# Contractibility of Fixed Point Sets of Mean-Type Mappings

Authors: S. Iampiboonvatana; P. Chaoha
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

We establish a convergence theorem and explore fixed point sets of certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces. We not only extend previous works by Matkowski to general normed linear spaces, but also obtain a new result on the structure of fixed point sets of quasi-nonexpansive mappings in a nonstrictly convex setting.


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## Body

## 1. Introduction

The theory of mean iteration has been studied long before the 19th century [1]. Johann Carl Friedrich Gauss observed the connection between the arithmetic-geometric mean iteration and an elliptic integral. Indeed, if we recursively define the following sequences of positive real numbers (1)an+1=an+bn2bn+1=anbn,we know that both (an) and (bn) converge to the same limit, say, M(a0,b0). He found that (2)1M1,2=2π∫01dt1-t4,which is later generalised to (3)1Ma,b=2π∫0π/2dθa2cos2⁡θ+b2sin2⁡θ.However, the convergence above is not coincidental as we will see in the next section.In 1999, Matkowski introduced the notion of mean-type mappings on a real interval and showed the convergence of its Picard iteration if at most one of its coordinate means is not strict. Later in 2009, he showed the same result for mappings with a weaker condition. The fixed point set of such a mapping is exactly the diagonal; however, the fixed point set of a general (continuous) mean-type mapping only covers the diagonal and may not be contractible. On the other hand, in 2012, Chaoha and Chanthorn introduced the concept of virtually stable (fixed point iteration) schemes to connect topological structures of the convergence set of a scheme to those of the fixed point set via a retraction. Many schemes for nonexpansive-type mappings have been proved to be virtually stable.With those results in mind, in this work, we first extend the concept of mean-type mappings to vector spaces and then explore their fixed point sets using the notion of virtually stable schemes developed in [2]. We are able to establish a convergence theorem for certain continuous quasi-nonexpansive mean-type mappings in general normed linear spaces (which immediately covers the result in [3]) and conclude the contractibility of their fixed point sets. This also gives a new result on the structure of fixed point sets of quasi-nonexpansive mappings outside the strict-convexity setting.


## 2. Preliminaries

We begin this section by recalling the notion of means and mean-type mappings from [3].Definition 1.
LetI be an interval in R and p≥2 an integer. A function M:Ip→I is said to be amean if (4)min⁡x1,…,xp≤Mx1,…,xp≤max⁡x1,…,xpfor all (x1,…,xp)∈Ip.
A meanM isstrict if, in addition, (5)min⁡x1,…,xp<Mx1,…,xp<max⁡x1,…,xpfor all (x1,…,xp)∈Ip-Δ, where Δ denotes thediagonal: (6)Δ=x1,…,xp∈Ip∣x1=⋯=xp.Definition 2.
A mappingM:Ip→Ip is said to be amean-type mapping if each coordinate function is a mean; i.e., Mi:Ip→I is a mean for all i=1,…,p, where (7)Mx=M1x,…,Mpx.Moreover, M isstrict if all coordinate means are strict.Definition 3.
LetM,N:Ip→I be means. We say that M and N arecomparable if one of the following holds: (i)
M(x)≤N(x) for all x∈Ip.(ii)
N(x)≤M(x) for all x∈Ip.We are ready to recall the classical and well-known convergence theorem for a2-dimensional mean iteration.Theorem 4 (see [1]).
LetM,N:I2→I be comparable continuous means. Suppose that M or N is strict. Define M:I2→I2 by (8)Mx=Mx,Nx.Then there exists a continuous mean K:I2→I such that (9)limn→∞⁡Mnx=Kx,Kxfor all x∈I2.Remark 5.
The convergence of the arithmetic-geometric mean iteration follows directly from Theorem4 by letting (10)Ma,b=a+b2,abfor all a,b∈(0,∞). In this case, (11)1Ka,b=2π∫0π/2dθa2cos2⁡θ+b2sin2⁡θ.In1999, Matkowski showed that the comparability between means is not necessary. He also extended the convergence theorem to a p-dimensional mean iteration as follows.Theorem 6 (see [4]).
