# Strong Convergence of Hybrid Algorithm for Asymptotically Nonexpansive Mappings in Hilbert Spaces

Authors: Juguo Su; Yuchao Tang; Liwei Liu
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

The hybrid algorithms for constructing fixed points of nonlinear mappings have been studied extensively in recent years. The advantage of this methods is that one can prove strong convergence theorems while the traditional iteration methods just have weak convergence. In this paper, we propose two types of hybrid algorithm to find a common fixed point of a finite family of asymptotically 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 for both iteration schemes.


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## Body

## 1. Introduction

LetH 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, T is said to be a Lipschitzian mapping if for each n≥1 there exists an nonnegative real number kn such that
(1.1)‖Tnx-Tny‖≤kn‖x-y‖,
for all x,y∈C. A Lipschitzian mapping is said to be nonexpansive mapping if kn=1 for all n≥1 and asymptotically nonexpansive mapping [1] if limn→∞kn=1, respectively. We use F(T) to denote the set of fixed points of T (i.e., F(T)={x∈C:Tx=x}). It is well known that if T is asymptotically nonexpansive mapping with F(T)≠∅, then F(T) is closed and convex.Iterative methods for finding fixed points of nonexpansive mappings are an important topic in the theory of nonexpansive mappings and have wide applications in a number of applied 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 the following:
(1.2)xn+1=αnxn+(1-αn)Txn,n≥0,
where x0∈C is chosen arbitrarily and {αn}⊆[0,1]. Reich [11] proved that if X is a uniformly convex Banach space with a Fréchet 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 fixed point of nonexpansive mapping T. However, we highlight that the Mann's iterations have only 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 nonexpansive mappings, in 2003, Nakajo and Takahashi [13] proved the following theorem in a Hilbert space by using an idea of the hybrid method in mathematical programming.Theorem 1.1 (see [13]).
LetC be a closed convex subset of a Hilbert space H and let T be a nonexpansive mapping of C into itself such that F(T) is nonempty. Let P be the metric projection of H onto F(T). Let x0∈C and
(1.3)yn=αnxn+(1-αn)Txn,Cn={z∈C:‖yn-z‖≤‖xn-z‖},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,
where {αn}⊆[0,1] satisfies supn≥0αn<1 and PCn∩Qnx0 is the metric projection of H 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 in the literature. We call it hybrid algorithm. Since then, the hybrid algorithm has been studied extensively by many authors (see, e.g., [14–18]). Specifically, Kim and Xu [19] extended the results of Nakajo and Takahashi [13] from nonexpansive mapping to asymptotically nonexpansive mapping; they proposed the following hybrid algorithm:
(1.4)x0∈Cischosenarbitrarily,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,
where θn=(1-αn)(kn2-1)(diamC)2. Zhang and Chen in [20], studied the following hybrid algorithm of Halpern's type for asymptotically nonexpansive mappings:
(1.5)x0∈Cbechosenarbitrarily,yn=αnx0+(1-αn)Tnxn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖2+θn},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,
