Applications of Fixed Point Theorems in the Theory of Generalized IFS

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Volume 2008, Article ID 312876,11pages doi:10.1155/2008/312876

Research Article

Applications of Fixed Point Theorems in the Theory of Generalized IFS

Alexandru Mihail and Radu Miculescu

Department of Mathematics, Bucharest University, Bucharest, Academiei Street 14, 010014 Bucharest, Romania

Correspondence should be addressed to Radu Miculescu,[email protected] Received 9 February 2008; Accepted 22 May 2008

Recommended by Hichem Ben-El-Mechaiekh

We introduce the notion of a generalized iterated function systemGIFS, which is a finite family of functionsfk:X^m→X, whereX, dis a metric space andm∈N. In case thatX, dis a compact metric space and the functionsfkare contractions, using some fixed point theorems for contractions fromX^mtoX, we prove the existence of the attractor of such a GIFS and its continuous dependence in thefk’s.

Copyrightq2008 A. Mihail and R. Miculescu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

We start with a short presentation of the notion of an iterated function systemIFS, one of the most common and most general ways to generate fractals. This will serve as a framework for our generalization of an iterated function system.

Then, we introduce the notion of a GIFS, which is a finite family of functionsfk:X^m→X, whereX, dis a metric space andm∈N. In case thatX, dis a compact metric space and the functionsf_kare contractions, using some fixed point theorems for contractions fromX^mtoX, we prove the existence of the attractor of such a GIFS and its continuous dependence in the fk’s.

IFSs were introduced in their present form by Hutchinsonsee 1 and popularized by Barnsleysee2. In the last period, IFSs have attracted much attention being used from researchers who work on autoregressive time series, engineer sciences, physics, and so forth.

For applications of IFSs in image processing theory, in the theory of stochastic growth models, and in the theory of random dynamical systems, one can consult3–5.

There is a current eﬀort to extend Hutchinson’s classical framework for fractals to more general spaces and infinite IFSs.

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Let us mention some papers containing results on this direction.

Results concerning infinite iterated function systems have been obtained for the case when the attractor is compactsee, e.g.,6where the case of a countable iterated function system on a compact metric space is considered. In 7, we provide a general framework where attractors are nonempty closed and bounded subsets of topologically complete metric spaces and where the IFSs may be infinite, in contrast with the classical theorysee2, where only attractors that are compact metric spaces and IFSs that are finite are considered.

Gw ´o´zd´z-Łukawska and Jachymski 8 discuss the Hutchinson-Barnsley theory for infinite iterated function systems.

Łozi ´nski et al.9introduce the notion of quantum iterated function systemsQIFSs which is designed to describe certain problems of nonunitary quantum dynamics.

K¨aenm¨aki 10 constructs a thermodynamical formalism for very general iterated function systems.

Le´sniak11presents a multivalued approach of infinite iterated function systems.

2. Preliminaries

Notations. LetX, dXandY, dYbe two metric spaces.

As usual,CX, Ydenotes the set of continuous functions fromXtoY, andd:CX, Y× CX, Y→RR∪ {∞}, defined by

df, g sup

x∈Xd_Y

fx, gx

, 2.1

is the generalized metric onCX, Y.

For a sequence fn_n of elements of CX, Y and f ∈ CX, Y, fn →−s f denotes the pointwise convergence,fn−−→u·c fdenotes the uniform convergence on compact sets, andfn−→u f denotes the uniform convergence, that is, the convergence in the generalized metricd.

Definition 2.1. LetX, dbe a complete metric space and letm ∈ N. For a functionf : X^m

×^m_k1X→X, the number inf

c:d f

x1, . . . , xm

, f

y1, . . . , ym

≤cmax d

x1, y1, . . . , d

xm, ym

, ∀x1, . . . , xm, y1, . . . , ym∈X 2.2 which is the same as

sup d

f

x₁, . . . , x_m , f

y₁, . . . , y_m : max

d x₁, y₁

, . . . , d

x_m, y_m

, 2.3

where the sup is taken overx1, . . . , xm, y1, . . . , ym∈Xsuch that max

d x1, y1

, . . . , d xm, ym

>0, 2.4 is denoted byLipfand is called the Lipschitz constant off.

A function f : X^m→X is called a Lipschitz function ifLipf < ∞and a Lipschitz contraction ifLipf<1.

