# Iterative Schemes for Fixed Point Computation of

Authors: Rudong Chen
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

Fixed point (especially, the minimum norm fixed point) computation is an interesting
topic due to its practical applications in natural science. The purpose of the
paper is devoted to finding the common fixed points of an infinite family of nonexpansive
mappings. We introduce an iterative algorithm and prove that suggested scheme
converges strongly to the common fixed points of an infinite family of nonexpansive
mappings under some mild conditions. As a special case, we can find the minimum
norm common fixed point of an infinite family of nonexpansive mappings.


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## Body

## 1. Introduction

In many problems, it is needed to find a solution with minimum norm. In an abstract way, we may formulate such problems as finding a pointx† with the property(1.1)x†∈C,‖x†‖2=min⁡x∈C‖x‖2,
where C is a nonempty closed convex subset of a real Hilbert space H. In other words, x† is the (nearest point or metric) projection of the origin onto C,(1.2)x†=PC(0),
where PC is the metric (or nearest point) projection from H onto C.A typical example is the least-squares solution to the constrained linear inverse problem [1](1.3)Ax=b,x∈C,
where A is a bounded linear operator from H to another real Hilbert space H1 and b is a given point in H1. The least-squares solution to (1.3) is the least-norm minimizer of the minimization problem(1.4)min⁡x∈C‖Ax-b‖2.
Recently, some authors consider the minimum norm solution problem by using the iterative algorithm. For some related works, please refer to [2–4]. Yao and Xu [3] introduced the following algorithm:(1.5)xn+1=PC(αnf(xn)+(1-αn)Txn),n≥0.
They proved that the sequence {xn} converges in norm to the unique solution x̃ of VI 〈(I-f)x̃,x-x̃〉≥0, x∈Fix(T). Particularly, the sequence {xn} defined by(1.6)xn+1=PC((1-αn)Txn),n≥0,
converges to the minimum norm fixed point of T. We note that the authors added an additional assumption, that is, limn→∞αn+1/αn=1. Iterative algorithm for finding the fixed points of nonexpansive mappings has been considered by many authors, see [5–20].The purpose of this paper is to extend Yao and Xu’s result to an infinite family of nonexpansive mappings{Tn}n=0∞. We suggest a new algorithm. Particularly, we drop the above additional assumption and prove the suggested algorithm converges strongly to the common fixed points of {Tn}n=0∞. As a special case, we can find the minimum norm fixed point of {Tn}n=0∞.


