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]
Mathematical Problems in Engineering
Special Issue on
Modeling Experimental Nonlinear Dynamics and Chaotic Scenarios
Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system. Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision. In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from “Qualitative Theory of Differential Equations,”
allowing more precise analysis and synthesis, in order to produce new vital products and services. Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Prob- lems in Engineering aims to provide a picture of the impor- tance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.
Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophis- ticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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 December 1, 2008 First Round of Reviews March 1, 2009 Publication Date June 1, 2009
Guest Editors
José Roberto Castilho Piqueira,Telecommunication and Control Engineering Department, Polytechnic School, The University of São Paulo, 05508-970 São Paulo, Brazil;
Elbert E. Neher Macau,Laboratório Associado de Matemática Aplicada e Computação (LAC), Instituto Nacional de Pesquisas Espaciais (INPE), São Josè dos Campos, 12227-010 São Paulo, Brazil ; [email protected] Celso Grebogi,Center for Applied Dynamics Research, King’s College, University of Aberdeen, Aberdeen AB24 3UE, UK; [email protected]
Hindawi Publishing Corporation http://www.hindawi.com
