strong convergence of hybrid algorithm for asymptotically ... · received 24 january 2012; ... the...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2012, Article ID 170540, 11 pagesdoi:10.1155/2012/170540

Research ArticleStrong Convergence of Hybrid Algorithmfor Asymptotically Nonexpansive Mappings inHilbert Spaces

Juguo Su,1 Yuchao Tang,2 and Liwei Liu2

1 Department of Basic Courses, Jiangxi Vocational and Technical College,Jiangxi Nanchang 330013, China

2 Department of Mathematics, Nanchang University, Nanchang 330031, China

Correspondence should be addressed to Yuchao Tang, [email protected]

Received 24 January 2012; Accepted 5 March 2012

Academic Editor: Rudong Chen

Copyright q 2012 Juguo Su et al. This is an open access article distributed under the CreativeCommons Attribution License, which permits unrestricted use, distribution, and reproduction inany medium, provided the original work is properly cited.

The hybrid algorithms for constructing fixed points of nonlinear mappings have been studiedextensively in recent years. The advantage of this methods is that one can prove strongconvergence theorems while the traditional iteration methods just have weak convergence. In thispaper, we propose two types of hybrid algorithm to find a common fixed point of a finite family ofasymptotically nonexpansive mappings in Hilbert spaces. One is cyclic Mann’s iteration scheme,and the other is cyclic Halpern’s iteration scheme. We prove the strong convergence theorems forboth iteration schemes.

1. Introduction

Let H be a real Hilbert space and C be a nonempty closed convex subset of H, 〈·, ·〉 and ‖ · ‖denote the inner product and norm in H, respectively. Let T be a self-mapping of C. Then, Tis said to be a Lipschitzian mapping if for each n ≥ 1 there exists an nonnegative real numberkn such that

∥∥Tnx − Tny

∥∥ ≤ kn

∥∥x − y

∥∥, (1.1)

for all x, y ∈ C. A Lipschitzian mapping is said to be nonexpansive mapping if kn = 1 for alln ≥ 1 and asymptotically nonexpansive mapping [1] if limn→∞ kn = 1, respectively. We useF(T) to denote the set of fixed points of T (i.e., F(T) = {x ∈ C : Tx = x}). It is well known thatif T is asymptotically nonexpansive mapping with F(T)/= ∅, then F(T) is closed and convex.

2 Journal of Applied Mathematics

Iterative methods for finding fixed points of nonexpansive mappings are an importanttopic in the theory of nonexpansive mappings and have wide applications in a number ofapplied areas, such as the convex feasibility problem [2–4], the split feasibility problem [5–7] and image recovery and signal processing [8–10]. The Mann’s iteration is defined by thefollowing:

xn+1 = αnxn + (1 − αn)Txn, n ≥ 0, (1.2)

where x0 ∈ C is chosen arbitrarily and {αn} ⊆ [0, 1]. Reich [11] proved that ifX is a uniformlyconvex Banach space with a Frechet differentiable norm and if {αn} is chosen such that∑∞

n=0 αn(1 − αn) = ∞, then the sequence {xn} defined by (1.2) converges weakly to a fixedpoint of nonexpansive mapping T . However, we highlight that the Mann’s iterations haveonly weak convergence even in a Hilbert space (see e.g., [12]).

In order to obtain the strong convergence theorem for the Mann iteration method (1.2)to nonexpansivemappings, in 2003, Nakajo and Takahashi [13] proved the following theoremin a Hilbert space by using an idea of the hybrid method in mathematical programming.

Theorem 1.1 (see [13]). Let C be a closed convex subset of a Hilbert space H and let T be anonexpansive mapping of C into itself such that F(T) is nonempty. Let P be the metric projectionof H onto F(T). Let x0 ∈ C and

yn = αnxn + (1 − αn)Txn,

Cn ={

z ∈ C :∥∥yn − z

∥∥ ≤ ‖xn − z‖},

Qn = {z ∈ C : 〈xn − z, x0 − xn〉 ≥ 0},xn+1 = PCn∩Qnx0,

(1.3)

where {αn} ⊆ [0, 1] satisfies supn≥0 αn < 1 and PCn∩Qnx0 is the metric projection ofH onto Cn ∩Qn.Then {xn} converges strongly to Px0 ∈ F(T).