## 2. Preliminaries

LetH be a real Hilbert space with inner product 〈·,·〉 and norm ∥·∥, respectively, and let C be a nonempty closed convex subset of H. We call f:C→H a κ-contraction if there exists a constant κ∈[0,1) such that ∥f(x)-f(y)∥≤κ∥x-y∥ for all x,y∈C. A bounded linear operator B is said to be strongly positive on H if there exists a constant α>0 such that(2.1)〈Bx,x〉≥α‖x‖2,∀x∈H.
Recall that the (nearest point or metric) projection from H onto C, denoted by PC, is defined in such a way that, for each x∈H, PCx is the unique point in C with the property(2.2)‖x-PCx‖=min⁡{‖x-y‖:y∈C}.
It is known that PC satisfies(2.3)〈x-y,PCx-PCy〉≥‖PCx-PCy‖2,∀x,y∈H.
Moreover, PC is characterized by the following properties:(2.4)〈x-PCx,y-PCx〉≤0,‖x-y‖2≥‖x-PCx‖2+‖y-PCx‖2,
for all x∈H and y∈C.We also need other sorts of nonlinear operators which are introduced below. LetT:H→H be a nonlinear operator.(a)
T is nonexpansive if ∥Tx-Ty∥≤∥x-y∥ for all x,y∈H.(b)
T is firmly nonexpansive if 2T-I is nonexpansive. Equivalently, T=(I+S)/2, where S:H→H is nonexpansive. Alternatively, T is firmly nonexpansive if and only if
(2.5)‖Tx-Ty‖2≤〈Tx-Ty,x-y〉,x,y∈H.(c)
T is averaged if T=(1-τ)I+τS, where τ∈(0,1) and S:H→H is nonexpansive. In this case, we also say that T is τ-averaged. A firmly nonexpansive mapping is 1/2-averaged.It is well known that bothPC and I-PC are firmly nonexpansive. We will need to use the following notation:(i)
Fix(T) stands for the set of fixed points of T;(ii)
xn⇀x stands for the weak convergence of {xn} to x;(iii)
xn→x stands for the strong convergence of {xn} to x.LetT1,T2,… be infinite mappings of C into itself, and let ξ1,ξ2,… be real numbers such that 0≤ξi≤1 for every i∈N. For any n∈N, define a mapping Wn of C into itself as follows:(2.6)Un,n+1=I,Un,n=ξnTnUn,n+1+(1-ξn)I,Un,n-1=ξn-1Tn-1Un,n+(1-ξn-1)I,⋮Un,k=ξkTkUn,k+1+(1-ξk)I,Un,k-1=ξk-1Tk-1Un,k+(1-ξk-1)I,⋮Un,2=ξ2T2Un,3+(1-ξ2)I,Wn=Un,1=ξ1T1Un,2+(1-ξ1)I.
Such Wn is called the W-mapping generated by Tn,Tn-1,…,T2,T1 and ξn,ξn-1,…,ξ2,ξ1. For the iterative algorithm for a finite family of nonexpansive mappings, we refer the reader to [21].We have the following crucial lemmas concerningWn which can be found in [22].Lemma 2.1.
LetC be a nonempty closed convex subset of a real Hilbert space  H. Let  T1,T2,… be nonexpansive mappings of  C into itself such that  ⋂n=1∞F(Tn) is nonempty, and let  ξ1,ξ2,… be real numbers such that  0<ξi≤b<1 for any  i∈N. Then, for every  x∈C and  k∈N, the limit  limn→∞Un,kx exists.Lemma 2.2.
LetC be a nonempty closed convex subset of a real Hilbert space  H. Let  T1,T2,… be nonexpansive mappings of  C into itself such that  ⋂n=1∞F(Tn) is nonempty, and let  ξ1,ξ2,… be real numbers such that  0<ξi≤b<1 for any  i∈N. Then,  F(W)=⋂n=1∞F(Tn).The following remark [23] is important to prove our main results.Remark 2.3.
Using Lemma2.1,  one can define a mapping W of C into itself as Wx=limn→∞Wnx=limn→∞Un,1x, for every x∈C. If {xn} is a bounded sequence in C, then one has
(2.7)lim⁡n→∞‖Wxn-Wnxn‖=0.Throughout this paper, we will assume that0<ξi≤b<1 for every i∈N.Lemma 2.4 (see [24]).
LetK be a nonempty closed convex subset of a real Hilbert space  H. Let  T:K→K be a nonexpansive mapping with  Fix(T)≠∅. Then  T is demiclosed on  K, that is, if  xn⇀x∈K weakly and  xn-Txn→0, then  x=Tx.Lemma 2.5 (see [25]).
Let{xn} and  {zn} be bounded sequences in a Banach space  X, and let  {βn} be a sequence in  [0,1] with  0<liminfn→∞βn≤limsupn→∞βn<1. Suppose  xn+1=(1-βn)zn+βnxn for all integers  n≥0 and  limsupn→∞(∥zn+1-zn∥-∥xn+1-xn∥)≤0. Then,  limn→∞∥zn-xn∥=0.Lemma 2.6 (see [26]).
Assume{an} is a sequence of nonnegative real numbers such that  an+1≤(1-γn)an+δn where  {γn} is a sequence in  (0,1) and  {δn} is a sequence such that (1)
∑n=1∞γn=∞;(2)
limsupn→∞δn/γn≤0 or ∑n=1∞|δn|<∞.Then limn→∞an=0.