LetM=(M1,…,Mp):Ip→Ip be a continuous mean-type mapping such that at most one of the coordinate means Mi is not strict. Then there exists a continuous mean K:Ip→I such that (12)limn→∞Mnx=Kx,…,Kxfor all x∈Ip.Again, in2009, he improved his earlier result to include the larger class of nonstrict mean-type mappings.Theorem 7 (see [3]).
LetM=(M1,…,Mp):Ip→Ip be a continuous mean-type mapping such that (13)max⁡M1x,…,Mpx-min⁡M1x,…,Mpx<max⁡x-min⁡xfor all x∈Ip-Δ. Then there exists a continuous mean K:Ip→I such that (14)limn→∞Mnx=Kx,…,Kxfor all x∈Ip.Remark 8.
If a mean-type mappingM satisfies the condition in Theorem 7, then (15)Fix⁡M=Δ.Next, we recall the concept of virtual stability of fixed point iteration schemes [2]. We use this to conclude the contractibility of the fixed point sets of nonexpansive-type mappings.LetS=(sn) be a sequence of self-maps on a Hausdorff space X. Define thefixed point set of S and theconvergence set of S as follows: (16)FS=⋂n∈NFix⁡snCS=x∈X∣lim⁡snxexists.Define a function r:C(S)→X by r(x)=limn→∞sn(x).Definition 9 (see [2]).
Let(fn) be a sequence of self-maps on X and (17)sn=∏i=1nfi=fn∘fn-1∘⋯∘f2∘f1.The sequence S=(sn) is called a(fixed point iteration) scheme if ∅≠F(S)=r(C(S)).Definition 10 (see [2]).
LetS=∏i=1n‍fi be a scheme. A fixed point p∈F(S) is calledvirtually stable if, for each neighbourhood U of p, there exist a neighbourhood V of p and a strictly increasing sequence (kn)⊆N such that ∏i=jkn‍fi(V)⊆U for all n∈N and j≤kn.
The schemeS is calledvirtually stable if all its common fixed points are virtually stable.Theorem 11 (see [2]).
IfX is a regular space and S is a virtually stable scheme having a subsequence consisting of continuous mappings, then the function r defined above is continuous and hence F(S) is a retract of C(S).Theorem 12 (see [5]).
A retract subspace of a contractible space is contractible.Lastly, we recall a very well-known fact about the structure of the fixed point sets of quasi-nonexpansive mappings.Definition 13.
A normed linear spaceX is calledstrictly convex if x+y/2<1 for all x,y∈X such that x≠y and x=y=1; equivalently the boundary of the unit ball does not contain any line segment.Definition 14.
LetX be a subset of a normed linear space. A mapping T:X→X is said to be (i)
nonexpansive if Tx-Ty≤x-y for all x,y∈X;(ii)
quasi-nonexpansive if Tx-q≤x-q for all x∈X and q∈Fix(T).It is easy to see that nonexpansive mappings are continuous and quasi-nonexpansive while continuous quasi-nonexpansive mappings may not be nonexpansive.Theorem 15 (see [6]).
The fixed point set of a quasi-nonexpansive mapping defined on a convex subset of a strictly convex space is convex.


## 3. Main Results

In this section, we extend the notions of means and mean-type mappings to general vector spaces. Then we prove a convergence theorem as well as the contractibility of fixed point sets for certain continuous quasi-nonexpansive mean-type mappings.LetX be a convex subset of a vector space (over R) and p an integer with p≥2. As usual, the diagonal in Xp is simply Δ=(x1,…,xp)∈Xp∣x1=⋯=xp.Definition 16.
A functionM:Xp→X is said to be amean if, for each i=1,…,p, there is a function αi:X→[0,1] such that (18)Mx=Mx1,…,xp=α1xx1+⋯+αpxxpwith α1(x)+⋯+αp(x)=1 for each x=(x1,…,xp)∈Xp. For simplicity, we usually write M(x)=∑n‍αnxn. We also call Mstrict if, in addition, αi(x)∈(0,1) for all x∈Xp and i=1,…,p.We note that Definition16 is equivalent to Definition 1 when X is an interval in R.Definition 17.
