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:Xm_→_{X, where}_{X, d}_{is a metric space and}_m_∈_N_{. In case that}_{X, d}_{is a compact} metric space and the functionsfkare contractions, using some fixed point theorems for contractions fromXm_to_{X, 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 :Xm→X, whereX, dis a metric space andm∈N. In case thatX, dis a compact metric space and the functionsfk are contractions, using some fixed point theorems for contractions fromXmtoX, 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 Hutchinson see 1 and popularized by Barnsley see2. 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.

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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 compact see, e.g., 6 where 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. 9 introduce 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→R R∪ {∞}, defined by

df, g sup x∈XdY

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, d be a complete metric space and letm ∈ N. For a functionf : Xm

×m

k1X→X, the number

infc:dfx1, . . . , xm, fy1, . . . , ym

≤cmaxdx1, y1, . . . , d

xm, ym

, ∀x1, . . . , xm, y1, . . . , ym ∈X

2.2

which is the same as

supdfx1, . . . , xm, fy1, . . . , ym: maxdx1, y1, . . . , dxm, ym, 2.3

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

maxdx1, y1, . . . , dxm, ym>0, 2.4

is denoted byLipfand is called the Lipschitz constant off.

A function f : Xm→X is called a Lipschitz function if Lipf < ∞ and a Lipschitz contraction ifLipf<1.

A functionf :Xm→Xis said to be a contraction if

dfx1, . . . , xm, fy1, . . . , ym<maxdx1, y1, . . . , dxm, ym, 2.5

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LConmXdenotes the set

f :Xm −→ X:Lipf<1 2.6

and ConmXdenotes the set

f :Xm −→ X:f is a contraction. 2.7

Remark 2.2. It is obvious that

LConmX⊆ConmX. 2.8

Notations. PXdenotes the family of all subsets of a given setX andP∗Xdenotes the set

PX\ {∅}.

For a subsetAofPX, byA∗we meanA\ {∅}.

Given a metric spaceX, d, KXdenotes the set of compact subsets of X 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 space X, d, one considers on P∗Xthe generalized

Hausdorﬀ-Pompeiu pseudometrich:P∗X× P∗X→0,∞defined by

hA, B maxdA, B, dB, A

infr ∈0,∞:A⊆BB, r, B⊆BA, r, 2.10

where

BA, r x∈X:dx, A< r,

dA, B sup

x∈Adx, B supx∈A

inf

y∈Bdx, 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

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iiifHi_i∈I andKi_i∈Iare 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

hfK, 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_k_1,n, consists of a finite family of functionsfk_k_1,n, fk :Xm_→_X_{such that}_{f1, . . . , fn}_∈_Con

mX.

Definition 2.9. Let f : Xm→X be a continuous function. The functionFf : K∗Xm→ K∗X

defined by

FfK1, K2, . . . , KmfK1×K2× · · · ×Km

fx1, x2, . . . , xm:xj ∈Kj, ∀j∈ {1, . . . , m} 2.15

is called the set function associated to the functionf.

Definition 2.10. GivenS X,fk_k_1,na generalized iterated function system onXof order

m, the functionFS :K∗Xm→ K∗Xdefined by

FSK1, K2, . . . , Km n

k1

FfkK1, K2, . . . , Km 2.16

is called the set function associated toS.

Lemma 2.11. For a sequencefn_nof elements ofCXm, Xandf ∈CXm, Xsuch thatfn→u fand forK1, K2, . . . , Km ∈ K∗X, one has

fnK1×K2× · · · ×Km−→fK1×K2× · · · ×Km 2.17

inK∗X, h.

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

hfnK1× · · · ×Km, fK1× · · · ×Km

≤ sup x1∈K1,...,xm∈Km

dfnx1, . . . , xm, fx1, . . . , xm, 2.18

which is valid for alln∈N.

Proposition 2.12. LetX, dXandY, dYbe two metric spaces and letfn, 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

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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

dYfx, fy< ε

3. 2.20 SinceKis compact, there existx1, x2, . . . , xn ∈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 all i ∈ {1, . . . , n}, limm→ ∞fmyi fyi, there exists mε ∈ N such that for everym∈N, m≥mε,we have

dYfmyi, fyi< ε

3, 2.22 for everyi∈ {1, . . . , n}.

