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Hindawi Publishing Corporation
Fixed Point Theory and Applications
Volume 2008, Article ID 312876, 11 pages
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,
Received 9 February 2008; Accepted 22 May 2008
Recommended by Hichem Ben-El-Mechaiekh
We introduce the notion of a generalized iterated function system GIFS, which is a finite family
of functions f
k
: X
m
→ X, where X, d is a metric space and m ∈ N.IncasethatX, d isacompact
metric space and the functions f
k
are contractions, using some fixed point theorems for contractions
from X
m
to X, we prove the existence of the attractor of such a GIFS and its continuous dependence
in the f
k
’s.
Copyright q 2008 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 system IFS, 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 functions f
k
: X
m
→ X,
where X, d is a metric space and m ∈
N.IncasethatX, d is a compact metric space and the
functions f
k
are contractions, using some fixed point theorems for contractions from X
m
to X,
we prove the existence of the attractor of such a GIFS and its continuous dependence in the
f
k
’s.
IFSs were introduced in their present form by Hutchinson see 1 and popularized
by Barnsley see 2. 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 consult 3–5.
There is a current effort to extend Hutchinson’s classical framework for fractals to more
general spaces and infinite IFSs.
2 Fixed Point Theory and Applications
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.In7, 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 theory see 2,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 systems QIFSs
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
´
sniak 11 presents a multivalued approach of infinite iterated function systems.
2. Preliminaries
Notations. Let X, d
X
and Y, d
Y
be two metric spaces.
As usual, CX, Y denotes the set of continuous functions from X to Y,and
d : CX, Y ×
CX, Y →
R
R
∪{∞}, defined by
df, gsup
x∈X
d
Y
fx,gx
, 2.1
is the generalized metric on CX, Y .
For a sequence f
n
n
of elements of CX, Y and f ∈ CX, Y ,f
n
s
−→ f denotes the
pointwise convergence, f
n
u·c
−−→ f denotes the uniform convergence on compact sets, and f
n
u
−→ f
denotes the uniform convergence, that is, the convergence in the generalized metric
d.
Definition 2.1. LetX, d be a complete metric space and let m ∈
N. For a function f : X
m
×
m
k1
X → X, the number
inf
c : d
f
x
1
, ,x
m
,f
y
1
, ,y
m
≤ c max
d
x
1
,y
1
, ,d
x
m
,y
m
, ∀x
1
, ,x
m
,y
1
, ,y
m
∈ X
2.2
which is the same as
sup
d
f
x
1
, ,x
m
,f
y
1
, ,y
m
:max
d
x
1
,y
1
, ,d
x
m
,y
m
, 2.3
where the sup is taken over x
1
, ,x
m
,y
1
, ,y
m
∈ X such that
max
d
x
1
,y
1
, ,d
x
m
,y
m
> 0, 2.4
is denoted by Lipf and is called the Lipschitz constant of f.
A function f : X
m
→ X is called a Lipschitz function if Lipf < ∞ and a Lipschitz
contraction if Lipf < 1.
A function f : X
m
→ X is said to be a contraction if
d
f
x
1
, ,x
m
,f
y
1
, ,y
m
< max
d
x
1
,y
1
, ,d
x
m
,y
m
, 2.5
for every x
1
,x
2
, ,x
m
,y
1
,y
2
, ,y
m
∈ X, such that x
i
/
y
i
for some i ∈{1, 2, ,n}.
A. Mihail and R. Miculescu 3
LCon
m
X denotes the set
f : X
m
−→ X : Lipf < 1
2.6
and Con
m
X denotes the set
f : X
m
−→ X : f is a contraction
. 2.7
Remark 2.2. It is obvious that
LCon
m
X ⊆ Con
m
X. 2.8
Notations. PX denotes the family of all subsets of a given set X and P
∗
X denotes the set
PX \{
∅}.
For a subset A of PX,byA
∗
we mean A \{∅} .
Given a metric space X, d, KX denotes the set of compact subsets of X and BX
denotes the set of closed bounded subsets of X.
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
∗
X the generalized Hausdorff-
Pompeiu pseudometric h : P
∗
X ×P
∗
X → 0, ∞ defined by
hA, Bmax
dA, B,dB, A
inf
r ∈ 0, ∞ : A ⊆ BB, r,B⊆ BA, r
,
2.10
where
BA, r
x ∈ X : dx, A <r
,
dA, Bsup
x∈A
dx, Bsup
x∈A
inf
y∈B
dx, y
.
