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Iterated function systems and well-posedness
Chaos, Solitons and Fractals 41 (2009) 1561–1568
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Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Iterated function systems and well-posedness Enrique Llorens-Fuster a,*, Adrian Petrusßel b, Jen-Chih Yao c a b c
Department of Mathematical Analysis, Faculty of Mathematics, University of Valencia, 46100 Burjassot, Valencia, Spain Department of Applied Mathematics, Babesß-Bolyai University Cluj-Napoca, Koga˘lniceanu 1, 400084 Cluj-Napoca, Romania Department of Applied Mathematics, National Sun Yat-sen University, 804 Kaohsiung, Taiwan, ROC
a r t i c l e
i n f o
Article history: Accepted 23 June 2008
Communicated by Prof. G. Iovane
a b s t r a c t Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems in several topics of applied sciences [see for example: El Naschie MS. Iterated function systems and the two-slit experiment of quantum mechanics. Chaos, Solitons & Fractals 1994;4:1965–8; Iovane G. Cantorian spacetime and Hilbert space: Part I-Foundations. Chaos, Solitons & Fractals 2006;28:857–78; Iovane G. Cantorian space-time and Hilbert space: Part II-Relevant consequences. Chaos, Solitons & Fractals 2006;29:1–22; Fedeli A. On chaotic set-valued discrete dynamical systems. Chaos, Solitons & Fractals 2005;23:13814; Shi Y, Chen G. Chaos of discrete dynamical systems in complete metric spaces. Chaos, Solitons & Fractals 2004;22:55571]. The purpose of this paper is twofold. First, some existence and uniqueness results for the self-similar sets of a mixed iterated function systems are given. Then, using the concept of well-posed fixed point problem, the well-posedness of the self-similarity problem for some classes of iterated multifunction systems is also studied. Well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Roughly speaking, fractals are very complicated subsets of some nice geometric spaces. Self-similar sets are special types of fractals. In few words, a self-similar set is a set consisting of retorts of itself. A self-similar set with a non-integer Hausdorff dimension is usually considered a fractal. The study of self-similar sets in connection with the mathematics of fractals is an important and actual topic in the research activity of the scientist international community. Some recent and very interesting works in the theory of black holes, two-split experiment in quantum mechanics, chaos in discrete dynamical systems (see for instance: [11– 14,16,18,19,23,24,27,44,47]) are directly connected with the theory of iterated function (or multifunction) systems. Throughout this paper, the standard notations and terminologies in nonlinear analysis are used. For the convenience of the reader we recall some of them. Let ðX; dÞ be a metric space. We will use the following symbols:
PðXÞ ¼ fY Xj Y is nonemptyg; Pb ðXÞ :¼ fY 2 PðXÞj Y is boundedg; Pcl ðXÞ :¼ fY 2 PðXÞj Y is closedg; Pb;cl ðXÞ :¼ Pb ðXÞ \ Pcl ðXÞ; P cp ðXÞ :¼ fY 2 PðXÞj Y is compactg:
* Corresponding author. E-mail addresses: [email protected] (E. Llorens-Fuster), [email protected] (A. Petrusßel), [email protected] (J.-C. Yao). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.06.019
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If T : X ! PðXÞ is a multivalued operator then
TðYÞ :¼
[
TðxÞ; Y 2 PðXÞ
x2Y
will denote the image of the set Y. The graph of the multivalued operator T is denoted by
GraphT :¼ fðx; yÞ 2 X Xjy 2 TðxÞg: Throughout the paper
FixT :¼ fx 2 Xj x 2 TðxÞg denotes the fixed point set of T, while
SFixT :¼ fx 2 Xj fxg ¼ TðxÞg is the strict fixed point set of the multivalued operator T. The following functionals are used in the main section of the paper. The gap functional. (1)
Dd : PðXÞ PðXÞ ! Rþ [ fþ1g; Dd ðA; BÞ :¼ inffdða; bÞj a 2 A; b 2 Bg:
In particular, we denote by Dd ðx; BÞ :¼ Dd ðfxg; BÞ the gap between the point x 2 X and the set B.The d generalized functional. (2)
dd : PðXÞ PðXÞ ! Rþ [ fþ1g; dd ðA; BÞ :¼ supfdða; bÞj a 2 A; b 2 Bg:
In particular, we denote by diamd A :¼ dd ðA; AÞ the diameter of the set A.The excess generalized functional. (3)
qd : PðXÞ PðXÞ ! Rþ [ fþ1g; qd ðA; BÞ :¼ supfDd ða; BÞj a 2 Ag:
The Pompeiu–Hausdorff generalized functional. (4)
Hd : PðXÞ PðXÞ ! Rþ [ fþ1g; Hd ðA; BÞ :¼ maxfqd ðA; BÞ; qd ðB; AÞg:
It is well-known that ðP b;cl ðXÞ; Hd Þ is a complete metric space provided ðX; dÞ is a complete metric space. 0
If ðX; dÞ; ðY; d Þ are metric spaces, then T : X ! PðYÞ is said to be (i) closed if GraphT is a closed set in X Y; (ii) upper semi-continuous on X (briefly u.s.c.) if for each open subset U of Y we have that T þ ðUÞ :¼ fx 2 XjTðxÞ Ug is open in X; (iii) k-Lipschitz if k P 0 and HðFðx1 Þ; Fðx2 ÞÞ 6 k dðx1 ; x2 Þ, for all x1 ; x2 2 X. In particular, if there exists k < 1, then F is said a multivalued k-contraction. Also, T : X ! P b;cl ðXÞ is u.s.c. in x0 2 X if and only if for each sequence ðxn Þn2N X such that limn!1 xn ¼ x0 we have limn!1 qd ðTðxn Þ; Tðx0 ÞÞ ¼ 0. For more details and basic results concerning the above notions see for example [25,34,40,42,44]. The purpose of this paper is twofold. First some existence and uniqueness results for the self-similar set of a mixed iterated functions systems are given. In this respect, our results extend and complement some previous results given in [1,2,6,36]. Second aim of this work is to study the well-posedness of the self-similarity problem for some classes of iterated multifunction systems, using the concept of well-posed fixed point problem. Notice that well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals. From this point of view, the results are in connection with some recent interesting results given in [45]. 2. Mixed iterated function systems We recall first some contractive-type conditions for a single-valued operator f : X ! X. Definition 2.1. The single-valued operator f : X ! X is said to be (i) a-contraction if a 2 ½0; 1½ and for each x; y 2 X we have dðf ðxÞ; f ðyÞÞ 6 adðx; yÞ. (ii) Meir–Keeler-type if for each g > 0 there exists d > 0 such that x; y 2 X, g 6 dðx; yÞ < g þ d we have dðf ðxÞ; f ðyÞÞ < g. (iii) contractive if for each x; y 2 X; x–y we have dðf ðxÞ; f ðyÞÞ < dðx; yÞ. Obviously, any contraction satisfies the Meir–Keeler type condition and any Meir–Keeler type operator is contractive (and hence continuous).
