Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions

Volume 2010, Article ID 948529,8pages doi:10.1155/2010/948529

Research Article

Iterative Algorithms with Variable Coefficients for Asymptotically Strict Pseudocontractions

Ci-Shui Ge,

Jin Liang,

and Ti-Jun Xiao

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

Correspondence should be addressed to Jin Liang,[email protected]

Received 8 October 2009; Revised 29 November 2009; Accepted 22 January 2010 Academic Editor: Anthony To Ming Lau

Copyrightq2010 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.

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

LetHbe a real Hilbert space,Ca nonempty closed convex subset ofH,T :C → Ca self- mapping ofCand FixT:{x∈C:Txx}.

Recall that a mappingT :C → Cis called to be nonexpansive if

Tx−Ty≤x−y, ∀x, y∈C. 1.1 T is called to be asymptotically nonexpansive1if there exists a sequence{k_n}withk_n ≥1 and lim_{n→ ∞}k_n1 such that

Tⁿx−Tⁿy≤k_nx−y, ∀x, y∈C, and all integers n≥1. 1.2 T is called to be an asymptoticallyκ-strict pseudocontraction, if there exist 0≤κ <1 and 0≤ γn → 0n → ∞such that

Tⁿx−Tⁿy²≤

1γnx−y²κI−Tⁿx−I−Tⁿy² 1.3 for allx, y∈Cand all integersn≥1.

Asκ0, asymptoticallyκ-strict pseudocontractionT is asymptotically nonexpansive.

In2, Nakajo and Takahashi studied the iterative approximation of fixed points of nonexpansive mappings and proved the following strong convergence theorem.

Theorem A. Let C be a nonempty closed convex subset of a Hilbert space H and let T be a nonexpansive mapping ofCinto itself such that FixT/∅. Suppose{x_n}is given by

x0∈Cchosen arbitrarily, ynαnxn 1−αnTxn, C_n

z∈C:y_n−z≤ xn−z , Q_n{z∈C:x_n−z, x0−x_n ≥0},

xn1PCn∩Qnx0, n∈N,

1.4

whereP_Cn∩Qn is the metric projection from C ontoC_n∩Q_nandα_nis chosen so that 0≤α_n ≤a <1.

Then,{x_n}converges strongly toP_FixTx0, whereP_FixTis the metric projection from C onto FixT. Such algorithm in1.4is referred to be theCQalgorithm in3, due to the fact that each iteratex_n1is obtained by projectingx0onto the intersection of the suitably constructed closed convex setsC_nandQ_n.It is known that theCQalgorithm in1.4is of independent interest, and theCQalgorithm has been extended to various mappings by many authors cf., e.g.,3–11.

Very recently, by extending theCQalgorithm, Takahashi et al.9studied 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 B. Let C be a closed convex subset of a Hilbert space H and let T : C → C be an asymptoticallyκ-strict pseudocontractions for some 0≤κ <1.Assume that the fixed point set FixT of T is nonempty and bounded. Let{x_n}^∞_n0be the sequence generated by the following (CQ) algorithm:

x0∈C, chosen arbitrarily, ynαnxn 1−αnTⁿxn, C_n

z∈C:y_n−z²≤ xn−z ² κ−α_n1−α_n x_n−Tx_n ²θ_n , Qn{z∈C:xn−z, x0−xn ≥0},

xn1PCn∩Qnx0,

1.5

where

θ_n Δ²_n1−α_nγ_n−→0 n−→ ∞, Δ_nsup{ x_n−z :z∈FixT}<∞. 1.6 Assume that control sequence{αn}^∞_n0is chosen so that lim sup_{n→ ∞}αn<1−κ.Then{xn}converges strongly toP_FixTx0.

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 Takahashi2, 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 of5.

2. Results and Proofs

Throughout this paper,

ixn xmeans that{xn}converges weakly tox.

iixn → xmeans that{xn}converges strongly tox.

iiiωwxn:{x:∃xnj x}, that is, the weakω-limit set of{xn}.

ivB_rx0:{x∈H: x−x0 ≤r}.

vNis the set of nonnegative integers.

The following lemmas are basiccf., e.g.,6forLemma 2.1, and5for Lemmas2.2- 2.3.

