# Fixed points in uniform spaces

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## Fixed points in uniform spaces

*

### and Sittichoke Songsa-ard

*Correspondence:

phichet.c@chula.ac.th Department of Mathematics and Computer Science, Faculty of Science, Chulalongkorn University, Bangkok, 10330, Thailand

Abstract

We improve Angelov’s ﬁxed point theorems of

### -contractions and

j-nonexpansive maps in uniform spaces and investigate their ﬁxed point sets using the concept of virtual stability. Some interesting examples and an application to the solution of a certain integral equation in locally convex spaces are also given.

Keywords: ﬁxed points;

### -contractions; uniform spaces

1 Introduction

In  [], Angelov introduced the notion of -contractions on Hausdorﬀ uniform spaces, which simultaneously generalizes the well-known Banach contractions on metric spaces as well asγ-contractions [] on locally convex spaces, and he proved the existence of their ﬁxed points under various conditions. Later in  [], he also extended the no-tion of-contractions toj-nonexpansive maps and gave some conditions to guarantee the existence of their ﬁxed points. However, there is a minor ﬂaw in his proof of Theorem  [] where the surjectivity of the mapjis implicitly used without any prior assumption. Additionally, we observe that such a map jcan be naturally replaced by a multi-valued mapJto obtain a more general, yet interesting, notion ofJ-nonexpansiveness. Therefore, in this work, we aim to correct and simplify the proof of Theorem  [] as well as extend the notion ofj-nonexpansive maps toJ-nonexpansive maps and investigate the existence of their ﬁxed points. Then we introduceJ-contractions, a special kind ofJ-nonexpansive maps, that play the similar role as Banach contractions in yielding the uniqueness of ﬁxed points. With the notion ofJ-contractions, we are able to recover results on-contractions proved in [] as well as present some new ﬁxed point theorems in which one of them nat-urally leads to a new existence theorem for the solution of a certain integral equation in locally convex spaces. Finally, we prove that, under a mild condition,J-nonexpansive maps are always virtually stable in the sense of [] and hence their ﬁxed point sets are retracts of their convergence sets. An example of a virtually stableJ-nonexpansive map whose ﬁxed point set is not convex is also given.

2 Fixed point theorems

For any setS, we will usePf(S) and|S|to denote the set of all nonempty ﬁnite subsets ofS

and the cardinality ofS, respectively. Let (E,A) be a Hausdorﬀ uniform space whose uni-formity is generated by a saturated family of pseudometricsA={:αA}indexed byA,

∅ =XE, andJ:APf(A). Interested readers should consult [] for general topological concepts of uniform spaces, and [] for the complete development of ﬁxed point theory in

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uniform spaces that motivates this work. We ﬁrst give the deﬁnition of aJ-nonexpansive map as follows:

Deﬁnition . A self-mapT:XXis said to beJ-nonexpansive if for eachαA,

(Tx,Ty)≤

βJ(α) (x,y),

for anyx,yX.

Example . Let  <p<∞,E=pbe equipped with the weak topology, andT:pp

be deﬁned by

T(x,x, . . . ) =

|x+x|

 ,

|x+x|

 ,x,x, . . .

,

for any (x,x, . . . )∈p. ThenA={|f|:fp}, where|f|(x) =|f(x)|for eachxp.

By Theorem . in [], we have

f(TxTy)≤f

 (x–y+x–y) +f

 (x–y+x–y)

+f(x–y)+f(x–y)+f(xy),

for eachfp, x= (x,x, . . . )∈p andy= (y,y, . . . )∈p. Here,f=sup{|f(x)|:xX,x ≤}.

By lettingJ:pPf(

p) be deﬁned byJ(f) ={|f|,|g|,|g|,|g|,|g|}, for eachfp,

where

g(x) = f

 (x+x), g(x) = f

 (x+x), g(x) =fx, g(x) =fx,

for eachx= (x,x, . . . )∈p, it follows thatTisJ-nonexpansive.