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We recall now the notion of comparison function. A function u : Rþ ! Rþ is said to be a comparison function (see [40]) if it is increasing and the sequence ðun ðtÞÞ converges to 0 for each t > 0. As consequence, we also have uðtÞ < t for each t P 0, t uð0Þ ¼ 0 and u is continuous in 0. In particular, uðtÞ ¼ at (where a 2 ½0; 1½), uðtÞ ¼ 1þt and uðtÞ ¼ lnð1 þ tÞ, t 2 Rþ are examples of comparison functions. In this setting, the operator f : X ! X is said to be u-contraction if u is a comparison function and dðf ðxÞ; f ðyÞÞ 6 uðdðx; yÞÞ, for each x; y 2 X. Theorem 2.2 ([9,31,39,40]). Let ðX; dÞ be a complete metric space and f : X ! X be such that f is an a-contraction (Banach, Caccioppoli) or f is a u- contraction (Matkowski, Rhoades, Rus) or f is a Meir–Keeler type operator (Meir, Keeler). Then Fixf ¼ fx g X and for each x 2 X the sequence ðf n ðxÞÞn2N of successive approximations for f starting from x converges to x as n ! þ1. The following notion was introduced by Lim in [29]. A function u : ½0; 1½! ½0; 1½ is said to be an L-function if uð0Þ ¼ 0, uðsÞ > 0 for all s > 0 and for every s > 0 there exists u > s such that
uðtÞ 6 s for t 2 ½s; u: In [29] Lim states that f : X ! X is a Meir–Keeler operator if and only if there exists a nondecreasing and right continuous L-function u such that dðf ðxÞ; f ðyÞÞ < uðdðx; yÞÞ for each x; y 2 X with x–y. Let us consider now the multivalued case. Let F 1 ; . . . ; F m : X ! Pcp ðXÞ be a finite family of upper semi-continuous multivalued operators. We define the multi-fractal operator T F generated by the iterated multifunctions system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ by the following relation
T F : Pcp ðXÞ ! Pcp ðXÞ; T F ðYÞ ¼
m [
F i ðYÞ:
i¼1
In this framework, a nonempty compact subset A of X is said to be a multivalued fractal with respect to the iterated multifunctions system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ if and only if it is a fixed point for the associated multi-fractal operator. In particular, if the operators F i :¼ fi are single-valued, then a fixed point for the fractal operator
T f : Pcp ðXÞ ! Pcp ðXÞ;
T f ðYÞ ¼
m [
fi ðYÞ
i¼1
generated by the iterated functions system f ¼ ðf1 ; f2 ; . . . ; fm Þ is said to be a self-similar set or a fractal (see [3,20,etc.]). Some contractive-type conditions for multivalued operators on a metric space ðX; dÞ are given now. Definition 2.3 ([7,32,46]). The multivalued operator F : X ! P cl ðXÞ is said to be: (i) a-contraction if a 2 ½0; 1½ and for each x; y 2 X we have HðFðxÞ; FðyÞÞ 6 a dðx; yÞ (ii) Meir–Keeler type operator if for each g > 0 there exists d > 0 such that for each x; y 2 X with g 6 dðx; yÞ < g þ d it follows HðFðxÞ; FðyÞÞ < g (iii) contractive if for each x; y 2 X; x–y we have HðFðxÞ; FðyÞÞ < dðx; yÞ. Any multivalued contraction satisfies the Meir–Keeler type condition and any multivalued Meir–Keeler type operator is contractive (and hence u.s.c.). The mapping u : Rþ ! Rþ is called a strict comparison function if (i) u is a continuous comparison function. (ii) limt!1 ðt uðtÞÞ ¼ 1. t and uðtÞ ¼ lnð1 þ tÞ are examples of strict comparison functions. In particular, uðtÞ ¼ at (where a 20; 1½), uðtÞ ¼ 1þt In this setting, the operator F : X ! P cl ðXÞ is called a multivalued u-strict contraction (see [1]) if u is a strict comparison function and for each x; y 2 X we have HðFðxÞ; FðyÞÞ 6 uðdðx; yÞÞ. If ðX; dÞ is a complete metric space and F i : X ! P cp ðXÞ are multivalued ui -strict contractions (for i 2 f1; . . . ; mg), then the multi-fractal operator T F is a single-valued u-strict contraction (with uðtÞ :¼ maxi2f1;...;mg ui ðtÞ) having a unique fixed point AF (Andres-Fišer [1]). In particular, if F i : X ! Pcp ðXÞ are multivalued ai -contractions (for i 2 f1; . . . ; mg), then the multi-fractal operator T F is a single-valued a-contraction (with a ¼ maxfa1 ; . . . ; am g, see Nadler [32]) having, by the well-known Banach-Caccioppoli contraction principle, a unique fixed point AF . The notion of multivalued Meir–Keeler type operator was introduced by Reich [37]. Lim also proved that the above characterization of Meir–Keeler operators, in terms of L-functions, remains true in the multivalued case (Theorem 2 in [29]). It is also known that if F i : X ! P cp ðXÞ are multivalued Meir–Keeler operators (for i 2 f1; . . . ; mg), then the multi-fractal operator T F is a single-valued Meir–Keeler having a unique fixed point AF (Petrusßel [33]), see also [15].
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See Chifu and Petrusßel [6] for other details and related questions. The main result of this section is the following theorem. Theorem 2.4. Let ðX; dÞ be a complete metric space and F i : X ! P cp ðXÞ, i 2 f1; 2; . . . ; mg be multivalued operators satisfying to one of the following conditions (A) Meir–Keeler-type condition. (B) ui -strict contraction condition. Then there exists a unique multivalued fractal with respect to the iterated multifunction system F ¼ ðF 1 ; F 2 ; . . . F m Þ, i.e. FixT F ¼ fA g and ðT nF ðAÞÞn2N converges to A , for each A 2 Pcp ðXÞ. Proof. Let us prove that the operator T F : P cp ðXÞ ! P cp ðXÞ; T F ðYÞ ¼ For arbitrary Y; Z 2 P cp ðXÞ we have
HðT F ðYÞ; T F ðZÞÞ ¼ H
m [
F i ðYÞ;
i¼1
m [
Sm
i¼1 F i ðYÞ
satisfies the conditions of Theorem 2.2.
! F i ðZÞ
i¼1
6 max HðF i ðYÞ; F i ðZÞÞ ¼ HðF k ðYÞ; F k ðZÞÞ; i2f1;...;mg
where k 2 fi 2 f1; . . . ; mg:
(a) Suppose first that F k is a u-strict contraction.Then we have: HðT F ðYÞ; T F ðZÞÞ 6 uk ðHðY; ZÞÞ. Hence T F is a (single-valued) uk -strict contraction, having, by Theorem 2.2, an unique fixed point A 2 Pcp ðXÞ, i.e. an unique multivalued fractal. (b) If F k is a multivalued Meir–Keeler operator then, by Lim’s characterization theorem for the case of a multivalued operator, there exists an L-function u such that HðF k ðxÞ; F k ðyÞÞ 6 uðdðx; yÞÞ, for each x; y 2 X. We have successively
HðFðYÞ; FðZÞÞ ¼ H
[ y2Y
FðyÞ;
[
! FðzÞ
¼ maxfqðFðYÞ; FðZÞÞ; qðFðZÞ; FðYÞÞg:
z2Z
Further, since u is nondecreasing and right continuous, we deduce
qðFðYÞ; FðZÞÞ ¼ sup Dðu; FðZÞÞ ¼ sup qðFðyÞ; FðZÞÞ 6 sup inf HðFðyÞ; FðzÞÞ 6 sup inf uðdðx; yÞÞ 6 uðqðY; ZÞÞ: u2FðYÞ
y2Y
y2Y
z2Z
y2Y
z2Z
Similarly we can prove that qðFðZÞ; FðYÞÞ 6 uðqðZ; YÞÞ. Finally, we get HðF k ðYÞ; F k ðZÞÞ 6 uðHðY; ZÞÞ. Then we have
HðT F ðYÞ; T F ðZÞÞ 6 uðHðY; ZÞÞ: Hence, using Lim’s characterization theorem for the single-valued case, we get that T F is a (single-valued) Meir–Keeler operator. By Theorem 2.2 we obtain again that here exists a unique multivalued fractal A 2 Pcp ðXÞ. The proof is complete. h In particular, when the operators F i are single-valued we have the following result. Corollary 2.5. Let ðX; dÞ be a complete metric space and fi : X ! X, i 2 f1; 2; . . . ; mg be operators satisfying to one of the following conditions (A) Meir–Keeler-type condition. (B) ui -strict contraction condition.Then there exists a unique self-similar set for the iterated function system f ¼ ðf1 ; f2 ; . . . fm Þ, i.e. FixT f ¼ fA g and ðT nf ðAÞÞn2N converges to A , for each A 2 P cp ðXÞ.