A functionf:X^m→Xis said to be a contraction if d

f

x₁, . . . , x_m , f

y₁, . . . , y_m

<max d

x₁, y₁ , . . . , d

x_m, y_m

, 2.5

for everyx1, x2, . . . , xm, y1, y2, . . . , ym∈X, such thatxi/yifor somei∈ {1,2, . . . , n}.

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LConmXdenotes the set

f:X^m −→ X:Lipf<1

2.6 and Con_mXdenotes the set

f:X^m −→ X:f is a contraction

. 2.7

Remark 2.2. It is obvious that

LCon_mX⊆Con_mX. 2.8

Notations.PXdenotes the family of all subsets of a given setXandP^∗Xdenotes the set PX\ {∅}.

For a subsetAofPX, byA^∗we meanA\ {∅}.

Given a metric space X, d, KXdenotes the set of compact subsets ofX andBX denotes the set of closed bounded subsets ofX.

Remark 2.3. It is obvious that

KX⊆ BX⊆ PX. 2.9

Definition 2.4. For a metric spaceX, d, one considers on P^∗Xthe generalized Hausdorﬀ- Pompeiu pseudometrich:P^∗X× P^∗X→0,∞defined by

hA, B max

dA, B, dB, A inf

r∈0,∞:A⊆BB, r, B⊆BA, r

, 2.10

where

BA, r

x∈X:dx, A< r , dA, B sup

x∈Adx, B sup

x∈A

y∈Binfdx, y

. 2.11

Remark 2.5. The Hausdorﬀ-Pompeiu pseudometric is a metric onB^∗Xand, in particular, on K^∗X.

Remark 2.6. The metric spacesB^∗X, handK^∗X, hare complete, provided thatX, dis a complete metric spacesee2,7,12. Moreover,K^∗X, his compact, provided thatX, d is a compact metric spacesee2.

The following proposition gives the important properties of the Hausdorﬀ-Pompeiu pseudometricsee2,13.

Proposition 2.7. LetX, dXandY, dYbe two metric spaces. Then iifHandKare two nonempty subsets ofX, then

hH, K h

H, K

; 2.12

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iiifHi_i∈I andKi_i∈I are two families of nonempty subsets ofX, then

h

i∈I

Hi,

i∈I

Ki

≤sup

i∈I h Hi, Ki

; 2.13

iiiifHandKare two nonempty subsets ofXandf:X→Xis a Lipschitz function, then h

fK, fH

≤LipfhK, H. 2.14

Definition 2.8. Let X, d be a complete metric space and let m ∈ N. A generalized iterated function systemin short a GIFSonXof orderm, denoted byS X,fk_k1,n, consists of a finite family of functionsfk_k1,n, f_k:X^m→Xsuch thatf₁, . . . , f_n∈Con_mX.

Definition 2.9. Letf :X^m→Xbe a continuous function. The functionF_f : K^∗X^m→ K^∗X defined by

F_f

K₁, K₂, . . . , K_m f

K₁×K₂× · · · ×K_m

f

x₁, x₂, . . . , x_m

:x_j ∈K_j, ∀j∈ {1, . . . , m} 2.15 is called the set function associated to the functionf.

Definition 2.10. GivenS X,fk_k1,na generalized iterated function system onX of order m, the functionF_S:K^∗X^m→ K^∗Xdefined by

F_S

K₁, K₂, . . . , K_m ⁿ

k1

F_f_k

K₁, K₂, . . . , K_m

2.16

is called the set function associated toS.

Lemma 2.11. For a sequencefn_nof elements ofCX^m, Xandf ∈CX^m, Xsuch thatfn→u fand forK1, K2, . . . , Km∈ K^∗X, one has

f_n

K₁×K₂× · · · ×K_m

−→f

K₁×K₂× · · · ×K_m

2.17

inK^∗X, h.

Proof. Indeed, the conclusion follows from the below inequality:

h fn

K1× · · · ×Km , f

K1× · · · ×Km

≤ sup

x1∈K1,...,xm∈Km

d f_n

x₁, . . . , x_m , f

x₁, . . . , x_m

, 2.18

which is valid for alln∈N.

Proposition 2.12. LetX, dXandY, dYbe two metric spaces and letf_n, f ∈CX, Ybe such that sup_n≥1Lipfn<∞andfn s

−

→fon a dense set inX.

Then

Lipf≤sup

n≥1 Lip fn

, fn u.c

−−−→f. 2.19

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Proof. SetM:sup_n≥1Lipfn.

