A SEMITOPOLOGICAL SEMIGROUP OF NON-LIPSCHITZIAN MAPPINGS
WITHOUT CONVEXITY
G. LI AND J. K. KIM Received 2 February 1999
LetGbe a semitopological semigroup,Ca nonempty subset of a real Hilbert spaceH, and = {Tt :t∈G}a representation ofGas asymptotically nonexpansive type map- pings ofCinto itself. LetL(x)= {z∈H:infs∈Gsupt∈GTtsx−z =inft∈GTtx−z}
for eachx∈CandL()=
x∈CL(x). In this paper, we prove that
s∈Gconv{Ttsx: t ∈ G}
L() is nonempty for each x ∈ C if and only if there exists a unique nonexpansive retraction P of C into L() such that P Ts = P for all s ∈G and P (x)∈conv{Tsx:s ∈G}for everyx∈C. Moreover, we prove the ergodic conver- gence theorem for a semitopological semigroup of non-Lipschitzian mappings without convexity.
1. Introduction and preliminaries
Let H be a Hilbert space with norm · and inner product(·,·). LetG be a semi- topological semigroup, that is, a semigroup with a Hausdorff topology such that for eachs∈G the mappings s→s·t ands →t·s ofGinto itself are continuous. Let Cbe a nonempty subset of H and let = {Tt :t ∈G}be a semigroup onC, that is, Tst(x)=TsTt(x)for alls,t∈Gandx∈C. Recall that a semigroupis said to be
(a) nonexpansive ifTtx−Tty ≤ x−yforx,y∈Candt ∈G.
(b) asymptotically nonexpansive [6] if there exists a functionk:G→ [0,∞)with infs∈Gsupt∈Gkts≤1 such thatTtx−Tty ≤ktx−yforx,y∈Candt∈G.
(c) of asymptotically nonexpansive type [6] if for eachxinC, there is a function r(·,x):G → [0,∞)with infs∈Gsupt∈Gr(ts,x)=0 such that Ttx−Tty ≤ x− y+r(t,x)for ally∈Candt∈G.
It is easily seen that (a)⇒(b)⇒(c) and that both the inclusions are proper (cf. [6, page 112]).
Baillon [1] proved the first nonlinear mean ergodic theorem for nonexpansive map- pings in a Hilbert space: letCbe a nonempty closed convex subset of a Hilbert space H andT a nonexpansive mapping ofCinto itself. If the setF (T )of fixed points ofT Copyright © 1999 Hindawi Publishing Corporation
Abstract and Applied Analysis 4:1 (1999) 49–59
1991 Mathematics Subject Classification: 47H09, 47H10, 47H20 URL: http://aaa.hindawi.com/volume-4/S1085337599000056.html
is nonempty, then for eachx∈C, the Cesáro means Sn(x)=1
n
n−1
k=0
Tkx (1.1)
converge weakly asn→ ∞to a point ofF (T ). In this case, puttingy=P xfor each x∈C,P is a nonexpansive retraction ofContoF (T )such thatP T =T P =P and P x∈conv{Tnx:n=0,1,2,...}for each x∈C, where convA is the closure of the convex hull ofA. The analogous results are given for nonexpansive semigroups onCby Baillon [2] and Bre´zis-Browder [3]. In [10], Mizoguchi-Takahashi proved a nonlinear ergodic retraction theorem for Lipschitzian semigroups by using the notion of submean.
Recently, Li and Ma [8, 9] proved the nonlinear ergodic retraction theorems for non- Lipschitzian semigroups in a Banach space without using the notion of submean. Also, in 1992, Takahashi [13] proved the ergodic theorem for nonexpansive semigroups on condition that
s∈Gconv{Tstx:t∈G} ⊂Cfor somex∈C.