## 3. Main Result

In this section, we introduce our algorithm and prove its strong convergence.Theorem 3.1.
LetC be a nonempty closed convex subset of a real Hilbert space  H. Let  {Tn}n=1∞ be a sequence of nonexpansive mappings from  C to  C such that the common fixed point set  F:=⋂n=1∞F(Tn)≠∅. Let  f:C→H be a  κ-contraction and  B:H→H be a self-adjoint, strongly positive bounded linear operator with coefficient  α>0. Let  σ be a constant such that  0<σκ<α. For an arbitrary initial point  x0 belonging to  C, one defines a sequence  {xn}n≥0 iteratively
(3.1)xn+1=PC[αnσf(xn)+(I-αnB)Wnxn],∀n≥0,
where {αn} is a real sequence in [0,1]. Assume the sequence {αn} satisfies the following conditions:(C1)
limn→∞αn=0;(C2)
∑n=0∞αn=∞.Then the sequence {xn} generated by (3.1) converges in norm to the unique solution x* which solves the following variational inequality:
(3.2)x*∈Fsuch that〈σf(x*)-Bx*,x̃-x*〉≤0,∀x̃∈F.Proof.
Letx̃∈F. From (3.1), we have
(3.3)‖xn+1-x̃‖=‖PC[αnσf(xn)+(I-αnB)Wnxn]-x̃‖≤‖αnσf(xn)+(I-αnB)Wnxn-x̃‖≤αnσ‖f(xn)-f(x̃)‖+‖I-αnB‖‖Wnxn-x̃‖+αn‖σf(x̃)-Bx̃‖≤αnσκ‖xn-x̃‖+(1-αnα)‖xn-x̃‖+αn‖σf(x̃)-Bx̃‖=[1-(α-σκ)αn]‖xn-x̃‖+(α-σκ)αn‖f(x̃)-Bx̃‖(α-σκ).
It follows by induction that
(3.4)‖xn+1-x̃‖≤max⁡{‖xn-x̃‖,‖f(x̃)-Bx̃‖(α-σκ)}≤max⁡{‖x0-x̃‖,‖f(x̃)-Bx̃‖(α-σκ)}.
This indicates that {xn} is bounded. It is easy to deduce that {f(xn)}, {Wnxn}, and {BWnxn} are also bounded.
SetS=2PC-I. It is known that S is nonexpansive. Note that Wn=ξ1T1Un,2xn+(1-ξ1)xn. Then, we can rewrite (3.1) as
(3.5)xn+1=I+S2[αnσf(xn)+(I-αnB)Wnxn]=1-αn2Wnxn+αn2(σf(xn)-BWnxn+Wnxn)+12S[αnσf(xn)+(I-αnB)Wnxn]=1-αn2[(1-ξ)I+ξT1Un,2]xn+αn2(σf(xn)-BWnxn+Wnxn)+12S[αnσf(xn)+(I-αnB)Wnxn]=(1-ξ)(1-αn)2xn+ξ(1-αn)2T1Un,2xn+αn2(σf(xn)-BWnxn+Wnxn)+12S[αnσf(xn)+(I-αnB)Wnxn].
Note that
(3.6)0<lim⁡n→∞(1-ξ)(1-αn)2=1-ξ2<1,ξ(1-αn)2+12=1+ξ2-ξ2αn.
From (3.5), we have
(3.7)xn+1=[1-(1+ξ2+1-ξ2αn)]xn+(1+ξ2+1-ξ2αn)×(ξ(1-αn)/2)T1Un,2xn+(αn/2)(σf(xn)-BWnxn+Wnxn)((1+ξ)/2)+((1-ξ)/2)αn+(1/2)S[αnσf(xn)+(I-αnB)Wnxn]((1+ξ)/2)+((1-ξ)/2)αn=[1-(1+ξ2+1-ξ2αn)]xn+(1+ξ2+1-ξ2αn)yn,
where