2.11
Remark 2.5. The Hausdorff-Pompeiu pseudometric is a metric on B
∗
X and, in particular, on
K
∗
X.
Remark 2.6. The metric spaces B
∗
X,h and K
∗
X,h are complete, provided that X, d is
a complete metric space see 2, 7, 12. Moreover, K
∗
X,h is compact, provided that X, d
is a compact metric space see 2.
The following proposition gives the important properties of the Hausdorff-Pompeiu
pseudometric see 2, 13.
Proposition 2.7. Let X, d
X
and Y, d
Y
be two metric spaces. Then
i if H and K are two nonempty subsets of X,then
hH, Kh
H,K
; 2.12
4 Fixed Point Theory and Applications
ii if H
i
i∈I
and K
i
i∈I
are two families of nonempty subsets of X,then
h
i∈I
H
i
,
i∈I
K
i
≤ sup
i∈I
h
H
i
,K
i
; 2.13
iii if H and K are two nonempty subsets of X and f : X → X is 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 system in short a GIFS on X of order m, denoted by S X, f
k
k1,n
, consists of a
finite family of functions f
k
k1,n
,f
k
: X
m
→ X such that f
1
, ,f
n
∈ Con
m
X.
Definition 2.9. Let f : X
m
→ X be a continuous function. The function F
f
: K
∗
X
m
→K
∗
X
defined by
F
f
K
1
,K
2
, ,K
m
f
K
1
× K
2
×···×K
m
f
x
1
,x
2
, ,x
m
: x
j
∈ K
j
, ∀ j ∈{1, ,m}
2.15
is called the set function associated to the function f.
Definition 2.10. Given S X, f
k
k1,n
a generalized iterated function system on X of order
m, the function F
S
: K
∗
X
m
→K
∗
X defined by
F
S
K
1
,K
2
, ,K
m
n
k1
F
f
k
K
1
,K
2
, ,K
m
2.16
is called the set function associated to S.
Lemma 2.11. For a sequence f
n
n
of elements of CX
m
,X and f ∈ CX
m
,X such that f
n
u
→ f and
for K
1
,K
2
, ,K
m
∈K
∗
X, one has
f
n
K
1
× K
2
×···×K
m
−→ f
K
1
× K
2
×···×K
m
2.17
in K
∗
X,h.
Proof. Indeed, the conclusion follows from the below inequality:
h
f
n
K
1
×···×K
m
,f
K
1
×···×K
m
≤ sup
x
1
∈K
1
, ,x
m
∈K
m
d
f
n
x
1
, ,x
m
,f
x
1
, ,x
m
,
2.18
which is valid for all n ∈
N.
Proposition 2.12. Let X, d
X
and Y, d
Y
be two metric spaces and let f
n
,f ∈ CX, Y be such that
sup
n≥1
Lipf
n
< ∞ and f
n
s
−→ f on a dense set in X.
Then
Lipf ≤ sup
n≥1
Lip
f
n
,f
n
u.c
−−−→ f. 2.19
A. Mihail and R. Miculescu 5
Proof. Set M : sup
n≥1
Lipf
n
.
Let us consider A {x ∈ X | f
m
x → fx}, which is a dense set in X,letK be a compact
set in X,andletε>0.
Since f is uniformly continuous on K, there exists δ ∈ 0,ε/3M1 such that if x, y ∈ K
and d
X
x, y <δ,then
d
Y
fx,fy
<
ε
3
. 2.20
Since K is compact, there exist x
1
,x
2
, ,x
n
∈ K such that
K ⊆
n
i1
B
x
i
,
δ
2
. 2.21
Taking into account the fact that A is dense in X, we can choose y
1
,y
2
, ,y
n
∈ A such
that y
1
∈ Bx
1
,δ/2, ,y
n
∈ Bx
n
,δ/2.
Since, for all i ∈{1, ,n}, lim
m →∞
f
m
y
i
fy
i
, there exists m
ε
∈ N such that for
every m ∈
N,m≥ m
ε
, we have
d
Y
f
m
y
i
,f
y
i
<
ε
3
, 2.22
for every i ∈{1, ,n}.