Example 2.6. Let X ¼ ½0; 1 [ f2; 3; 4; 5; . . . ; 2n; 2n þ 1; . . .g and f1 ; f2 : X ! X given by
8x x 2 ½0; 1; > <3; x ¼ 2n; n 2 N ; f1 ðxÞ ¼ 0; > : 1 1 ; x ¼ 2n þ 1; n 2 N ; nþ2 8 x > < 1 3 ; x 2 ½0; 1; f2 ðxÞ ¼
> :
0;
1 ; nþ2
x ¼ 2n; n 2 N ; x ¼ 2n þ 1; n 2 N :
Then f ¼ ðf1 ; f2 Þ is an iterated function system of Meir–Keeler type operators having a unique fractal Y 2 Pcp ðXÞ, namely the well-known Cantor set.
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Example 2.7. Let X ¼ ½0; 1 [ f3; 4; 5; . . . ; 2n; 2n þ 1; . . .g and f1 ; f2 : X ! X given by:
8x x 2 ½0; 1; > <3; x ¼ 2n; n 2 N ; n P 2; f1 ðxÞ ¼ 0; > : 1 1 ; x ¼ 2n þ 1; n 2 N ; nþ2 1 3x ; x 2 ½0; 1; f2 ðxÞ ¼ 0; x 2 N; n P 3: Then f ¼ ðf1 ; f2 Þ is an mixed iterated function system (i. e. f1 is a Meir–Keeler type operator, while f2 is a 13-contraction) having a unique fractal Y 2 P cp ðXÞ, Again here Y is the Cantor set. Remark 2.8. Notice that the above examples show that one can obtain the Cantor set as the self-similar set of a iterated function system not only composed by contraction mappings, but also using much general operators such as Meir–Keeler operators or u-contractions. It is of interest to remark that, using the above results, relevant applications of such iterated function systems given in: [21,22] (see also [10–14,18,21–24]) can be established.
3. Well-posedness for self-similar problems Let us present first the notion of well-posedness in the generalized sense for a fixed point problem. Definition 3.1 (Petrusßel et al. [36]). Let ðX; dÞ be a metric space, Y 2 PðXÞ and T : Y ! P cl ðXÞ be a multivalued operator. Then the fixed point problem for T with respect to Dd is well-posed in the generalized sense (respectively, well-posed [35]) iff ða1 Þ FixT–; (respectively, FixT ¼ fx g). ðb1 Þ If xn 2 Y, n 2 N and Dd ðxn ; Tðxn ÞÞ ! 0 as n ! þ1, then there exists a subsequence ðxni Þ of ðxn Þ such that xni converges to x 2 FixT as i ! þ1 (respectively, xn ! x as n ! þ1).
Definition 3.2 (Petrusßel et al. [36]). Let ðX; dÞ be a metric space, Y 2 PðXÞ and T : Y ! Pcl ðXÞ be a multivalued operator. Then the fixed point problem for T with respect to Hd is well-posed in the generalized sense (respectively, well-posed [35]) iff ða2 Þ SFixT–; (respectively, SFixT ¼ fx g) ðb2 Þ If xn 2 Y, n 2 N and Hd ðxn ; Tðxn ÞÞ ! 0 as n ! þ1, then there exists a subsequence ðxni Þ of ðxn Þ such that xni converges to x 2 SFixT as i ! þ1 (respectively, xn ! x as n ! þ1). If in the above definitions we consider the setting of a normed space, then a fixed point problem is weakly well-posed in the generalized sense (respectively. weakly well-posed) if the convergence of the subsequence ðxni Þ (respectively, of the sequence ðxn Þ) to x is weakly. Remark 3.3. It’s easy to see that if the fixed point problem is well-posed (in the generalized sense) for T with respect to Dd and FixT ¼ SFixT, then the fixed point problem is well-posed (in the generalized sense) for T with respect to Hd . Also, when the operator F is single-valued, then Definitions 3.1 and 3.2 coincide and we obtain the concept given by De Blasi and Myjak [4], Lemaire [28], Reich and Zaslavski [38]. See also I.A. Rus [41]. For other similar definitions see [35,48,49]. Notice that well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals. The first purpose of this section is to study the weakly well-posedness in the generalized sense of some fixed point problems. Some of them are concerned with multivalued mappings satisfying a kind of convexity, of course in the framework of Banach spaces. It is well-known that there exists in the literature several different concepts of convexity for multivalued operators (see for instance, [8,43]). In [30] (see also [5]) it was considered the following one. Let X be a Banach space and C 2 P cl;cv ðXÞ. Given a multivalued operator T : C ! Pb;cl ðXÞ, we can consider the associate functionals J T : C ! R and HT : C ! R, respectively, defined by
J T ðxÞ :¼ Dðx; TðxÞÞ ¼ inffkx yk : y 2 TðxÞg; and
HT ðxÞ :¼ Hðfxg; TðxÞÞ ¼ supfkx yk : y 2 TðxÞg:
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Notice that, since the set TðxÞ is closed for every x 2 C, one has that J T ðxÞ ¼ 0 if and only if x 2 TðxÞ, i.e. if x is a fixed point of T. Moreover, if diamTðxÞ > 0 then HT ðxÞ > 0 and HT ðxÞ ¼ 0 if and only if fxg ¼ TðxÞ. Definition 3.4. For a continuous strictly increasing function a : Rþ ! Rþ with að0Þ ¼ 0, we say that the multivalued operator T : C ! P b;cl ðXÞ is a-almost convex if for all x; y 2 C and all k 2 ½0; 1 we have
J T ðkx þ ð1 kÞyÞ 6 aðmaxfJ T ðxÞ; J T ðyÞgÞ: Although the above class of multivalued operators seems to be the natural generalization of the analogous concept for single-valued operators (see [17]), we also will be concerned with another class of multivalued operators. Definition 3.5. We say that the multivalued operator T : C ! Pb;cl ðXÞ is strongly a-almost convex if for all x; y 2 C and all k 2 ½0; 1 we have
HT ðkx þ ð1 kÞyÞ 6 aðmaxfHT ðxÞ; HT ðyÞgÞ: In [30] was shown that if C is a closed convex subset of a Banach space X then every multivalued contraction T : C ! Pb;cl ðXÞ is strongly almost convex. Even more, the same statement holds for many generalized contractions in the sense of Kannan and C´iric´. On the other hand, a multivalued operator T : C ! P b;cl ðXÞ is said to be semiconvex on C (see for instance, [26]), if for any x; y 2 X; k 2 ½0; 1, and any x1 2 TðxÞ and y1 2 TðyÞ, there exists z 2 Tðkx þ ð1 kÞyÞ such that