In this paper, without using the concept of submean, we prove nonlinear ergodic theorem for semitopological semigroup of non-Lipschitzian mappings without convex- ity in a Hilbert space. We first prove that ifCis a nonempty subset of a Hilbert space H,G a semitopological semigroup, and = {Tt : t ∈G} a representation of G as asymptotically nonexpansive type mappings of C into itself, then
s∈Gconv{Ttsx: t ∈G}
L()is nonempty for eachx ∈Cif and only if there exists a unique non- expansive retractionP of C intoL() such thatP Ts =P for alls ∈G and P x is in the closed convex hull of{Tsx:s∈G}, whereL(x)= {z:infs∈Gsupt∈GTtsx− z =inft∈GTtx−z} and L()=
x∈CL(x). By using this result, we also prove the ergodic convergence theorem for semitopological semigroup of non-Lipschitzian mapping without convexity. Our results are generalizations and improvements of the previously known results of Brézis-Browder [3], Hirano-Takahashi [4], Mizoguchi- Takahashi [10], Takahashi-Zhang [14], and Takahashi [11, 12, 13] in many directions.
Further, it is safe to say that in the results [1, 2, 3, 4, 5, 7, 10, 11, 12, 13, 14], many key conditions are not necessary.
2. Ergodic convergence theorems
Throughout this paper, we assume thatCis a nonempty subset of a real Hilbert space H,Ga semitopological semigroup, and = {Tt :t∈G}an asymptotically nonexpan- sive type semigroup onC.For eachx∈C, defineL(x)andL()by
L(x)=
z:inf
s∈G sup
t∈G
Ttsx−z=inf
t∈GTtx−z
, L()=
x∈C
L(x), (2.1)
respectively. We denoteF ()by the set{x∈C:Ts(x)=xfor alls∈G}of common fixed point of.We begin with the following lemma.
Lemma2.1. LetCbe a nonempty subset of a Hilbert spaceH and = {Tt :t∈G}an asymptotically nonexpansive type semigroup onC.ThenF ()⊂L().
Proof. Letx∈Candf ∈F (). Sinceis asymptotically nonexpansive type, for an arbitraryε >0,there existss0∈Gsuch that for allt∈G
r ts0,f
< ε. (2.2)
Hence, for eacha∈G,
s∈Ginf sup
t∈G
Ttsx−f≤sup
t∈G
Tts0ax−f≤sup
t∈G
Tax−f+r ts0,f
≤Tax−f+ε. (2.3)
Sinceε >0 is arbitrary, we have infs∈Gsupt∈GTtsx−f ≤inft∈GTtx−f.There-
fore,f ∈L(x). This completes the proof.
Remark 2.2. It is not easy to prove that F ()is nonempty whenC is not a convex subset. However, we can show thatL()is nonempty under some conditions and it is important for the ergodic convergence theorem.
The following proposition plays a crucial role in the proof of our main theorems in this paper.
Proposition2.3. LetG be a semitopological semigroup,C a nonempty subset of a Hilbert spaceH, and = {Tt :t∈G}an asymptotically nonexpansive type semigroup onC. Then, for everyx∈C,the set
s∈G
conv Ttsx:t∈G
L(x), (2.4)
consists of at most one point.
Proof. Let u,v∈
s∈Gconv{Ttsx:t ∈G}
L(x), without loss of generality, we as- sume that
tinf∈GTtx−u2≤inf
t∈GTtx−v2. (2.5)
Now, for eacht,s∈G, since u−v2+2
Ttsx−u,u−v
=Ttsx−v2−Ttsx−u2, (2.6) we have
u−v2+2 inf
t∈G
Ttsx−u,u−v
≥inf
t∈GTtsx−v2−sup
t∈G
Ttsx−u2
≥inf
t∈GTtx−v2−sup
t∈G
Ttsx−u2. (2.7) Fromu∈L(x), we have
u−v2+2 sup
s∈Ginf
t∈G
Ttsx−u,u−v
≥inf
t∈GTtx−v2−inf
s∈Gsup
t∈G
Ttsx−u2
=inf
t∈GTtx−v2−inf
t∈GTtx−u2≥0. (2.8)
Therefore, forε >0 there is ans1∈Gsuch that u−v2+2
Tts1x−u,u−v
>−ε ∀t∈G. (2.9)
Fromv∈conv{Tts1x:t∈G}, we have
u−v2+2(v−u,u−v)≥ −ε. (2.10) This inequality implies thatu−v2≤ε. Sinceε >0 is arbitrary, we haveu=v.This
completes the proof.