For x ∈ K, there exists i ∈{1, ,n}, such that x ∈ Bx
i
,δ/2 and therefore
d
X
x, y
i
≤ d
X
x, x
i
d
X
x
i
,y
i
<
δ
2
δ
2
<δ, 2.23
so
d
Y
f
y
i
,fx
<
ε
3
. 2.24
Hence, for m ≥ m
ε
,wehave
d
Y
f
m
x,fx
≤ d
Y
f
m
x,f
m
y
i
d
Y
f
m
y
i
,f
y
i
d
Y
f
y
i
,fx
≤ Md
X
x, y
i
ε
3
ε
3
≤ M
ε
3M 1
2ε
3
<ε.
2.25
Consequently, as x was arbitrarily chosen in K,weinferthatf
n
u
→ f on K,so
f
n
u·c
−−→ f. 2.26
The inequality Lipf ≤ sup
n≥1
Lipf
n
is obvious.
From Lemma 2.11 and Proposition 2.12, using Proposition 2.7ii we obtain the follow-
ing lemma.
Lemma 2.13. Let X, d
X
be a complete metric space, let m ∈ N,letS
j
X, f
j
k
k1,n
,wherej ∈ N
∗
,
and let S X, f
k
k1,n
be generalized iterated function systems of order m, such that, for all k ∈
{1, ,n},f
j
k
s
−→ f
k
on a dense subset of X
m
.
Then, for every K
1
,K
2
, ,K
m
∈K
∗
X,
F
S
j
K
1
,K
2
, ,K
m
−→ F
S
K
1
,K
2
, ,K
m
, 2.27
in K
∗
X,h.
6 Fixed Point Theory and Applications
3. The existence of the attractor of a GIFs for contractions
In this section, m is a natural number, X, d is a compact metric space, and S X, f
k
k1,n
is a generalized iterated function system on X of order m.
First, we prove that F
S
: K
∗
X
m
→K
∗
X is a contraction Proposition 3.1, then, using
some results concerning the fixed points of contractions from X
m
to X Theorem 3.4,weprove
the existence of the attractor of S Theorem 3.5 and its continuous dependence in the f
k
’s
Theorem 3.7.
The following proposition is crucial.
Proposition 3.1. F
S
: K
∗
X
m
→K
∗
X is a contraction.
Proof. By Proposition 2.7,wehave
h
F
S
K
1
,K
2
, ,K
m
,F
S
H
1
,H
2
, ,H
m
h
n
k1
f
k
K
1
× K
2
×···×K
m
,
n
k1
f
k
H
1
× H
2
×···×H
m
h
n
k1
F
f
k
K
1
,K
2
, ,K
m
,
n
k1
F
f
k
H
1
,H
2
, ,H
m
≤ max
h
f
1
K
1
×···×K
m
,f
1
H
1
×···×H
m
, ,h
f
n
K
1
×···×K
m
,
f
n
H
1
×···×H
m
≤ max
h
H
1
,K
1
, ,h
H
m
,K
m
,
3.1
for all K
1
, ,K
m
,H
1
, ,H
m
∈K
∗
X.
It remains to prove that the above inequality is strict.
Let K
1
,K
2
, ,K
m
,H
1
,H
2
, ,H
m
∈K
∗
X be fixed such that K
i
/
H
i
for some i ∈
{1, 2, ,m}.
Since
h
F
S
K
1
, ,K
m
,F
S
H
1
, ,H
m
max
d
F
S
K
1
, ,K
m
,F
S
H
1
, ,H
m
,d
F
S
H
1
, ,H
m
,F
S
K
1
, ,K
m
,
3.2
we can suppose, by using symmetry arguments, that
h
F
S
K
1
, ,K
m
,F
S
H
1
, ,H
m
d
F
S
K
1
, ,K
m
,F
S
H
1
, ,H
m
, 3.3
that is,
h
n
k1
f
k
K
1
×···×K
m
,
n
k1
f
k
H
1
×···×H
m
d
n
k1
f
k
K
1
×···×K
m
,
n
k1
f
k
H
1
×···×H
m
.
3.4
A. Mihail and R. Miculescu 7
LetusnotethatforeveryK
1
,K
2
, ,K
m
∈K
∗
X, since f
1
, ,f
n
are continuous
functions, F
S
K
1
,K
2
, ,K
m
n
k1
f
j
K
1
,K
2
, ,K
m
is a compact set.