kzk 6 maxfkx1 k; ky1 kg: Lemma 2 of [26] immediately yields that if C is a bounded, closed and convex subset of X, F : C ! P b;cl ðXÞ and I : X ! X is the identity operator, if I F is semiconvex, then F is almost convex. Moreover, Lemma 3 of the same work [26] furnishes a kind of converse, in the sense that if F : C ! PðXÞ is compact-valued and almost convex, then I F is semiconvex. Now, we are able to give some results concerning the well-posedness of the fixed point problem for multivalued operators. Theorem 3.6. Let X be a Banach space and Y X be weakly compact and convex. If T : Y ! P b;cl ðXÞ is an u.s.c. a-almost convex multivalued operator such that inf x2X Dðx; TðxÞÞ ¼ 0, then the fixed point problem for T with respect to D is weakly well-posed in the generalized sense. Proof. We follow the method given in the proof of Theorem 1 in [30]. Let ðxn Þn2N 2 Y such that Dðxn ; Tðxn ÞÞ ! 0 as n ! þ1. Since Y is weakly compact we get that there exists a subsequence ðxni Þ * x as i ! þ1. Denote this subsequence also by ðxn Þ. Let us consider the functional J T ðxÞ :¼ Dðx; TðxÞÞ; x 2 Y. Let e > 0. Since JT is lower semicontinous on Y, the set
n eo S :¼ y 2 Y : J T ðyÞ > J T ðx Þ 2
is open. Since x 2 S there exists d > 0 such that if y 2 Y and ky x k < d then
e J T ðyÞ > J T ðx Þ : 2 On the other hand, since a is a continuous function at 0 and að0Þ ¼ 0 there exists n1 2 N such that
0 < aðJ T ðxn1 ÞÞ <
e : 2
Since J T ðxn Þ ! 0 and aðJ T ðxn ÞÞ ! 0 as n ! þ1, we can find n2 > n1 such that
0 < J T ðxn2 Þ < minfJ T ðxn1 Þ; aðJ T ðxn1 ÞÞg and
0 < aðJ T ðxn2 ÞÞ < minfJ T ðxn1 Þ; aðJ T ðxn1 ÞÞg: By induction we obtain a subsequence ðxnk Þ of ðxn Þ satisfying for all k 2 N the following relations
0 < J T ðxnkþ1 Þ < minfJ T ðxnk Þ; aðJ T ðxnk ÞÞg and
0 < aðJ T ðxnkþ1 ÞÞ < minfJ T ðxnk Þ; aðJ T ðxnk ÞÞg: By Mazur’s theorem there exists a convex combination
Pp
k¼1 kk xnk
of the elements xni , i 2 f1; 2 . . . ; pg such that
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p X k x x
JT
! kk xnk
k¼1
e > J T ðx Þ : 2
By induction we can prove now that if p X
JT
Pp
k¼1 kk xnk
is a convex combination of xni , i 2 f1; 2 . . . ; pg then
! kk xnk
6 aðJ T ðxn1 ÞÞ;
for each p 2 N; p P 2:
k¼1
Therefore
J T ðx Þ < J T
p X
! kk xnk
k¼1
þ
e < e: 2
Hence x 2 FixT. The proof is now complete.
h
Theorem 3.7. Let X be a Banach space and Y X be weakly compact and convex. Suppose that T : Y ! P b;cl ðXÞ is a strongly aalmost convex and k-Lipschitz multivalued operator such that inf x2X Hðx; TðxÞÞ ¼ 0. Then the fixed point problem for T with respect to H is weakly well-posed in the generalized sense. Proof. From Theorem 2 in [30] we have that SFixT–;. Let ðxn Þn2N 2 Y such that Hðxn ; Tðxn ÞÞ ! 0 as n ! þ1. Since Y is weakly compact we get that there exists a subsequence ðxni Þ * x as i ! þ1. In a similar way to the proof of Theorem 3.6, replacing J T by HT , we get that x 2 SFixT. h The second aim of this section is to give partial answers to the following problem. Open question. If the fixed point problem is well-posed for the finite family of continuous operators fi : X ! X (respectively, for the finite family of u.s.c. multivalued operators F i : X ! Pcl ðXÞ), then is the fixed point problem well-posed for the S Barnsley-Hutchinson operator T f : ðPcp ðXÞ; HÞ ! ðP cp ðXÞ; HÞ, T f ðYÞ ¼ m i¼1 fi ðYÞ (respectively, for T F : ðP cp ðXÞ; HÞ ! ðP cp ðXÞ; HÞ, Sm T F ðYÞ ¼ i¼1 F i ðYÞ)? If the answer is affirmative, then we say that the self-similar problem is well-posed for the iterated function system f ¼ ðf1 ; f2 ; . . . ; fm Þ (respectively, for F ¼ ðF 1 ; F 2 ; . . . ; F m Þ). An answer to the above problem is the following theorem. Theorem 3.8. Let ðX; dÞ be a complete metric space and F i : X ! P cl ðXÞ be a finite family of multivalued strict ui -contractions for each i 2 f1; . . . ; mg. Then the self-similar problem for the iterated function system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ is well-posed. H
Proof. We will prove that FixT f ¼ fX g and if ðX n Þn2N 2 P cp ðXÞ is such that HðX n ; T F ðX n ÞÞ ! 0, then X n ! X as n ! þ1. Since F i : X ! Pcl ðXÞ is a strict ui -contraction for each i 2 f1; . . . ; mg then, following Andres and Fišer [1], we have that T F is a strict maxfu1 ; . . . ; um g-contraction, having (from Matkowski-Rus theorem) a unique fixed point X 2 P cp ðXÞ. Denote by u :¼ maxfu1 ; . . . ; um g and by wðtÞ :¼ t uðtÞ, for t 2 Rþ . Obvious w is a continuous bijection on Rþ and w1 ðgÞ & 0 as g & 0. Next, we have HðX n ; X Þ 6 HðX n ; T F ðX n ÞÞ þ HðT F ðX n Þ; T F ðX ÞÞ 6 HðX n ; T F ðX n ÞÞ þ uðHðX n ; X ÞÞ. Hence HðX n ; X Þ 6 1 w ðHðX n ; T F ðX n ÞÞÞ. The conclusion follows now from the properties of w. h In particular, if F i are multivalued ai -contractions we have the following result. Corollary 3.9. Let ðX; dÞ be a complete metric space and F i : X ! Pcl ðXÞ be a finite family of multivalued ai -contractions for each i 2 f1; . . . ; mg. Then the self-similar problem for the iterated function system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ is well-posed. A second answer is: Theorem 3.10. Let ðX; dÞ be a complete metric space and F i : X ! Pcp ðXÞ be a finite family of multivalued Meir–Keeler operators for each i 2 f1; . . . ; mg. Then: (i) T F : Pcp ðXÞ ! Pcp ðXÞ is a Meir–Keeler type operator. (ii) if, additionally, the corresponding L-function u given by Lim’s characterization theorem for T F satisfies the condition
t uðtÞ ! 0 implies t ! 0; then the self-similar problem for the iterated function system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ is well-posed.
