Vol. 22, No. 2 (1999) 377–386 S 0161-17129922377-0 ©Electronic Publishing House
COMMON FIXED POINT THEOREMS FOR SEMIGROUPS
ON METRIC SPACES
YOUNG-YE HUANG and CHUNG-CHIEN HONG
(Received 21 December 1995 and in revised form 16 February 1998)
Abstract.This paper consists of two main results. The first one shows that ifSis a left reversible semigroup of selfmaps on a complete metric space(M,d)such that there is a gauge functionϕfor whichd(f (x),f (y))≤ϕ(δ(Of(x,y)))forf∈Sandx,yinM, whereδ(Of(x,y))denotes the diameter of the orbit ofx,yunderf, thenShas a unique common fixed pointξinM and, moreover, for anyf inSandxinM, the sequence of iterates{fn(x)}converges toξ. The second result is a common fixed point theorem for a left reversible uniformly Lipschitzian semigroup of selfmaps on a bounded hyperconvex metric space(M,d).
Keywords and phrases. Fixed point, left reversible, upper semicontinuous.
1991 Mathematics Subject Classification. 47H09, 47H10.
1. Introduction. A gauge function is an upper semicontinuous functionϕ:[0,∞)→ [0,∞)such thatϕ(0)=0 andϕ(t) < tfort >0. A selfmapfon a metric space(M,d) is said to beϕ-contractive if it satisfies
df (x),f (y)≤ϕd(x,y), x,y∈M. (1.1)
In 1969, Boyd and Wong [1] showed that aϕ-contractive selfmapf on a complete metric space(M,d) has a unique fixed pointξ inM and that, for anyx in M, the sequence of iteratives{fn(x)}converges to ξ. This result was generalized recently
by Huang and Hong [6], where they showed that if S is a left reversible semigroup ofϕ-contractive selfmaps on a complete metric space (M,d)for which there is an x0inM with bounded orbitO(x0), thenS has a unique common fixed pointξinM and, furthermore, for anyf inS and anyxinM, the sequence of iterates{fn(x)}
converges toξ. In Section 2 of this paper, we deal with the same common fixed point problem for semigroups with (1.1) replaced by
df (x),f (y)≤ϕδO(x,y), x,y∈M, f∈S, (1.2)
or
df (x),f (y)≤ϕδOf(x,y), x,y∈M, f∈S, (1.3)
whereδ(O(x,y))denotes the diameter of the orbit ofx,yunderS andδ(Of(x,y))
denotes the diameter of the orbit ofx,yunderf.
metric space version of Casini and Maluta’s theorem in Banach spaces, (cf. Casini and Maluta [2]). In the setting of bounded hyperconvex metric space, we extend Lim and Xu’s theorem in Section 3 by showing a common fixed point theorem for left reversible uniformlyk-Lipschitzian semigroups.
2. Fixed point theorems for contractions. LetS be a semigroup of selfmaps on a metric space(M,d). For anyxinM, the orbit ofxunderSstarting atxis the setO(x) defined to be{x}∪Sx, whereSxis the setg(x):g∈S. Iff∈S, then the orbit ofx underfstarting atxis the setOf(x):=fnx:n∈N∪{0}, wheref0x:=x. Forx,y
inM, the setO(x,y)is the union ofO(x)andO(y), andOf(x,y):=Of(x)∪Of(y).
A subsetKofMis said to be bounded if its diameterδ(K), defined to be sup{d(x,y): x,y∈K}, is finite. It is easy to check thatO(x,y)is bounded provided that bothO(x) andO(y)are bounded.
We say that a semigroupSis near-commutative if, for anyf ,ginS, there istinSsuch that f g =gt. Examples of near-commutative semigroups include all commutative semigroups and all groups.
Also, recall that a semigroupSis said to be left reversible if, for anyf ,ginS, there area,b such thatf a=gb. It is obvious that left reversibility is equivalent to the statement that any two right ideals ofS have nonempty intersection. Clearly, every near-commutative semigroup is left reversible.
Ifϕ:[0,∞)→[0,∞)is a gauge function, then Chang [3] constructed a strictly in-creasing continuous functionα:[0,∞)→[0,∞)such thatα(0)=0 andϕ(t)≤α(t)<t fort >0. This result is used in what follows.
It is well known that ifα:[0,∞)→[0,∞)is a strictly increasing continuous function such thatα(t) < t(t >0), then, for any t >0, one has limn→∞αn(t)=0, (cf. Huang and Hong [6]).
Theorem2.1. Suppose thatSis a near-commutative semigroup of continuous self-maps on a complete metric space(M,d)such that the following conditions(i)and(ii) are satisfied
(i) For anyxinM, its orbitO(x)is bounded.
