# Robustness of Krasnoselski-Mann's Algorithm for Asymptotically Nonexpansive Mappings

Authors: Yu-Chao Tang; Li-Wei Liu
Journal: ISRN Mathematical Analysis
 (2011)

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## Abstract

Iterative approximation of fixed points of nonexpansive mapping is a very active theme in many aspects of mathematical and engineering areas, in particular, in image recovery and signal processing. Because the errors usually occur in few places, it is necessary to show that whether the iterative algorithm is robust or not. In the present work, we prove that Krasnoselski-Mann's algorithm is robust for asymptotically nonexpansive mapping in a Banach space setting. Our results generalize the corresponding results existing in the literature.


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## Body

## 1. Introduction

Many practical problems can be formulated as the fixed point problem ofx=Tx, where T is a nonexpansive mapping. Iterative methods as a powerful tool are often used to approximate the fixed points of such mapping. It has been show that the methods used to find fixed points of nonexpansive mapping covered a widely applied mathematics problems, such as the convex feasibility problem [1–3] and the split feasibility problem [4–6]. It is recommended for interested reader to [7] for an extensive study on the theory about iterative fixed point theory.LetX be a real Banach space. T:X→X is called a nonexpansive mapping if for any x,y∈X, ∥Tx-Ty∥≤∥x-y∥. Krasnoselski-Mann's iteration method for finding fixed points of T is defined by
(1.1)foranyinitialx0∈X,xn+1=(1-αn)xn+αnTxn,n≥0,
where {αn} is a sequence in [0,1].In 2001, Combettes [8] considered a parallel projection method algorithm in signal synthesis problems in a real Hilbert space H as follows:(1.2)xn+1=xn+λn(∑i=1mωi(Pixn+ci,n)-xn),
where {λn}⊆(0,2), {ωi}i=1m are positive weights such that ∑i=1mωi=1, Pi is the projection of a signal x∈H onto a closed convex subset Si of H, and ci,n stands for the error made in computing the projection onto Si at each iteration n. He firstly proved that the sequence {xn} generated by (1.2) converges weakly to a point in G, where G:=⋂i=1mSi.Kim and Xu [9] generalized the results of Combettes [8] from Hilbert spaces to uniformly convex Banach spaces and obtained its equivalent form as follows:
(1.3)xn+1=(1-αn)xn+αn(Txn+en),n≥0,
where αn:=λn/2∈(0,1), en:=2∑i=1mωici,n, and T is nonexpansive. They proved that the weak convergence of the (1.3) in a uniformly convex Banach space. More precisely, they proved that the following main theorems.Theorem 1.1 (see [9]).
Assume thatX is a uniformly convex Banach space. Assume, in addition, that either X* has the Kadec-Klee property or X satisfies Opial's property. Let T:X→X be a nonexpansive mapping such that F(T)≠∅(F(T) denotes the set of fixed points of  T, that is, F(T)={x∈X:Tx=x}). Given an initial guess x0∈X. Let {xn} be generated by (1.3) and satisfy the following properties:(i)
∑n=0∞αn(1-αn)=∞,(ii)
∑n=0∞αn∥en∥<∞.Then the sequence {xn} converges weakly to a fixed point of  T.Theorem 1.2 (see [9]).
LetC be a nonempty closed convex subset of a Hilbert space H and T:C→C a nonexpansive mapping with F(T)≠∅. Given an initial guess x0∈X. Let {xn} be generated by either
(1.4)xn+1=(1-αn)xn+αnPC(Txn+en),n≥0,
or
(1.5)xn+1=PC[(1-αn)xn+αn(Txn+en)],n≥0,
where the sequences {αn} and {en} are such that(i)
∑n=0∞αn(1-αn)=∞,(ii)
∑n=0∞αn∥en∥<∞.Then {xn} converges weakly to a fixed point of  T.Very recently, Ceng et al. [10] extended the algorithm (1.3) of Kim and Xu [9] to Krasnoselski-Mann's algorithm with perturbed mapping defined by the following:
(1.6)xn+1=λnxn+(1-λn)(Txn+en)-λnμnF(xn),n≥0,
where λn,μn∈[0,1], and F is a strongly accretive and strictly pseudocontractive mapping.An important generalization of the class of nonexpansive mapping is asymptotically nonexpansive mapping (i.e., forT:C→C, if there exists a sequence {un}⊂[0,+∞), limn→∞un=0 such that
(1.7)‖Tnx-Tny‖≤(1+un)‖x-y‖,
for all x,y∈C and n≥0), which was introduced by Goebel and Kirk [11]; they proved that if C is a nonempty closed, convex, and bounded subset of a uniformly convex Banach space, then every asymptotically nonexpansive self-mapping has a fixed point. The class of asymptotically nonexpansive mapping has been studied by many authors and some recent results can be found in [12–17] and references cited therein.Inspired and motivated by the above works, the purpose of this paper is to extend the results of Kim and Xu [9] from nonexpansive mapping to asymptotically nonexpansive mapping. We prove that the Krasnoselski-Mann iterative sequence converges weakly to the fixed point of asymptotically nonexpansive mapping.