Remark 2.4. In the Takahashi-Zhang’s result [14], it is assumed thatCis a closed convex subset,G a reversible semigroup, and an asymptotically nonexpansive semigroup.
Proposition 2.3 shows those key conditions are not necessary.
Let m(G) be the Banach space of all bounded real-valued functions on a semi- topological semigroupG with the supremum norm and letXbe a subspace ofm(G) containing constants. Then, an elementµofX∗(the dual space ofX) is called a mean onXifµ =µ(1)=1. Letµbe a mean onXandf ∈X.Then, according to time and circumstances, we useµt(f (t)) instead ofµ(f ).For eachs∈Gandf ∈m(G), we define elementslsf andrsf inm(G)given by(lsf )(t)=f (st)and(rsf )(t)=f (ts) for allt∈G, respectively.
Throughout the rest of this section, letXbe a subspace ofm(G)containing constants invariant underlsandrsfor eachs∈G. Furthermore, suppose that for eachx∈Cand y∈H,a functionf (t)= Ttx−y2is inX. Forµ∈X∗, we define the valueµt(Ttx,y) ofµat this function. By Riesz theorem, there exists a unique elementµxinXsuch that
µt Ttx,y
= µx,y
∀y∈H. (2.11)
Lemma2.5. Suppose thatXhas an invariant meanµ. Then we have
s∈G
conv Ttsx:t∈G
L(x)= µx
for everyx∈C. (2.12) Further, ifTt is continuous for eacht∈Gand
s∈Gconv{Tstx:t∈G} ⊂Cfor some x∈C, thenµx∈F ().
Proof. Sinceµis an invariant mean, it is easy to show thatµx∈
s∈Gconv{Ttsx: t ∈G}for eachx∈C. By Proposition 2.3, it is enough to prove thatµx∈L(x)for each x∈C. To this end, let ε >0, since is an asymptotically nonexpansive type semigroup, for eacht∈Gthere is anht ∈Gsuch that for eachh∈G,
r
hht,Ttx
< ε. (2.13)
PutM=supt,s∈GTtx−Tsx, then we have
Thhttx−µx2−Ttx−µx2=µsThhttx−Tsx2−Ttx−Tsx2
=µsThhttx−Thhtsx2−Ttx−Tsx2
≤2Mε for eachh∈G.
(2.14)
Hence, we have
s∈Ginf sup
h∈G
Thsx−µx2≤Ttx−µx2+2Mε ∀t∈G. (2.15) Sinceε >0 is arbitrary, we haveµx∈L(x). Finally, suppose that
s∈Gconv{Tstx: t ∈G} ⊂Cand each Tt is continuous fromC into itself. Then, we can easily prove thatµx∈
s∈Gconv{Tstx:t ∈G}and hence we haveµx∈C.For eachh∈Gand ε∈(0,1),there exists 0< δ < εsuch thatThy−Thµx< ε whenevery∈Cand y− µx ≤δ. Since is an asymptotically nonexpansive type semigroup, there is s0∈Gsuch that
r
ts0,µx
< 1 2
M1+1δ2 ∀t∈G, (2.16)
whereM1=supt∈GTtx−µx. Then for eacht,s∈G, we have Tss0µx−µx2+2
Ttx−µx,µx−Tss0µx
=Ttx−Tss0µx2−Ttx−µx2
=Tss0tx−Tss0µx2−Ttx−µx2−Tss0tx−Tss0µx2+Ttx−Tss0µx2
≤δ2−Tss0tx−Tss0µx2+Ttx−Tss0µx2.
(2.17) It follows that
Tss0µx−µx≤δ ∀s∈G. (2.18) This implies that
Thµx−µx≤Thµx−ThTss0µx+Thss0µx−µx<2ε. (2.19)
Sinceε >0 is arbitrary, we haveThµx= µx.This completes the proof.