Let us considerA{x∈X|fmx→fx}, which is a dense set inX, letKbe a compact set inX, and letε >0.

Sincefis uniformly continuous onK, there existsδ∈0, ε/3M1such that ifx, y∈K anddXx, y< δ, then

dY

fx, fy

< ε

3. 2.20

SinceKis compact, there existx₁, x₂, . . . , x_n∈Ksuch that K⊆ ⁿ

i1

B

xi,δ 2

. 2.21

Taking into account the fact thatAis dense inX, we can choosey1, y2, . . . , yn ∈Asuch thaty1∈Bx1, δ/2, . . . , yn∈Bxn, δ/2.

Since, for alli ∈ {1, . . . , n}, lim_m_{→ ∞}f_myi fyi, there exists m_ε ∈ Nsuch that for everym∈N, m≥m_ε,we have

dY

fm

yi

, f yi

< ε

3, 2.22

for everyi∈ {1, . . . , n}.

Forx∈K, there existsi∈ {1, . . . , n}, such thatx∈Bxi, δ/2and therefore dX

x, yi

≤dX x, xi

dX xi, yi

< δ 2δ

2 < δ, 2.23

so

d_Y f

y_i , fx

< ε

3. 2.24

Hence, form≥mε, we have d_Y

f_mx, fx

≤d_Y

f_mx, fm

y_i d_Y

f_m y_i

, f y_i

d_Y f

y_i , fx

≤MdX

x, yi

ε 3ε

3

≤M ε

3M12ε 3 < ε.

2.25

Consequently, asxwas arbitrarily chosen inK, we infer thatf_n→^u fonK, so

f_n−−→^u·c f. 2.26

The inequalityLipf≤sup_n≥1Lipfnis obvious.

FromLemma 2.11andProposition 2.12, usingProposition 2.7iiwe obtain the following lemma.

Lemma 2.13. LetX, dXbe a complete metric space, letm∈N, letSj X,f_k^j_k1,n, wherej∈N^∗, and let S X,fk_k1,nbe generalized iterated function systems of orderm, such that, for allk ∈ {1, . . . , n}, f_k^j→−^s f_kon a dense subset ofX^m.

Then, for everyK₁, K₂, . . . , K_m∈ K^∗X, F_S_j

K₁, K₂, . . . , K_m

−→F_S

K₁, K₂, . . . , K_m

, 2.27

inK^∗X, h.

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3. The existence of the attractor of a GIFs for contractions

In this section,mis a natural number,X, dis a compact metric space, andS X,fk_k1,n is a generalized iterated function system onXof orderm.

First, we prove thatFS :K^∗X^m→ K^∗Xis a contractionProposition 3.1, then, using some results concerning the fixed points of contractions fromX^mtoXTheorem 3.4, we prove the existence of the attractor of S Theorem 3.5and its continuous dependence in the f_k’s Theorem 3.7.

The following proposition is crucial.

Proposition 3.1. FS:K^∗X^m→ K^∗Xis a contraction.

Proof. ByProposition 2.7, we have h

FS

K₁, K₂, . . . , K_m , FS

H₁, H₂, . . . , H_m h

_n

k1

f_k

K₁×K₂× · · · ×K_m ,

n k1

f_k

H₁×H₂× · · · ×H_m

h _n

k1

Ffk

K1, K2, . . . , Km

,

n k1

Ffk

H1, H2, . . . , Hm

≤max h

f1

K1× · · · ×Km

, f1

H1× · · · ×Hm

, . . . , h fn

K1× · · · ×Km

,

f_n

H₁× · · · ×H_m

≤max h

H1, K1

, . . . , h Hm, Km

,

3.1

for allK1, . . . , Km, H1, . . . , Hm∈ K^∗X.

It remains to prove that the above inequality is strict.

Let K₁, K₂, . . . , K_m, H₁, H₂, . . . , H_m ∈ K^∗X be fixed such that K_i/H_i for some i ∈ {1,2, . . . , m}.

Since h

FS

K1, . . . , Km , FS

H1, . . . , Hm max

d FS

K₁, . . . , K_m , FS

H₁, . . . , H_m , d

FS

H₁, . . . , H_m , FS

K₁, . . . , K_m , 3.2 we can suppose, by using symmetry arguments, that

h FS

K1, . . . , Km

, FS

H1, . . . , Hm

d FS

K1, . . . , Km

, FS

H1, . . . , Hm

, 3.3

that is,

h _n

k1

f_k

K₁× · · · ×K_m ,

n k1

f_k

H₁× · · · ×H_m

d _n

k1

fk

K1× · · · ×Km

,

n k1

fk

H1× · · · ×Hm

.

