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, miculesc@yahoo.com 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.IncasethatX, 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.IncasethatX, 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.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 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, CX, Y  denotes the set of continuous functions from X to Y,and d : CX, Y × CX, Y  → R   R  ∪{∞}, defined by df, gsup x∈X d Y  fx,gx  , 2.1 is the generalized metric on CX, Y . For a sequence f n  n of elements of CX, Y and f ∈ CX, 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. LetX, d be a complete metric space and let m ∈ N. For a function f : X m  × m k1 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 Lipf and is called the Lipschitz constant of f. A function f : X m → X is called a Lipschitz function if Lipf < ∞ and a Lipschitz contraction if Lipf < 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 : Lipf < 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. PX denotes the family of all subsets of a given set X and P ∗ X denotes the set PX \{ ∅}. For a subset A of PX,byA ∗ we mean A \{∅} . Given a metric space X, d, KX denotes the set of compact subsets of X and BX denotes the set of closed bounded subsets of X. Remark 2.3. It is obvious that KX ⊆BX ⊆PX. 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 hA, Bmax  dA, B,dB, A   inf  r ∈ 0, ∞ : A ⊆ BB, r,B⊆ BA, r  , 2.10 where BA, r  x ∈ X : dx, A <r  , dA, Bsup x∈A dx, Bsup x∈A  inf y∈B dx, 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 hH, Kh  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  fK,fH  ≤ LipfhK, 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  k1,n , consists of a finite family of functions f k  k1,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  k1,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  k1 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 CX m ,X and f ∈ CX 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 ∈ CX, Y be such that sup n≥1 Lipf n  < ∞ and f n s −→ f on a dense set in X. Then Lipf ≤ 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 Lipf n . Let us consider A  {x ∈ X | f m x → fx}, 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/3M1 such that if x, y ∈ K and d X x, y <δ,then d Y  fx,fy  < ε 3 . 2.20 Since K is compact, there exist x 1 ,x 2 , ,x n ∈ K such that K ⊆ n  i1 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 ∈ Bx 1 ,δ/2, ,y n ∈ Bx n ,δ/2. Since, for all i ∈{1, ,n}, lim m →∞ f m y i fy 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 ∈ Bx 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  ,fx  < ε 3 . 2.24 Hence, for m ≥ m ε ,wehave d Y  f m x,fx  ≤ d Y  f m x,f m  y i   d Y  f m  y i  ,f  y i   d Y  f  y i  ,fx  ≤ Md X  x, y i   ε 3  ε 3 ≤ M ε 3M  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 Lipf ≤ sup n≥1 Lipf n  is obvious. From Lemma 2.11 and Proposition 2.12, using Proposition 2.7ii 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  k1,n ,wherej ∈ N ∗ , and let S X, f k  k1,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  k1,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  k1 f k  K 1 × K 2 ×···×K m  , n  k1 f k  H 1 × H 2 ×···×H m    h  n  k1 F f k  K 1 ,K 2 , ,K m  , n  k1 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  k1 f k  K 1 ×···×K m  , n  k1 f k  H 1 ×···×H m    d  n  k1 f k  K 1 ×···×K m  , n  k1 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 k1 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 j1 K j ,where{1, 2, ,n} is endowed with the discrete topology, is compact and the function t : {1, 2, ,n}×× m j1 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  k1 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 j1 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 j1 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 yd  x k ,y  , 3.8 is continuous, H k is a compact set, and dx k ,H k inf{dx 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  k1 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 fαα. For every x 0 ∈ X, the sequence x k  k≥0 , defined by x k1  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 fαα and, for any x 0 ∈ X, the sequence x k  k≥0 defined by x k1  fx 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 k1  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 npj x 0  n≥0 is convergent to α. It follows that the sequence x k  k≥0 , defined by x k1  fx 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 fα,α, ,αα. 3.17 For every x 0 ,x 1 , ,x m−1 ∈ X, the sequence x k  k≥0 defined by x km  f  x km−1 ,x km−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 gxfx,x, ,x, g j xf 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  gαfα,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 l1 , ,x lm−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  k1,n ,thereexists a unique AS ∈K ∗ X such that F S  AS,AS, ,AS   AS. 3.28 Moreover, for any H 0 ,H 1 , ,H m−1 ∈K ∗ X, the sequence H n  n≥0 , defined by H nm  F S  H nm−1 ,H nm−2 , ,H n  , 3.29 for all n ∈ N, is convergent to AS. Definition 3.6. Let m be a fixed natural number, let X, d be a compact metric space, and let S X, f k  k1,n  be a generalized iterated function system on X of order m . TheuniquesetAS given by the previous theorem is called the attractor of the GIFS S. Theorem 3.7. If S X, f k  k1,n  and S j X, f j k  k1,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  −→ AS. 3.30 [...]... Scientiarium Fennicæ Mathematica, vol 29, no 2, pp 419–458, 2004 11 K Le´ niak, In nite iterated function systems: a multivalued approach,” Bulletin of the Polish Academy s of Sciences Mathematics, vol 52, no 1, pp 1–8, 2004 12 K J Falconer, The Geometry of Fractal Sets, vol 85 of Cambridge Tracts in Mathematics, Cambridge University Press, Cambridge, UK, 1986 13 K J Falconer, Fractal Geometry: Mathematical Foundations... d´ -Łukawska and J Jachymski, The Hutchinson-Barnsley theory for in nite iterated ´z z function systems,” Bulletin of the Australian Mathematical Society, vol 72, no 3, pp 441–454, 2005 ˙ 9 A Łozinski, K Zyczkowski, and W Słomczynski, “Quantum iterated function systems,” Physical ´ ´ Review E, vol 68, no 4, Article ID 046110, 9 pages, 2003 10 A K¨ enm¨ ki, “On natural invariant measures on generalised... optimal control models,” Annals of Operations Research, vol 88, pp 183–197, 1999 6 N.-A Secelean, “Countable iterated function systems,” Far East Journal of Dynamical Systems, vol 3, no 2, pp 149–167, 2001 7 R Miculescu and A Mihail, “Lipscomb’s space ωA is the attractor of an in nite IFS containing affine transformations of l2 A ,” Proceedings of the American Mathematical Society, vol 136, no 2, pp 587–592,... arising from recursive estimation problems,” Probability Theory and Related Fields, vol 91, no 1, pp 103–114, 1992 4 B Forte and E R Vrscay, “Solving the inverse problem for function/image approximation using iterated function systems—I: theoretical basis,” Fractals, vol 2, no 3, pp 325–334, 1994 5 L Montrucchio and F Privileggi, “Fractal steady states in stochastic optimal control models,” Annals of. .. 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 References 1 J E Hutchinson, “Fractals and self-similarity,” Indiana University Mathematics Journal, vol 30, no 5, pp 713–747, 1981 2 M F Barnsley, Fractals Everywhere, Academic Press Professional, Boston,... Mathematics, Cambridge University Press, Cambridge, UK, 1986 13 K J Falconer, Fractal Geometry: Mathematical Foundations and Applications , John Wiley & Sons, Chichester, UK, 1990 14 M Edelstein, “On fixed and periodic points under contractive mappings,” Journal of the London Mathematical Society, vol 37, pp 74–79, 1962 . 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. 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. 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