Now, we prove a nonlinear ergodic theorem for asymptotically nonexpansive type semigroups without convexity. Before doing this, we give a definition concerning means. Let {µα :α ∈A} be a net of means on X,where A is a directed set. Then {µα:α∈A}is said to be asymptotically invariant if for eachf ∈Xands∈G,
µα(f )−µα lsf
−→0, µα(f )−µα rsf
−→0. (2.20)
Theorem 2.6. Let C be a nonempty subset of a Hilbert space H, X an invariant subspace ofm(G)containing constants, and = {Tt :t ∈G}an asymptotically non- expansive type semigroup on C.If for eachx ∈C andy∈H, the functionf on G defined by f (t)= Ttx−y2 belong to X,then for an asymptotically invariant net {µα :α ∈A} on X, the net {µαx}α∈A converges weakly to an element x0∈L(x).
Further, ifTt is continuous for each t ∈G and
s∈Gconv{Tstx:t ∈G} ⊂C, then x0∈F ().
Proof. LetW be the set of all weak limit points of subnet of the net {µαx:α∈A}. By Proposition 2.3, it is enough to prove that
W ⊂
s∈G
conv Ttsx:t ∈G
L(x). (2.21)
To show this, let z∈W and let {uαβx} be a subnet of{µαx} such that {µαβx}
converges weakly toz. Now, without loss of generality, we can suppose that{µαβx}
converges weakly* toµ∈X∗. It is easily seen thatµis an invariant mean onXand then Lemma 2.5 implies thatz= µx∈
s∈Gconv{Ttsx:t ∈G}
L(x). This completes
the proof.
LetC(G)be the Banach space of all bounded continuous real-valued functions on Gand letRUC(G)be the space of all bounded right uniformly continuous functions on G,that is, allf ∈C(G)such that the mappings→rsf is continuous. ThenRUC(G) is a closed subalgebra ofC(G)containing constants and invariant underlsandrs.
As a direct consequence of Theorem 2.6, we obtain the following corollary.
Corollafry2.7 (see [13]). LetC be a nonempty subset of a Hilbert space H and letG be a semitopological semigroup such that RUC(G)has an invariant mean. Let = {Tt :t ∈G}be a nonexpansive semigroup onCsuch that{Ttx:t ∈G}is bounded and
s∈Gconv{Tstx : t ∈G} ⊂ C for some x ∈C. Then, F ()= ∅. Further, for an asymptotically invariant net {µα}α∈A of means on RUC(G), the net {µα}α∈A, converges weakly to an elementx0∈F ().
Remark 2.8. For the proof of Corollary 2.7, Takahashi [13] used the condition
s∈G
conv{Tstx:t ∈G} ⊂C. But, from Theorem 2.6, we can prove the result without this condition except proving the fact that the weak limit of{µαx}is inF ().
3. Nonexpansive retractions
In this section, we prove an ergodic retraction theorem for a semitopological semigroup of asymptotically nonexpansive type mappings without convexity.
Theorem3.1. LetCbe a nonempty subset of a Hilbert spaceHand let = {Tt :t∈G}
be a semitopological semigroup of asymptotically nonexpansive type mappings onC such thatL()= ∅. Then the following statements are equivalent:
(a)
s∈Gconv{Ttsx:t∈G}
L()= ∅for eachx∈C.
(b)There is a unique nonexpansive retractionP ofCintoL()such thatP Tt=P for everyt∈GandP x∈conv{Ttx:t∈G}for everyx∈C.
Proof. (b)⇒(a). Letx∈C, thenP x∈L().AlsoP x∈
s∈Gconv{Ttsx:t∈G}.In fact, for eachs∈G, P x=P Tsx∈conv{TtTsx:t∈G} =conv{Ttsx:t∈G}.
(a)⇒(b). Let x ∈C. Then by Proposition 2.3,
s∈Gconv{Ttsx :t ∈G}
L() contains exactly one pointP x. For eacha∈G, we have
{P Tax} =
s∈G
conv Ttsax:t∈G L()
⊇
s∈G
conv Ttsx:t∈G
L()= {P x} (3.1) and hence we haveP Ta=P for everya∈G.
