FIXED POINTS OF SEMIGROUPS OF LIPSCHITZIAN MAPPINGS DEFINED ON NONCONVEX DOMAINS
KOK-KEONG TAN AND HONG-KUN XU
Abstract. Certain fixed point theorems are established for nonlinear semigroups of Lipschitzian mappings defined on nonconvex domains in Hilbert and Banach spaces. Some known results are thus general-ized.
1. Introduction
LetX be a Banach space andCbe a nonempty subset ofX. A mapping T :C→C is said to be a Lipschitzian mapping if, for each integern≥1, there exists a constantkn≥0 such thatkTnx−Tnyk ≤knkx−yk ∀x, y∈C.
A Lipschitzian mapping T is said to be uniformly Lipschitzian if kn = k
for all n ≥ 1, nonexpansive if kn = 1 for all n ≥ 1, and asymptotically
nonexpansive if limn→∞kn= 1, respectively. Goebel and Kirk [1] initiated
in 1973 the study of the fixed point theory for Lipschitzian mappings. They showed that if X is uniformly convex and C is a bounded closed convex subset ofX, then every uniformlyk-Lipschitzian mappingT :C→C with k < γ has a fixed point, whereγ >1 is the unique solution of the equation
γ
1−δX
1
γ
= 1,
with δX the modulus of convexity of X. Since then, much effort has been
devoted to the existence theory for fixed points of Lipschitzian mappings in both Hilbert and Banach spaces; see [2], [4], [5], [7], [10], [11], and refereces cited there. Usually the domainC on whichT is defined is assumed to be convex. Recently, Ishihara [3] and Takahashi [9] studied in Hilbert spaces the existence theory for fixed points of Lipschitzian mappings which are defined on nonconvex domains. However, their methods do not work outside Hilbert spaces.
1991Mathematics Subject Classification. 47H10, 47H09 .
Key words and phrases. Fixed point, semigroup, Lipschitzian mapping, nonconvex domain.
547
The purpose of the present paper is to investigate the existence theory for the fixed point theory of semigroups of Lipschitzian mappings defined on nonconvex domains in both Hilbert and Banach spaces.
2. Preliminaries
Let G be a semitopological semigroup, i.e., G is a semigroup with a Hausdorff topology such that for each a ∈ G, the mappings t → at and t →ta from G into itself are continuous. Let C be a nonempty subset of a Banach space X. Then a family F ={Tt:t∈G} of self-mappings ofC
is said to be a Lipschitzian semigroupon C if the following properties are satisfied:
(1) Ttsx=TtTsxfor allt, s∈Gandx∈C;
(2) for eachx∈C, the mappingt→Ttxis continuous onG;
(3) for eacht∈G, there is a constantkt>0 such thatkTtx−Ttyk ≤
ktkx−yk ∀x, y ∈C.
A Lipschitzian semigroup F is calleduniformlyk-Lipschitzianifkt=kfor
allt ∈Gand in particular,nonexpansive ifkt= 1 for allt ∈G. We shall
denote byF(F) the set of common fixed points ofF ={Tt:t∈G}.
Recall that a semitopological semigroupGis said to be left reversibleif any two closed right ideals of G have nonvoid intersection. In this case, (G,≤) is a directed system when the binary relation “≤” onGis defined by a≤bif and only if{a} ∪aG⊇ {b} ∪bG. LetB(G) be the Banach space of all bounded real-valued functions onGwith the supremum norm and letX be a subspace ofB(G) containing constants. Then an elementµofX∗, the dual space of X, is said to be ameanonX ifkµk=µ(1) = 1. It is known thatµ∈X∗ is a mean onX if and only if the inequalty
inf{f(t) :t∈G} ≤µ(f)≤sup{f(t) :t∈G}
holds for all f ∈ X. For a mean µon X∗ and an elementf ∈ X, we use either µt(f(t)) or µ(f) to denote the value of µat f. For each s∈G, we
define the left transformationℓs from B(G) into itself by (ℓsf)(t) =f(st),
t∈G, for allf ∈B(G). The right transformationrsis defined similarly. Let
X be a subspace of B(G) containing constants which is ℓG-invariant (rG
-invariant), i.e.,ℓs(X)⊆X(rs(X)⊆X) for alls∈G. Then a meanµonX
is said to beleft invariant(right invariant) ifµ(f) =µ(ℓsf) (µ(f) =µ(rsf))
for allf ∈X ands∈G. An invariant mean is a mean that is both left and right invariant.
