Fixed points in uniform spaces

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R E S E A R C H

Open Access

Fixed points in uniform spaces

Phichet Chaoha

*

_{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 of}_S

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

∅ =X⊆E, andJ:A→Pf(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:X→Xis said to beJ-nonexpansive if for eachα∈A,

dα(Tx,Ty)≤

β∈J(α) dβ(x,y),

for anyx,y∈X.

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

be deﬁned by

T(x,x, . . . ) =

|x+x|

 ,

|x+x|

 ,x,x, . . .

,

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

By Theorem . in [], we have

f(Tx–Ty)≤f

 (x–y+x–y) +f

 (x–y+x–y)

+f(x–y)+f(x–y)+f(x–y),

for eachf ∈∗_p, x= (x,x, . . . )∈p andy= (y,y, . . . )∈p. Here,f=sup{|f(x)|:x∈ X,x ≤}.

By lettingJ:∗_p→Pf₍∗

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

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 α∈Aandn∈N, we let

An(α) =(α, . . . ,αn) :α∈J(α) andαk∈J(αk–) for  <k≤n

and

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

.

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

πndenotes thenth coordinate projection (αk)→αn.

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Proof SupposeT:X→XisJ-nonexpansive. Letx∈Xand (xγ) be a net inXconverging

tox. Then for eachα∈A, we have

dα(Txγ,Tx)≤

β∈J(α)

dβ(xγ,x).

Since (xγ) converges to x, (dβ(xγ,x)) converges to  for anyβ∈A, and this proves the

continuity ofT.

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

(i) the sequence(Tn_x

)has a convergence subsequence,and

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

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

(⇐): Suppose that (Tni_x

) converges to somez∈X. Letα∈Aand (αk)∈A(α). Then

limi→∞dα(z,Tnix) =  andlimn→∞dαn(x,Tx) = . We can choose N∈Nsuﬃciently

large so thatdα(z,Tnix) <anddα_ni(x,Tx) <, for alli≥N. It follows that

dα

z,Tni+_x

 ≤dα

z,Tni_x

 +dα

Tni_x

,Tni(Tx)

≤dα

z,Tni_x

 +

(αk)∈Ani(α)

dα_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:X→X be a j-nonexpansive map.If there exists x∈X such that

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

(ii) for everyα∈A,lim_n_→∞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 callφ∈subadditive ifφ(t+t)≤φ(t) +φ(t) for all t,t≥. Also, for a subfamily{φα}α∈Aof,α∈A, (αk)∈An(α) andi≤n, we let

φi₍_α

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Deﬁnition . A self-mapT:X→Xis said to be aJ-contraction if for eachα∈A, there existsφα∈such that

dα(Tx,Ty)≤

β∈J(α)

φα

dβ(x,y) ,

for anyx,y∈X, 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:X→XandT:X→Xas deﬁned []. Then

there existj,j:A→A, and for eachα∈A, there existφ,α,φ,α∈such that

dα(Tx,Ty)≤φ,α

dj(α)(x,y) and dα(Tx,Ty)≤φ,α

dj(α)(x,y),

for anyα∈Aandx,y∈X. If for eachα∈A,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 anyα∈A.

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

dα

Tnx,Tny ≤

(αk)∈An(α)

φα◦φ₍nα–k)

dαn(x,y) ,

for anyα∈A,n≥and x,y∈X.

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

n≥ andx,y∈X, we clearly have

dα

Tnx,Tny ≤

α∈J(α) φα

dα

Tn–x,Tn–y

≤

α∈J(α)

φα α∈J(α)

φα

dα

Tn–x,Tn–y

≤

α∈J(α)

α∈J(α)

φα◦φα

dα

Tn–x,Tn–y

.. .

≤

α∈J(α)

α∈J(α)

· · ·

αn∈J(αn–)

φα◦φα◦ · · · ◦φαn–

dαn(x,y)

=

(αk)∈An(α)

φα◦φ₍nα–k)

dαn(x,y) .