Finally, we have to show thatP is nonexpansive. Letx,y∈Cand 0< λ <1. Then for anyε >0, there existss1∈Gsuch that
sup
t∈G
Tts1x−Py≤inf
t∈GTtx−Py+ε, (3.2) fromPy∈L(). Hence, we have
λTtss1x+(1−λ)P x−Py2
=λ
Ttss1x−Py
+(1−λ)(P x−Py)2
=λTtss1x−Py2+(1−λ)P x−Py2−λ(1−λ)Ttss1x−P x2
≤λTabx−Py+ε2
+(1−λ)P x−Py2−λ(1−λ)inf
t∈GTtx−P x2, (3.3) for eacht,s,a,b∈G. Sinceε >0 is arbitrary, this implies
s∈Ginfsup
t∈G
λTtsx+(1−λ)P x−Py2
≤λTabx−Py2+(1−λ)P x−Py2−λ(1−λ)inf
t∈GTtx−P x2
=λTabx+(1−λ)P x−Py2+λ(1−λ)Tabx−P x2−λ(1−λ)inf
t∈GTtx−P x2. (3.4) Then it is easily seen that
s∈Ginfsup
t∈G
λTtsx+(1−λ)P x−Py2−λ(1−λ)inf
b∈Gsup
a∈G
Tabx−P x2
≤sup
b∈Ginf
a∈G
λTabx+(1−λ)P x−Py2−λ(1−λ)inf
t∈GTtx−P x2. (3.5) SinceP x∈L(), we have
s∈Ginfsup
t∈G
λTtsx+(1−λ)P x−Py2≤sup
s∈Ginf
t∈GλTtsx+(1−λ)P x−Py2. (3.6) Let
h(λ)= inf
s∈Gsup
t∈G
λTtsx+(1−λ)P x−Py2. (3.7)
Then for anyε >0, there existss2∈Gsuch that for allt∈G,
λTts2x+(1−λ)P x−Py2≤h(λ)+ε (3.8) and hence
λTts2x+(1−λ)P x−Py,P x−Py
≤
h(λ)+ε1/2
P x−Py ∀t∈G. (3.9) FromP x∈conv{Tts2x:t∈G}, we have
λP x+(1−λ)P x−Py,P x−Py
≤
h(λ)+ε1/2
P x−Py. (3.10)
Sinceε >0 is arbitrary, this yields that
P x−Py2≤h(λ). (3.11)
That is,
P x−Py2≤inf
s∈Gsup
t∈G
λTtsx+(1−λ)P x−Py2. (3.12) Now, one can choose ans3 ∈G such thatTts3x−P x ≤ M for all t ∈G, where M=1+inft∈GTtx−P x. Then, we have
λTtss3x+(1−λ)P x−Py2
=λ
Ttss3x−P x
+(P x−Py)2
=λ2Ttss3x−P x2+P x−Py2+2λ
Ttss3x−P x,P x−Py
≤M2λ2+P x−Py2+2λ
Ttss3x−P x,P x−Py .
(3.13)
It then follows from (3.6) and (3.12) that 2λsup
s∈Ginf
t∈G
Ttsx−P x,P x−Py
≥2λsup
s∈Ginf
t∈G
Ttss3x−P x,P x−Py
≥sup
s∈Ginf
t∈G
λTtss3x+(1−λ)P x−Py2−P x−Py2−M2λ2
=sup
s∈Ginf
t∈G
λTtsTs3x+(1−λ)P Ts3x−Py2−P x−Py2−M2λ2
≥P Ts3x−Py2−P x−Py2−M2λ2
= −M2λ2.
(3.14)
Hence, we have
s∈Gsupinf
t∈G
Ttsx−P x,P x−Py
≥ −1
2M2λ. (3.15)
Lettingλ→0,then we have sup
s∈Ginf
t∈G
Ttsx−P x,P x−Py
≥0. (3.16)
Letε >0,then there iss4∈Gsuch that r
ts4,x
< ε ∀t∈G. (3.17)
For such ans4∈G, from (3.16), we have
s∈Gsupinf
t∈G
TtsTs4x−P Ts4x,P Ts4x−Py
≥0 (3.18)
and hence there iss5∈Gsuch that
t∈Ginf
Tts5Ts4x−P Ts4x,P Ts4x−Py
>−ε. (3.19)
Then, fromP Ts4x=P x,we have
tinf∈G
Tts5s4x−P x,P x−Py
>−ε. (3.20)
Similarly, from (3.16), we also have sups∈Ginf
t∈G
TtsTs5s4y−P Ts5s4y,P Ts5s4y−P x
≥0, (3.21)
and there existss6∈Gsuch that
t∈Ginf
Tts6s5s4y−P Ts5s4y,P Ts5s4y−P x
≥ −ε, (3.22) that is,
t∈Ginf
Py−Tts6s5s4y,P x−Py
≥ −ε. (3.23) On the other hand, from (3.20)
tinf∈G
Tts6s5s4x−P x,P x−Py
>−ε. (3.24)
Combining (3.23) and (3.24), we have
−2ε <
Tts6s5s4x−Tts6s5s4y,P x−Py
−P x−Py2
≤Tts6s5s4x−Tts6s5s4y·P x−Py−P x−Py2
≤ r
ts6s5s4,x)+x−y
·P x−Py−P x−Py2
≤
ε+x−y
·P x−Py−P x−Py2.