(3.8)yn=(ξ(1-αn)/2)T1Un,2xn+(αn/2)(σf(xn)-BWnxn+Wnxn)((1+ξ)/2)+((1-ξ)/2)αn+(1/2)S[αnσf(xn)+(I-αnB)Wnxn]((1+ξ)/2)+((1-ξ)/2)αn=ξ(1-αn)T1Un,2xn+αn(σf(xn)-BWnxn+Wnxn)+S[αnσf(xn)+(I-αnB)Wnxn]1+ξ+(1-ξ)αn.
Set zn=σf(xn)-BWnxn+Wnxn and z̃n=αnσf(xn)+(I-αnB)Wnxn for all n. Then
(3.9)yn=ξ(1-αn)T1Un,2xn+αnzn+Sz̃n1+ξ+(1-ξ)αn,∀n≥0.
It follows that
(3.10)yn+1-yn=ξ(1-αn+1)T1Un+1,2xn+1+αn+1zn+1+Sz̃n+11+ξ+(1-ξ)αn+1-ξ(1-αn)T1Un,2xn+αnzn+Sz̃n1+ξ+(1-ξ)αn=ξ(1-αn+1)1+ξ+(1-ξ)αn+1(T1Un+1,2xn+1-T1Un,2xn)+(ξ(1-αn+1)1+ξ+(1-ξ)αn+1-ξ(1-αn)1+ξ+(1-ξ)αn)T1Un,2xn+αn+1zn+11+ξ+(1-ξ)αn+1-αnzn1+ξ+(1-ξ)αn+Sz̃n+1-Sz̃n1+ξ+(1-ξ)αn+1+(11+ξ+(1-ξ)αn+1-11+ξ+(1-ξ)αn)Sz̃n.
Thus,
(3.11)‖yn+1-yn‖≤ξ(1-αn+1)1+ξ+(1-ξ)αn+1‖T1Un+1,2xn+1-T1Un,2xn‖+|ξ(1-αn+1)1+ξ+(1-ξ)αn+1-ξ(1-αn)1+ξ+(1-ξ)αn|‖T1Un,2xn‖+αn+11+ξ+(1-ξ)αn+1‖zn+1‖+αn1+ξ+(1-ξ)αn‖zn‖+11+ξ+(1-ξ)αn+1‖Sz̃n+1-Sz̃n‖+|11+ξ+(1-ξ)αn+1-11+ξ+(1-ξ)αn|‖Sz̃n‖.
From the nonexpansivity of S, we get
(3.12)‖Sz̃n+1-Sz̃n‖≤‖z̃n+1-z̃n‖=‖αn+1σf(xn+1)+(I-αn+1B)Wn+1xn+1-(αnσf(xn)+(I-αnB)Wnxn)‖≤αn+1‖σf(xn+1)-BWn+1xn+1‖+αn‖σf(xn)-BWnxn‖+‖Wn+1xn+1-Wnxn‖≤αn+1‖σf(xn+1)-BWn+1xn+1‖+αn‖σf(xn)-BWnxn‖+‖Wn+1xn+1-Wn+1xn‖+‖Wn+1xn-Wnxn‖≤αn+1‖σf(xn+1)-BWn+1xn+1‖+αn‖σf(xn)-BWnxn‖+‖xn+1-xn‖+‖Wn+1xn-Wnxn‖.
Since Ti and Un,i are nonexpansive, we have
(3.13)∥T1Un+1,2xn-T1Un,2xn∥≤∥Un+1,2xn-Un,2xn∥=∥ξ2T2Un+1,3xn-ξ2T2Un,3xn∥≤ξ2∥Un+1,3xn-Un,3xn∥≤⋯≤ξ2⋯ξn∥Un+1,n+1xn-Un,n+1xn∥≤M∏i=2nξi,
where M>0 is a constant such that ∥Un+1,n+1xn-Un,n+1xn∥≤M for all n≥0. So,
(3.14)‖T1Un+1,2xn+1-T1Un,2xn‖≤‖T1Un+1,2xn+1-T1Un+1,2xn‖+‖T1Un+1,2xn-T1Un,2xn‖≤‖xn+1-xn‖+M∏i=2nξi.
Hence,
(3.15)‖yn+1-yn‖≤ξ(1-αn+1)1+ξ+(1-ξ)αn+1‖xn+1-xn‖+M∏i=2nξi+|ξ(1-αn+1)1+ξ+(1-ξ)αn+1-ξ(1-αn)1+ξ+(1-ξ)αn|‖T1Un,2xn‖+αn+11+ξ+(1-ξ)αn+1‖zn+1‖+αn1+ξ+(1-ξ)αn‖zn‖+11+ξ+(1-ξ)αn+1×(αn+1‖σf(xn+1)-BWn+1xn+1‖+αn‖σf(xn)-BWnxn‖+‖xn+1-xn‖)+‖Wn+1xn-Wnxn‖+|11+ξ+(1-ξ)αn+1-11+ξ+(1-ξ)αn|‖Sz̃n‖=1+ξ-ξαn+11+ξ+(1-ξ)αn+1‖xn+1-xn‖+|ξ(1-αn+1)1+ξ+(1-ξ)αn+1-ξ(1-αn)1+ξ+(1-ξ)αn|‖T1Un,2xn‖+αn+11+ξ+(1-ξ)αn+1‖zn+1‖+αn1+ξ+(1-ξ)αn‖zn‖+11+ξ+(1-ξ)αn+1×(αn+1‖σf(xn+1)-BWn+1xn+1‖+αn‖σf(xn)-BWnxn‖)+‖Wn+1xn-Wnxn‖+|11+ξ+(1-ξ)αn+1-11+ξ+(1-ξ)αn|‖Sz̃n‖.
Since αn→0, we have ξ(1-αn+1)/(1+ξ+(1-ξ)αn+1)-ξ(1-αn)/(1+ξ+(1-ξ)αn)→0, ∥Wn+1xn-Wnxn∥→0, and 1/(1+ξ+(1-ξ)αn+1)-1/(1+ξ+(1-ξ)αn)→0. Therefore,
(3.16)limsup⁡n→∞(‖yn+1-yn‖-‖xn+1-xn‖)≤0.