The above deﬁnition of a J-nonexpansive map clearly extends the deﬁnition of a

j-nonexpansive map in []. Before giving general existence criteria for ﬁxed points of

J-nonexpansive maps, we need the following notations. For each αAandnN, we let

An(α) =(α, . . . ,αn) :α∈J(α) andαkJk–) for  <kn

and

A(α) =(α,α, . . . ) :α∈J(α) andαkJk–) fork> 

.

When there is no ambiguity, we will denote an element of bothAn(α) andA(α) simply by (αk). Notice that for eachαAandnN, the setsAn(α) andπn(A(α)) are ﬁnite, where

πndenotes thenth coordinate projection (αk)→αn.

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Proof SupposeT:XXisJ-nonexpansive. LetxXand () be a net inXconverging

tox. Then for eachαA, we have

(Txγ,Tx)≤

βJ(α)

(,x).

Since () converges to x, ((,x)) converges to  for anyβA, and this proves the

continuity ofT.

Theorem . Let T :XX be J-nonexpansive whose A(α)is ﬁnite for anyαA.Then T has a ﬁxed point in X if and only if there exists x∈X such that

(i) the sequence(Tnx

)has a convergence subsequence,and

(ii) for eachαAandk)∈A(α),limn→∞dαn(x,Tx) = .

Proof (⇒): It is obvious by lettingxbe a ﬁxed point ofT.

(⇐): Suppose that (Tnix

) converges to somezX. LetαAand (αk)∈A(α). Then

limi→∞(z,Tnix) =  andlimn→∞dαn(x,Tx) = . We can choose NNsuﬃciently

large so that(z,Tnix) <andni(x,Tx) <, for alliN. It follows that

z,Tni+x

 ≤

z,Tnix

 +

Tnix

,Tni(Tx)

z,Tnix

 +

(αk)∈Ani(α)

ni(x,Tx)

≤ +A(α) .

Sinceαis arbitrary, (Tni+x

) converges toz. By the continuity ofT, we havez=Tzand

hencezis a ﬁxed point ofT.

As a corollary of the previous theorem, we immediately obtain Theorem  [], with a corrected and simpliﬁed proof, as follows:

Corollary . Let T:XX be a j-nonexpansive map.If there exists x∈X such that

(i) the sequence(Tnx)has a convergence subsequence,and

(ii) for everyαA,limn→∞djn(α)(x,Tx) = ,

then T has a ﬁxed point.

Proof The proof follows directly from the previous theorem by considering the mapJ:

α→ {j(α)}. Notice thatA(α) ={(jn(α))}which is ﬁnite.

We will now consider a special kind ofJ-nonexpansive maps that resemble Banach con-tractions in yielding the uniqueness of ﬁxed points. Letdenote the family of all functions φ: [,∞)→[,∞) satisfying the following conditions:

() φis non-decreasing and continuous from the right, and () φ(t) <tfor anyt> .

Notice thatφ() = , and we will callφsubadditive ifφ(t+t)≤φ(t) +φ(t) for all t,t≥. Also, for a subfamily{φα}αAofA, (αk)∈An(α) andin, we let

φi(α

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Deﬁnition . A self-mapT:XXis said to be aJ-contraction if for eachαA, there existsφαsuch that

(Tx,Ty)≤

βJ(α)

φα

(x,y) ,

for anyx,yX, andφαis subadditive whenever|J(α)|> .

Clearly, a-contraction as deﬁned in [] is aJ-contraction and aJ-contraction is always

J-nonexpansive. A natural example of aJ-contraction can be obtained by adding (ﬁnitely many) appropriate-contractions as shown in the following example.

Example . Given two-contractionsT:XXandT:XXas deﬁned []. Then

there existj,j:AA, and for eachαA, there existφ,α,φ,αsuch that

(Tx,Ty)≤φ,α

dj(α)(x,y) and (Tx,Ty)≤φ,α

dj(α)(x,y),

for anyαAandx,yX. If for eachαA,j(α)=j(α) and there is a subadditiveφ,α

so that φ,α(t)≤φ,α(t) andφ,α(t)≤φ,α(t) for anyt≥, then the mapH=T+T is

clearly aJ-contraction with respect toJ(α) ={j(α),j(α)}andφH,α=φ,αfor anyαA.