The iterative algorithm (1.3) is often referred to as hybrid algorithm or CQ algorithm inthe literature. We call it hybrid algorithm. Since then, the hybrid algorithm has been studiedextensively by many authors (see, e.g., [14–18]). Specifically, Kim and Xu [19] extendedthe results of Nakajo and Takahashi [13] from nonexpansive mapping to asymptoticallynonexpansive mapping; they proposed the following hybrid algorithm:

x0 ∈ C is chosen arbitrarily,

yn = αnxn + (1 − αn)Tnxn,

Cn ={

z ∈ C :∥∥yn − z

∥∥2 ≤ ‖xn − z‖2 + θn

}

,

Qn = {z ∈ C : 〈xn − z, x0 − z〉 ≥ 0},xn+1 = PCn∩Qnx0,

(1.4)

Journal of Applied Mathematics 3

where θn = (1 − αn)(k2n − 1)(diamC)2. Zhang and Chen in [20], studied the following hybrid

algorithm of Halpern’s type for asymptotically nonexpansive mappings:

x0 ∈ C be chosen arbitrarily,

yn = αnx0 + (1 − αn)Tnxn,

Cn ={

z ∈ C :∥∥yn − z

∥∥2 ≤ ‖xn − z‖2 + θn

}

,


(1.5)

where θn = (1 − αn)(k2n − 1)(diamC)2. Some other related works can be found in [21–25].

The hybrid algorithm of (1.3)–(1.5) just considered a single nonexpansive andasymptotically nonexpansive mapping. In order to extend them to a finite family ofmappings. Recall that in 1996, Bauschke [26] investigated the following cyclic Halpern’s typealgorithm for a finite family of nonexpansive mappings {Tj}N−1

j=0 :

C x0 �−→ x1 := α0u + (1 − α0)T0x0 �−→ · · ·�−→ xN := αN−1u + (1 − αN−1)TN−1xN−1

�−→ xN+1 := αNu + (1 − αN)T0xN �−→ · · · ,(1.6)

or, more compactly,

u, x0 ∈ C,

xn+1 = αnu + (1 − αn)T[n]xn, n ≥ 0,(1.7)

where T[n] := Tn mod N , and the modN function takes values in {0, 1, . . . ,N − 1}.If αn = 0 and each nonexpansive mapping {Tj}N−1

j=0 is a projection onto a closed convexset, then (1.7) reduces to the famous Algebraic Reconstruction Technique (ART), which hasnumerous applications from computer tomograph to image reconstruction.

For the cyclic Mann’s type algorithm, a finite family of asymptotically nonexpansivemappings was introduced by Qin et al. [17] and Osilike and Shehu [14], independently. Let{Tj}N−1

j=0 be a finite family of asymptotically nonexpansive self-mappings of C. For a givenx0 ∈ C, and a real sequence {αn}∞n=0 ⊆ (0, 1), the sequence {xn}∞n=0 is generated as follows:

x1 = α0x0 + (1 − α0)T0x0,

x2 = α1x1 + (1 − α1)T1x1,

...

xN = αN−1xN−1 + (1 − αN−1)TN−1xN−1,


xN+1 = αNxN + (1 − αN)T20xN,

xN+2 = αN+1xN+1 + (1 − αN+1)T21xN+1,

...

x2N = α2N−1x2N−1 + (1 − α2N−1)T2N−1x2N−1,

x2N+1 = α2Nx2N + (1 − α2N)T30x2N,

x2N+2 = α2N+1x2N+1 + (1 − α2N+1)T31x2N+1,

...

(1.8)

The algorithm can be expressed in a compact form as

xn+1 = αnxn + (1 − αn)Tk(n)i(n) xn, n ≥ 0, (1.9)

where n = (k − 1)N + i, i = i(n) ∈ J = {0, 1, 2, . . . ,N − 1}, k = k(n) ≥ 1 positive integerand limn→∞ k(n) = ∞. Similarly, we can define the cyclic Halpern’s type algorithm forasymptotically nonexpansive mappings as follows:

xn+1 = αnx0 + (1 − αn)Tk(n)i(n) xn, n ≥ 0. (1.10)

The purpose of this paper is to extend the hybrid algorithms (1.4) and (1.5) tothe cyclic Mann’s type (1.9) and the cyclic Halpern’s type (1.10). Our results generalizethe corresponding results of Kim and Xu [19] and Zhang and Chen [20] from a singleasymptotically nonexpansive mapping to a finite family of asymptotically nonexpansivemappings, respectively.

2. Preliminaries

In this section, we collect some useful results which will be used in the following section.We use the following notations:

(i) ⇀ for weak convergence and → for strong convergence;

(ii) ωw(xn) = {x : ∃xnj ⇀ x} denotes the weak ω-limit set of {xn}.It is well known that a Hilbert space H satisfies the Opial’s condition [27]; that is, for

each sequence {xn} inH which converges weakly to a point x ∈ H, we have

lim supn→∞

‖xn − x‖ < lim supn→∞

∥∥xn − y

∥∥, (2.1)

for all y ∈ H, y /=x.