where θn=(1-αn)(kn2-1)(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 and asymptotically nonexpansive mapping. In order to extend them to a finite family of mappings. Recall that in 1996, Bauschke [26] investigated the following cyclic Halpern's type algorithm for a finite family of nonexpansive mappings {Tj}j=0N-1:
(1.6)C∋x0⟼x1≔α0u+(1-α0)T0x0⟼⋯⟼xN≔αN-1u+(1-αN-1)TN-1xN-1⟼xN+1≔αNu+(1-αN)T0xN⟼⋯,
or, more compactly,
(1.7)u,x0∈C,xn+1=αnu+(1-αn)T[n]xn,n≥0,
where T[n]:=TnmodN, and the modN function takes values in {0,1,…,N-1}.Ifαn=0 and each nonexpansive mapping {Tj}j=0N-1 is a projection onto a closed convex set, then (1.7) reduces to the famous Algebraic Reconstruction Technique (ART), which has numerous applications from computer tomograph to image reconstruction.For the cyclic Mann's type algorithm, a finite family of asymptotically nonexpansive mappings was introduced by Qin et al. [17] and Osilike and Shehu [14], independently. Let {Tj}j=0N-1 be a finite family of asymptotically nonexpansive self-mappings of C. For a given x0∈C, and a real sequence {αn}n=0∞⊆(0,1), the sequence {xn}n=0∞ is generated as follows:
(1.8)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)T02xN,xN+2=αN+1xN+1+(1-αN+1)T12xN+1,⋮x2N=α2N-1x2N-1+(1-α2N-1)TN-12x2N-1,x2N+1=α2Nx2N+(1-α2N)T03x2N,x2N+2=α2N+1x2N+1+(1-α2N+1)T13x2N+1,⋮
The algorithm can be expressed in a compact form as
(1.9)xn+1=αnxn+(1-αn)Ti(n)k(n)xn,n≥0,
where n=(k-1)N+i, i=i(n)∈J={0,1,2,…,N-1}, k=k(n)≥1 positive integer and limn→∞k(n)=∞. Similarly, we can define the cyclic Halpern's type algorithm for asymptotically nonexpansive mappings as follows:
(1.10)xn+1=αnx0+(1-αn)Ti(n)k(n)xn,n≥0.The purpose of this paper is to extend the hybrid algorithms (1.4) and (1.5) to the cyclic Mann's type (1.9) and the cyclic Halpern's type (1.10). Our results generalize the corresponding results of Kim and Xu [19] and Zhang and Chen [20] from a single asymptotically nonexpansive mapping to a finite family of asymptotically nonexpansive mappings, 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 spaceH satisfies the Opial's condition [27]; that is, for each sequence {xn} in H which converges weakly to a point x∈H, we have
(2.1)limsup⁡n→∞‖xn-x‖<limsup⁡n→∞‖xn-y‖,
for all y∈H,y≠x.Recall that given a closed convex subset ofC of a real Hilbert space H, the nearest point projection PC form H onto C assigns to each x∈C its nearest point denoted PCx in C from x to C; that is, PCx is the unique point in X with the property
(2.2)‖x-PCx‖≤‖x-y‖,∀y∈C.
The following Lemmas 2.1 and 2.2 are well known.Lemma 2.1.
LetC be a closed convex subset of a real Hilbert space H. Given x∈H and z∈C, then z=PCx if and only if there holds the relation
(2.3)〈x-z,y-z〉≤0,∀y∈C.Lemma 2.2.
LetH be a real Hilbert space, then for all x,y∈H(2.4)‖x-y‖2=‖x‖2-‖y2‖-2〈x-y,y〉.Lemma 2.3 (see [28]).
LetX be a uniformly convex Banach space, C a nonempty closed convex subset of X, and T:C→C an asymptotically nonexpansive mapping. Then (I-T) is demiclosed at zero, that is, if xn⇀x and xn-Txn→0, then x∈F(T).Lemma 2.4 (see [22]).
LetC be a closed convex subset of a real Hilbert space H. Let {xn} be sequences in H and u∈H. Let q=PCu. If {xn} is such that ωw(xn)⊂C and satisfies the condition
(2.5)‖xn-u‖≤‖u-q‖,∀n≥1,
then {xn} converges strongly to q.Lemma 2.5 (see [22]).
LetC be a closed convex subset of a real Hilbert space H. For any x,y,z∈H and real number a∈ℝ, the set
(2.6){v∈C:‖y-v‖2≤‖x-v‖2+〈z,v〉+a}
is convex and closed.