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

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(i) for eachα∈A,there existscα∈such that

φαi(t)≤cα(t),

for any(αk)∈A(α),i∈N,t≥,and

(ii) there existsx∈Xsuch that for eachα∈A,(αk)∈A(α),i∈Nandn,m∈N,we have

dαi

Tnx,Tmx ≤Mα(x),

for someMα(x)∈R,

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

dαi(x,y)≤Fα(x,y),

for all(αk)∈A(α)and i∈N,then the ﬁxed point of T is unique.

Proof For eachα∈Aandn,m,N∈N, sinceφαis non-decreasing, we have

dα

Tnx,Tmx ≤

α∈J(α)

φα

dα

Tn–x,Tm–x

≤

α∈J(α) φα

supdα

Tn–x,Tm–x :n,m≥N ,

and by lettinghα

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

hα_N≤

α∈J(α) φα

supdα

Tn–x,Tm–x :n,m≥N

=

α∈J(α) φα

hα

N–

≤

α∈J(α)

α∈J(α) φα

φα

hα

N–

.. .

≤

(αk)∈AN–(α)

φα◦φ(Nα–k)

hαN–



≤

(α_k)∈AN–(α)

cN_αMα(x)

≤A(α)cNα

Mα(x) . ()

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

for somerα≥. Sincecαis right continuous, we havelimN→∞cα(cNα–(t)) =cα(rα), and

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

ar-bitrary, (Tk_x

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Now suppose that for each x,y∈X and α ∈A, there existsFα(x,y)∈R+_ such that

dαi(x,y)≤Fα(x,y) for all (αk)∈A(α) and i∈N. If x, y are ﬁxed points ofT, then by Lemma ., we have for eachα∈Aandn∈N,

dα(x,y) =dα

Tnx,Tny

≤

(αk)∈An(α)

φα◦φn(α–k)

dαn(x,y)

≤

(αk)∈An(α) cnα

dαn(x,y)

≤A(α)cn_αFα(x,y) .

Sincelim_n_→∞cn

α(Fα(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:X→ X is-contraction.If

(i) for eachα∈A,there existscα∈such thatφjn₍_α₎(t)≤c_α(t)for alln∈Nandt≥,

(ii) for eachn∈N,sup{djn₍_α₎(x,y) :x,y∈X} ≤p(α) :=sup{d_α(x,y) :x,y∈X},

then there exists a unique ﬁxed point x∈X of T.

Proof For eachx,x,y∈X,α∈A, (αk)∈A(α) andi,m,n∈N, by lettingJ(α) ={j(α)}and Mα(x) =p(α) =Fα(x,y), we haveA(α) ={(α,j(α),j(α), . . . ,jk(α), . . . )}, dαi(T

m_x

,Tnx) = dji₍_α₎(Tmx_,Tnx_)≤Mα(x) and dαi(x,y)≤Fα(x,y). Hence, by Theorem ., T has a

unique ﬁxed point.

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

for eachα∈A and k∈N,there existφα,k∈,a ﬁnite set Dα,kand a map Pα,k:Dα,k→A such that

dα

Tkx,Tky ≤

γ∈Dα,k φα,k

dP_α_,k(γ)(x,y) ,

for any x,y∈X.

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

k∈N|Dα,k|φα,k(Mα(x)) <∞and

dP_α,k(γ)(x,Tx)≤Mα(x),

for allk∈Nandγ∈Dα,k,thenT has a ﬁxed point inX.

. If for each α∈Aandx,y∈X,there existsFα(x,y)∈R+such that

k∈N|Dα,k|φα,k(Fα(x,y)) <∞and

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for allk∈Nandγ∈Dα,k,thenT has a unique ﬁxed point inXand,for anyx∈X, the sequence(Tn_x₎_{converges to the ﬁxed point of}_T_.

Proof First notice thatTis clearly aJ-contraction. . For anyα∈Aandm>n∈N, we have

dα

Tnx,Tmx ≤

n≤i<m dα

Tix,Ti+x

≤

n≤i<m

γ∈Dα,i

φα,i

dPα,i(γ)(x,Tx)

≤

n≤i<m

|Dα,i|φα,i

Mα(x).