(ii) There exists a gauge functionϕ:[0,∞)→[0,∞)with the property that, for any f inS, there existsnf ∈Nsuch thatd(fn(x),fn(y))≤ϕ(δ(O(x,y)))for all n≥nf andx,yinM.
ThenShas a unique common fixed pointξinMand, moreover, for anyfinSand any xinM, the sequence of iteratesfn(x)converges toξ.
Proof. Choose a strictly increasing continuous functionα:[0,∞)→[0,∞)such thatα(0)=0 andϕ(t)≤α(t) < tfort >0. Then (ii) implies that
dfn(x),fn(y)≤αδO(x,y), for alln≥n
f andx,yinM. (2.1)
We now show that, forn≥nf,
δOfn(x),fn(y)≤αδO(x,y). (2.2)
(a) u,v∈O(fn(x)),
(b) u,v∈O(fn(y)),
(c) u∈O(fn(x)),v∈O(fn(y)),
(d) u∈O(fn(y)),v∈O(fn(x)).
In case (a), if u = sfn(x) and v = tfn(x) for some s,t ∈ S, then, by the
near-commutativity ofS, there area,binSsuch thatsfn=fnaandtfn=fnb, and so
d(u,v)=dsfn(x),tfn(x) =dfna(x),fnb(x) ≤αδOa(x),b(x) ≤αδO(x)
≤αδO(x,y).
(2.3)
Also, ifu=fn(x)andv=tfn(x)for some t∈S, then choosing b∈S such that tfn=fnb, we get
d(u,v)=dfn(x),tfn(x) =dfn(x),fnb(x) ≤αδOx,b(x) ≤αδO(x,y).
(2.4)
Likewise, in either the case of (b), (c), or (d), we also haved(u,v)≤α(δ(O(x,y))). Taking supremum over allu,vinO(fn(x),fn(y)), we conclude that (2.2) holds.
Next, we show that, for anyf inS, there isξf inM such that, for anyxinM, the
sequence of iterates{fn(x)}converges toξ
f. For this, letybe a member inM and,
for anyk∈N, letak=δ(O(fknf(x),fknf(y))). We obtain from (2.2) that
0≤ak=δOfknf(x),fknf(y) ≤αδOf(k−1)nf(x),f(k−1)nf(y) =αak−1≤ak−1.
(2.5)
Repeating the procedure (2.5)ktimes, we get
ak≤αkδO(x,y). (2.6)
It then follows from (2.5) and (2.6) in conjunction with limn→∞αk(δ(O(x,y)))=0 that limk→∞ak=0. In particular, limk→∞δ(O(fknf(x)))=0. Now, forn≥nf, choose the
largestk∈Nsuch thatknf ≤n < (k+1)nf. Then once noticing thatδ(O(fn(x)))≤ δ(O(fknf(x)))andk→ ∞asn→ ∞, we see that
0≤nlim→∞δOfn(x)≤ lim n→∞δ
Ofknf(x)=0 (2.7)
and so,
lim
n→∞δ
Hence,{fn(x)}is Cauchy and thus, lim
n→∞fn(x)=ξf for someξf ∈M. For the
same reasons, we have limn→∞fn(y)=ξf for anyyinM.
Thatξf is a fixed point offfollows from the continuity off. Indeed,
dfξf,ξf=nlim→∞dffnξf,fnξf =nlim→∞dfn+1ξ
f,fnξf
=dξf,ξf=0.
(2.9)
Finally, to complete our proof, it remains to show that, for any f ,g in S, one has ξf =ξg. For this, letmbe the least common multiple ofnf andng, and asSis left
reversible, for anyk∈N, chooseakandbkinSsuch that
fkma
k=gkmbk. (2.10)
Then, for anyxinM, we have
dξf,ξg=lim k→∞d
fkm(x),gkm(x)
≤lim
k→∞d
fkm(x),fkma
k(x)+lim k→∞d
gkm(x),gkmb k(x)
≤lim
k→∞α
kδOx,a
k(x)+lim k→∞α
kδOx,b k(x)
≤2lim
k→∞α
kδO(x)=0,
(2.11)
which shows thatξf=ξg.
Here, we like to give two concrete examples for the above theorem.
Example2.1. LetM=Rwith the usual metricd(x,y)= |x−y|and letS be the semigroup generated by
f:M →M:f (x)=
2
3x, ifx≥0,
0, ifx <0 (2.12)
and
g:M →M:g(x)=
−2
3x, ifx≥0,
0, ifx <0. (2.13)
In addition, putϕ:[0,∞)→[0,∞):ϕ(t)=1/2t. ThenS is commutative and, for any x,y in M and any h in S, d(h(x),h(y)) ≤d(f (x),f (y))=d(g(x),g(y))≤ ϕ(δ(O(x,y))).
