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.
References
[1] J.-B. Baillon,Un théorème de type ergodique pour les contractions non linéaires dans un es- pace de Hilbert, C. R. Acad. Sci. Paris Sér. A-B280(1975), no. 22, 1511–1514 (French).
MR 51#11205. Zbl 307.47006.
[2] ,Quelques propriétés de convergence asymptotique pour les semi-groupes de con- tractions impaires, C. R. Acad. Sci. Paris Sér. A-B283(1976), no. 3, 75–78 (French).
MR 54#13655. Zbl 339.47028.
[3] H. Brézis and F. E. Browder,Remarks on nonlinear ergodic theory, Advances in Math.25 (1977), no. 2, 165–177. MR 57#1218. Zbl 399.47058.
[4] N. Hirano and W. Takahashi, Nonlinear ergodic theorems for nonexpansive mappings in Hilbert spaces, Kodai Math. J. 2 (1979), no. 1, 11–25 (English). MR 80j:47064.
Zbl 404.47031.
[5] H. Ishihara and W. Takahashi,A nonlinear ergodic theorem for a reversible semigroup of Lipschitzian mappings in a Hilbert space, Proc. Amer. Math. Soc. 104(1988), no. 2, 431–436 (English). MR 90g:47120. Zbl 692.47010.
[6] W. A. Kirk and R. Torrejón,Asymptotically nonexpansive semigroups in Banach spaces, Nonlinear Anal.3(1979), no. 1, 111–121 (English). MR 82a:47062. Zbl 411.47035.
[7] A. T. M. Lau, K. Nishiura, and W. Takahashi,Nonlinear ergodic theorems for semigroups of nonexpansive mappings and left ideals, Nonlinear Anal.26(1996), no. 8, 1411–1427 (English). MR 97b:47074. Zbl 880.47048.
[8] G. Li, Weak convergence and non-linear ergodic theorems for reversible semigroups of non-Lipschitzian mappings, J. Math. Anal. Appl.206(1997), no. 2, 451–464 (English).
MR 98k:47139. Zbl 888.47046.
[9] G. Li and J. Ma, Nonlinear ergodic theorem for semitopological semigroups of non- Lipschitzian mappings in Banach spaces, Chinese Sci. Bull. 42 (1997), no. 1, 8–11 (English). MR 98e:47110. Zbl 904.47063.
[10] N. Mizoguchi and W. Takahashi,On the existence of fixed points and ergodic retractions for Lipschitzian semigroups in Hilbert spaces, Nonlinear Anal.14(1990), no. 1, 69–80 (English). MR 91h:47071. Zbl 695.47063.
[11] W. Takahashi,A nonlinear ergodic theorem for an amenable semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc.81(1981), no. 2, 253–256 (English).
MR 82f:47079. Zbl 456.47054.
[12] ,A nonlinear ergodic theorem for a reversible semigroup of nonexpansive mappings in a Hilbert space, Proc. Amer. Math. Soc.97(1986), no. 1, 55–58. MR 88f:47051.
[13] ,Fixed point theorem and nonlinear ergodic theorem for nonexpansive semigroups without convexity, Canad. J. Math.44(1992), no. 4, 880–887 (English). MR 93j:47091.
Zbl 786.47047.
[14] W. Takahashi and P.-J. Zhang,Asymptotic behavior of almost-orbits of reversible semigroups of Lipschitzian mappings, J. Math. Anal. Appl.142(1989), no. 1, 242–249 (English).
MR 90g:47121. Zbl 695.47062.
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]
Mathematical Problems in Engineering
Special Issue on
Time-Dependent Billiards
Call for Papers
This subject has been extensively studied in the past years for one-, two-, and three-dimensional space. Additionally, such dynamical systems can exhibit a very important and still unexplained phenomenon, called as the Fermi acceleration phenomenon. Basically, the phenomenon of Fermi accelera- tion (FA) is a process in which a classical particle can acquire unbounded energy from collisions with a heavy moving wall.
This phenomenon was originally proposed by Enrico Fermi in 1949 as a possible explanation of the origin of the large energies of the cosmic particles. His original model was then modified and considered under different approaches and using many versions. Moreover, applications of FA have been of a large broad interest in many different fields of science including plasma physics, astrophysics, atomic physics, optics, and time-dependent billiard problems and they are useful for controlling chaos in Engineering and dynamical systems exhibiting chaos (both conservative and dissipative chaos).
We intend to publish in this special issue papers reporting research on time-dependent billiards. The topic includes both conservative and dissipative dynamics. Papers dis- cussing dynamical properties, statistical and mathematical results, stability investigation of the phase space structure, the phenomenon of Fermi acceleration, conditions for having suppression of Fermi acceleration, and computational and numerical methods for exploring these structures and applications are welcome.
To be acceptable for publication in the special issue of Mathematical Problems in Engineering, papers must make significant, original, and correct contributions to one or more of the topics above mentioned. Mathematical papers regarding the topics above are also welcome.
Authors should follow the Mathematical Problems in Engineering manuscript format described at http://www .hindawi.com/journals/mpe/. Prospective authors should submit an electronic copy of their complete manuscript through the journal Manuscript Tracking System athttp://
mts.hindawi.com/according to the following timetable:
Manuscript Due March 1, 2009 First Round of Reviews June 1, 2009 Publication Date September 1, 2009
Guest Editors
Edson Denis Leonel,Department of Statistics, Applied Mathematics and Computing, Institute of Geosciences and Exact Sciences, State University of São Paulo at Rio Claro, Avenida 24A, 1515 Bela Vista, 13506-700 Rio Claro, SP, Brazil; [email protected]
Alexander Loskutov,Physics Faculty, Moscow State University, Vorob’evy Gory, Moscow 119992, Russia;
Hindawi Publishing Corporation http://www.hindawi.com