## 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 Opial's condition [18]; that is, for each sequence {xn} in H which converges weakly to a point x∈H, one has
(2.1)liminf⁡n→∞‖xn-x‖<liminf⁡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 from H onto C assigns to each x∈C its nearest point denoted by 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.A Banach spaceX is said to have the Kadec-Klee property [19] if for any sequence {xn} in X, xn⇀x and ∥xn∥→∥x∥ imply that xn→x.A mappingT is said to be demiclosed at zero if whenever {xn} is a sequence in D(T) such that {xn} converges weakly to x∈D(T) and {Txn} converges strongly to zero, then Tx=0.Lemma 2.1 (see [20]).
LetX be a real uniformly convex Banach space, let C be a nonempty closed convex subset of X, and let T:C→X be an asymptotically nonexpansive mapping with a sequence {un}⊂[0,∞) and limn→∞un=0; then (I-T) is demiclosed at zero.Lemma 2.2 (see [21]).
Given a numberr>0, a real Banach space is uniformly convex if and only if there exists a continuous strictly increasing function ϕ:[0,∞)→[0,∞), ϕ(0)=0, such that
(2.3)‖λx+(1-λ)y‖2≤λ‖x‖2+(1-λ)‖y‖2-λ(1-λ)ϕ(‖x-y‖),
for all λ∈[0,1] and x,y∈X such that ∥x∥≤r and ∥y∥≤r.Lemma 2.3 (see [22]).
LetX be a real uniformly convex Banach space such that its dual X* has Kadec-Klee property. Let {xn} be a bounded sequence in X and q1,q2∈ωw({xn}). Suppose that that
(2.4)lim⁡n→∞‖αxn+(1-α)q1-q2‖
exists for all α∈[0,1]. Then q1=q2.Lemma 2.4 (see [23]).
Let{an}, {bn}, and {cn} be sequences of nonnegative real numbers satisfying the inequality
(2.5)an+1≤(1+cn)an+bn,n≥1.
If ∑n=1∞cn<+∞,∑n=1∞bn<+∞, then (i) limn→∞an exists. (ii) In particular, if liminfn→∞an=0, one has limn→∞an=0.