Let C(G) be the Banach space of all bounded continuous real-valued functions on G, let RU C(G) be the space of all bounded right uniformly continuous functions onG, i.e., allf ∈C(G) for which the mappings→rsf
is continuous, and letAP(G) be the space of allf ∈C(G) for which {ℓsf :
C(G) containing constants and bothℓG- andrG-invariant; see [7] for details.
If{xs:s∈G}is a bounded family of elements of a Banach space E, then
for allx∈E andp≥1, the functionsg(s) :=kxs−xkp andh(s) :=hxs, xi
(ifE is a Hilbert space) are inRU C(G).
3. The Hilbert Space Setting
In this section, we prove fixed point theorems for Lipschitzian semigroups defined on nonconvex domains in Hilbert spaces.
Theorem 3.1. Let C be a nonempty subset of a real Hilbert space H, letGbe a semitopological semigroup such thatRU C(G)has a left invariant mean µ, and let F ={Tt :t ∈G} be a Lipschitzian semigroup on C such
that µt(k2t)<2. Suppose that {Ttx:t∈G} is bounded and ∩s∈Gco{Tstx:
t∈G} is contained in C for allx∈C. Then there exists a point z∈Cfor whichTtz=z for allt∈G.
Proof. Letx0∈C. It is easily seen that the functionalµthTtx0, xi, x∈H,
is a continuous linear functional onH. By Riesz’s representation theorem, there is a unique elementx1∈H satisfyingµthTtx0, xi=hx1, xi, ∀x∈H.
By a routine argument (cf. [3] and [9]) via the separation theorem, we have x1 ∈ ∩
s∈Gco{Tstx0 : t ∈ G} and hence by assumption, x1 does remain in
C. Therefore, we can continue the above procedure to obtain a sequence
{xn}∞n=1 inC satisfying the following property:
µthTtxn−1, xi=hxn, xi, ∀x∈H, ∀n≥1. (3.1)
Noting the fact that for allu, v ∈H, the function h(t) :=kTtu−vk2 is in
RU C(G), it follows from (3.1) that
µtkTtxn−1−xk2=µtk(Ttxn−1−xn) + (xn−x)k2=µt(kTtxn−1−xnk2+
+kxn−xk2+ 2hTtxn−1−xn, xn−xi) =µtkTtxn−1−xnk2+
+kxn−xk2+ 2µthTtxn−1−xn, xn−xi=µtkTtxn−1−xnk2+
+kxn−xk2+ 2hxn−xn, xn−xi=µtkTtxn−1−xnk2+kxn−xk2.
This shows thatxnis the unique minimizer of the convex functionµtkTtxn−1 −xk2 over H and in particular, takingx=T
sxn and noting thatµis left
invariant, we get
µtkTtxn−1−xnk2+kxn−Tsxnk2=µtkTtxn−1−Tsxnk2=
=µtkTstxn−1−Tsxnk2=µtkTsTtxn−1−Tsxnk2≤ks2µtkTtxn−1−xnk2.
It follows that
SetA=µs(k2s−1),rn=µtkTtxn−xn+1k2andRn =µtkTtxn−xnk2.Then
from (3.2) we have
Rn≤Arn−1≤ARn−1≤ · · · ≤AnR0. (3.3)
It then follows from (3.3) that
kxn+1−xnk2=µtk(xn+1−Ttxn) + (Ttxn−xn)k2≤ ≤2µt(kTtxn−xn+1k2+kTtxn−xnk2) =
= 2(rn+Rn)≤4Rn≤4AnR0.