Since for all K
1
,K
2
, ,K
m
,H
1
,H
2
, ,H
m
∈K
∗
X, the product topological space
{1, 2, ,n}××
m
j1
K
j
,where{1, 2, ,n} is endowed with the discrete topology, is compact
and the function t : {1, 2, ,n}××
m
j1
K
j
→ R, given by
t
k, x
1
,x
2
, ,x
m
d
f
k
x
1
,x
2
, ,x
m
,F
S
H
1
,H
2
, ,H
m
, 3.5
is continuous and
d
F
S
K
1
,K
2
, ,K
m
,F
S
H
1
,H
2
, ,H
m
d
n
k1
f
j
K
1
,K
2
, ,K
m
,F
S
H
1
,H
2
, ,H
m
sup
j,x
1
,x
2
, ,x
m
∈{1,2, ,n}××
m
j1
K
j
d
f
j
x
1
,x
2
, ,x
m
,F
S
H
1
,H
2
, ,H
m
sup
j,x
1
,x
2
, ,x
m
∈{1,2, ,n}××
m
j1
K
j
t
k, x
1
,x
2
, ,x
m
,F
S
H
1
,H
2
, ,H
m
,
3.6
it follows that there exist
k ∈{1, 2, ,n}, x
1
∈ K
1
, x
2
∈ K
2
, ,and x
m
∈ K
m
such that
d
f
k
x
1
, ,x
m
,F
S
H
1
, ,H
m
d
F
S
K
1
, ,K
m
,F
S
H
1
, ,H
m
h
F
S
K
1
, ,K
m
,F
S
H
1
, ,H
m
.
3.7
Let us also note that since for all k ∈{1, ,n}, the function t
k
: H
k
→ R, given by
t
k
yd
x
k
,y
, 3.8
is continuous, H
k
is a compact set, and dx
k
,H
k
inf{dx
k
,y : y ∈ H
k
}, it follows that there
exists
y
k
∈ H
k
such 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
F
S
K
1
,K
2
, ,K
m
,F
S
H
1
,H
2
, ,H
m
< max
h
H
1
,K
1
, ,h
H
m
,K
m
, 3.11
for all K
1
,K
2
, ,K
m
,H
1
,H
2
, ,H
m
∈K
∗
X such that K
i
/
H
i
for some i ∈{1, 2, ,m}.
8 Fixed Point Theory and Applications
Indeed, we have
h
F
S
K
1
,K
2
, ,K
m
,F
S
H
1
,H
2
, ,H
m
d
f
k
x
1
, x
2
, ,x
m
,F
S
H
1
,H
2
, ,H
m
d
f
k
x
1
, x
2
, ,x
m
,
n
k1
f
k
H
1
× H
2
×···×H
m
inf
d
f
k
x
1
, ,x
m
,f
k
y
1
, ,y
m
: k ∈{1, 2, ,n},y
1
∈ H
1
, ,y
m
∈ H
m
≤ d
f
k
x
1
, ,x
m
,f
k
y
1
, ,y
m
.
3.12
If
x
k
y
k
, for all k ∈{1, 2, ,n},then
h
F
S
K
1
,K
2
, ,K
m
,F
S
H
1
,H
2
, ,H
m
0, 3.13
so the above claim is true.
Otherwise, we have
h
F
S
K
1
,K
2
, ,K
m
,F
S
H
1
,H
2
, ,H
m
≤ d
f
k
x
1
, ,x
m
,f
k
y
1
, ,y
m
< max
d
x
1
, y
k
, ,d
x
m
, y
m
max
d
x
1
,H
1
, ,d
x
m
,H
m
≤ max
d
K
1
,H
1
, ,d
K
m
,H
m
≤ max
h
K
1
,H
1
, ,h
K
m
,H
m
,
3.14
for all K
1
,K
2
, ,K
m
,H
1
,H
2
, ,H
m
∈K
∗
X such that K
i
/
H
i
for some i ∈{1, 2, ,m}.
Let us recall the following result.
Theorem 3.2. For a contraction f : X → X, there exists a unique α ∈ X such that fαα.
For every x
0
∈ X, the sequence x
k
k≥0
, defined by
x
k1
f
x
k
, 3.15
for all k ∈
N, is convergent to α.
Moreover, if f
j
: X → X,wherej ∈ N, are contractions having the fixed points α
j
, such that
f
j
s
−→ f on a dense subset of X,then
lim
j →∞
α
j
α. 3.16
Let us mention that the first part of Theorem 3.2 is due to Edelstein see 14.