Recall that given a closed convex subset of C of a real Hilbert space H, the nearestpoint projection PC form H onto C assigns to each x ∈ C its nearest point denoted PCx in Cfrom x to C; that is, PCx is the unique point in X with the property

‖x − PCx‖ ≤ ∥∥x − y

∥∥, ∀y ∈ C. (2.2)

The following Lemmas 2.1 and 2.2 are well known.

Lemma 2.1. Let C be a closed convex subset of a real Hilbert spaceH. Given x ∈ H and z ∈ C, thenz = PCx if and only if there holds the relation

⟨

x − z, y − z⟩ ≤ 0, ∀y ∈ C. (2.3)

Lemma 2.2. LetH be a real Hilbert space, then for all x, y ∈ H

∥∥x − y

∥∥2 = ‖x‖2 −

∥∥∥y2

∥∥∥ − 2〈x − y, y〉. (2.4)

Lemma 2.3 (see [28]). Let X be a uniformly convex Banach space, C a nonempty closed convexsubset of X, and T : C → C an asymptotically nonexpansive mapping. Then (I − T) is demiclosed atzero, that is, if xn ⇀ x and xn − Txn → 0, then x ∈ F(T).

Lemma 2.4 (see [22]). Let C be a closed convex subset of a real Hilbert space H. Let {xn} besequences inH and u ∈ H. Let q = PCu. If {xn} is such that ωw(xn) ⊂ C and satisfies the condition

‖xn − u‖ ≤ ∥∥u − q

∥∥, ∀n ≥ 1, (2.5)

then {xn} converges strongly to q.

Lemma 2.5 (see [22]). Let C be a closed convex subset of a real Hilbert spaceH. For any x, y, z ∈ Hand real number a ∈ R, the set

{

v ∈ C :∥∥y − v

∥∥2 ≤ ‖x − v‖2 + 〈z, v〉 + a

}

(2.6)

is convex and closed.

3. Main Results

In this section, we consider a finite family of asymptotically nonexpansive mappings {Tj}N−1j=0 ;

that is, there exists {ujn} ⊆ [0,∞), j ∈ J := {0, 1, 2, . . . ,N − 1}with limn→∞ ujn = 0, for all j ∈ Jsuch that

∥∥∥Tn

j x − Tnj y

∥∥∥ ≤ (

1 + ujn

)∥∥x − y

∥∥, (3.1)


for all n ≥ 1 and x, y ∈ C. Let un := maxj∈J{ujn}, then limn→∞ un = 0, and

∥∥∥Tn

j x − Tnj y

∥∥∥ ≤ (1 + un)

∥∥x − y

∥∥, (3.2)

for all n ≥ 1, and for all x, y ∈ C and j ∈ J .We prove the following theorems.

Theorem 3.1. Let C be a bounded closed convex subset of a Hilbert space H, and let {Tj}N−1j=0 : C →

C be a finite family of asymptotically nonexpansive mappings with F :=⋂N−1

j=0 F(Tj)/= ∅. Assume that{αn} ⊆ (0, 1) such that limn→∞ αn = 0. Suppose the sequence {xn} generated by

x0 ∈ C is chosen arbitrary,

yn = αnx0 + (1 − αn)Tk(n)i(n) xn,

Cn ={

z ∈ C :∥∥yn − z

∥∥2 ≤ ‖xn − z‖2 + αn

(

‖x0‖2 + 2〈xn − x0, z〉)

+ θn}

,


(3.3)

where θn = (2 + un)un(1 − αn)(diamC)2 → 0, as n → ∞. Then {xn} converges strongly to PFx0.

Proof. By Lemma 2.5, we conclude that Cn is closed and convex. It is obvious that Qn and Fare closed and convex. Then, the projection mappings PCn∩Qnx0 and PFx0 are well defined.We divide the proof into several steps.

Step 1. We show that F ⊆ Cn ∩ Qn, for all n. Let p ∈ F. By the hybrid algorithm (3.3)and that ‖ · ‖2 is convex, we have

∥∥yn − p

∥∥2 ≤ αn

∥∥x0 − p

∥∥2 + (1 − αn)

∥∥∥T

k(n)i(n) − p

∥∥∥

2

≤ αn

∥∥x0 − p

∥∥2 + (1 − αn)(1 + un)2

∥∥xn − p

∥∥2

=∥∥xn − p

∥∥2 + αn

(∥∥x0 − p

∥∥2 − ∥

∥xn − p∥∥2)

+ (1 − αn)(2 + un)un

∥∥xn − p

∥∥2

≤ ∥∥xn − p

∥∥2 + αn

(

‖x0‖2 + 2⟨

xn − x0, p⟩)

+ θn,

(3.4)

where θn = (2 + un)un(1 − αn)(diamC)2. Hence, p ∈ Cn, that is, F ⊆ Cn, for all n.Next, we prove that F ⊆ Qn, for all n ≥ 0. Indeed, for n = 0, Q0 = C, then F ⊆ Q0.