By Lemma 2.5, we get
(3.17)lim⁡n→∞‖yn-xn‖=0.
Hence, from (3.7), we deduce
(3.18)lim⁡n→∞‖xn+1-xn‖=lim⁡n→∞(1+ξ2+1-ξ2αn)‖yn-xn‖=0.
Observe that
(3.19)‖xn+1-xn‖=‖PC[αnσf(xn)+(I-αnB)Wn]xn-Wxn‖≤αk‖σf(xn)-BWnxn‖+‖Wnxn-Wxn‖→0.
From (3.18) and (3.19), we deduce
(3.20)lim⁡k→∞‖Wxn-xn‖=0.
Next we prove
(3.21)limsup⁡k→∞〈σf(x*)-Bx*,xk-x*〉≤0,
where x* is the unique solution of VI (3.2).
Indeed, we can choose a subsequence{xni} of {xn} such that
(3.22)limsup⁡n→∞〈σf(x*)-Bx*,xn-x*〉=lim⁡i→∞〈σf(x*)-Bx*,xni-x*〉.
Since {xni} is bounded, there exists a subsequence of {xni} which converges weakly to a point x̃. Without loss of generality, we may assume that {xni} converges weakly to x̃. Therefore, from (3.20) and Lemma 2.4, we have xni⇀x̃∈Fix(W)=F. Therefore,
(3.23)limsup⁡n→∞〈σf(x*)-Bx*,xn-x*〉=lim⁡i→∞〈σf(x*)-Bx*,xni-x*〉=〈σf(x*)-Bx*,x̃-x*〉≤0.
Finally, we show that xn→x*. We observe that
(3.24)‖xn+1-x*‖2=〈xn+1-z̃n,xn+1-x*〉+〈z̃n-x*,xn+1-x*〉.
Since 〈xn+1-z̃n,xn+1-x*〉≤0, we get
(3.25)‖xn+1-x*‖2≤〈z̃n-x*,xn+1-x*〉=〈αnσ(f(xn)-f(x*))+(I-αnB)(Wnxn-x*),xn+1-x*〉+αn〈σf(x*)-Bx*,xn+1-x*〉≤(αnσ‖f(xn)-f(x*)‖+‖I-αnB‖‖Wnxn-x*‖)‖xn+1-x*‖+αn〈σf(x*)-Bx*,xn+1-x*〉≤(1-αn(α-σκ))‖xn-x*‖‖xn+1-x*‖+αn〈σf(x*)-Bx*,xn+1-x*〉≤[1-αn(α-σκ)]22‖xn-x*‖2+12‖xn+1-x*‖2+αn〈σf(x*)-Bx*,xn+1-x*〉.
It follows that
(3.26)‖xn+1-x*‖2≤[1-αn(α-σκ)]‖xn-x*‖2+2αn〈σf(x*)-Bx*,xn+1-x*〉.
Hence, all conditions of Lemma 2.6 are satisfied. Therefore, we immediately deduce that xk→x*. This completes the proof.From (3.1) and Theorem 3.1, we can deduce easily the following result.Corollary 3.2.
LetC be a nonempty closed convex subset of a real Hilbert space  H. Let  {Tn}n=1∞ be a sequence of nonexpansive mappings from  C to  C such that the common fixed point set  F:=⋂n=1∞F(Tn)≠∅. For an arbitrary initial point  x0, one defines a sequence  {xn}n≥0 iteratively
(3.27)xn+1=PC[(1-αn)Wnxn],n≥0,
where {αn} is a real sequence in [0,1]. Assume the sequence {αn} satisfies the following conditions:(C1)
limn→∞αn=0;(C2)
∑n=0∞αn=∞. Then the sequence {xn} generated by (3.27) converges to the minimum norm common fixed point x* of {Tn}.



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