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

dXx, yi≤dXx, xidXxi, yi< δ 2

δ

2 < δ, 2.23

so

dYfyi, fx< ε

3. 2.24 Hence, form≥mε, we have

dYfmx, fx≤dYfmx, fmyidYfmyi, fyidYfyi, fx

≤MdXx, yi ε 3

ε 3

≤M ε 3M1

2ε 3 < ε.

2.25

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

fn−−→u·c f. 2.26

The inequalityLipf≤sup_n≥1Lipfnis obvious.

FromLemma 2.11andProposition 2.12, usingProposition 2.7iiwe obtain the follow-ing lemma.

Lemma 2.13. LetX, dXbe a complete metric space, letm∈N, letSj X,f_kj_k_1,n, wherej∈N∗,

and let S X,fk_k_1,nbe generalized iterated function systems of orderm, such that, for allk ∈

{1, . . . , n}, f_kj→−s fk on a dense subset ofXm_.

Then, for everyK1, K2, . . . , Km∈ K∗X,

FSjK1, K2, . . . , Km−→FSK1, K2, . . . , Km, 2.27

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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_k_1,n is a generalized iterated function system onXof orderm.

First, we prove thatFS:K∗Xm→ K∗Xis a contractionProposition 3.1, then, using some results concerning the fixed points of contractions fromXm_to_X_{Theorem 3.4}_{, we prove} the existence of the attractor of S Theorem 3.5 and its continuous dependence in the fk’s Theorem 3.7.

The following proposition is crucial.

Proposition 3.1. FS :K∗Xm→ K∗Xis a contraction.

Proof. ByProposition 2.7, we have

hFSK1, K2, . . . , Km, FSH1, H2, . . . , Hm

h

n

k1 fk

K1×K2× · · · ×Km

, n

k1 fk

H1×H2× · · · ×Hm

h

_n

k1

FfkK1, K2, . . . , Km, n

k1

FfkH1, H2, . . . , Hm

≤maxhf1K1× · · · ×Km, f1H1× · · · ×Hm, . . . , hfnK1× · · · ×Km,

fn

H1× · · · ×Hm

≤maxhH1, K1, . . . , hHm, Km,

3.1

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

It remains to prove that the above inequality is strict.

Let K1, K2, . . . , Km, H1, H2, . . . , Hm ∈ K∗X be fixed such that Ki/Hi for some i ∈

{1,2, . . . , m}. Since

hFSK1, . . . , Km, FSH1, . . . , Hm

maxdFSK1, . . . , Km, FSH1, . . . , Hm, dFSH1, . . . , Hm, FSK1, . . . , Km, 3.2

we can suppose, by using symmetry arguments, that

hFSK1, . . . , Km, FSH1, . . . , HmdFSK1, . . . , Km, FSH1, . . . , Hm, 3.3

that is,

h

_n

k1

fkK1× · · · ×Km, n

k1

fkH1× · · · ×Hm

d

_n

k1

fkK1× · · · ×Km, n

k1

fkH1× · · · ×Hm

.

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Let us note that for every K1, K2, . . . , Km ∈ K∗X, since f1, . . . , fn are continuous functions,FSK1, K2, . . . , Km nk1fjK1, K2, . . . , Kmis a compact set.

Since for all K1, K2, . . . , Km, H1, H2, . . . , Hm ∈ K∗X, the product topological space

{1,2, . . . , n} ××m_j₁Kj, where{1,2, . . . , n}is endowed with the discrete topology, is compact and the functiont:{1,2, . . . , n} ××m_j₁Kj→R, given by

tk, x1, x2, . . . , xmdfkx1, x2, . . . , xm, FSH1, H2, . . . , Hm, 3.5

is continuous and

dFSK1, K2, . . . , Km, FSH1, H2, . . . , Hm

d

_n

k1

fjK1, K2, . . . , Km, FSH1, H2, . . . , Hm

sup

j,x1,x2,...,xm∈{1,2,...,n}××mj1Kj

dfjx1, x2, . . . , xm, FSH1, H2, . . . , Hm

sup

j,x1,x2,...,xm∈{1,2,...,n}××m_j₁Kj

tk, x1, x2, . . . , xm, FSH1, H2, . . . , Hm,

3.6

it follows that there existk∈ {1,2, . . . , n}, x1∈K1, x2∈K2, . . . ,andxm∈Kmsuch that

df_kx1, . . . , xm, FSH1, . . . , HmdFSK1, . . . , Km, FSH1, . . . , Hm

hFSK1, . . . , Km, FSH1, . . . , Hm.