Lemma . If T:XX is a J-contraction.Then we have

Tnx,Tny

(αk)∈An(α)

φαφ(–k)

dαn(x,y) ,

for anyαA,n≥and x,yX.

Proof Recall thatφαis assumed to be subadditive whenever|J(α)|> . Then, for anyαA,

n≥ andx,yX, we clearly have

Tnx,Tny

α∈J(α) φα

Tn–x,Tn–y

α∈J(α)

φα α∈J(α)

φα

Tn–x,Tn–y

α∈J(α)

α∈J(α)

φαφα

Tn–x,Tn–y

.. .

α∈J(α)

α∈J(α)

· · ·

αnJ(αn–)

φαφα◦ · · · ◦φαn–

dαn(x,y)

=

(αk)∈An(α)

φαφ(–k)

dαn(x,y) .

We now obtain some general criteria for the existence of ﬁxed points ofJ-contractions.

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(i) for eachαA,there existscαsuch that

φαi(t)≤(t),

for anyk)∈A(α),iN,t≥,and

(ii) there existsx∈Xsuch that for eachαA,(αk)∈A(α),iNandn,mN,we have

dαi

Tnx,Tmx ≤(x),

for someMα(x)∈R,

then T has a ﬁxed point.Moreover,if for eachαA and x,yX,there exists Fα(x,y)∈R+ such that

dαi(x,y)≤(x,y),

for allk)∈A(α)and iN,then the ﬁxed point of T is unique.

Proof For eachαAandn,m,NN, sinceφαis non-decreasing, we have

Tnx,Tmx ≤

α∈J(α)

φα

Tn–x,Tm–x

α∈J(α) φα

sup

Tn–x,Tm–x :n,mN ,

and by letting

N:=sup{(Tnx,Tmx) :n,mN}, it follows that

N

α∈J(α) φα

sup

Tn–x,Tm–x :n,mN

=

α∈J(α) φα

N–

α∈J(α)

α∈J(α) φα

φα

N–

.. .

(αk)∈AN–(α)

φαφ(–k)

hαN–

(αk)∈AN–(α)

cNα(x)

A(α)cNα

(x) . ()

Also, for a givent≥, since ≤cNα(t) =(cNα–(t)) <cNα–(t), we havelimN→∞cNα(t) =

for some≥. Sinceis right continuous, we havelimN→∞(cNα–(t)) =(), and

hence() =. Therefore,= . By (), it follows thatlimN→∞hαN = . Sinceα is

ar-bitrary, (Tkx

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Now suppose that for each x,yX and αA, there exists(x,y)∈R+ such that

dαi(x,y)≤(x,y) for all (αk)∈A(α) and iN. If x, y are ﬁxed points ofT, then by Lemma ., we have for eachαAandnN,

(x,y) =

Tnx,Tny

(αk)∈An(α)

φαφn(α–k)

dαn(x,y)

(αk)∈An(α) cnα

dαn(x,y)

A(α)cnα(x,y) .

Sincelimn→∞cn

α((x,y)) = , we must havex=y.

As a corollary of the previous theorem, we immediately obtain Theorem  in [] as fol-lows.

Corollary . Suppose X is a bounded and sequentially complete subset of E and T:XX is-contraction.If

(i) for eachαA,there existscαsuch thatφjn(α)(t)≤cα(t)for allnNandt≥,

(ii) for eachnN,sup{djn(α)(x,y) :x,yX} ≤p(α) :=sup{dα(x,y) :x,yX},

then there exists a unique ﬁxed point xX of T.