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Proof. (i)–(ii) We will prove that FixT F ¼ fX g and if ðX n Þn2N 2 P cp ðXÞ is such that HðX n ; T F ðX n ÞÞ ! 0, then X n ! X as n ! þ1. Since F i : X ! P cp ðXÞ is a multivalued Meir–Keeler operator, for each i 2 f1; . . . ; mg then, we have that T F is a (single-valued) Meir–Keeler type operator having (from Petrusßel [33]) a unique fixed point X 2 P cp ðXÞ. Next, we have HðX n ; X Þ 6 HðX n ; T F ðX n ÞÞ þ HðT F ðX n Þ; T F ðX ÞÞ < HðX n ; T F ðX n ÞÞ þ uðHðX n ; X ÞÞ, where u is the L-function given in Lim’s characterization theorem. Hence HðX n ; X Þ uðHðX n ; X ÞÞ < HðX n ; T F ðX n ÞÞ and so HðX n ; X Þ uðHðX n ; X ÞÞ ! 0 as n ! þ1. This proves that HðX n ; X Þ ! 0 as n ! þ1. h Acknowledgement The authors are thankful to anonymous reviewer whose suggestions improved the quality of the paper in a significant way. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49]
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Contents lists available at ScienceDirect
Chaos, Solitons and Fractals journal homepage: www.elsevier.com/locate/chaos
Iterated function systems and well-posedness Enrique Llorens-Fuster a,*, Adrian Petrusßel b, Jen-Chih Yao c a b c
Department of Mathematical Analysis, Faculty of Mathematics, University of Valencia, 46100 Burjassot, Valencia, Spain Department of Applied Mathematics, Babesß-Bolyai University Cluj-Napoca, Koga˘lniceanu 1, 400084 Cluj-Napoca, Romania Department of Applied Mathematics, National Sun Yat-sen University, 804 Kaohsiung, Taiwan, ROC
a r t i c l e
i n f o
Article history: Accepted 23 June 2008
Communicated by Prof. G. Iovane
a b s t r a c t Fractals and multivalued fractals play an important role in biology, quantum mechanics, computer graphics, dynamical systems, astronomy and astrophysics, geophysics, etc. Especially, there are important consequences of the iterated function (or multifunction) systems in several topics of applied sciences [see for example: El Naschie MS. Iterated function systems and the two-slit experiment of quantum mechanics. Chaos, Solitons & Fractals 1994;4:1965–8; Iovane G. Cantorian spacetime and Hilbert space: Part I-Foundations. Chaos, Solitons & Fractals 2006;28:857–78; Iovane G. Cantorian space-time and Hilbert space: Part II-Relevant consequences. Chaos, Solitons & Fractals 2006;29:1–22; Fedeli A. On chaotic set-valued discrete dynamical systems. Chaos, Solitons & Fractals 2005;23:13814; Shi Y, Chen G. Chaos of discrete dynamical systems in complete metric spaces. Chaos, Solitons & Fractals 2004;22:55571]. The purpose of this paper is twofold. First, some existence and uniqueness results for the self-similar sets of a mixed iterated function systems are given. Then, using the concept of well-posed fixed point problem, the well-posedness of the self-similarity problem for some classes of iterated multifunction systems is also studied. Well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals. Ó 2008 Elsevier Ltd. All rights reserved.
1. Introduction Roughly speaking, fractals are very complicated subsets of some nice geometric spaces. Self-similar sets are special types of fractals. In few words, a self-similar set is a set consisting of retorts of itself. A self-similar set with a non-integer Hausdorff dimension is usually considered a fractal. The study of self-similar sets in connection with the mathematics of fractals is an important and actual topic in the research activity of the scientist international community. Some recent and very interesting works in the theory of black holes, two-split experiment in quantum mechanics, chaos in discrete dynamical systems (see for instance: [11– 14,16,18,19,23,24,27,44,47]) are directly connected with the theory of iterated function (or multifunction) systems. Throughout this paper, the standard notations and terminologies in nonlinear analysis are used. For the convenience of the reader we recall some of them. Let ðX; dÞ be a metric space. We will use the following symbols:
PðXÞ ¼ fY Xj Y is nonemptyg; Pb ðXÞ :¼ fY 2 PðXÞj Y is boundedg; Pcl ðXÞ :¼ fY 2 PðXÞj Y is closedg; Pb;cl ðXÞ :¼ Pb ðXÞ \ Pcl ðXÞ; P cp ðXÞ :¼ fY 2 PðXÞj Y is compactg:
* Corresponding author. E-mail addresses: [email protected] (E. Llorens-Fuster), [email protected] (A. Petrusßel), [email protected] (J.-C. Yao). 0960-0779/$ - see front matter Ó 2008 Elsevier Ltd. All rights reserved. doi:10.1016/j.chaos.2008.06.019
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If T : X ! PðXÞ is a multivalued operator then
TðYÞ :¼
[
TðxÞ; Y 2 PðXÞ
x2Y
will denote the image of the set Y. The graph of the multivalued operator T is denoted by
GraphT :¼ fðx; yÞ 2 X Xjy 2 TðxÞg: Throughout the paper
FixT :¼ fx 2 Xj x 2 TðxÞg denotes the fixed point set of T, while
SFixT :¼ fx 2 Xj fxg ¼ TðxÞg is the strict fixed point set of the multivalued operator T. The following functionals are used in the main section of the paper. The gap functional. (1)
Dd : PðXÞ PðXÞ ! Rþ [ fþ1g; Dd ðA; BÞ :¼ inffdða; bÞj a 2 A; b 2 Bg:
In particular, we denote by Dd ðx; BÞ :¼ Dd ðfxg; BÞ the gap between the point x 2 X and the set B.The d generalized functional. (2)
dd : PðXÞ PðXÞ ! Rþ [ fþ1g; dd ðA; BÞ :¼ supfdða; bÞj a 2 A; b 2 Bg:
In particular, we denote by diamd A :¼ dd ðA; AÞ the diameter of the set A.The excess generalized functional. (3)
qd : PðXÞ PðXÞ ! Rþ [ fþ1g; qd ðA; BÞ :¼ supfDd ða; BÞj a 2 Ag:
The Pompeiu–Hausdorff generalized functional. (4)
Hd : PðXÞ PðXÞ ! Rþ [ fþ1g; Hd ðA; BÞ :¼ maxfqd ðA; BÞ; qd ðB; AÞg:
It is well-known that ðP b;cl ðXÞ; Hd Þ is a complete metric space provided ðX; dÞ is a complete metric space. 0
If ðX; dÞ; ðY; d Þ are metric spaces, then T : X ! PðYÞ is said to be (i) closed if GraphT is a closed set in X Y; (ii) upper semi-continuous on X (briefly u.s.c.) if for each open subset U of Y we have that T þ ðUÞ :¼ fx 2 XjTðxÞ Ug is open in X; (iii) k-Lipschitz if k P 0 and HðFðx1 Þ; Fðx2 ÞÞ 6 k dðx1 ; x2 Þ, for all x1 ; x2 2 X. In particular, if there exists k < 1, then F is said a multivalued k-contraction. Also, T : X ! P b;cl ðXÞ is u.s.c. in x0 2 X if and only if for each sequence ðxn Þn2N X such that limn!1 xn ¼ x0 we have limn!1 qd ðTðxn Þ; Tðx0 ÞÞ ¼ 0. For more details and basic results concerning the above notions see for example [25,34,40,42,44]. The purpose of this paper is twofold. First some existence and uniqueness results for the self-similar set of a mixed iterated functions systems are given. In this respect, our results extend and complement some previous results given in [1,2,6,36]. Second aim of this work is to study the well-posedness of the self-similarity problem for some classes of iterated multifunction systems, using the concept of well-posed fixed point problem. Notice that well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals. From this point of view, the results are in connection with some recent interesting results given in [45]. 2. Mixed iterated function systems We recall first some contractive-type conditions for a single-valued operator f : X ! X. Definition 2.1. The single-valued operator f : X ! X is said to be (i) a-contraction if a 2 ½0; 1½ and for each x; y 2 X we have dðf ðxÞ; f ðyÞÞ 6 adðx; yÞ. (ii) Meir–Keeler-type if for each g > 0 there exists d > 0 such that x; y 2 X, g 6 dðx; yÞ < g þ d we have dðf ðxÞ; f ðyÞÞ < g. (iii) contractive if for each x; y 2 X; x–y we have dðf ðxÞ; f ðyÞÞ < dðx; yÞ. Obviously, any contraction satisfies the Meir–Keeler type condition and any Meir–Keeler type operator is contractive (and hence continuous).