Assuming that F ⊆ Qm, we show that F ⊆ Qm+1. Since xm+1 is the projection of x0 ontoCm ∩Qm, it follows from Lemma 2.1 that

〈xm+1 − z, x0 − xm+1〉 ≥ 0, ∀z ∈ Cm ∩Qm. (3.5)


As F ⊆ Cm ∩Qm, in particular, we have

⟨

xn+1 − p, x0 − xm+1⟩ ≥ 0, ∀p ∈ F. (3.6)

Thus, F ⊆ Qm+1. Therefore, F ⊆ Cn ∩Qn, for all n ≥ 0.Step 2. We prove that

∥∥xn+j − xn

∥∥ −→ 0 as n −→ ∞, ∀j = 0, 1, 2, . . . ,N − 1. (3.7)

Since the definition of Qn implies that xn = PQnx0, we have

‖xn − x0‖ ≤ ∥∥x0 − y

∥∥, ∀y ∈ Qn. (3.8)

By Step 1, F ⊆ Qn, we have

‖xn − x0‖ ≤ ∥∥x0 − p

∥∥, ∀p ∈ F. (3.9)

In particular,

‖xn − x0‖ ≤ ∥∥x0 − q

∥∥, q = PFx0. (3.10)

Since xn+1 ∈ Qn, we have 〈xn+1 − xn, xn − x0〉 ≥ 0 and ‖xn − x0‖ ≤ ‖xn+1 − x0‖. The secondinequality shows that the sequence {‖xn − x0‖} is nondecreasing. Since C is bounded, weobtain that the limn→∞‖xn − x0‖ exists.

With the help of Lemma 2.2, we obtain

‖xn+1 − xn‖2 = ‖(xn+1 − x0) − (xn − x0)‖2

= ‖xn+1 − x0‖2 − ‖xn − x0‖2 − 2〈xn+1 − xn, xn − x0〉

≤ ‖xn+1 − x0‖2 − ‖xn − x0‖2 −→ 0 as n −→ ∞.

(3.11)

Consequently,

∥∥xn+j − xn

∥∥ −→ 0 as n −→ ∞, ∀j ∈ J. (3.12)

Step 3. We now claim that ‖xn − Tjxn‖ → 0, as n → ∞, for all j ∈ J . Notice that for alln > N, n = (n−N)(mod N), since n = (k(n)−1)N+i(n), we obtain n−N = (k(n)−1)N+i(n)−N = (k(n−N)−1)N+i(n−N). So that n−N = [(k(n)−1)−1]N+i(n) = (k(n−N)−1)N+i(n−N).Hence k(n) − 1 = k(n −N) and i(n) = i(n −N).

By the hybrid algorithm (3.3) and the condition limn→∞αn = 0, we get

∥∥∥yn − T

k(n)i(n) xn

∥∥∥ = αn

∥∥∥x0 − T

k(n)i(n) xn

∥∥∥ −→ 0 as n −→ ∞. (3.13)


It follows from the fact xn+1 ∈ Cn that we have

∥∥yn − xn+1

∥∥2 ≤ ‖xn − xn+1‖2 + αn

(

‖x0‖2 + 2〈xn − x0, xn+1〉)

+ θn

−→ 0 as n −→ ∞,

∥∥∥T

k(n)i(n) xn − xn

∥∥∥ ≤

∥∥∥T

k(n)i(n) xn − yn

∥∥∥ +

∥∥yn − xn+1

∥∥ + ‖xn+1 − xn‖

−→ 0 as n −→ ∞.