A mappingM:Xp→Xp is said to be amean-type mapping if each coordinate function is a mean; that is (19)M=M1,…,Mp,where Mi:Xp→X is a mean for all i=1,…,p. Moreover, M isstrict if each Mi is strict.Remark 18.
For any meanM:Xp→X, we clearly have M(x,…,x)=x, for each x∈X. Consequently, for any mean-type mapping M:Xp→Xp, we have Δ⊆Fix⁡(M).Example 19.
DefineM:(0,∞)2→(0,∞)2 by (20)Mx,y=x+y2,xy=x+y2,xyx+y+yxx+y.Clearly, M is a nonlinear mean-type mapping and it is straightforward to verify that M is strict.Theorem 20.
IfM:Xp→Xp is a strict mean-type mapping, then Fix⁡(M)=Δ.Proof.
Letx∈Fix(M). We can form the following system of linear equations: (21)x1=M1x=α11x1+α12x2+⋯+α1p-1xp-1+1-∑j=1p-1α1jxpx2=M2x=α21x1+α22x2+⋯+α2p-1xp-1+1-∑j=1p-1α2jxp⋮xp=Mpx=αp1x1+αp2x2+⋯+αpp-1xp-1+1-∑j=1p-1αpjxp,which is equivalent to (22)1-α11-α12-α13⋯-α1p-1-α211-α22-α23⋯-α2p-1-α31-α321-α33⋯-α3p-1⋮⋮⋮⋱⋮-αp-11-αp-12-αp-13⋯1-αp-1p-1-αp1-αp2-αp3⋯-αpp-1x1-xpx2-xpx3-xp⋮xp-1-xp=0¯.
SinceM is strict, we can apply Gauss-Jordan elimination to the coefficient matrix to obtain (23)100⋯0010⋯0001⋯0⋮⋮⋮⋱⋮000⋯1000⋯0x1-xpx2-xpx3-xp⋮xp-1-xp=0¯,which implies x1=⋯=xp.WhenM is nonstrict, Fix⁡(M) may still be the diagonal Δ, or even the whole space Xp.Example 21.
ConsiderId,M:Xp→Xp defined by (24)Idx1,…,xp=x1,x2,…,xp,Mx1,…,xp=xp,x1,…,xp-1.Clearly, we have Fix⁡(Id)=Xp and Fix⁡(M)=Δ.WhenX is also a metric space, the next theorem surprisingly gives an explicit construction of a continuous mean-type mapping, whose fixed point set is any closed subset of Xp containing the diagonal.Recall that the distance between a pointx in the metric space (X,d) and ∅≠A⊆X is defined to be (25)dx,A=infa∈A⁡dx,a.When A is closed, we also have d(x,A)=0 iff x∈A.Theorem 22.
Suppose further that(X,d) is a metric space. For any closed subset F of Xp such that Δ⊆F, there exists a continuous mean-type mapping M:Xp→Xp such that Fix⁡(M)=F.Proof.
Definet:Xp→[0,1] and M:Xp→Xp by (26)tx=dx,F1+dx,F,Mx=x1,txx1+1-txx2,…,txx1+1-txxp.
It is easy to verify thatt(x)=0 iff x∈F, and M is continuous with Fix⁡(M)=F.From now on, letX be a convex subset of a normed linear space (E,), and we will always use the maximum norm (27)x1,…,xp=maxn⁡xnon Xp. Notice that Xp together with maximum norm may not be strictly convex. This prevents us from using Theorem 15 to conclude the convexity (and hence contractibility) of fixed point sets of quasi-nonexpansive mean-type mappings.The following theorem shows that, under a simple condition, a mean-type mapping is always quasi-nonexpansive.Theorem 23.