Since A < 1, we see that {xn} is Cauchy and hence convergent in norm.
Letz be the limit of{xn}. We claim that z is a common fixed point ofF.
In fact, for anys∈G, sinceµis left invariant, we have
kTsz−zk2=µt(k(Tsz−Tstxn) + (Tstxn−z)k2≤ ≤2µt(kTstxn−Tszk2+kTstxn−zk2)≤ ≤2(k2
sµtkTtxn−zk2+µtkTstxn−zk2) =
= 2(1 +k2
s)µtkTtxn−zk2=
= 2(1 +k2
s)µtk(Ttxn−xn) + (xn−z)k2≤ ≤4(1 +k2
s)(µtkTtxn−xnk2+kxn−zk2) =
= 4(1 +ks)(Rn+kxn−zk2)→0 asn→ ∞.
Therefore,Tsz=z.
Corollary 3.1 (Theorem 3.5 [4]). If F = {Tt : t ∈ G} is a
non-expansive semigroup on a closed convex subset C of a Hilbert space H, RU C(G) has a left invariant mean, and there exists an x ∈ C such that
{Ts(x) :s∈G} is bounded, then F has a common fixed point in C.
LetX be a subspace ofB(G) containing constants. Following Mizoguchi and Takahashi [7T], we say that a real valued function on X is a submean onX if the following conditions are fulfilled:
(1) µ(f+g)≤µ(f) +µ(g), ∀f, g∈X; (2) µ(αf)≤α µ(f), ∀f ∈X, ∀α≥0; (3) ∀f, g∈X, f≤g=⇒µ(f)≤µ(g); (4) µ(c) =cfor all constantsc.
Theorem 3.2. Let H be a real Hilbert space, C a nonempty subset of H, X an ℓG-invariant subspace of B(G) containing constants that has a
left invariant submean µ on X, and F = {Tt : t ∈ G} a Lipschitzian
semigroup on C. Suppose that {Ttx: t ∈ G} is bounded for some x∈ C
and ∩s∈Gco{Tstx : t ∈ G} ⊂ C for all x ∈ C. Suppose also that for all
hon G defined by h(t) = k2
t belong to X andµt(k2t)<2. Then there is a
point z∈C such thatTtz=z for allt∈G.
Proof. Letx0∈Cand define the functionr0onH byr0(x) =µtkTtx0−xk2,
x ∈ H. Note that r0 is well defined since by assumption, the function
t 7→ kTtx0−xk2 is in X for every x ∈ H. As r0 is strictly convex and
continuous andr0(x)→ ∞as kxk → ∞, there is a unique elementx1∈H
such thatr0:=r0(x1) = inf{r0(x) :x∈H}. We claim that thisx1belongs
to ∩s∈Gco{Tstx0 :t ∈G} and thus toC by our hypothesis. Indeed, if we
denote byPsthe nearest point projection ofH onto the setco{Tstx0;t∈G},
then, asPsis nonexpansive andµis left invariant, we get
r0(Psx1) =µtkTtx0−Psx1k2=µtkTstx0−Psx1k2=
=µtkPsTstx0−Psx1k2≤µtkTstx0−x1k2=
=µtkTtx0−x1k2=r0,
which shows that Psx1 is also a minimizer of r0 and hence by uniqueness,
Psx1 = x1, i.e., x1 ∈ ∩s∈Gco{Tstx0 : t ∈ G}. This proves the claim.