3.4

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Let us note that for every K1, K2, . . . , Km ∈ K^∗X, since f1, . . . , fn are continuous functions,FSK1, K2, . . . , Km _n

k1fjK1, K2, . . . , Kmis a compact set.

Since for all K₁, K₂, . . . , K_m, H₁, H₂, . . . , H_m ∈ K^∗X, the product topological space {1,2, . . . , n} ××^m_j1K_j, where{1,2, . . . , n}is endowed with the discrete topology, is compact and the functiont:{1,2, . . . , n} ××^m_j1Kj→R, given by

t

k, x1, x2, . . . , xm

d fk

x1, x2, . . . , xm

, FS

H1, H2, . . . , Hm

, 3.5

is continuous and d

FS

K1, K2, . . . , Km , FS

H1, H2, . . . , Hm d

_n

k1

f_j

K₁, K₂, . . . , K_m , F_S

H₁, H₂, . . . , H_m

sup

j,x1,x2,...,xm∈{1,2,...,n}××^m_j1Kj

d f_j

x₁, x₂, . . . , x_m , FS

H₁, H₂, . . . , H_m

sup

j,x1,x2,...,xm∈{1,2,...,n}××^m_j1Kj

t

k, x1, x2, . . . , xm

, FS

H1, H2, . . . , Hm

,

3.6

it follows that there existk∈ {1,2, . . . , n}, x1∈K₁, x₂∈K₂, . . . ,andx_m∈K_msuch that d

f_kx1, . . . , xm

, FS

H1, . . . , Hm

d FS

K1, . . . , Km

, FS

H1, . . . , Hm

h

FS

K1, . . . , Km

, FS

H1, . . . , Hm

.

3.7

Let us also note that since for allk∈ {1, . . . , n}, the functiontk:Hk→R, given by tky d

xk, y

, 3.8

is continuous,H_kis a compact set, anddxk, H_k inf{dxk, y : y∈H_k}, it follows that there existsy_k∈Hksuch that

d x_k, y_k

d x_k, H_k

, 3.9

thus

d x_k, y_k

d x_k, H_k

≤d

K_k, H_k

≤h

K_k, H_k

. 3.10

Now we are able to prove that h

FS

K1, K2, . . . , Km

, FS

H1, H2, . . . , Hm

<max h

H1, K1

, . . . , h

Hm, Km

, 3.11

for allK1, K2, . . . , Km, H1, H2, . . . , Hm∈ K^∗Xsuch thatKi/Hifor somei∈ {1,2, . . . , m}.

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Indeed, we have h

F_S

K₁, K₂, . . . , K_m , F_S

H₁, H₂, . . . , H_m d

f_k

x1, x2, . . . , xm

, FS

H1, H2, . . . , Hm

d

f_k

x₁, x₂, . . . , x_m ,

n k1

f_k

H₁×H₂× · · · ×H_m

inf d

f_k

x1, . . . , xm , fk

y1, . . . , ym

:k∈ {1,2, . . . , n}, y1∈H1, . . . , ym∈Hm

≤d f_k

x₁, . . . , x_m , f_k

y₁, . . . , y_m .

3.12

Ifx_ky_k, for allk∈ {1,2, . . . , n}, then h

F_S

K₁, K₂, . . . , K_m , F_S

H₁, H₂, . . . , H_m

0, 3.13 so the above claim is true.

Otherwise, we have h

FS

K1, K2, . . . , Km , FS

H1, H2, . . . , Hm

≤d f_k

x1, . . . , xm , f_k

y₁, . . . , y_m

<max d

x1, y_k , . . . , d

xm, y_m max

d x₁, H₁

, . . . , d

x_m, H_m

≤max d

K₁, H₁ , . . . , d

K_m, H_m

≤max h

K1, H1 , . . . , h

Km, Hm ,

3.14

for allK₁, K₂, . . . , K_m, H₁, H₂, . . . , H_m∈ K^∗Xsuch thatK_i/H_ifor somei∈ {1,2, . . . , m}.

Let us recall the following result.

Theorem 3.2. For a contractionf:X→X,there exists a uniqueα∈Xsuch thatfα α.