LetM:Xp→Xp be a mean-type mapping. If Fix⁡(M)=Δ, then M is quasi-nonexpansive. In particular, strict mean-type mappings are always quasi-nonexpansive.Proof.
Letx=(x1,…,xp)∈Xp, q^=(q,…,q)∈Δ=Fix⁡(M), and write M=(M1,…,Mp). Consider, for each i=1,…,p, (28)Mix-q=∑nαinxn-∑nαinq≤∑nαinxn-q≤maxn⁡xn-q∑nαin=maxn⁡xn-q,and hence, (29)Mx-q^=maxi⁡Mix-q≤maxn⁡xn-q=x-q^.Unfortunately, the converse of the previous theorem is not true (see Example36 below). Moreover, the last line in the proof of the previous theorem shows that any mean-type mapping is continuous on the diagonal. However, the continuity may not hold at other points as in the next example.Example 24.
DefineM:R2→R2 by (30)Mx,y=2x+y3,x+2y3;x,y∈Q2,x+2y3,2x+y3;x,y∉Q2.It is easy to see that M is a strict mean-type mapping. However, M is not continuous at each (x,y)∉Δ.Definition 25.
For eachk∈N, λ=(λ0,λ1,…,λk)∈[0,1]k+1 with ∑i=0k‍λi=1, and T:X→X, we define the λ-combination of T as follows: (31)Tλ=λ0Id+λ1T+⋯+λkTk=∑i=0kλiTi,where T0 denotes the identity mapping (Id). The λ-combination of a mean-type mapping M:Xp→Xp is defined similarly.From the above definition, it is easy to verify that theλ-combination of a mean-type mapping M:Xp→Xp is also a mean-type mapping, and it is continuous if M is continuous. Before we establish a convergence theorem, let us recall some basic facts about quasi-nonexpansive mappings and the distance between points in a convex hull.Lemma 26.
LetX be a convex subset of a normed linear space, T:X→X a quasi-nonexpansive mapping, p∈Fix⁡(T), and x0,y,z∈X. For a given λ=(λ0,λ1,…,λk)∈[0,1]k+1 with ∑i=0k‍λi=1, define a sequence (xn)⊆X by (32)xn=Tλxn-1.(1)
Ify and z are limit points of (xn), then y-p=z-p.(2)
Ifp is a limit point of (xn), then limn→∞xn=p.Proof.
SinceT is quasi-nonexpansive, we have for each n∈N,(33)xn+1-p=∑i=0kλiTixn-p=∑i=0kλiTixn-p≤∑i=0kλiTixn-p=xn-p.(1)
Ify and z are limit points of (xn), then there exist subsequences xnk,(xnl) of (xn) such that limk→∞xnk=y and liml→∞xnl=z. From (33), for each l∈N, there exist k0,l0∈N such that nl≤nk0≤nl0, and hence (34)xnl0-p≤xnk0-p≤xnl-p.This implies, as l→∞, that z-p≤y-p≤z-p.(2)
Ifp is a limit point of (xn), then there exists a subsequence (xnk) of (xn) such that limk→∞xnk=p. Again, by (33), we must have limn→∞xn=p.For eachx=(x1,…,xp)∈Xp and A⊆X, recall that theconvex hull of x, thealgebraic interior of the convex hull of x, and thediameter of A are, respectively, defined as follows (see [7] for details):(35)cox=α1x1+⋯+αpxp:αi∈0,1,α1+⋯+αp=1,algintcox=α1x1+⋯+αpxp:αi∈0,1,α1+⋯+αp=1,diam⁡A=supx-y:x,y∈A.Note thatdiam⁡(A) always exists when A is nonempty and bounded. Otherwise, let diam⁡(A)=∞.Lemma 27.
LetX be a subset of a normed linear space. co(x) is compact for each x∈Xp.Proof.
LetH=(α1,…,αp)∈[0,1]p:α1+⋯+αp=1. Define T:H→co(x) by (36)Tα1,…,αp=α1x1+⋯+αpxp.It is easy to verify that co(x) is the image of the compact set H under the continuous function T.Lemma 28.