Repeating the above process, we obtain a sequence {xn} in C with the
following property:
xn∈
\
s∈G
co{Tstxn−1:t∈G} ∀n≥1
and xn is the unique minimizer over H of the functional rn(·) defined by
rn(x) =µtkTtxn−1−xk2,x∈H. Now by the same argument as in the proof
of Theorem 3.1, we conclude that{xn}converges strongly to a common fixed
pointz∈C.
Corollary 3.2 (Theorem 1 [8]). Let C be a closed convex subset of a Hilbert space H and X be an ℓG-invariant subspace of B(G) containing
constants which has a left invariant submean µ. Let F={Tt:t∈G} be a
Lipschitzian semigroup onC such that {Tsx:s∈G} is bounded for some
x∈C. If for eachu, v∈C, the functionf(t) :=kTtu−vk2and the function
g(t) :=k2
t (t ∈ G) belong to X and µs(ks2) <2, then there is z ∈C such
that Tsz=z for alls∈G.
We now extend Theorem 3.3 of Lau [4] to a wider class of Lipschitzian semigroups which are defined on nonconvex domains.
Theorem 3.3. SupposeHis a real Hilbert space,Cis a nonempty subset ofH, andF ={Tt:t∈G}is a Lipschitzian semigroup onC. Suppose also
AP(G)has a left invariant meanµ. Ifµt(k2t)≤1and if there exists anx∈
C such that{Ttx:t∈G} is relatively compact in norm and∩s∈Gco{Tstx:
Proof. Since{Ttx:t∈G} is relatively compact, by Lemma 3.1 of Lau [4],
for ally ∈ H, the functions h and g defined on Gby h(t) =hy, Ttxi and
g(t) =ky−Ttxk2are both inAP(G). So we have a uniquez∈Hsuch that
µthTtx, yi=hz, yi, ∀y∈H. As seen before, we have (i)z∈ ∩s∈Gco{Tstx:
t∈G} and hence z ∈C and (ii)µtkTtx−yk2 =µtkTtx−zk2+ky−zk2 ∀y∈H. In particular, we have for alls∈G,
µtkTtx−Tszk2=µtkTtx−zk2+kTsz−zk2.
Noting that
µtkTtx−Tszk2=µtkTstx−Tszk2=µtkTsTtx−Tszk2≤ks2µtkTtx−zk2,
we get for alls∈G,kTsz−zk2≤(k2s−1)µtkTtx−zk2. Hence
µskTsz−zk2≤(µs(k2s−1))µtkTtx−zk2≤0
forµs(k2s)≤1; namely,µskTsz−zk2= 0. Now fora∈G, we have
kTaz−zk2=µsk(Taz−Tsz) + (Tsz−z)k2≤2µs(kTaz−Tszk2+
+kTsz−zk2) = 2(µskTsz−Tazk2+µskTsz−zk2) =
= 2µskTasz−Tazk2= 2µkTaTsz−Tazk2≤
≤2ka2µskTsz−zk2= 0.
ThereforeTaz=z.
Corollary 3.3 (Theorem 3.2 [4]). If C is a closed convex subset of a Hilbert space H, F = {Tt : t ∈ G} is a nonexpansive semigroup on C,
AP(G) has a left invariant mean, and x∈ C such that {Tsx : s ∈ G} is
relatively compact, thenC contains a common fixed point for F.
By the same proof as in Theorem 3.1, we get immediately the following result.
Theorem 3.4. SupposeH andCare as in Theorem3.3, supposeAP(G) has a left invariant meanµ, and suppose F={Tt:t∈G}is a Lipschitzian
semigroup on C such that µt(kt2)<2. If, for every x∈C, {Ttx: t∈ G}
is relatively compact in norm and ∩s∈Gco{Tstx:t∈G} is contained in C,
4. The Banach Space Setting
In this section we study the existence of fixed points for Lipschitzian mappings defined on nonconvex domains in Banach spaces. So supposeC is a nonempty subset of a Banach space X and F = {Tt : t ∈ G} is a
Lipschitzian semigroup defined on C. (Here G is as in Section 3 a semi-topological semigroup.) We will employ the following condition introduced by Goebel, Kirk, and Thele in [2]: A nonempty subset E of C is said to satisfy the property
(P): For every x ∈ E and ε > 0, there exists s ∈ G such that dist(Ttx, E)< εfor allt≥s.