Proof For eachx,x,yX,αA, (αk)∈A(α) andi,m,nN, by lettingJ(α) ={j(α)}and (x) =p(α) =(x,y), we haveA(α) ={(α,j(α),j(α), . . . ,jk(α), . . . )}, dαi(T

mx

,Tnx) = dji(α)(Tmx,Tnx)≤(x) and dαi(x,y)≤(x,y). Hence, by Theorem ., T has a

unique ﬁxed point.

Theorem . Suppose X is sequentially complete and T:XX is a self-map satisfying:

for eachαA and kN,there existφα,k,a ﬁnite set Dα,kand a map Pα,k:,kA such that

Tkx,Tky

γ,k φα,k

dPα,k(γ)(x,y) ,

for any x,yX.

. If there exists x∈Xsuch that for eachαAthere existsMα(x)∈R+so that

kN|,k|φα,k((x)) <∞and

dPα,k(γ)(x,Tx)≤(x),

for allkNandγ,k,thenT has a ﬁxed point inX.

. If for each αAandx,yX,there existsFα(x,y)∈R+such that

k∈N|,k|φα,k((x,y)) <∞and

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for allkNandγ,k,thenT has a unique ﬁxed point inXand,for anyxX, the sequence(Tnx)converges to the ﬁxed point ofT.

Proof First notice thatTis clearly aJ-contraction. . For anyαAandm>nN, we have

Tnx,Tmx ≤

ni<m

Tix,Ti+x

ni<m

γ,i

φα,i

dPα,i(γ)(x,Tx)

ni<m

|,i|φα,i

(x).

Also, sincek∈N|,k|φα,k((x)) <∞, (Tkx) is a Cauchy sequence and converges to a

ﬁxed point ofT by the sequential completeness ofXand the continuity ofT. . For anyxX,αAandm>nN, we have

Tnx,Tmx

ni<m

Tix,Ti+x

ni<m

γDα,i

φα,i

dPα,i(γ)(x,Tx)

ni<m

|,i|φα,i

(x,Tx).

Also, sincekN|,k|φα,k((x,Tx)) <∞, (Tkx) is a Cauchy sequence and converges to a

ﬁxed point ofT by the sequential completeness ofXand the continuity ofT. Now, since for eachαA,kNandx,yF(T),

(x,y) =

Tkx,Tky

γ,k φα,k

dPα,k(γ)(x,y)

γ,k φα,k

(x,y)

=|,k|φα,k

(x,y),

andlimk→∞|,k|φα,k((x,y)) = , we have the uniqueness. Corollary .(Theorem  in []) Let us suppose

(i) for eachαAandn> ,there existφα,nandj(α,n)∈Asuch that

Tnx,Tnyφα,n

dj(α,n)(x,y),

for anyx,yX,

(ii) there exists x∈Xsuch thatdj(α,n)(x,Tx)≤p(α) <∞(n= , , . . .),

nφα,n(p(α)) <∞andj:A×NA.

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Proof By letting,k={j(α,k)}for anyαAandkNand,k=πk|,k. Then for each

iN, we have|,i|=  and(x) =p(α). By Theorem .(),Thas a ﬁxed point.

Theorem . Suppose X is sequentially complete and T:XX is a J-contraction whose A(α)is ﬁnite for eachαA.If,for eachαA,there exists cαsatisfying:

(i) (t)/tis non-decreasing int,

(ii) φαn(t)≤(t)for anyk)∈A(α),nNandt∈[,∞),and

(iii) there existx∈XandMα(x)∈R+such thatdαn(x,Tx)≤(x)for any

k)∈A(α)andnN, then T has a ﬁxed point in X.

Proof Let,i=Ai(α),,i((αk)) =αi, andφα,i(t) =ciα(t) for anyiN,αA, (αk)∈Ai(α),

andt∈[,∞). Then for anyαAandx,yX, we have, by Lemma .,

Tix,Tiy

(αk)∈Ai(α)

φαφ(i–αk)

dαi(x,y)

(αk)∈Ai(α)

ciα

dαi(x,y)

=

(αk)∈,i φα,i

dPα,i((αk))(x,y) .