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We recall now the notion of comparison function. A function u : Rþ ! Rþ is said to be a comparison function (see [40]) if it is increasing and the sequence ðun ðtÞÞ converges to 0 for each t > 0. As consequence, we also have uðtÞ < t for each t P 0, t uð0Þ ¼ 0 and u is continuous in 0. In particular, uðtÞ ¼ at (where a 2 ½0; 1½), uðtÞ ¼ 1þt and uðtÞ ¼ lnð1 þ tÞ, t 2 Rþ are examples of comparison functions. In this setting, the operator f : X ! X is said to be u-contraction if u is a comparison function and dðf ðxÞ; f ðyÞÞ 6 uðdðx; yÞÞ, for each x; y 2 X. Theorem 2.2 ([9,31,39,40]). Let ðX; dÞ be a complete metric space and f : X ! X be such that f is an a-contraction (Banach, Caccioppoli) or f is a u- contraction (Matkowski, Rhoades, Rus) or f is a Meir–Keeler type operator (Meir, Keeler). Then Fixf ¼ fx g X and for each x 2 X the sequence ðf n ðxÞÞn2N of successive approximations for f starting from x converges to x as n ! þ1. The following notion was introduced by Lim in [29]. A function u : ½0; 1½! ½0; 1½ is said to be an L-function if uð0Þ ¼ 0, uðsÞ > 0 for all s > 0 and for every s > 0 there exists u > s such that
uðtÞ 6 s for t 2 ½s; u: In [29] Lim states that f : X ! X is a Meir–Keeler operator if and only if there exists a nondecreasing and right continuous L-function u such that dðf ðxÞ; f ðyÞÞ < uðdðx; yÞÞ for each x; y 2 X with x–y. Let us consider now the multivalued case. Let F 1 ; . . . ; F m : X ! Pcp ðXÞ be a finite family of upper semi-continuous multivalued operators. We define the multi-fractal operator T F generated by the iterated multifunctions system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ by the following relation
T F : Pcp ðXÞ ! Pcp ðXÞ; T F ðYÞ ¼
m [
F i ðYÞ:
i¼1
In this framework, a nonempty compact subset A of X is said to be a multivalued fractal with respect to the iterated multifunctions system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ if and only if it is a fixed point for the associated multi-fractal operator. In particular, if the operators F i :¼ fi are single-valued, then a fixed point for the fractal operator
T f : Pcp ðXÞ ! Pcp ðXÞ;
T f ðYÞ ¼
m [
fi ðYÞ
i¼1
generated by the iterated functions system f ¼ ðf1 ; f2 ; . . . ; fm Þ is said to be a self-similar set or a fractal (see [3,20,etc.]). Some contractive-type conditions for multivalued operators on a metric space ðX; dÞ are given now. Definition 2.3 ([7,32,46]). The multivalued operator F : X ! P cl ðXÞ is said to be: (i) a-contraction if a 2 ½0; 1½ and for each x; y 2 X we have HðFðxÞ; FðyÞÞ 6 a dðx; yÞ (ii) Meir–Keeler type operator if for each g > 0 there exists d > 0 such that for each x; y 2 X with g 6 dðx; yÞ < g þ d it follows HðFðxÞ; FðyÞÞ < g (iii) contractive if for each x; y 2 X; x–y we have HðFðxÞ; FðyÞÞ < dðx; yÞ. Any multivalued contraction satisfies the Meir–Keeler type condition and any multivalued Meir–Keeler type operator is contractive (and hence u.s.c.). The mapping u : Rþ ! Rþ is called a strict comparison function if (i) u is a continuous comparison function. (ii) limt!1 ðt uðtÞÞ ¼ 1. t and uðtÞ ¼ lnð1 þ tÞ are examples of strict comparison functions. In particular, uðtÞ ¼ at (where a 20; 1½), uðtÞ ¼ 1þt In this setting, the operator F : X ! P cl ðXÞ is called a multivalued u-strict contraction (see [1]) if u is a strict comparison function and for each x; y 2 X we have HðFðxÞ; FðyÞÞ 6 uðdðx; yÞÞ. If ðX; dÞ is a complete metric space and F i : X ! P cp ðXÞ are multivalued ui -strict contractions (for i 2 f1; . . . ; mg), then the multi-fractal operator T F is a single-valued u-strict contraction (with uðtÞ :¼ maxi2f1;...;mg ui ðtÞ) having a unique fixed point AF (Andres-Fišer [1]). In particular, if F i : X ! Pcp ðXÞ are multivalued ai -contractions (for i 2 f1; . . . ; mg), then the multi-fractal operator T F is a single-valued a-contraction (with a ¼ maxfa1 ; . . . ; am g, see Nadler [32]) having, by the well-known Banach-Caccioppoli contraction principle, a unique fixed point AF . The notion of multivalued Meir–Keeler type operator was introduced by Reich [37]. Lim also proved that the above characterization of Meir–Keeler operators, in terms of L-functions, remains true in the multivalued case (Theorem 2 in [29]). It is also known that if F i : X ! P cp ðXÞ are multivalued Meir–Keeler operators (for i 2 f1; . . . ; mg), then the multi-fractal operator T F is a single-valued Meir–Keeler having a unique fixed point AF (Petrusßel [33]), see also [15].
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See Chifu and Petrusßel [6] for other details and related questions. The main result of this section is the following theorem. Theorem 2.4. Let ðX; dÞ be a complete metric space and F i : X ! P cp ðXÞ, i 2 f1; 2; . . . ; mg be multivalued operators satisfying to one of the following conditions (A) Meir–Keeler-type condition. (B) ui -strict contraction condition. Then there exists a unique multivalued fractal with respect to the iterated multifunction system F ¼ ðF 1 ; F 2 ; . . . F m Þ, i.e. FixT F ¼ fA g and ðT nF ðAÞÞn2N converges to A , for each A 2 Pcp ðXÞ. Proof. Let us prove that the operator T F : P cp ðXÞ ! P cp ðXÞ; T F ðYÞ ¼ For arbitrary Y; Z 2 P cp ðXÞ we have
HðT F ðYÞ; T F ðZÞÞ ¼ H
m [
F i ðYÞ;
i¼1
m [
Sm
i¼1 F i ðYÞ
satisfies the conditions of Theorem 2.2.
! F i ðZÞ
i¼1
6 max HðF i ðYÞ; F i ðZÞÞ ¼ HðF k ðYÞ; F k ðZÞÞ; i2f1;...;mg
where k 2 fi 2 f1; . . . ; mg:
(a) Suppose first that F k is a u-strict contraction.Then we have: HðT F ðYÞ; T F ðZÞÞ 6 uk ðHðY; ZÞÞ. Hence T F is a (single-valued) uk -strict contraction, having, by Theorem 2.2, an unique fixed point A 2 Pcp ðXÞ, i.e. an unique multivalued fractal. (b) If F k is a multivalued Meir–Keeler operator then, by Lim’s characterization theorem for the case of a multivalued operator, there exists an L-function u such that HðF k ðxÞ; F k ðyÞÞ 6 uðdðx; yÞÞ, for each x; y 2 X. We have successively
HðFðYÞ; FðZÞÞ ¼ H
[ y2Y
FðyÞ;
[
! FðzÞ
¼ maxfqðFðYÞ; FðZÞÞ; qðFðZÞ; FðYÞÞg:
z2Z
Further, since u is nondecreasing and right continuous, we deduce
qðFðYÞ; FðZÞÞ ¼ sup Dðu; FðZÞÞ ¼ sup qðFðyÞ; FðZÞÞ 6 sup inf HðFðyÞ; FðzÞÞ 6 sup inf uðdðx; yÞÞ 6 uðqðY; ZÞÞ: u2FðYÞ
y2Y
y2Y
z2Z
y2Y
z2Z
Similarly we can prove that qðFðZÞ; FðYÞÞ 6 uðqðZ; YÞÞ. Finally, we get HðF k ðYÞ; F k ðZÞÞ 6 uðHðY; ZÞÞ. Then we have
HðT F ðYÞ; T F ðZÞÞ 6 uðHðY; ZÞÞ: Hence, using Lim’s characterization theorem for the single-valued case, we get that T F is a (single-valued) Meir–Keeler operator. By Theorem 2.2 we obtain again that here exists a unique multivalued fractal A 2 Pcp ðXÞ. The proof is complete. h In particular, when the operators F i are single-valued we have the following result. Corollary 2.5. Let ðX; dÞ be a complete metric space and fi : X ! X, i 2 f1; 2; . . . ; mg be operators satisfying to one of the following conditions (A) Meir–Keeler-type condition. (B) ui -strict contraction condition.Then there exists a unique self-similar set for the iterated function system f ¼ ðf1 ; f2 ; . . . fm Þ, i.e. FixT f ¼ fA g and ðT nf ðAÞÞn2N converges to A , for each A 2 P cp ðXÞ.