Example2.2. LetM= {1,3,5,7}with the usual Euclidean metricd. Letα,β,γ,θ be the selfmaps onM defined byα(1)=α(3)=1,α(5)=3,α(7)=5, andβ(1)= β(3)=β(5)=1,β(7)=3, andγ(1)=γ(3)=1,γ(5)=γ(7)=3, andθ(1)=θ(3)= θ(5)=θ(7)=1. ThenS:= {α,β,γ,θ}is a near-commutative semigroup under com-position. Sinceαγ=θandγα=β,Sis not commutative. Puttingnα=3,nβ=nγ=2, nθ=1 andϕ:[0,∞)→[0,∞):ϕ(t)=1/2t, it is easy to check that condition (ii) of
LetSbe the semigroup generated byf, where
f:[0,1]→[0,1]:f (x)=
11, forx=0, 2x, for 0< x≤1.
(2.14)
Then [13, Ex. 2] showed that the continuity hypothesis on each member of S in Theorem 2.1 cannot be dropped in general.
However, when condition (ii) of Theorem 2.1 is replaced by
df (x),f (y)≤ϕδOf(x,y), (2.15)
the continuity condition on each member ofScan be dropped and the semigroupS itself can be relaxed to the case for which it is left reversible.
Theorem2.2. SupposeS is a left reversible semigroup of selfmaps on a complete metric space(M,d)such that the following conditions(i)and(ii)are satisfied
(i) For anyxinMand anyfinS, the orbitOf(x)is bounded;
(ii) There exists a gauge functionϕ :[0,∞)→[0,∞) such that d(f (x),f (y))≤ ϕ(δ(Of(x,y)))for anyfinSand anyx,yinM. ThenShas a unique common fixed
pointξinMand, for anyfinSandxinM, the sequence of iterates{fn(x)}converges
toξ.
Proof. It follows from [13, Thm. 2] that eachf inS has a unique fixed pointξf
inM and, for anyx inM, the sequence of iterates{fn(x)}converges toξ
f. So, to
complete the proof, it suffices to show thatξf =ξg for anyf,ginS. Letnbe any
positive integer. The left reversibility ofSshows that there areanandbn inS such
thatfna
n=gnbn. Also, condition (i) implies that, for anyfinSandxinM,
supδOfx,h(x):h∈S<∞. (2.16)
Thus, once we choose a strictly increasing continuous function α:[0,∞)→[0,∞) such thatα(0)=0 andϕ(t)≤α(t) < tfort >0, we then have
dξf,ξg=nlim→∞dfn(x),gn(x) ≤nlim→∞dfn(x),fna
n(x)+nlim→∞dgn(x),gnbn(x)
≤nlim→∞αnδO
fx,an(x)+nlim→∞αnδOgx,bn(x)
≤lim
n→∞α
nsup δO
fx,h(x):h∈S
+nlim→∞αnsup δO
gx,h(x):h∈S
=0,
(2.17)
which shows thatξf=ξg.
fact thatd(h(x),h(y))≤ϕ(δ(O(x,y)))for anyhinS and x,yinM, we have, for x=0 andy >0,d(f (x),f (y))=2/3yandδ(Of(x,y))=yand so, we do not have d(f (x),f (y))≤ϕ(δ(Of(x,y))).
Example2.4. For anyn∈N, letfn:[0,1]→[0,1]:
fn(x)=
1
n+1x, ifxis rational,
0, ifxis irrational, (2.18)
and putS= {fn:n∈N}. ThenSis a commutative semigroup under composition and
each member inS is discontinuous. Also, for anyn∈Nandx,yin[0,1],
dfn(x),fn(y)≤12δOfn(x,y)
. (2.19)
So, if we putϕ:[0,∞)→[0,∞):ϕ(t)=1/2t, thend(fn(x),fn(y))≤ϕ(δ(Ofn(x,y))). In this case, 0 is the unique common fixed point ofS.