Lemma 2.1. LetKbe a closed convex subset of a real Hilbert spaceH. Givenx∈ H, z∈K. Then zPKxif and only if

x−z, y−z ≤0, ∀y∈K, 2.1

whereP_Kxis the unique point inKwith the property

x−P_Kx ≤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 qPKu. Suppose that{xn}satisfies

x_n−u ≤u−q, ∀n∈N, 2.3 andω_wx_n⊂K. Thenx_n → q.

Lemma 2.3. LetCbe a closed convex subset of a Hilbert spaceHandT:C → Can asymptotically κ-strict pseudocontraction. Then

Ifor eachn≥1,Tⁿsatisfies the Lipschitz condition:

Tⁿx−Tⁿy≤Lnx−y, ∀x, y∈C, 2.4

where

Ln κ 1γn1−κ

1−κ ,

γn

is as in1.3; 2.5

IIif{xn}is a sequence inCsuch thatxnxand lim sup

m→ ∞ lim sup

n→ ∞ x_n−T^mx_n 0, 2.6

then

I−Tx_n−→0⇒I−Tx0. 2.7 In particular,

x_nx, I−Tx_n−→0⇒I−Tx0. 2.8 IIIFixTis closed and convex so that the projectionP_FixTis well defined.

Theorem 2.4. LetCbe a closed convex subset of a Hilbert spaceH,T :C → Can asymptotically κ-strict pseudocontraction for some 0≤κ <1, and FixT/∅.Let{xn}be the sequence generated by the following CQ-type algorithm with variable coefficients:

x0∈Cchosen arbitrarily, yn

1−βn

xnβnTⁿxn,

Cn

z∈C:yn−z² ≤ xn−z ²βn

κβn−1

xn−Tⁿxn 2θn

,

Q_n{z∈C:x_n−z, x0−x_n ≥0}, x_n1P_Cn∩Qnx0, n∈N,

2.9

where

βn βn

1 xn−x0 2, βn∈ 1

2,1

, θn2 1r₀²

βnγn, 2.10

the sequence{β_n}is chosen so thatβ_n → 1n → ∞, the positive real numberr0 is chosen so that Br0x0∩FixT/∅, and{γn}is as in1.3. Then{xn}converges strongly toPFixTx0.

Proof. We divide the proof into five steps.

Step 1. We prove thatC_n∩Q_nis nonempty, convex and closed.

Clearly, bothQnandCn are convex and closed, so isCn∩Qn. SinceT :C → Cis an asymptoticallyκ-strict pseudocontraction, we have by1.3,

Tⁿx−p²≤

1γ_nx−p²κI−Tⁿx−I−Tⁿp²

≤

1γnx−p²κ x−Tⁿx ², 2.11 for allx∈C,p∈FixT, and all integersn≥1.

By2.9and2.11, we deduce that for eachp∈Br0x0∩FixT, n∈N, yn−p²

1−βn xn−p

βn

Tⁿxn−p²

1−βnxn−p²βnTⁿxn−p²−βn

1−βn

xn−Tⁿxn 2

1−β_nx_n−p²β_n

1γ_nx_n−p²κ x_n−Tⁿx_n ²

−β_n 1−β_n

x_n−Tⁿx_n ²

≤xn−p²βn

κβn−1

xn−Tⁿxn 2βnγn

2

x_n−x0 2x0−p² 1 x_n−x0 2

≤x_n−p²β_n

κβ_n−1

xn−Tⁿx_n ²2 1r₀²

β_nγ_n xn−p²βn

κβn−1

xn−Tⁿxn 2θn.

2.12

Therefore,

Brx0∩FixT⊂Cn, ∀n∈N. 2.13

Next, we prove by induction that

B_r0x0∩FixT⊂Q_n, ∀n∈N. 2.14

Obviously,B_r0x0∩FixT⊂ C Q0, that is,2.14holds forn 0. Assume thatB_r0x0∩ FixT ⊂ Qn for somen ∈ N.Then, 2.13implies thatBr0x0∩FixT ⊂ Cn∩Qn/∅and xn1PCn∩Qnx0is well defined.

ByLemma 2.1, we getx_n1−z, x0−x_n1 ≥ 0,∀ z∈ C_n∩Q_n.In particular, for each z ∈ Br0x0∩FixT,we havexn1−z, x0−xn1 ≥ 0.This together with the definition of Qn1, the inequality2.14holds forn1. So2.14is true.

Step 2. We prove that lim_{n→ ∞} x_n1−x_n 0.