(3.14)

Putting L = supn≥0{1 + un}, we deduce that

∥∥xn+1 − Ti(n)xn

∥∥ ≤

∥∥∥xn+1 − T

k(n)i(n) xn

∥∥∥ +

∥∥∥T

k(n)i(n) xn − Ti(n)xn

∥∥∥

≤ ‖xn+1 − xn‖ +∥∥∥xn − T

k(n)i(n) xn

∥∥∥ + L

∥∥∥T

k(n)−1i(n) xn − xn

∥∥∥

≤ ‖xn+1 − xn‖ +∥∥∥xn − T

k(n)i(n) xn

∥∥∥

+ L(∥∥∥T

k(n)−1i(n) xn − T

k(n)−1i(n−N)xn−N

∥∥∥ +

∥∥∥T

k(n)−1i(n−N)xn−N − xn−N−1

∥∥∥ + ‖xn−N−1 − xn‖

)

≤ ‖xn+1 − xn‖ +∥∥∥xn − T

k(n)i(n) xn

∥∥∥ + L2‖xn − xn−N‖

+ L∥∥∥T

k(n)−1i(n−N)xn−N − xn−N−1

∥∥∥ + L‖xn−N−1 − xn‖

−→ 0 as n −→ ∞.

(3.15)

Hence,

∥∥xn − Ti(n)xn

∥∥ ≤ ‖xn − xn+1‖ +

∥∥xn+1 − Ti(n)xn

∥∥ −→ 0 as n −→ ∞. (3.16)

Consequently, for all j = 0, 1, . . . ,N − 1, we have

∥∥xn − Tn+jxn

∥∥ ≤ ∥

∥xn − xn+j∥∥ +

∥∥xn+j − Tn+jxn+j

∥∥ + L

∥∥xn+j − xn

∥∥

−→ 0 as n −→ ∞.(3.17)

Thus, ‖xn − Tjxn‖ → 0, as n → ∞, for all j ∈ J .Step 4. Since {xn} is bounded, then {xn} has a weakly convergent subsequence {xnj}.

Suppose {xnj} converges weakly to p. Since C is weakly closed and {xnj} ⊂ C, we have p ∈ C.By Lemma 2.3, I −Tj is demiclosed at 0 for all j ∈ J , and we get p−Tjp = 0(j ∈ J), that is p ∈ F.Suppose {xn} does not converge weakly to p, then there exists another subsequence {xnk}of{xn} which converges weakly to some p1. Similarly we can prove that p1 ∈ F. It follows from


the proof of above that we know that limn→∞ ‖xn−p‖ and limn→∞ ‖xn−p1‖ exist. Since everyHilbert space satisfies Opial’s condition, we have

limn→∞

∥∥xn − p

∥∥ = lim

j→∞

∥∥∥xnj − p

∥∥∥ < lim

j→∞

∥∥∥xnj − p1

∥∥∥

= limn→∞

∥∥xn − p1

∥∥ = lim

k→∞

∥∥xnk − p1

∥∥

< limk→∞

∥∥xnk − p

∥∥ < lim

n→∞∥∥xn − p

∥∥.

(3.18)

This is a contradiction. Hence, ωw(xn) ⊆ F. Then by virtue of (3.10) and Lemma 2.4, weconclude that xn → q as n → ∞, where q = PFx0.

Recall that a mapping T is said to be asymptotically strictly pseudocontractive [29], ifthere exist λ ∈ [0, 1) and a sequence {un} ⊆ [0,∞) with limn→∞ un = 0 such that

∥∥Tnx − Tny

∥∥2 ≤ (1 + un)2

∥∥x − y

∥∥2 + λ

∥∥(I − Tn)x − (I − Tn)y

∥∥2, (3.19)

for all n and x, y ∈ C.

Theorem 3.2. Let C be a bounded closed convex subset of a Hilbert space H, and let {Tj}N−1j=0 : C →

C be a finite family of asymptotically nonexpansive mappings with F :=⋂N−1

j=0 F(Tj)/= ∅. Assume that{αn} ⊆ (0, a), for some 0 < a < 1. Suppose the sequence {xn} generated by

x0 ∈ C is chosen arbitrary,

yn = αnxn + (1 − αn)Tk(n)i(n) xn,

Cn ={

z ∈ C :∥∥yn − z

∥∥2 ≤ ‖xn − z‖2 + θn

}

,


(3.20)

where θn = (2 + un)un(diamC)2. Then {xn} converges strongly to PFx0.

Proof. Since Tj is asymptotically nonexpansive if and only if Tj is asymptotically strictlypseudocontractive mapping with λ = 0. Then, the rest of proof follows from Theorem 3.2of Osilike and Shehu [14] and Theorem 2.2 of Qin et al. [17] directly by letting λ = 0.

Acknowledgments

The authors are deeply grateful to Professor Rudong Chen (Editor) for managing the reviewprocess. This work was supported by the Natural Science Foundations of Jiangxi Province(2009GZS0021, CA201107114) and the Youth Science Funds of the Education Department ofJiangxi Province (GJJ12141).


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