3.7

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

tky dxk, y, 3.8

is continuous,Hkis a compact set, anddxk, Hk inf{dxk, y : y∈Hk}, it follows that there existsy_k ∈Hk such that

dxk, yk

dxk, Hk

, 3.9

thus

dxk, y_kdxk, Hk≤dKk, Hk≤hKk, Hk. 3.10

Now we are able to prove that

hFSK1, K2, . . . , Km, FSH1, H2, . . . , Hm<maxhH1, K1, . . . , hHm, Km, 3.11

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Indeed, we have

hFSK1, K2, . . . , Km, FSH1, H2, . . . , Hm

df_kx1, x2, . . . , xm, FSH1, H2, . . . , Hm

d

f_kx1, x2, . . . , xm, n

k1

fkH1×H2× · · · ×Hm

infdf_kx1, . . . , xm, fky1, . . . , ym:k∈ {1,2, . . . , n}, y1∈H1, . . . , ym∈Hm

≤df_kx1, . . . , xm, f_ky₁, . . . , y_m.

3.12

Ifxk yk, for allk∈ {1,2, . . . , n}, then

hFSK1, K2, . . . , Km, FSH1, H2, . . . , Hm0, 3.13

so the above claim is true. Otherwise, we have

hFSK1, K2, . . . , Km, FSH1, H2, . . . , Hm≤dfk

x1, . . . , xm, f_ky₁, . . . , y_m <maxdx1, y_k, . . . , dxm, y_m maxdx1, H1, . . . , dxm, Hm

≤maxdK1, H1, . . . , dKm, Hm

≤maxhK1, H1, . . . , hKm, Hm,

3.14

for allK1, K2, . . . , Km, H1, H2, . . . , Hm ∈ K∗Xsuch thatKi/Hifor 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, if fj : X→X, wherej ∈ N, are contractions having the fixed points αj, such that

fj→−s fon a dense subset ofX, then

lim

j→ ∞αj α. 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 thatfp is a contraction.

Then there exists a uniqueα∈Xsuch thatfα αand, for anyx0∈X, the sequencexk_k≥0

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Proof. It is clear thatfp_{has a unique fixed point}_α _∈ _X _{and, for every}_y0 _∈ _X_{, the sequence}

yk_k≥1defined byyk1fpykis convergent toα.

In particular for y₀j fjx0, where x0 ∈ X and j ∈ {0,1, . . . , p −1}, the sequence ynj fnpjx0n≥0is convergent toα.

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

Since every fixed point off is a fixed point offp_{, it follows that}_α_{is the unique fixed} point off.

Theorem 3.4. Given a contractionf :Xm→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 :Xm→Xis a contraction andαjis the unique point ofXhaving

the property that

fjαj, αj, . . . , αjαj, 3.19

then

lim

j→ ∞αj α, 3.20

provided thatfj→−s fon a dense subset ofXm.

Proof. Letg:X→Xandgj :X→Xbe the functions defined by

gx fx, x, . . . , x,

gjx fjx, x, . . . , x, 3.21

for everyx∈X.

Thengandgj are contractions.

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

αgα fα, α, . . . , α, αj gαjfαj, αj, . . . , αj,

lim

j→ ∞αj α.

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The functionh:Xm_→_Xm_{, given by}

hx0, 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∈Xm_{such that}

hβ1, β2, . . . , βm

β1, β2, . . . , βm

, 3.24

so

β1β2· · ·βmfβ1, β2, . . . , βm. 3.25

Hence,

β1β2· · ·βm α. 3.26

Then,

lim l→ ∞h

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_k_1,n, there exists

a uniqueAS∈ K∗Xsuch that

FSAS, AS, . . . , ASAS. 3.28

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

HnmFSHnm−1, Hnm−2, . . . , Hn, 3.29

for alln∈N, is convergent toAS.

Definition 3.6. Let mbe a fixed natural number, let X, dbe a compact metric space, and let

S X,fk_k_1,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_k_1,nandSj X,f_kjk1,n, wherej ∈ N, are GIFS of ordermsuch

that, for everyk∈ {1,2, . . . , n},f_kj→−s fk on a dense set inXm_{, then}

ASj

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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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