LetX be a subset of a normed linear space and x∈Xp. (1)
diam⁡(co(x))=maxi,jxi-xj.(2)
Ifx∉Δ, y∈algint⁡(co(x)) and z∈co(x), then y-z<diam⁡(co(x)).Proof.
(1) It is clear thatmaxi,jxi-xj≤diam⁡(co(x)). Conversely, let y,z∈co(x); say y=∑n=1p‍αnxn and z=∑n=1p‍βnxn. Then (37)y-z=∑nαnxn-∑nβnxn=∑nβn∑mαmxm-∑nβnxn≤∑nβn∑mαmxm-xn=∑nβn∑mαmxm-∑mαmxn≤∑nβn∑mαmxm-xn≤∑nβn∑mαmmaxi,jxi-xj=maxi,jxi-xj.
(2) From the proof of (1), also notice that(38)y-z≤∑nβn∑mαmxm-xn=∑nβn∑m≠nαmxm-xn≤∑nβn∑m≠nαmmaxi,jxi-xj
Ify∈algint(co(x)), we have αi∈(0,1) for all i=1,…,p, which implies ∑m≠n‍αm<1, and hence (39)y-z<maxi,jxi-xj=diam⁡cox.Lemma28(1) naturally leads us to the notion ofdiameter of x=(x1,…,xp)∈Xp: (40)diam⁡x=diam⁡x1,…,xp=maxi,jxi-xj,which will replace diam⁡(co(x)) until the end of this paper.Remark 29.
From the above notation, we clearly have(41)diam⁡x=maxixi-xj0for some j0∈{1,…,p}, and diam⁡(x)=0 iff x∈Δ.Definition 30.
A mean-type mappingM:Xp→Xp is calleddiametrically contractive if (42)diam⁡Mx<diam⁡xfor all x∈Xp-Fix⁡(M).Example 31.
It is easy to see that the mean-type mappingM:Xp→Xp defined by (43)Mx1,…,xp=xp,x1,…,xp-1is not diametrically contractive.Theorem 32.
SupposeX is a convex subset of a normed linear spaces, and M:Xp→Xp is a continuous quasi-nonexpansive mean-type mapping such that Mλ is diametrically contractive for some λ=(λ0,λ1,…,λk)∈[0,1]k+1 with ∑i=0k‍λi=1 and Fix⁡(Mλ)=Fix⁡(M).
Letx0∈Xp and define a sequence (xn)⊆Xp by (44)xn+1=Mλxn.Then (xn) converges to a fixed point of M.Proof.
LetK=co(x0)⊆X. Then x0∈Kp and xn∈Kp for all n∈N because Mλ is a mean-type mapping. Since Kp is compact by Lemma 27, (xn) has a subsequence (xnk) converging to some z∈Kp. It follows that xnk+1=Mλxnk→Mλz, and hence both z and Mλz are limit points of (xn). By Lemma 26(1), we have (45)Mλz-q^=z-q^for all q^=(q,…,q)∈Fix(Mλ). By writing z=(z1,…,zp) and Mλ=(Mλ1,…,Mλp), the above equation becomes(46)maxiMλiz-q=maxizi-q,for all q∈K.
By Remark29, there exists j0∈{1,…,p} such that maxizi-zj0=diam⁡(z).
By lettingq=zj0 in (46), we have (47)maxiMλiz-zj0=diam⁡z,and hence (48)Mλi0z-zj0=diam⁡z,for some i0∈{1,…,p}.
Again, by lettingq=Mλi0z in (46), we have (49)diam⁡Mλz≥maxiMλiz-Mλi0z=maxizi-Mλi0z≥zj0-Mλi0z=diam⁡z,which contradicts the diametrical contractivity unless z∈Fix⁡(Mλ)=Fix⁡(M).
Therefore, by Lemma26(2), we have limn→∞xn=z.We now combine the above convergence theorem with virtual stability to obtain the contractibility of fixed point sets of certain continuous quasi-nonexpansive mean-type mappings.Corollary 33.