Since

|,i+|φα,i+((x))

|,i|φα,i((x))

=|Ai+(α)|c

i+

α ((x))

|Ai(α)|ci

α((x)) ≤

(ciα((x))) ci

α((x)) ≤

((x)) (x)

< ,

for anyiN, we havei∈N|,i|φα,i((x)) <∞. Then by Theorem .(),Thas a ﬁxed

point.

Corollary .(Theorem  in []) Let us suppose

(i) the operatorT:XXis a-contraction,

(ii) for eachαAthere exists a-functioncαsuch thatφjn(α)(t)≤cα(t)for allnN

andcα(t)/tis non-decreasing,

(iii) there exists an elementx∈Xsuch thatdjn(α)(x,Tx)≤p(α) <∞(n= , , . . .).

Then T has at least one ﬁxed point in X.

Proof By lettingJ(α) ={j(α)}for anyαAand(x) =p(α). Then|A(α)|= , and, by

Theorem .,Thas a ﬁxed point.

Example . Given a sequentially complete locally convex space X, and two -contractionsT,T:XX;i.e., there existj,j:AA, and for eachαA, there exist

φ,α,φ,αsuch that

(Tx,Ty)≤φ,α

dj(α)(x,y) and (Tx,Ty)≤φ,α

dj(α)(x,y),

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(i) jn+=jnjandjn◦j=jn+for anynN,

(ii) for eachαA,φ,α(t) =c(α)tandφ,α(t) =c(α)tfor somec(α) +c(α)∈(, ),

and

(iii) there existsx∈Xsuch thatdjn(α)(x,Tx)≤p(x,α) <∞and djn

(α)(x,Tx)≤p(x,α) <∞for anyαAandn= , , . . .. ThenH=T+T

 is aJ-contraction withJ(α) ={j(α),j(α)}andφH,α(t) = (c(α) +c(α))t.

Also, by (i) and (iii), we have|A(α)|=  <∞and

dαn(x,Hx)≤

dαn(x,Tx) +dαn(x,Tx)

 ≤

p(x,α) +p(x,α)

 .

Hence,Hsatisﬁes all conditions in Theorem ., and it has a ﬁxed point inX. Notice that

H may not be a-contraction, by choosingj andjso thatdj(α)+dj(α)∈/ Afor some αA, and hence Theorem  in [] cannot be applied.

We now end this section by giving an application to the solution of a certain integral equation in locally convex spaces.

Example . Following terminologies in [], letXbe an S-space topologized by the family of seminorms{| · |α:αA}andC([,T];X) the space of all continuous functions

from [,T] intoXtopologized by the family of seminorms{ · α:αA}, where:=

supt∈[,T]|x(t)|α for anyxC([,T];X). LetL(X) denote the set of all continuous linear

operators onX,

L(X) =

lL(X) :∀αA,∃M(α) > ,∀xX,|lx|αM(α)|x|α

,

and let{S(t)}t≥be aC-semigroup onXsuch thatS: [,∞)→L(X) is locally bounded.

Now, we replace H and H in [] by conditions (N), (N) and (N) as follows:

(N) B:C([,T];X)→C([,T];X)is an operator such that there existsJB:APf(A)

so that for anyαA, there is,BLloc([,T]; [,∞))such that

Bx(t) –By(t)α,B(t)

βJB(α)

x(t) –y(t)β,

for anyx,yC([,T];X).

(N) f : [,TX×XXis continuous and there existJf:APf(A)and KfLloc([,T]; [,∞))such that for eachαA,

f(t,u,v) –f(t,u,v)αKf(t) βJf(α)

|u–u|β+|v–v|α

,

for anyt∈[,T]andu,u,v,v∈X,

(N) Kf ·,BLloc([,T]; [,∞)).