For everyx₀∈X, the sequencexk_k≥0, defined by xk1f

xk

, 3.15

for allk∈N, is convergent toα.

Moreover, iffj : X→X, wherej ∈ N, are contractions having the fixed points αj, such that f_j→−^s fon a dense subset ofX, then

jlim→ ∞αjα. 3.16

Let us mention that the first part ofTheorem 3.2is due to Edelsteinsee14.

Theorem 3.3. Letf:X→Xbe a function having the property that there existsp∈N^∗such thatf^p is a contraction.

Then there exists a uniqueα∈Xsuch thatfα αand, for anyx₀ ∈X, the sequencexk_k≥0 defined byxk1fxkis convergent toα.

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Proof. It is clear thatf^p has a unique fixed pointα∈X and, for everyy0 ∈X, the sequence yk_k≥1defined byyk1f^pykis convergent toα.

In particular fory₀^j f^jx0, where x₀ ∈ X and j ∈ {0,1, . . . , p−1}, the sequence yn^jf^npjx0_n≥0is convergent toα.

It follows that the sequencexk_k≥0, defined byxk1fxk, is convergent toα.

Since every fixed point offis a fixed point off^p, it follows thatαis the unique fixed point off.

Theorem 3.4. Given a contractionf:X^m→X, there exists a uniqueα∈Xsuch that

fα, α, . . . , α α. 3.17

For everyx0, x1, . . . , xm−1∈X, the sequencexk_k≥0defined by

xkmf

xkm−1, xkm−2, . . . , xk

, 3.18

for allk∈N, is convergent toα.

Moreover, if for everyj∈N, fj:X^m→Xis a contraction andα_jis the unique point ofXhaving the property that

fj

αj, αj, . . . , αj

αj, 3.19

then

jlim→ ∞α_jα, 3.20

provided thatfj→−s fon a dense subset ofX^m.

Proof. Letg:X→Xandgj:X→Xbe the functions defined by gx fx, x, . . . , x,

g_jx f_jx, x, . . . , x, 3.21

for everyx∈X.

Thengandgjare contractions.

It follows, usingTheorem 3.2, that there exist uniqueα∈Xandαj∈Xsuch that

αgα fα, α, . . . , α, α_jg

α_j f

α_j, α_j, . . . , α_j ,

jlim→ ∞α_jα.

3.22

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The functionh:X^m→X^m, given by h

x0, x1, . . . , xm−1

x1, x2, . . . , xm−1, f

x0, x1, . . . , xm−1

x1, x2, . . . , xm−1, xm

, 3.23

for allx0, x1, . . . , xm−1∈X, fulfills the conditions ofTheorem 3.3takingpm.

Therefore, there existsβ1, β2, . . . , βm∈X^msuch that h

β1, β2, . . . , βm

, 3.24

so

β₁β₂· · ·β_mf

β₁, β₂, . . . , β_m

. 3.25

Hence,

β₁β₂· · ·β_mα. 3.26

Then,

l→ ∞limh^l

x0, x1, . . . , xm−1 lim

l→ ∞

xl, xl1, . . . , xlm−1

α, α, . . . , α, 3.27 so we conclude our claim.

Using Proposition 3.1, Theorem 3.4, and Lemma 2.13, we obtain the following two results.

Theorem 3.5. Given a generalized iterated function system of ordermS X,fk_k1,n, there exists a uniqueAS∈ K^∗Xsuch that

FS

AS, AS, . . . , AS

AS. 3.28 Moreover, for anyH0, H1, . . . , Hm−1∈ K^∗X, the sequenceHn_n≥0, defined by

HnmFS

Hnm−1, Hnm−2, . . . , Hn

, 3.29

for alln∈N, is convergent toAS.

Definition 3.6. Letmbe a fixed natural number, letX, dbe a compact metric space, and let S X,fk_k1,nbe a generalized iterated function system onXof orderm.

The unique setASgiven by the previous theorem is called the attractor of the GIFSS.

Theorem 3.7. IfS X,fk_k1,nandSj X,f_k^j_k1,n, wherej ∈N, are GIFS of ordermsuch that, for everyk∈ {1,2, . . . , n},f_k^j→−^s fkon a dense set inX^m, then

A Sj

−→AS. 3.30

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Acknowledgments

The authors want to thank the referees whose generous and valuable remarks and comments brought improvements to the paper and enhanced clarity. The work was supported by GAR 30/2007.

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