3. Fixed point theorems for Lipschitzian mappings. A nonempty familyᏲof sub-sets of a metric space(M,d)is said to define a convexity structure onMif it is stable under intersection. A subset ofM is said to be admissible if it is an intersection of closed balls. We denote, byᏭ(M), the family of all admissible subsets ofM. Obviously, Ꮽ(M)defines a convexity structure onM. Forr >0 andxinMand a bounded subset DofM, we adopt the following notations
B(x,r )is the closed ball with centerxand radiusr , (3.1)
r (x,D)=sup{d(x,y):y∈D}, (3.2)
δ(D)=sup{r (x,D):x∈D} = the diameter ofD, (3.3) R(D)=inf{r (x,D):x∈D} = the Chebyshev radius ofDrelative toD. (3.4)
Following Khamsi [7], a metric space(M,d)is said to have a uniform normal struc-ture if there exist a convexity strucstruc-tureᏲ onM and a constantc∈(0,1)such that R(D)≤cδ(D)for any bounded subsetDinᏲwithδ(D) >0. We also say thatᏲis uniformly normal. The uniform normal structure coefficient ˜N(M)ofMrelative toᏲ is the number supR(D)/δ(D):D∈Ᏺis bounded andδ(D) >0.
A metric space(M,d)is said to be hyperconvex if any family{B(xα,rα)}of closed
balls inMsatisfyingd(xα,xβ)≤rα+rβhas nonempty intersection.
The following results are well known, (cf. Goebel and Kirk [4] and Kirk [8]). Lemma3.1. Suppose that(M,d)is a hyperconvex metric space. Then
(i) Mis complete,
(ii) Ꮽ(M)is a uniformly normal convexity structure such that the uniform normal structure coefficientN(M)˜ ofMrelative to it is1/2,
(iii) for any subfamilyofᏭ(M)which has the finite intersection property, one has
A∈A≠∅.
For a bounded subsetDofM, the admissible hull ofD, denoted by ad(D), is the set
The following definition is a net version of [10, Def. 5].
Definition3.1. A metric space(M,d) is said to have the property(P)if given any two bounded nets{xi}i∈I and{zi}i∈I, one can find somez∈i∈Iad{zj:j≥i}
such that limi∈Id(z,xi)≤limj∈Ilimi∈Id(zj,xi), where limi∈Id(z,xi)is defined to be
infβ∈Isupi≥βd(z,xi).
Using property (iii) in Lemma 3.1 to conclude thati∈IAi≠φ for any decreasing
net{Ai}i∈Iof admissible subsets ofM, the proof for Lemma 3.2 below is the same as
that in Lim and Xu [10].
Lemma3.2. Let(M,d)be a bounded hyperconvex metric space with property(P). Then, for any net{xi}i∈I inM and any constantc >1/2, there exists a pointzinM
satisfying
(i) limi∈Id(z,xi)≤cδ({xi}i∈I), and
(ii) d(z,y)≤limi∈Id(xi,y)for allyinX.
LetSbe a left reversible semigroup. Fora,binS, we say thata≥bifa∈bS∪{b}. Then(S,≥)is a directed set. In what follows in this section, we deal only with this order.
Lemma3.3. LetS be a left reversible semigroup acting on a metric space(M,d). Then, forx,yinM, one haslimt∈Sd(x,ty)=infs∈Ssupt∈Sd(x,sty).
Proof. By definition, limt∈Sd(x,ty)=infs∈Ssupt≥sd(x,ty). Since, for anys in S, supr≥sd(x,r y)≥supt∈Sd(x,sty), we see that infs∈Ssupr≥sd(x,r y)≥infs∈S×
supt∈Sd(x,sty), that is limt∈Sd(x,ty)≥infs∈Ssupt∈Sd(x,sty). On the other hand,
if a∈ S then, for any s ∈S, we have supt≥sad(x,ty)≤supt∈Sd(x,sty) and so,
infb∈Ssupt≥bd(x,ty)≤supt∈Sd(x,sty)for anys∈S. Therefore, limt∈Sd(x,ty)≤
infs∈Ssupt∈Sd(x,sty).
To prove our main result in this section, we need the following lemma.
Lemma3.4. LetSbe a left reversible semigroup acting on a metric space(M,d)and letx,ybe two points inM. Then, for anya∈S, one has
lim
t∈Sd(x,aty)=limt∈Sd(x,ty). (3.6)
Proof. By Lemma 3.3, limt∈Sd(x,aty)=infs∈Ssupt∈Sd(x,saty). But, since
inf
s∈Ssupt∈Sd(x,saty)≥infs∈Ssupt∈Sd(x,sty)=limt∈Sd(x,ty), (3.7)
we get limt∈Sd(x,aty)≥limt∈Sd(x,ty). On the other hand, ifs∈S, then, for anyr
insS∩aS, we have
sup
Hence, supt∈Sd(x,sty) ≥ infu∈Ssupt∈Sd(x,auty) for any s ∈ S. Consequently,
lettingsvary overS, we obtain that lim
t∈Sd(x,ty)=infs∈Ssupt∈Sd(x,sty) ≥inf
u∈Ssupt∈Sd(x,auty) =lim
t∈Sd(x,aty).