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

ﬁxed point ofT by the sequential completeness ofXand the continuity ofT. . For anyx∈X,α∈Aandm>n∈N, we have

dα

Tnx,Tmx ≤

n≤i<m dα

Tix,Ti+x

≤

n≤i<m

γ∈Dα,i

φα,i

dPα,i(γ)(x,Tx)

≤

n≤i<m

|Dα,i|φα,i

Fα(x,Tx).

Also, since_k_∈N|Dα,k|φα,k(Fα(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 eachα∈A,k∈Nandx,y∈F(T),

dα(x,y) =dα

Tkx,Tky

≤

γ∈Dα,k φα,k

dP_α,k(γ)(x,y)

≤

γ∈Dα,k φα,k

Fα(x,y)

=|Dα,k|φα,k

Fα(x,y),

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

(i) for eachα∈Aandn> ,there existφα,n∈andj(α,n)∈Asuch that

dα

Tnx,Tny ≤φα,n

dj(α,n)(x,y),

for anyx,y∈X,

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

nφα,n(p(α)) <∞andj:A×N→A.

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Proof By lettingDα,k={j(α,k)}for anyα∈Aandk∈NandPα,k=πk|Dα,k. Then for each

i∈N, we have|Dα,i|=  andMα(x) =p(α). By Theorem .(),Thas a ﬁxed point.

Theorem . Suppose X is sequentially complete and T:X→X is a J-contraction whose A(α)is ﬁnite for eachα∈A.If,for eachα∈A,there exists cα∈satisfying:

(i) cα(t)/tis non-decreasing int,

(ii) φαn(t)≤cα(t)for any(αk)∈A(α),n∈Nandt∈[,∞),and

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

(αk)∈A(α)andn∈N, then T has a ﬁxed point in X.

Proof LetDα,i=Ai(α),Pα,i((αk)) =αi, andφα,i(t) =ciα(t) for anyi∈N,α∈A, (αk)∈Ai(α),

andt∈[,∞). Then for anyα∈Aandx,y∈X, we have, by Lemma .,

dα

Tix,Tiy ≤

(αk)∈Ai(α)

φα◦φ₍i–αk)

dαi(x,y)

≤

(αk)∈Ai(α)

ciα

dαi(x,y)

=

(αk)∈Dα,i φα,i

dPα,i((αk))(x,y) .

Since

|Dα,i+|φα,i+(Mα(x))

|Dα,i|φα,i(Mα(x))

=|Ai+(α)|c

i+

α (Mα(x))

|Ai(α)|ci

α(Mα(x)) ≤

cα(ciα(Mα(x))) ci

α(Mα(x)) ≤

cα(Mα(x)) Mα(x)

< ,

for anyi∈N, we have_i_∈N|Dα,i|φα,i(Mα(x)) <∞. Then by Theorem .(),Thas a ﬁxed

point.

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

(i) the operatorT:X→Xis a-contraction,

(ii) for eachα∈Athere exists a-functioncαsuch thatφjn₍_α₎(t)≤c_α(t)for alln∈N

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 anyα∈AandMα(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:X→X;i.e., there existj,j:A→A, and for eachα∈A, there exist

φ,α,φ,α∈such that

dα(Tx,Ty)≤φ,α

dj(α)(x,y) and dα(Tx,Ty)≤φ,α

dj(α)(x,y),

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(i) jn_+=jn_◦jandjn◦j=jn+for anyn∈N,

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

and

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

(α)(x,Tx)≤p(x,α) <∞for anyα∈Aandn= , , . . .. 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,Tx) +dαn(x,Tx)

 ≤

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 α∈A, 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:α∈A}andC([,T];X) the space of all continuous functions

from [,T] intoXtopologized by the family of seminorms{ · α:α∈A}, wherexα:=

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

operators onX,

L(X) =

l∈L(X) :∀α∈A,∃M(α) > ,∀x∈X,|lx|α≤M(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:A→Pf(A)

so that for anyα∈A, there iskα,B∈L_loc([,T]; [,∞))such that

Bx(t) –By(t)_α≤kα,B(t)

β∈JB(α)

x(t) –y(t)_β,

for anyx,y∈C([,T];X).