Theorem 3.3. Let f : X → X be a function having the property that there exists p ∈
N
∗
such that f
p
is a contraction.
Then there exists a unique α ∈ X such that fαα and, for any x
0
∈ X, the sequence x
k
k≥0
defined by x
k1
fx
k
is convergent to α.
A. Mihail and R. Miculescu 9
Proof. It is clear that f
p
has a unique fixed point α ∈ X and, for every y
0
∈ X, the sequence
y
k
k≥1
defined by y
k1
f
p
y
k
is convergent to α.
In particular for y
j
0
f
j
x
0
,wherex
0
∈ X and j ∈{0, 1, ,p − 1}, the sequence
y
j
n
f
npj
x
0
n≥0
is convergent to α.
It follows that the sequence x
k
k≥0
, defined by x
k1
fx
k
, is convergent to α.
Since every fixed point of f is a fixed point of f
p
, it follows that α istheuniquefixed
point of f.
Theorem 3.4. Given a contraction f : X
m
→ X, there exists a unique α ∈ X such that
fα,α, ,αα. 3.17
For every x
0
,x
1
, ,x
m−1
∈ X, the sequence x
k
k≥0
defined by
x
km
f
x
km−1
,x
km−2
, ,x
k
, 3.18
for all k ∈
N, is convergent to α.
Moreover, if for every j ∈
N,f
j
: X
m
→ X is a contraction and α
j
is the unique point of X having
the property that
f
j
α
j
,α
j
, ,α
j
α
j
, 3.19
then
lim
j →∞
α
j
α, 3.20
provided that f
j
s
−→ f on a dense subset of X
m
.
Proof. Let g : X → X and g
j
: X → X be the functions defined by
gxfx,x, ,x,
g
j
xf
j
x,x, ,x,
3.21
for every x ∈ X.
Then g and g
j
are contractions.
It follows, using Theorem 3.2, that there exist unique α ∈ X and α
j
∈ X such that
α gαfα,α, ,α,
α
j
g
α
j
f
α
j
,α
j
, ,α
j
,
lim
j →∞
α
j
α.
3.22
10 Fixed Point Theory and Applications
The function h : X
m
→ X
m
, given by
h
x
0
,x
1
, ,x
m−1
x
1
,x
2
, ,x
m−1
,f
x
0
,x
1
, ,x
m−1
x
1
,x
2
, ,x
m−1
,x
m
,
3.23
for all x
0
,x
1
, ,x
m−1
∈ X, fulfills the conditions of Theorem 3.3 taking p m.
Therefore, there exists β
1
,β
2
, ,β
m
∈ X
m
such that
h
β
1
,β
2
, ,β
m
β
1
,β
2
, ,β
m
, 3.24
so
β
1
β
2
··· β
m
f
β
1
,β
2
, ,β
m
. 3.25
Hence,
β
1
β
2
··· β
m
α. 3.26
Then,
lim
l →∞
h
l
x
0
,x
1
, ,x
m−1
lim
l →∞
x
l
,x
l1
, ,x
lm−1
α,α, ,α,
3.27
so we conclude our claim.
Using Proposition 3.1, Theorem 3.4 ,andLemma 2.13, we obtain the following two
results.
Theorem 3.5. Given a generalized iterated function system of order mS X, f
k
k1,n
,thereexists
a unique AS ∈K
∗
X such that
F
S
AS,AS, ,AS
AS. 3.28
Moreover, for any H
0
,H
1
, ,H
m−1
∈K
∗
X, the sequence H
n
n≥0
, defined by
H
nm
F
S
H
nm−1
,H
nm−2
, ,H
n
, 3.29
for all n ∈
N, is convergent to AS.
Definition 3.6. Let m be a fixed natural number, let X, d be a compact metric space, and let
S X, f
k
k1,n
be a generalized iterated function system on X of order m .
TheuniquesetAS given by the previous theorem is called the attractor of the GIFS S.
Theorem 3.7. If S X, f
k
k1,n
and S
j
X, f
j
k
k1,n
,wherej ∈ N, are GIFS of order m such
that, for every k ∈{1, 2, ,n}, f
j
k
s
−→ f
k
on a dense set in X
m
,then
A
S
j
−→ AS. 3.30
A. Mihail and R. Miculescu 11
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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