Consider the integral equation

x(t) =S(t)x+

t

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whose solution is closely related to the mild solution of the diﬀerential equation

dx

dt =ax+f

t,x(t),Bx(t) ,

We now deﬁne an operatorGonCx([,T];X) ={xC([,T];X) :x() =x}by

(Gx)(t) =S(t)x+

t

S(ts)fs,x(s),Bx(s) ds,

for any xCx([,T];X). Following the proof of Theorem  in [] and for eacht> ,

S(t)∈L(X), then we can show that, for anyαA, there existsM(α) >  such that

GxGyα βJf(α)

x+

βJB(α)

x

,

where=max{M(α)

T

Kf(s)ds,M(α)

T

Kf(s),B(s)ds}. It is easy to see that if for each

αA,∈(, ) andJf(α)∩JB(α) =∅, thenGis aJ-contraction withJG(α) =Jf(α)∪JB(α).

In particular, if we assume further that for eachαA,Jf(α) ={α},|JB(α)|=  such that

JBJB=JBand=HJB(α)<

. Then for anykNandx,yCx([,T];X), we have

GkxGkyαHαkx+

k

i=

(HJB(α))

kiHi

α

xyJB(α)

=Hαkx+

k

i=

kiHαk

xyJB(α)

≤k–Hαk

x+

k

i=

xyJB(α)

.

Now, by lettingφα,k(t) = k–Hαkt,,k={(,α), (,JB(α))(,JB(α)), . . . , (k,JB(α))},,k(γ) =

π(γ), and(x,y) =max{x,xyJB(α)}, we have

(i) xyPα,k(γ)≤(x,y)for anyx,yCx([,T];X),kN,αA, andγ,k,

(ii) kN|,k|φα,k((x,y)) =

kNk+ ()kFα(x,y) <∞for anyx,yCx([,T];X)

andαA.

Therefore, by Theorem .(),Ghas a unique ﬁxed point, so the integral equation () has a unique solution.

3 Fixed point sets

In this section, we will show that, under a mild condition, aJ-nonexpansive map is always virtually stable. This immediately gives a connection between the ﬁxed point set and the convergence set of aJ-nonexpansive map. Recall that a continuous self-mapT :XX, whose ﬁxed point setF(T) is nonempty, on a Hausdorﬀ spaceXis said to be virtually stable [] if for eachxF(T) and each neighborhoodUofx, there exist a neighborhood

V ofxand an increasing sequence (kn) of positive integers such thatTkn(V)Ufor all

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nonexpansive self-map, whose ﬁxed point set is nonempty, on a metric space is always uniformly virtually stable with respect to the sequence (n) of all natural numbers. An im-portant feature of a virtually stable map is the connection between its ﬁxed point set and its convergence set as given in the following theorem.

Theorem .([], Theorem .) Suppose X is a regular space.If T :XX is virtually stable,then F(T)is a retract of C(T),where C(T)is the(Picard)convergence set of T deﬁned as follows:

C(T) =xX:the sequenceTnx converges.

As in the previous section, let (E,A) be a Hausdorﬀ uniform space whose uniformity is generated by a saturated family of pseudometricsA={:αA}indexed byAand∅ =

XE. The following theorem gives a general criterion for a self-map onXto be virtually stable.

Theorem . Let T:XX be a self-map whose ﬁxed point set F(T)is nonempty,and which satisﬁes the following conditions:

(i) for eachαAandkN,there exist a ﬁnite setDα,kand a mapPα,k:,kAsuch that

Tkx,Tky

γ,k

dPα,k(γ)(x,y),

for anyx,yX,

(ii) there existsNNsuch that|,n| ≤ |,N|andPα,n(,n)⊆,N(,N)for any

nNandαA.

Then T is uniformly virtually stable with respect to the sequence of all natural numbers.

Proof LetzF(T) and letUbe a neighborhood ofz. We may assume thatU=mi={wX:dαi(w,z) <}for some>  andα, . . . ,αmA. For eachnN, let

Vn=

m

i=

γDαi,n

wX:dPαi,n(γ)(w,z) < |Dαi,n|

.