(3.25)
Sinceε >0 is arbitrary, this impliesP x−Py ≤ x−y. The proof is completed.
Using Lemma 2.1, we have the following ergodic retraction theorem for asymptoti- cally nonexpansive type semigroups.
Theorem 3.2. Let C be a nonempty subset of a real Hilbert space H and let = {Tt:t∈G}be a semitopological semigroup of asymptotically nonexpansive type map- pings onCsuch thatF ()= ∅. Then the following statements are equivalent:
(a)
s∈Gconv{Ttsx:t∈G}
F ()= ∅for eachx∈C.
(b)There is a unique nonexpansive retractionP ofC ontoF ()such thatP Tt = TtP =P for everyt∈GandP x∈conv{Ttx:t∈G}for everyx∈C.
We denote byB(G)the Banach space of all bounded real-valued functions onGwith supremum norm. LetXbe a subspace ofB(G)containing constants. Then, according to Mizoguchi-Takahashi [10], a real-valued functionµonXis called a submean onX if the following conditions are satisfied:
(1)µ(f+g)≤µ(f )+µ(g)for everyf,g∈X; (2)µ(αf )=αµ(f )for everyf ∈Xandα≥0;
(3) forf,g∈X, f ≤gimpliesµ(f )≤µ(g); (4)µ(c)=cfor every constantc.
The following corollaries are immediately deduced from Theorem 3.2.
Corollafry3.3 (see [10]). LetC be a closed convex subset of a Hilbert space H and let X be an rs-invariant subspace of B(G) containing constants which has a right invariant submean. Let = {Tt :t ∈G}be a Lipschitzian semigroup onCwith infssuptkts2 ≤1 and F ()= ∅, where kt is the Lipschitzian constants. If for each x,y∈C, the functionf onGdefined by
f (t)=Ttx−y2 ∀t∈G (3.26) and the functiongonGdefined by
g(t)=k2t ∀t∈G (3.27)
belong toX, then the following statements are equivalent:
(a)
s∈Gconv{Ttsx:t∈G}
F ()= ∅for eachx∈C.
(b)There is a nonexpansive retractionP ofContoF ()such thatP Tt=TtP=P for everyt∈GandP x∈conv{Ttx:t∈G}for everyx∈C.
Corollafry3.4 (see [7]). LetC be a nonempty closed convex subset of a Hilbert spaceH and let = {Tt :t∈G}be a continuous representation of a semitopological semigroup as nonexpansive mappings from C into itself. If for each x∈C, the set s∈Gconv{Ttsx:t∈G}
F ()= ∅,then there exists a nonexpansive retraction P of C ontoF ()such thatP Tt =TtP =P for everyt ∈GandP x∈conv{Ttx:t ∈G}
for everyx∈C.
Remark 3.5. By Theorem 3.2, many key conditions, in Corollaries 3.3 and 3.4, such as C is convex closed subset and is continuous Lipschitzian semigroup, are not necessary.
Acknowledgement
The authors wish to acknowledge the financial support of the Korea Research Founda- tion made in the program year of 1998.
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G. Li: Department of Mathematics, Yangzhou University, Yangzhou225002, China E-mail address: [email protected]
J. K. Kim: Department of Mathematics, Kyungnam University, Masan, Kyungnam631- 701, Korea
E-mail address: [email protected]