(3.9)
Definition3.2. A semigroupS acting on a metric space(M,d) is said to be a uniformlyk-Lipschitzian semigroup if
d(sx,sy)≤kd(x,y) (3.10)
for allsinS and allx,yinM.
IfS is a left reversible semigroup, then(S,≥)is a totally ordered set if any x,y inS satisfy eitherx≤y ory≤x. For example, if᐀= {Ts:s∈[0,∞)}is a family of
selfmaps onRsuch thatTs+h(x)=TsTh(x)for alls,hin[0,∞)andx∈R, then(᐀,≥)
is a totally ordered left reversible semigroup.
We are now in a position to prove our main result in this section.
Theorem3.1. Let(M,d)be a bounded hyperconvex metric space with property(P) and letSbe a left reversible uniformlyk-Lipschitzian semigroup of selfmaps onMsuch thatk <√2and(S,≥)is a totally ordered set. ThenShas a common fixed pointξinM. Proof. Choose a constantcsuch that 1/2< c <1 and k <1/√c. Letx0 be any point inM. FortinS, denotetx0byx0,t. Then{x0,t}t∈Sis a net inM. By Lemma 3.2,
we can inductively construct a sequence{xj}inMsuch that, for eachj∈N∪{0},
(a) limt∈Sd(xj+1,xj,t)≤cδ(Sxj), and
(b) d(xj+1,y)≤limt∈Sd(xj,t,y)for allyinM.
Write Dj = limt∈Sd(xj+1,xj,t) and h= ck2 <1. For s,t ∈S with s ≥ t, we have d(sxj,txj)=0 if s =t, and d(sxj,txj)= d(taxj,txj)if s =ta for somea∈ S.
Then
dtaxj,txj≤kdaxj,xj ≤klim
t∈Sd
xj−1,t,axj by (b) =kinf
s∈Ssupt∈S d
stxj−1,axj by Lemma 3.3
=kinf
s∈Ssupt∈S d
astxj−1,axj by Lemma 3.4
≤k2inf
s∈Ssupt∈S d
stxj−1,xj
=k2D
j−1.
(3.11)
Taking supremum fors,tover S and noting that(S,≥)is a totally ordered set, we then obtain that
δSxj≤k2Dj−1. (3.12)
Hence,
Therefore, for anyj∈N∪{0}and anyt∈S,
dxj+1,xj≤dxj+1,xj,t+dxj,t,xj ≤dxj+1,xj,t+lim
r∈Sd
xj−1,r,xj,t by (b) =dxj+1,xj,t+inf
s∈Ssupr∈S d
sr xj−1,txj
=dxj+1,xj,t+inf s∈Ssupr∈Sd
tsr xj−1,txj by Lemma 3.4
≤dxj+1,xj,t+kinf s∈Ssupr∈S d
sr xj−1,xj
=dxj+1,xj,t+kDj−1,
(3.14)
which implies that
dxj+1,xj≤lim t∈Sd
xj+1,xj,t+kDj−1 =Dj+kDj−1
≤hj+khj−1D 0.
(3.15)
Sinceh∈(0,1), we conclude that{xj}is a Cauchy sequence inM. By Lemma 3.1,M
is complete. Thus, there isξinMsuch that limj→∞xj=ξ.
Finally, we show thatξis a common fixed point ofS. For anytinS,
dξ,tξ≤dξ,xj+1+dxj+1,xj,t+dxj,t,tξ ≤dξ,xj+1+dxj+1,xj,t+kdxj,ξ ≤dxj+1,ξ+lim
r∈Sd
xj,r,xj,t+kdxj,ξ
=dxj+1,ξ+kdxj,ξ+lim r∈Sd
tr xj,txj by Lemma 3.4
≤dxj+1,ξ+kdxj,ξ+klim r∈Sd
r xj,xj
≤dxj+1,ξ+kdxj,ξ+klim r∈Slima∈Sd
xj−1,a,r xj by (b).
(3.16)
But,
lim
a∈Sd
xj−1,a,r xj=lim a∈Sd
r axj−1,r xj by Lemma 3.4
≤klim
a∈Sd
axj−1,xj ≤kcδSxj−1 by (a) ≤kck2D
j−2 by (3.12) =khDj−2≤ ··· ≤khj−1D0.
(3.17)
So,d(ξ,tξ)≤d(xj+1,ξ)+kd(xj,ξ)+k2hj−1D0, which shows thattξ=ξonce we let j→ ∞.
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