Recall that the modulus of convexity of a Banach spaceX is defined as the function
δX(ε) = inf
1−12kx+yk:x, y∈BX withkx−yk ≥ε
, 0≤ε≤2,
where BX is the closed unit ball of X. A Banach space X is said to be
uniformly convexifδX(ε)>0 for all 0< ε≤2. It is said to bep-uniformly
convex for somep≥2 if there exists a constantd >0 such thatδX(ε)≥d εp
for 0≤ε≤2. It is known that a Hilbert space is 2-uniformly convex and anLp (1< p <
∞) space is max(2,p)-uniformly convex. We shall need the following characterization of ap-uniformly convex Banach space.
Proposition ([cf. [12]). Given a numberp≥2. A Banach spaceX is p-uniformly convex if and only if there exists a constant d=dp >0 such
that
kλx+ (1−λ)ykp
≤λkxkp+ (1
−λ)kykp
−dWp(λ)kx−ykp (4.1)
for allx, y∈X and 0≤λ≤1, whereWp(λ) =λp(1−λ) +λ(1−λ)p.
Theorem 4.1. Suppose X is a p-uniformly convex Banach space for somep≥2,Cis a nonempty subset ofX,Gis a semitopological semigroup that is left reversible, and F ={Tt :t ∈G} is a uniformly k-Lipschitzian
semigroup onC withk <(1 +d)p1,dbeing the constant appearing in (4.1).
Suppose also there exist an x¯∈ C such that {Ttx¯ :t ∈G} is bounded and
a nonempty bounded closed convex subsetE of C with Property (P). Then there exists a pointz∈E such that Ttz=z for allt∈G.
Proof. Since{Ttx¯:t ∈G} is bounded, it is easily seen that for all x∈C, {Ttx:t ∈G} is bounded. Now choose anyx0∈E and define a functional
f on E by f(x) = infssupt≥skTtx0−xkp, x ∈ E. By Lemma 3 of [11],
Continuing this procedure, we construct a sequence{xn}∞n=1inEsuch that
for every integern≥1,
inf
s supt≥skTtxn−1−xnk p
≤inf
s supt≥skTtxn−1−xk p
−
−dkx−xnkp,
∀x∈E. (4.2)
Now by Property (P), we can find for eachn≥1 an an ∈Gsuch that
dist(Taxn, E)< An for alla≥an, (4.3)
whereA= kp−1
d <1. For each fixeda≥an, from (4.3) we can thus find a
yn ∈E such that
kTaxn−ynk< An. (4.4)
Using the mean value theorem, it is easy to see that for allx, y∈X,
|kykp− kxkp| ≤p[max{kxk,kyk}]p−1|kyk − kxk| ≤p(kxk+kyk)p−1ky−xk.
Since all the involved sequences are bounded, we can find a constantM big enough so that for allt, a∈Gandn≥1,
p(kTtxn−1−ynk+kTtxn−1−Taxnk)p−1≤M/2
and
p(kTaxn−xnk+kyn−xnk)p−1≤M/2.
It thus follows from (4.2) and (4.4) that
inf
s supt≥skTtxn−1−xnk p
≤inf
s supt≥skTtxn−1−ynk p
−dkyn−xnkp≤
≤inf
s supt≥skTtxn−1−Taxnk p
−dkTaxn−xnkp+ sup t
(kTtxn−1−
−ynkp− kTtxn−1−Taxnkp) +d(kTaxn−xnkp− kyn−xnkp)≤ ≤inf
s supt≥skTtxn−1−Taxnk p
−dkTaxn−xnkp+MkTaxn−ynk ≤
≤inf
s supt≥skTtxn−1−Taxnk p
−dkTaxn−xnkp+M An≤
≤inf
s supt≥skTatxn−1−Taxnk p
−dkTaxn−xnkp+M An ≤
≤kp a infs sup
t≥sk
Ttxn−1−xnkp−dkTaxn−xnkp+M An.
