝑧 ∈ F i x ( 𝑇 ) ‖ ‖ 𝑥 𝑛 ‖ ‖ − 𝑧 2 ̂ 𝛽 𝑛 𝛾 𝑛 , ( 2 . 2 9 ) the sequence { 𝛽 𝑛 } is chosen so that 𝛽 𝑛 → 1 ( 𝑛 → ∞ ) , and { 𝛾 𝑛 } is as in (1.3). Then { 𝑥 𝑛 } converges strongly to 𝑃 F i x ( 𝑇 ) 𝑥 0 .

Proof. It is easy to see that 𝜃 𝑛 → 0 in Theorem 2.6. Following the reasoning in the proof of Theorem 2.4 and using F i x ( 𝑇 ) instead of 𝐵 𝑟 0 ( 𝑥 0 ) ∩ F i x ( 𝑇 ) , we deduce the conclusion of Theorem 2.6.

Acknowledgments

The authors are very grateful to the referee for his/her valuable suggestions and comments. The work was supported partly by the NSF of China (10771202), the Research Fund for Shanghai Key Laboratory for Contemporary Applied Mathematics (08DZ2271900), and the Specialized Research Fund for the Doctoral Program of Higher Education of China (2007035805). This work is dedicated to W. Takahashi.

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