Example 2.6. Let X ¼ ½0; 1 [ f2; 3; 4; 5; . . . ; 2n; 2n þ 1; . . .g and f1 ; f2 : X ! X given by
8x x 2 ½0; 1; > <3; x ¼ 2n; n 2 N ; f1 ðxÞ ¼ 0; > : 1 1 ; x ¼ 2n þ 1; n 2 N ; nþ2 8 x > < 1 3 ; x 2 ½0; 1; f2 ðxÞ ¼
> :
0;
1 ; nþ2
x ¼ 2n; n 2 N ; x ¼ 2n þ 1; n 2 N :
Then f ¼ ðf1 ; f2 Þ is an iterated function system of Meir–Keeler type operators having a unique fractal Y 2 Pcp ðXÞ, namely the well-known Cantor set.
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Example 2.7. Let X ¼ ½0; 1 [ f3; 4; 5; . . . ; 2n; 2n þ 1; . . .g and f1 ; f2 : X ! X given by:
8x x 2 ½0; 1; > <3; x ¼ 2n; n 2 N ; n P 2; f1 ðxÞ ¼ 0; > : 1 1 ; x ¼ 2n þ 1; n 2 N ; nþ2 1 3x ; x 2 ½0; 1; f2 ðxÞ ¼ 0; x 2 N; n P 3: Then f ¼ ðf1 ; f2 Þ is an mixed iterated function system (i. e. f1 is a Meir–Keeler type operator, while f2 is a 13-contraction) having a unique fractal Y 2 P cp ðXÞ, Again here Y is the Cantor set. Remark 2.8. Notice that the above examples show that one can obtain the Cantor set as the self-similar set of a iterated function system not only composed by contraction mappings, but also using much general operators such as Meir–Keeler operators or u-contractions. It is of interest to remark that, using the above results, relevant applications of such iterated function systems given in: [21,22] (see also [10–14,18,21–24]) can be established.
3. Well-posedness for self-similar problems Let us present first the notion of well-posedness in the generalized sense for a fixed point problem. Definition 3.1 (Petrusßel et al. [36]). Let ðX; dÞ be a metric space, Y 2 PðXÞ and T : Y ! P cl ðXÞ be a multivalued operator. Then the fixed point problem for T with respect to Dd is well-posed in the generalized sense (respectively, well-posed [35]) iff ða1 Þ FixT–; (respectively, FixT ¼ fx g). ðb1 Þ If xn 2 Y, n 2 N and Dd ðxn ; Tðxn ÞÞ ! 0 as n ! þ1, then there exists a subsequence ðxni Þ of ðxn Þ such that xni converges to x 2 FixT as i ! þ1 (respectively, xn ! x as n ! þ1).
Definition 3.2 (Petrusßel et al. [36]). Let ðX; dÞ be a metric space, Y 2 PðXÞ and T : Y ! Pcl ðXÞ be a multivalued operator. Then the fixed point problem for T with respect to Hd is well-posed in the generalized sense (respectively, well-posed [35]) iff ða2 Þ SFixT–; (respectively, SFixT ¼ fx g) ðb2 Þ If xn 2 Y, n 2 N and Hd ðxn ; Tðxn ÞÞ ! 0 as n ! þ1, then there exists a subsequence ðxni Þ of ðxn Þ such that xni converges to x 2 SFixT as i ! þ1 (respectively, xn ! x as n ! þ1). If in the above definitions we consider the setting of a normed space, then a fixed point problem is weakly well-posed in the generalized sense (respectively. weakly well-posed) if the convergence of the subsequence ðxni Þ (respectively, of the sequence ðxn Þ) to x is weakly. Remark 3.3. It’s easy to see that if the fixed point problem is well-posed (in the generalized sense) for T with respect to Dd and FixT ¼ SFixT, then the fixed point problem is well-posed (in the generalized sense) for T with respect to Hd . Also, when the operator F is single-valued, then Definitions 3.1 and 3.2 coincide and we obtain the concept given by De Blasi and Myjak [4], Lemaire [28], Reich and Zaslavski [38]. See also I.A. Rus [41]. For other similar definitions see [35,48,49]. Notice that well-posedness is closely related to the approximation of the solution of a fixed point equation, which is an important aspect of the construction of the fractals using the so-called pre-fractals. The first purpose of this section is to study the weakly well-posedness in the generalized sense of some fixed point problems. Some of them are concerned with multivalued mappings satisfying a kind of convexity, of course in the framework of Banach spaces. It is well-known that there exists in the literature several different concepts of convexity for multivalued operators (see for instance, [8,43]). In [30] (see also [5]) it was considered the following one. Let X be a Banach space and C 2 P cl;cv ðXÞ. Given a multivalued operator T : C ! Pb;cl ðXÞ, we can consider the associate functionals J T : C ! R and HT : C ! R, respectively, defined by
J T ðxÞ :¼ Dðx; TðxÞÞ ¼ inffkx yk : y 2 TðxÞg; and
HT ðxÞ :¼ Hðfxg; TðxÞÞ ¼ supfkx yk : y 2 TðxÞg:
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Notice that, since the set TðxÞ is closed for every x 2 C, one has that J T ðxÞ ¼ 0 if and only if x 2 TðxÞ, i.e. if x is a fixed point of T. Moreover, if diamTðxÞ > 0 then HT ðxÞ > 0 and HT ðxÞ ¼ 0 if and only if fxg ¼ TðxÞ. Definition 3.4. For a continuous strictly increasing function a : Rþ ! Rþ with að0Þ ¼ 0, we say that the multivalued operator T : C ! P b;cl ðXÞ is a-almost convex if for all x; y 2 C and all k 2 ½0; 1 we have
J T ðkx þ ð1 kÞyÞ 6 aðmaxfJ T ðxÞ; J T ðyÞgÞ: Although the above class of multivalued operators seems to be the natural generalization of the analogous concept for single-valued operators (see [17]), we also will be concerned with another class of multivalued operators. Definition 3.5. We say that the multivalued operator T : C ! Pb;cl ðXÞ is strongly a-almost convex if for all x; y 2 C and all k 2 ½0; 1 we have
HT ðkx þ ð1 kÞyÞ 6 aðmaxfHT ðxÞ; HT ðyÞgÞ: In [30] was shown that if C is a closed convex subset of a Banach space X then every multivalued contraction T : C ! Pb;cl ðXÞ is strongly almost convex. Even more, the same statement holds for many generalized contractions in the sense of Kannan and C´iric´. On the other hand, a multivalued operator T : C ! P b;cl ðXÞ is said to be semiconvex on C (see for instance, [26]), if for any x; y 2 X; k 2 ½0; 1, and any x1 2 TðxÞ and y1 2 TðyÞ, there exists z 2 Tðkx þ ð1 kÞyÞ such that