IfM:Xp→Xp is a continuous quasi-nonexpansive mean-type mapping such that (1-α)Id+αM is diametrically contractive for some α∈(0,1], then there is a continuous mean-type mapping r:Xp→Fix(M)⊆Xp such that (50)rx=limn→∞1-αId+αMnx,and hence Fix(M) is contractible.Proof.
For eachx0∈Xp and n∈N, let xn+1=Mλxn, where λ=(1-α,α). Notice that Fix(M)=Fix(Mλ) and xn=sn(x0), where sn=Mλn=(1-α)Id+αMn. Then, by Theorem 32, r(x0)=limn→∞sn(x0)=limn→∞xn∈Fix(M), and hence S=(sn) is a scheme consisting of continuous mappings, where F(S)=Fix(M) and C(S)=Xp, which is contractible. It is easy to see that (1-α)Id+αM is a mean-type mapping. So is r. Moreover, for each x∈Xp and q∈Fix(M), we have (51)s1x-q=1-αx+αMx-1-αq-αq≤1-αx-q+αMx-q≤x-qand hence (52)snx-q=s1nx-q≤s1n-1x-q≤⋯≤x-q.Therefore, S is virtually stable. By Theorem 11, r is continuous and Fix(M) is a retract of Xp. Hence, by Theorem 12, Fix(M) is contractible.Remark 34.
The previous corollary immediately extends Theorem7 to general normed linear spaces because when X is an interval I⊆R, the condition (53)maxM1x,…,Mpx-minM1x,…,Mpx<maxx-minx,for all x∈Ip-Δ, implies Fix(M)=Δ. Hence, M is quasi-nonexpansive and diametrically contractive, and Theorem 7 follows (with α=1). Notice that, in this case, the existence of K in Theorem 7 is the consequence of the fact that r(x)∈Δ for each x∈Ip.
Moreover, it should be pointed out here that, for any mean-type mappingM:X2→X2 and α∈(0,1), the combination (1-α)Id+αM is always diametrically contractive. If x=(x1,x2)∈X2 such that x∉Fix(M), we have (54)1-αx1+αM1x≠x1or1-αx2+αM2x≠x2.Since α≠1, we must have (55)1-αx1+αM1x∈algintcoxor1-αx2+αM2x∈algintcox.Then, by Lemma 28(2), diam⁡1-αx+αMx<diam⁡(x).The next example shows that, without quasi-nonexpansiveness, the diametrical contractivity alone may not be sufficient to obtain the contractibility of fixed point sets.Example 35.
DefineM:[0,1]2→[0,1]2 by (56)Mx,y=x,x1-yx+1-x1-yy=x,y+x2-xy+xy2-x2y.Clearly, M is a continuous mean-type mapping and it is diametrically contractive by the previous remark. However, M is not quasi-nonexpansive because (57)M14,12-14,1=14,1532-14,1=1732>12=14,12-14,1,and Fix(M)=(t,1),(0,t),(t,t)∣t∈[0,1] is not contractible.We now end this paper by giving an example showing that, under the condition of Theorem32, we only have the contractibility of fixed point sets but not the convexity.Example 36.
LetR2 be equipped with the maximum norm. Define f:R2→R2 by (58)fx,y=x,x;y≥x,y,y;x≥y,x,-x;y≤-x,-y,y;x≤-y.It is easy to see that f is a nonexpansive, hence quasi-nonexpansive, mean-type mapping. The previous remark assures that (1-α)Id+αf is diametrically contractive for any α∈(0,1). We note that Fix(f)=(t,t),t,-t∣t∈R which is not convex but still contractible.
Moreover, with a slight modification off, we obtain (59)gx,y=x,xy;x≤y≤1,x,x;x≤y≤x.Then g:[-1,1]2→[-1,1]2 is a continuous quasi-nonexpansive mean-type mapping and Fix(g)=Fix(f)∩[-1,1]2. We note that g is not nonexpansive because (60)g0,1-g14,1=0,0-14,12=12>14=0,1-14,1.



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