Hence for alla≥an,
dkTaxn−xnkp≤(kp−1) inf
s supt≥skTtxn−1−xnk
which results in the conclusion
inf
s supt≥skTtxn−xnk p
≤Ainf
s supt≥skTtxn−1−xnk p
+M′An,
where M′ = M/d. Write R
n = infssupt≥skTtxn −xnkp and rn =
infssupt≥skTtxn −xn+1kp. Thenrn≤Rn by (4.2) and
Rn ≤Arn−1+M′An≤ARn−1+M′An≤A(ARn−2+M′An−1) +
+M′An =A2R
n−2+ 2M′An ≤ · · · ≤(R0+nM′)An.
Therefore,
kxn−xn−1kp= inf
s supt≥sk(xn−Ttxn−1) + (Ttxn−1−xn−1)k p
≤
≤2p−1inf
s supt≥s(kxn−Ttxn−1k p+
kTtxn−1−xn−1kp)≤
≤2p−1(r
n−1+Rn−1)≤2pRn−1≤2p(R0+(n−1)M′)An−1,
which shows that{xn}is Cauchy and hence convergent to somezstrongly.
We now show that thiszis a common fixed point ofF. In fact, noting that the inequality (cf. [11])
inf
s supt≥skTtx−yk p
≤inf
s supt≥skTatx−yk p
is valid for allx, y∈C anda∈G, we have for alla∈G,
kz−Tazkp≤inf
s supt≥s(kz−xnk+kxn−Ttxnk+
+kTtxn−Taxnk+kTaxn−Tazk)p≤4p inf
s supt≥s((1 +k p)
kz−xnkp+
+kxn−Ttxnkp+kTtxn−Taxnkp)≤4p((1 +kp)kz−xnkp+Rn+
+ inf
s supt≥skTatxn−Taxnk p)
≤4p(1 +kp)(
kz−xnkp+Rn)→0 (n→ ∞).
HenceTaz=z.
Theorem 4.2. Let X be a p-uniformly convex Banach space for some p≥2,Ca nonempty subset of X,Ga semitopological semigroup for which RU C(G) has a left invariant meanµ, and F ={Tt :t ∈ G} a uniformly
k-Lipschitzian semigroup on C such that k < (1 +d)p1 with d being the
constant appearing in (4.1). Suppose there is a point x¯ ∈ C for which the orbit {Ttx¯ : t ∈ G} is bounded. Suppose also there exists a nonempty
bounded closed convex subsetE of C with the following property: (P∗) For everyx∈Eandε >0, there existss∈Gsuch that dist(T
stx, E)
< ε,∀t∈G.