By the definition ofQnandLemma 2.1, we getxn PQnx0.Hence,

x_n−x0 ≤p−x0, ∀p∈B_r0x0∩FixT. 2.15 DenotingM: x0 p−x0 , we have x_n ≤M,for alln∈N,and

xn−x0 ≤q−x0, ∀n∈N, 2.16

whereq PFixTx0 ⊂ Br0x0∩FixT.The definition ofxn1 shows thatxn1 ∈ Qn, that is, x_n1−x_n, x_n−x0 ≥0.This implies that

x_n1−x_n ² x_n1−x0 2− xn−x0 2−2x_n1−x_n, x_n−x0

≤ xn1−x0 2− xn−x0 2. 2.17

Thus{ xn−x0 }is increasing. Since{xn}is bounded, lim_{n→ ∞} xn−x0 exists and

nlim→ ∞ xn1−xn 0. 2.18

Step 3. We prove that lim_{n→ ∞} x_n−Tⁿx_n 0.

The definition ofxn1shows thatxn1∈Cn, that is, yn−xn1²≤ xn−xn1 2βn

κβn−1

xn−Tⁿxn 2θn. 2.19

By2.19and the definition ofynin2.9, we deduce that β²_n xn−Tⁿxn 2yn−xn²

≤y_n−x_n1² x_n1−x_n ²2y_n−x_n1· xn1−x_n

≤βn

κβn−1

xn−Tⁿxn 2θn2 xn1−xn 22yn−xn1· xn1−xn . 2.20

Further, we have

1−κβn xn−Tⁿxn 2≤2 xn1−xn 22 xn1−xn ·yn−xn1θn. 2.21 Thus,2.19and2.21imply that

1−κβ_n x_n−Tⁿx_n ²≤4 x_n1−x_n ²2 x_n1−x_n · x_n−Tⁿx_n

β_nκβ_n−1 2 x_n1−x_n θ_nθ_n.

2.22

Noticing xn ≤M, βn∈1/2,1, we get β_n β_n

1 x_n ² ≥ 1

21M²>0. 2.23

From lim_{n→ ∞} xn1−xn 0, lim_{n→ ∞}θn0,and2.22, it follows that

nlim→ ∞ xn−Tⁿxn 0. 2.24

Step 4. We prove that

nlim→ ∞ x_n−Tx_n 0. 2.25

ByLemma 2.3and the definition ofT, we obtain

x_n−Tx_n ≤ x_n−x_n1 x_n1−Tⁿ¹x_n1Tⁿ¹x_n1−Tⁿ¹x_nTⁿ¹x_n−Tx_n

≤1L_n1 x_n1−x_n x_n1−Tⁿ¹x_n1L1 x_n−Tⁿx_n ,

2.26

where

Ln κ 1γ_n1−κ

1−κ ,

γn

is as in1.3. 2.27

By2.18,2.24, and2.26, we know that2.25holds.

Step 5. Finally, byLemma 2.3and2.25, we haveωwxn⊂ FixT. Furthermore, it follows from2.16andLemma 2.2that the sequence{x_n}converges strongly toqP_FixTx0. Remark 2.5. Theorem 2.4 improves 5, Theorem 4.1 since the condition that θn → 0 is satisfied and the boundedness of FixTis dropped off.

Theorem 2.6. LetCbe a closed convex subset of a Hilbert spaceH,T :C → Can asymptotically κ-strict pseudocontraction for some 0≤κ < 1, and FixTbe nonempty and bounded. Let{xn}the sequence generated by the following CQ-type algorithm with variable coefficients:

x0∈Cchosen arbitrarily, yn

1−βn

xnβnTⁿxn,

C_n

z∈C:y_n−z²≤ xn−z ²β_n

κβ_n−1

x_n−Tⁿx_n ²θ_n , Qn{z∈C:xn−z, x0−xn ≥0},

x_n1P_Cn∩Qnx0, n∈N,

2.28

where

βn βn

1 xn−x0 2, βn∈ 1

2,1

, θn

sup

z∈FixT xn−z ₂

βnγn, 2.29

the sequence{βn}is chosen so thatβn → 1n → ∞,and{γn}is as in1.3. Then{xn}converges strongly toP_FixTx0.

Proof. It is easy to see that θn → 0 in Theorem 2.6. Following the reasoning in the proof of Theorem 2.4and using FixT instead ofB_r0x0∩FixT, 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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