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
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.
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 lettingxbe 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}
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(Tx,Ty)≤φ,α
dj(α)(x,y) and dα(Tx,Ty)≤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.
(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_{α}Mα(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}
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_{α}Fα(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
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.
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(Tx,Ty)≤φ,α
dj(α)(x,y) and dα(Tx,Ty)≤φ,α
dj(α)(x,y),
(i) jn_{}+=jn_{}◦jandjn◦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,Tx)≤p(x,α) <∞and djn
(α)(x,Tx)≤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,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 andjso 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 ∈Lloc([,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∈Lloc([,T]; [,∞)).
Consider the integral equation
x(t) =S(t)x+
t
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_{H}i
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}
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|
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,
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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