## 3. Main Results

We state our first theorem as follows.Theorem 3.1.
Suppose thatX is a uniformly convex Banach space, and X* has the Kadec-Klee property or X satisfies Opial's property. Let T:X→X be an asymptotically nonexpansive mapping with ∑n=0∞un<∞. For any x0∈X, the sequence {xn} is generated by the following Krasnoselski-Mann's algorithm:
(3.1)xn+1=(1-αn)xn+αn(Tnxn+en),n≥0,
where {αn} and {en} satisfy the following conditions:(i)
0<a<αn<b<1, for some a,b∈(0,1) and for all n≥0;(ii)
∑n=0∞αn∥en∥<∞.If F(T)≠∅, then the sequence {xn} converges weakly to a fixed point of  T.For the sake of convenience, we need the following lemmas.Lemma 3.2.
LetX be a real normed linear space and let T:X→X be an asymptotically nonexpansive mapping with ∑n=0∞un<∞. Let {xn} be the sequence as defined in (3.1) and satisfy the conditions in Theorem 3.1. Suppose that F(T)≠∅; then the limit limn→∞∥xn-p∥ exists for p∈F(T).Proof.
By (3.1), one has
(3.2)‖xn+1-p‖=‖(1-αn)(xn-p)+αn(Tnxn-p+en)‖≤(1-αn)‖xn-p‖+αn‖Tnxn-p‖+αn‖en‖≤(1-αn)‖xn-p‖+αn(1+un)‖xn-p‖+αn‖en‖≤(1+un)‖xn-p‖+αn‖en‖.
Since ∑n=0∞un<∞ and ∑n=0∞αn∥en∥<∞, we obtain from Lemma 2.4 that the limit limn→∞∥xn-p∥ exists. Furthermore, the sequence {xn} is bounded.Lemma 3.3.
LetX be a real uniformly convex Banach space and let T:X→X be an asymptotically nonexpansive mapping with ∑n=0∞un<∞. Let {xn} be the sequence as defined in (3.1) and satisfy the conditions in Theorem 3.1. Suppose that F(T)≠∅; then limn→∞∥txn+(1-t)p-q∥ exists for all t∈[0,1] and p,q∈F(T).Proof.
Letdn(t)=∥txn+(1-t)p-q∥; then limn→∞dn(0)=∥p-q∥ exists. It follows from Lemma 3.2 that limn→∞dn(1)=limn→∞∥xn-q∥ exists. Next, we show that limn→∞dn(t) exists for any t∈(0,1).
LetTnx:=(1-αn)x+αnTnx+αnen, for all x∈X. For any x,z∈X, one has
(3.3)‖Tnx-Tnz‖=‖(1-αn)(x-z)+αn(Tnx-Tnz)‖≤(1-αn)‖x-z‖+αn‖Tnx-Tnz‖≤(1-αn)‖x-z‖+αn(1+un)‖x-z‖≤(1+un)‖x-z‖.
Set Sn,m=Tn+m-1Tn+m-2⋯Tn, m≥1. The rest of the proof is the same as Lemma  3.3 of [14, 16]. This completes the proof of Lemma 3.3.Now, we give the proof of Theorem3.1.Proof.
Letp∈F(T). With the help of Lemma 2.2 and the inequality ∥a+b∥2≤∥a∥2+2∥a∥·∥b∥+∥b∥2, one has
(3.4)‖xn+1-p‖2=‖(1-αn)(xn-p)+αn(Tnxn-p)+αnen‖2≤‖(1-αn)(xn-p)+αn(Tnxn-p)‖2+2αn‖en‖⋅‖(1-αn)(xn-p)+αn(Tnxn-p)‖+αn2‖en‖2≤(1-αn)‖xn-p‖2+αn‖Tnxn-p‖2-αn(1-αn)ϕ(‖xn-Tnxn‖)+2αn‖en‖((1-αn)‖xn-p‖+αn(1+un)‖xn-p‖)+αn2‖en‖2≤(1-αn)‖xn-p‖2+αn(1+un)2‖xn-p‖2-αn(1-αn)ϕ(‖xn-Tnxn‖)+2αn‖en‖(1+un)‖xn-p‖+αn2‖en‖2≤(1+un)2‖xn-p‖2-αn(1-αn)ϕ(‖xn-Tnxn‖)+2αn‖en‖(1+un)‖xn-p‖+αn2‖en‖2,
which follows that
(3.5)a(1-b)ϕ(‖xn-Tnxn‖)≤αn(1-αn)ϕ(‖xn-Tnxn‖)≤(1+un)2‖xn-p‖2-‖xn+1-p‖2+2αn‖en‖(1+un)‖xn-p‖+αn2‖en‖2.
This implies that
(3.6)∑n=0∞ϕ(‖xn-Tnxn‖)<∞.
Therefore limn→∞ϕ(∥xn-Tnxn∥)=0. Since ϕ is strictly increasing and continuous function with ϕ(0)=0, then limn→∞∥xn-Tnxn∥=0. Also, one has the following inequalities:
(3.7)‖xn+1-xn‖≤αn‖Tnxn-xn‖+αn‖en‖⟶0asn⟶∞,(3.8)‖Tnxn+1-xn+1‖=‖Tnxn+1-(1-αn)xn-αn(Tnxn+en)‖=‖(Tnxn+1-Tnxn)+(1-αn)(Tnxn-xn)-αnen‖≤‖Tnxn+1-Tnxn‖+(1-αn)‖Tnxn-xn‖+αn‖en‖≤(1+un)‖xn+1-xn‖+(1-αn)‖Tnxn-xn‖+αn‖en‖=(1+un)‖αn(Tnxn-xn)+αnen‖+(1-αn)‖Tnxn-xn‖+αn‖en‖≤(1+un)αn‖Tnxn-xn‖+(1-αn)‖Tnxn-xn‖+(1+un)αn‖en‖+αn‖en‖≤(1+un)‖Tnxn-xn‖+(1+un)αn‖en‖+αn‖en‖⟶0asn⟶∞.
On the other hand,
(3.9)‖xn+1-Txn+1‖≤‖xn+1-Tn+1xn+1‖+‖Tn+1xn+1-Txn+1‖≤‖xn+1-Tn+1xn+1‖+(1+u1)‖Tnxn+1-xn+1‖⟶0asn⟶∞.
By (3.7)–(3.9), we obtain
(3.10)‖xn-Txn‖≤‖xn-xn+1‖+‖xn+1-Txn+1‖+‖Txn+1-Txn‖≤(2+u1)‖xn-xn+1‖+‖xn+1-Txn+1‖⟶0asn⟶∞,
that is, limn→∞∥xn-Txn∥=0.
From Lemma3.2, we know that {xn} is bounded. Since X is a uniformly convex Banach space, {xn} has a convergent subsequence {xnj}. By the demiclosedness principle of I-T, we obtain ωw⊆F(T). The rest of proof is followed by the standard argument in Theorem  3.3 of Kim and Xu [9]. This completes the proof.Theorem 3.4.
LetC be a nonempty closed convex subset of a Hilbert space H and let T:C→C be an asymptotically nonexpansive mapping with ∑n=0∞un<∞. For any x0∈X, the sequence {xn} is generated by either
(3.11)xn+1=(1-αn)xn+αnPC(Tnxn+en),n≥0,
or
(3.12)xn+1=PC[(1-αn)xn+αn(Tnxn+en)],n≥0,
where the sequences {αn} and {en} are such that(i)
0<a<αn<b<1, for some a,b∈(0,1) and for all n≥0;(ii)
∑n=0∞αn∥en∥<∞.If F(T)≠∅, then {xn} converges weakly to a fixed point of  T.Proof.
Letp∈F(T). By (3.11), one has
(3.13)‖xn+1-p‖=‖[(1-αn)xn+αnPC(Tnxn+en)]-p‖≤(1-αn)‖xn-p‖+αn‖PC(Tnxn+en)-p‖≤(1-αn)‖xn-p‖+αn‖Tnxn+en-p‖≤(1-αn)‖xn-p‖+αn(1+un)∥xn-p∥+αn‖en‖≤(1+un)‖xn-p‖+αn‖en‖.
Notice the condition (ii) and ∑n=0∞un<∞; by Lemma 2.4, limn→∞∥xn-p∥ exists. Hence, {xn} is bounded.
By the well-known inequality∥tx+(1-t)y∥2=t∥x∥2+(1-t)∥y∥2-t(1-t)∥x-y∥2, for all x,y∈H and t∈[0,1], we obtain
(3.14)∥xn+1-p∥2=‖[(1-αn)xn+αnPC(Tnxn+en)]-p‖2=‖(1-αn)(xn-p)+αn(Tnxn-p)+αn[PC(Tnxn+en)-Tnxn]‖2≤‖(1-αn)(xn-p)+αn(Tnxn-p)‖2+‖αn[PC(Tnxn+en)-Tnxn]‖2+2αn‖(1-αn)(xn-p)+αn(Tnxn-p)‖⋅‖PC(Tnxn+en)-Tnxn‖≤(1-αn)‖xn-p‖2+αn‖Tnxn-p‖2-αn(1-αn)‖xn-Tnxn‖2+2αn[(1-αn)‖xn-p‖+αn(1+un)‖xn-p‖]‖en‖≤(1+un)2‖xn-p‖2-αn(1-αn)‖xn-Tnxn‖2+2(1+un)αn‖en‖‖xn-p‖.
That is,
(3.15)a(1-b)‖xn-Tnxn‖2≤αn(1-αn)‖xn-Tnxn‖2≤(1+un)2‖xn-p‖2-‖xn+1-p‖2+2(1+un)αn‖en‖‖xn-p‖.
This implies that
(3.16)∑n=0∞‖xn-Tnxn‖2<∞.
Therefore limn→∞∥xn-Tnxn∥=0. We also have
(3.17)‖xn+1-xn‖=‖(1-αn)xn+αnPC(Tnxn+en)-xn‖≤αn‖Tnxn-xn‖+αn‖en‖⟶0asn⟶∞,∥Tnxn+1-xn+1∥=‖Tnxn+1-(1-αn)xn-αnPC(Tnxn+en)‖=‖(Tnxn+1-Tnxn)+(Tnxn-xn)+αn(xn-PC(Tnxn+en))‖≤(1+un)‖xn+1-xn‖+(1+αn)‖Tnxn-xn‖+αn‖en‖⟶0asn⟶∞.
It follows from (3.9) and (3.10) that limn→∞∥xn-Txn∥=0. Since a Hilbert space H must be a uniformly convex Banach space and satisfy Opial's property, then the rest of proof is the same as Theorem 3.1. So it is omitted.



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