Proof. Choose an arbitraryx0∈ E. As the function t 7→ kTtu−vkp is in
RU C(G) for anyu∈Candv∈X, we can inductively construct a sequence
{xn}∞n=1 in E in the following way: For each integer n ≥ 1, xn ∈ E is
the unique minimizer of the functional µtkTtxn−1−xk2 over E. Then by
Lemma 2 of [11], we have
µtkTtxn−1−xnkp≤µtkTtxn−1−xkp−dkx−xnkp, ∀x∈E. (4.5)
Write rn = inf{µtkTtxn −xkp : x ∈ E} = µtkTtxn −xn+1kp and Rn =
µtkTtxn−xnkp. Then for a fixed integern ≥1, condition (P∗) yields an
sn∈Gsuch that
dist(Tsnrxn, E)< A
n,
∀r∈G, (4.6)
whereA= (kp−1)/d <1. From (4.6) we can find ay
n∈E (depending on
r) such thatkTsnrxn−ynk< A
n. It then follows from (4.5) that
rn−1≤µtkTtxn−1−ynkp−dkyn−xnkp=µtk(Ttxn−1−Tsnrxn) +
+ (Tsnrxn−yn)k
p
−dk(yn−Tsnrxn) + (Tsnrxn−xn)k
p ≤ ≤µtkTtxn−1−Tsnrxnk
p
−dkTsnrxn−xnk+M A
n
= =µtkTsnrtxn−1−Tsnrxnk
p
−dkTsnrxn−xnk
p+M An ≤ ≤kpµtkTtxn−1−xnkp−dkTsnrxn−xnk
p
+M An,
where M > 0 is some appropriate constant independent of r ∈ G, which can be found similarly to the proof of Theorem 4.2. Hence
kTsnrxn−xnk
p ≤k
p−1
d rn−1+ M
dA
n=Ar
n−1+M′An,
whereM′ =M/d, and
Rn=µtkTtxn−xnkp=µrkTsnrxn−xnk
p ≤ ≤Arn−1+M′An≤ · · · ≤(R0+nM′)An.
Therefore, we have
kxn−xn−1kp≤µtk(xn−Ttxn−1) + (Ttxn−1−xn−1)kp≤ ≤2pµ(kxn−Ttxn−1kp+kTtxn−1−xn−1kp) =
= 2p(r
showing that{xn} is Cauchy and hence strongly convergent. Let z be the
limit. Now for alla∈G, we have
kz−Tazkp≤µt(kz−xnk+kxn−Ttxnk+kTtxn−Taxnk+
+kTaxn−Tazk)p≤4pµt(kz−xnkp+kxn−Ttxnkp+kTtxn−Taxnkp+
+kTaxn−Tazkp)≤4p((1 +kp)kxn−zkp+Rn+µtkTatxn−Taznkp)≤ ≤4p(1 +kp)(kxn−zkp+Rn) →0 (n→ ∞).
HenceTaz=z.
InLp (1< p <∞), we have the following inequalities (cf. [5], [6], [12]):
kλx+ (1−λ)ykq
≤λkxkq+ (1
−λ)kykq
−dpWq(λ)kx−ykq
for allx, y ∈Lp andλ∈[0,1], where q= max{2, p},W
q(λ) =λq(1−λ) +
λ(1−λ)q, and
dp=
((1+tp−1
p )
(1+tp)p−1, if 2< p <∞;
p−1, if 1< p≤2.
tp is the unique solution of the equation (p−2)tp−1+ (p−1)tp−2−1 = 0,
t∈(0,1). Thus we have the following consequence of Theorems 4.1 and 4.2.
Corollary 4.1. Let C be a nonempty subset of Lp (1 < p <
∞), G a semitopological semigroup which is left reversible or for which the space RU C(G) has an invariant mean, and F = {Tt : t ∈ G} a uniformly
k-Lipschitzian semigroup onC withk <√pif1< p≤2 ork <1 + (1 +tp−1 p )
(1+tp)1−p]
1
p if2< p <∞. Suppose there exists anx∈Csuch that the orbit
{Ttx : t ∈ G} is bounded. Suppose also there exists a nonempty bounded
closed convex subset of C which possesses Property (P) in the case where Gis left reversible or Property (P∗) in the caseRU C(G)has an invariant mean. Then there exists az∈E such that Tsz=z for alls∈G.
Acknowledgement
The authors would like to thank the anonymous referee for a careful reading of the manuscript and helpful suggestions.
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(Received 15.07.1994)
Authors’ addresses: Kok-Keong Tan
Department of Mathematics, Statistics and Computing Science Dalhousie University
Halifax, Nova Scotia Canada B3H 3J5 Hong-Kun Xu
Institute of Applied Mathematics,
East China University of Science and Technology, Shanghai 200237,