(N) f : [,T]×X×X→Xis continuous and there existJf:A→Pf₍_A₎_and Kf ∈Lloc([,T]; [,∞))such that for eachα∈A,

f(t,u,v) –f(t,u,v)_α≤Kf(t) β∈Jf(α)

|u–u|β+|v–v|α

,

for anyt∈[,T]andu,u,v,v∈X,

(N) Kf ·kα,B∈L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) ,

whereadenotes the inﬁnitesimal generator of{S(t)}t≥.

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

(Gx)(t) =S(t)x+

t



S(t–s)fs,x(s),Bx(s) ds,

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

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

Gx–Gyα≤Hα β∈Jf(α)

x–yβ+

β∈JB(α)

x–yβ

,

whereHα=max{M(α)

T

 Kf(s)ds,M(α)

T

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

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

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

JB◦JB=JBandHα=HJB(α)<



. Then for anyk∈Nandx,y∈Cx([,T];X), we have

Gkx–Gky_α≤Hαkx–yα+

_k

i=

(HJB(α))

k–i_Hi

α

x–yJB(α)

=Hαkx–yα+

_k

i=

k–iHαk

x–yJB(α)

≤k–Hαk

x–yα+

k

i=

x–yJB(α)

.

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

π(γ), andFα(x,y) =max{x–yα,x–yJB(α)}, we have

(i) x–yPα,k(γ)≤Fα(x,y)for anyx,y∈Cx([,T];X),k∈N,α∈A, andγ ∈Dα,k,

(ii) _k_∈N|Dα,k|φα,k(Fα(x,y)) =

k∈Nk+ (Hα)kFα(x,y) <∞for anyx,y∈Cx([,T];X)

andα∈A.

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 :X→X, whose fixed point setF(T) is nonempty, on a Hausdorff spaceXis said to be virtually stable [] if for eachx∈F(T) and each neighborhoodUofx, there exist a neighborhood

V ofxand an increasing sequence (kn) of positive integers such thatTkn₍_V₎⊆_U_{for 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 :X→X 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) =x∈X: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={dα:α∈A}indexed byAand∅ =

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

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

(i) for eachα∈Aandk∈N,there exist a ﬁnite setDα,kand a mapPα,k:Dα,k→Asuch that

dα

Tkx,Tky ≤

γ∈Dα,k

dP_α_,k(γ)(x,y),

for anyx,y∈X,

(ii) there existsN∈Nsuch that|Dα,n| ≤ |Dα,N|andPα,n(Dα,n)⊆Pα,N(Dα,N)for any

n≥Nandα∈A.

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

Proof Letz∈F(T) and letUbe a neighborhood ofz. We may assume thatU=m_i_={w∈ X:dαi(w,z) <}for some>  andα, . . . ,αm∈A. For eachn∈N, let

Vn=

m

i=

γ∈Dαi,n

w∈X:dPαi,n(γ)(w,z) < |Dαi,n|

.

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

subset ofX,y∈V,l∈Nandi∈ {, . . . ,m}. It follows that

dαi

Tly,z =dαi

Tly,Tlz ≤

γ∈Dαi,l

dPα_i,l(γ)(y,z).

Ifl<N, then

dαi

Tly,z <

γ∈Dα_i,l |Dαi,l|

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

|Dα_i,N|for each γ ∈Dαi,l, and hence

dαi

Tly,z <

γ∈Dαi,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 N∈Nsuch that|An(α)| ≤ |AN(α)|andπn(An(α))⊆πN(AN(α))for any n≥N andα∈A,then T is uniformly virtually stable with respect to the sequence of all natural numbers.

Proof By lettingDα,n=An(α) andPα,n=πn|An(α)for anyn∈Nandα∈A, we have

dα

Tlx,Tly ≤

γ∈Dα,l

dPα,l(γ)(x,y),

for anyx,y∈X. 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

fTnx–Tny

≤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(x–y),

for eachf ∈,n∈N,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,m∈N,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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