By (ii), there existsNNsuch that|Dαi,n| ≤ |Dαi,N|andPαi,n(Dαi,n)⊆Pαi,N(Dαi,N) for anynNandi= , . . . ,m. LetV=V∩V∩ · · · ∩VN which is clearly a nonempty open

subset ofX,yV,lNandi∈ {, . . . ,m}. It follows that

dαi

Tly,z =dαi

Tly,Tlz

γDαi,l

dPαi,l(γ)(y,z).

Ifl<N, then

dαi

Tly,z <

γi,l |Dαi,l|

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IflN, sincePαi,l(γ)∈Pαi,l(Dαi,l)⊆Pαi,N(Dαi,N), we havedPαi,l(γ)(y,z) <

|i,N|for each γDαi,l, and hence

dαi

Tly,z <

γDαi,l |Dαi,N|

=|Dαi,l| |Dαi,N|

.

Hence,T is uniformly virtually stable with respect to the sequence of all natural

num-bers.

Corollary . Suppose that T is J-nonexpansive with F(T)=∅.If there exists NNsuch that|An(α)| ≤ |AN(α)|andπn(An(α))⊆πN(AN(α))for any nN andαA,then T is uniformly virtually stable with respect to the sequence of all natural numbers.

Proof By letting,n=An(α) and,n=πn|An(α)for anynNandαA, we have

Tlx,Tly

γ,l

dPα,l(γ)(x,y),

for anyx,yX. The result then follows from Theorem ..

Example . LetE=equipped with the weak topology andT:→be deﬁned by

T(x,x, . . . ) =

|

x+x|

 ,

|x+x|

 ,x,x, . . .

,

for any (x,x, . . . )∈. ThenA={|f|:f}, and by Lemma . and Theorem . in [],

we have

fTnxTny

≤f

√  

|x–y+x–y|+|x–y+x–y|

+ √

(|x–y|+|x–y|+|x–y+x–y|+|x–y+x–y|)

 – √

+f

|x–y|+|x–y+x–y|+ 

|x–y|+|x–y+x–y|

+f|x–y|+f|x–y|+f(xy),

for eachf,nN,x= (x,x, . . . ) andy= (y,y, . . . )∈.

By lettingJ:→P() be deﬁned byJ(f) ={|f|,|g|,|g|,|g|,|g|}for eachf, where

g(x) =f

√

 + √  – √+ 

(x+x),

g(x) =f

√

 + √  – √+ 

(x+x),

g(x) =f

√  – √ +

 

x, g(x) =f

√  – √+

 

x,

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Notice that (, , . . . ) is a ﬁxed point ofT, and for eachfandn,mN,πn(A(|f|)) =

πm(A(|f|)). Then, by Theorem .,Tis virtually stable and hence the ﬁxed point set ofT

is a retract of the convergence set ofT. Moreover, the ﬁxed point set is not convex be-causex= (, , , , , . . . ) andy= (, , –, –, , . . . ) are ﬁxed points ofT, while the convex combination x+y= (, , –, –, , . . . ) is not.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the ﬁnal manuscript.

Acknowledgements

The authors are grateful to the anonymous referee(s) for their valuable comments and suggestions for improving this manuscript. This research is (partially) supported by the Centre of Excellence in Mathematics, the Commission on Higher Education, Thailand.

Received: 23 January 2014 Accepted: 23 May 2014 Published:02 Jun 2014

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39(3), 581-592 (1971)

3. Angelov, VG:J-Nonexpansive mappings in uniform spaces and applications. Bull. Aust. Math. Soc.43, 331-339 (1991) 4. Chaoha, P, Atiponrat, W: Virtually stable maps and their ﬁxed point sets. J. Math. Anal. Appl.359, 536-542 (2009) 5. Weil, A: Sur les Espaces à Structure Uniforme et la Topologie Générale. Hermann, Paris (1938)

6. Angelov, VG: Fixed Points in Uniform Spaces and Applications. Cluj University Press, Cluj-Napoca (2009) 7. Chaoha, P, Songsa-ard, S: Fixed points of functionally Lipschitzian maps. J. Nonlinear Convex Anal.15(4), 665-679

(2014)

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10.1186/1687-1812-2014-134

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