## 3. Main Results

In this section, we consider a finite family of asymptotically nonexpansive mappings{Tj}j=0N-1; that is, there exists {ujn}⊆[0,∞), j∈J:={0,1,2,…,N-1} with limn→∞ujn=0, for all j∈J such that
(3.1)‖Tjnx-Tjny‖≤(1+ujn)‖x-y‖,
for all n≥1 and x,y∈C. Let un:=maxj∈J{ujn}, then limn→∞un=0, and
(3.2)‖Tjnx-Tjny‖≤(1+un)‖x-y‖,
for all n≥1, and for all x,y∈C and j∈J.We prove the following theorems.Theorem 3.1.
LetC be a bounded closed convex subset of a Hilbert space H, and let {Tj}j=0N-1:C→C be a finite family of asymptotically nonexpansive mappings with F:=⋂j=0N-1‍F(Tj)≠∅. Assume that {αn}⊆(0,1) such that limn→∞αn=0. Suppose the sequence {xn} generated by
(3.3)x0∈Cischosenarbitrary,yn=αnx0+(1-αn)Ti(n)k(n)xn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖2+αn(‖x0‖2+2〈xn-x0,z〉)+θn},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,
where θn=(2+un)un(1-αn)(diamC)2→0, as n→∞. Then {xn} converges strongly to PFx0.Proof.
By Lemma2.5, we conclude that Cn is closed and convex. It is obvious that Qn and F are closed and convex. Then, the projection mappings PCn∩Qnx0 and PFx0 are well defined. We divide the proof into several steps.
Step1. 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
(3.4)‖yn-p‖2≤αn‖x0-p‖2+(1-αn)‖Ti(n)k(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,
where θn=(2+un)un(1-αn)(diamC)2. Hence, p∈Cn, that is, F⊆Cn, for all n.
Next, we prove thatF⊆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 onto Cm∩Qm, it follows from Lemma 2.1 that
(3.5)〈xm+1-z,x0-xm+1〉≥0,∀z∈Cm∩Qm.
As F⊆Cm∩Qm, in particular, we have
(3.6)〈xn+1-p,x0-xm+1〉≥0,∀p∈F.
Thus, F⊆Qm+1. Therefore, F⊆Cn∩Qn, for all n≥0.
Step2. We prove that
(3.7)‖xn+j-xn‖⟶0asn⟶∞,∀j=0,1,2,…,N-1.
Since the definition ofQn implies that xn=PQnx0, we have
(3.8)‖xn-x0‖≤‖x0-y‖,∀y∈Qn.
By Step 1, F⊆Qn, we have
(3.9)‖xn-x0‖≤‖x0-p‖,∀p∈F.
In particular,
(3.10)‖xn-x0‖≤‖x0-q‖,q=PFx0.
Since xn+1∈Qn, we have 〈xn+1-xn,xn-x0〉≥0 and ∥xn-x0∥≤∥xn+1-x0∥. The second inequality shows that the sequence {∥xn-x0∥} is nondecreasing. Since C is bounded, we obtain that the limn→∞∥xn-x0∥ exists.
With the help of Lemma2.2, we obtain
(3.11)‖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⟶0asn⟶∞.
Consequently,
(3.12)‖xn+j-xn‖⟶0asn⟶∞,∀j∈J.
Step3. We now claim that ∥xn-Tjxn∥→0, as n→∞, for all j∈J. Notice that for all n>N, n=(n-N)(modN), 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
(3.13)‖yn-Ti(n)k(n)xn‖=αn‖x0-Ti(n)k(n)xn‖⟶0asn⟶∞.
It follows from the fact xn+1∈Cn that we have
(3.14)‖yn-xn+1‖2≤‖xn-xn+1‖2+αn(‖x0‖2+2〈xn-x0,xn+1〉)+θn⟶0asn⟶∞,‖Ti(n)k(n)xn-xn‖≤‖Ti(n)k(n)xn-yn‖+‖yn-xn+1‖+‖xn+1-xn‖⟶0asn⟶∞.
Putting L=supn≥0{1+un}, we deduce that
(3.15)‖xn+1-Ti(n)xn‖≤‖xn+1-Ti(n)k(n)xn‖+‖Ti(n)k(n)xn-Ti(n)xn‖≤‖xn+1-xn‖+‖xn-Ti(n)k(n)xn‖+L‖Ti(n)k(n)-1xn-xn‖≤‖xn+1-xn‖+‖xn-Ti(n)k(n)xn‖+L(‖Ti(n)k(n)-1xn-Ti(n-N)k(n)-1xn-N‖+‖Ti(n-N)k(n)-1xn-N-xn-N-1‖+‖xn-N-1-xn‖)≤‖xn+1-xn‖+‖xn-Ti(n)k(n)xn‖+L2‖xn-xn-N‖+L‖Ti(n-N)k(n)-1xn-N-xn-N-1‖+L‖xn-N-1-xn‖⟶0asn⟶∞.
Hence,
(3.16)‖xn-Ti(n)xn‖≤‖xn-xn+1‖+‖xn+1-Ti(n)xn‖⟶0asn⟶∞.
Consequently, for all j=0,1,…,N-1, we have (3.17)‖xn-Tn+jxn‖≤‖xn-xn+j‖+‖xn+j-Tn+jxn+j‖+L‖xn+j-xn‖⟶0asn⟶∞.
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 every Hilbert space satisfies Opial’s condition, we have
(3.18)lim⁡n→∞⁡‖xn-p‖=lim⁡j→∞⁡‖xnj-p‖<lim⁡j→∞⁡‖xnj-p1‖=lim⁡n→∞⁡‖xn-p1‖=lim⁡k→∞⁡‖xnk-p1‖<lim⁡k→∞⁡‖xnk-p‖<lim⁡n→∞⁡‖xn-p‖.
This is a contradiction. Hence, ωw(xn)⊆F. Then by virtue of (3.10) and Lemma 2.4, we conclude that xn→q as n→∞, where q=PFx0.Recall that a mappingT is said to be asymptotically strictly pseudocontractive [29], if there exist λ∈[0,1) and a sequence {un}⊆[0,∞) with limn→∞un=0 such that
(3.19)‖Tnx-Tny‖2≤(1+un)2‖x-y‖2+λ‖(I-Tn)x-(I-Tn)y‖2,
for all n and x,y∈C.Theorem 3.2.
LetC be a bounded closed convex subset of a Hilbert space H, and let {Tj}j=0N-1:C→C be a finite family of asymptotically nonexpansive mappings with F:=⋂j=0N-1‍F(Tj)≠∅. Assume that {αn}⊆(0,a), for some 0<a<1. Suppose the sequence {xn} generated by
(3.20)x0∈Cischosenarbitrary,yn=αnxn+(1-αn)Ti(n)k(n)xn,Cn={z∈C:‖yn-z‖2≤‖xn-z‖2+θn},Qn={z∈C:〈xn-z,x0-xn〉≥0},xn+1=PCn∩Qnx0,
where θn=(2+un)un(diamC)2. Then {xn} converges strongly to PFx0.Proof.
SinceTj is asymptotically nonexpansive if and only if Tj is asymptotically strictly pseudocontractive mapping with λ=0. Then, the rest of proof follows from Theorem 3.2 of Osilike and Shehu [14] and Theorem 2.2 of Qin et al. [17] directly by letting λ=0.



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