kzk 6 maxfkx1 k; ky1 kg: Lemma 2 of [26] immediately yields that if C is a bounded, closed and convex subset of X, F : C ! P b;cl ðXÞ and I : X ! X is the identity operator, if I F is semiconvex, then F is almost convex. Moreover, Lemma 3 of the same work [26] furnishes a kind of converse, in the sense that if F : C ! PðXÞ is compact-valued and almost convex, then I F is semiconvex. Now, we are able to give some results concerning the well-posedness of the fixed point problem for multivalued operators. Theorem 3.6. Let X be a Banach space and Y X be weakly compact and convex. If T : Y ! P b;cl ðXÞ is an u.s.c. a-almost convex multivalued operator such that inf x2X Dðx; TðxÞÞ ¼ 0, then the fixed point problem for T with respect to D is weakly well-posed in the generalized sense. Proof. We follow the method given in the proof of Theorem 1 in [30]. Let ðxn Þn2N 2 Y such that Dðxn ; Tðxn ÞÞ ! 0 as n ! þ1. Since Y is weakly compact we get that there exists a subsequence ðxni Þ * x as i ! þ1. Denote this subsequence also by ðxn Þ. Let us consider the functional J T ðxÞ :¼ Dðx; TðxÞÞ; x 2 Y. Let e > 0. Since JT is lower semicontinous on Y, the set
n eo S :¼ y 2 Y : J T ðyÞ > J T ðx Þ 2
is open. Since x 2 S there exists d > 0 such that if y 2 Y and ky x k < d then
e J T ðyÞ > J T ðx Þ : 2 On the other hand, since a is a continuous function at 0 and að0Þ ¼ 0 there exists n1 2 N such that
0 < aðJ T ðxn1 ÞÞ <
e : 2
Since J T ðxn Þ ! 0 and aðJ T ðxn ÞÞ ! 0 as n ! þ1, we can find n2 > n1 such that
0 < J T ðxn2 Þ < minfJ T ðxn1 Þ; aðJ T ðxn1 ÞÞg and
0 < aðJ T ðxn2 ÞÞ < minfJ T ðxn1 Þ; aðJ T ðxn1 ÞÞg: By induction we obtain a subsequence ðxnk Þ of ðxn Þ satisfying for all k 2 N the following relations
0 < J T ðxnkþ1 Þ < minfJ T ðxnk Þ; aðJ T ðxnk ÞÞg and
0 < aðJ T ðxnkþ1 ÞÞ < minfJ T ðxnk Þ; aðJ T ðxnk ÞÞg: By Mazur’s theorem there exists a convex combination
Pp
k¼1 kk xnk
of the elements xni , i 2 f1; 2 . . . ; pg such that
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p X k x x
JT
! kk xnk
k¼1
e > J T ðx Þ : 2
By induction we can prove now that if p X
JT
Pp
k¼1 kk xnk
is a convex combination of xni , i 2 f1; 2 . . . ; pg then
! kk xnk
6 aðJ T ðxn1 ÞÞ;
for each p 2 N; p P 2:
k¼1
Therefore
J T ðx Þ < J T
p X
! kk xnk
k¼1
þ
e < e: 2
Hence x 2 FixT. The proof is now complete.
h
Theorem 3.7. Let X be a Banach space and Y X be weakly compact and convex. Suppose that T : Y ! P b;cl ðXÞ is a strongly aalmost convex and k-Lipschitz multivalued operator such that inf x2X Hðx; TðxÞÞ ¼ 0. Then the fixed point problem for T with respect to H is weakly well-posed in the generalized sense. Proof. From Theorem 2 in [30] we have that SFixT–;. Let ðxn Þn2N 2 Y such that Hðxn ; Tðxn ÞÞ ! 0 as n ! þ1. Since Y is weakly compact we get that there exists a subsequence ðxni Þ * x as i ! þ1. In a similar way to the proof of Theorem 3.6, replacing J T by HT , we get that x 2 SFixT. h The second aim of this section is to give partial answers to the following problem. Open question. If the fixed point problem is well-posed for the finite family of continuous operators fi : X ! X (respectively, for the finite family of u.s.c. multivalued operators F i : X ! Pcl ðXÞ), then is the fixed point problem well-posed for the S Barnsley-Hutchinson operator T f : ðPcp ðXÞ; HÞ ! ðP cp ðXÞ; HÞ, T f ðYÞ ¼ m i¼1 fi ðYÞ (respectively, for T F : ðP cp ðXÞ; HÞ ! ðP cp ðXÞ; HÞ, Sm T F ðYÞ ¼ i¼1 F i ðYÞ)? If the answer is affirmative, then we say that the self-similar problem is well-posed for the iterated function system f ¼ ðf1 ; f2 ; . . . ; fm Þ (respectively, for F ¼ ðF 1 ; F 2 ; . . . ; F m Þ). An answer to the above problem is the following theorem. Theorem 3.8. Let ðX; dÞ be a complete metric space and F i : X ! P cl ðXÞ be a finite family of multivalued strict ui -contractions for each i 2 f1; . . . ; mg. Then the self-similar problem for the iterated function system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ is well-posed. H
Proof. We will prove that FixT f ¼ fX g and if ðX n Þn2N 2 P cp ðXÞ is such that HðX n ; T F ðX n ÞÞ ! 0, then X n ! X as n ! þ1. Since F i : X ! Pcl ðXÞ is a strict ui -contraction for each i 2 f1; . . . ; mg then, following Andres and Fišer [1], we have that T F is a strict maxfu1 ; . . . ; um g-contraction, having (from Matkowski-Rus theorem) a unique fixed point X 2 P cp ðXÞ. Denote by u :¼ maxfu1 ; . . . ; um g and by wðtÞ :¼ t uðtÞ, for t 2 Rþ . Obvious w is a continuous bijection on Rþ and w1 ðgÞ & 0 as g & 0. Next, we have HðX n ; X Þ 6 HðX n ; T F ðX n ÞÞ þ HðT F ðX n Þ; T F ðX ÞÞ 6 HðX n ; T F ðX n ÞÞ þ uðHðX n ; X ÞÞ. Hence HðX n ; X Þ 6 1 w ðHðX n ; T F ðX n ÞÞÞ. The conclusion follows now from the properties of w. h In particular, if F i are multivalued ai -contractions we have the following result. Corollary 3.9. Let ðX; dÞ be a complete metric space and F i : X ! Pcl ðXÞ be a finite family of multivalued ai -contractions for each i 2 f1; . . . ; mg. Then the self-similar problem for the iterated function system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ is well-posed. A second answer is: Theorem 3.10. Let ðX; dÞ be a complete metric space and F i : X ! Pcp ðXÞ be a finite family of multivalued Meir–Keeler operators for each i 2 f1; . . . ; mg. Then: (i) T F : Pcp ðXÞ ! Pcp ðXÞ is a Meir–Keeler type operator. (ii) if, additionally, the corresponding L-function u given by Lim’s characterization theorem for T F satisfies the condition
t uðtÞ ! 0 implies t ! 0; then the self-similar problem for the iterated function system F ¼ ðF 1 ; F 2 ; . . . ; F m Þ is well-posed.
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Proof. (i)–(ii) We will prove that FixT F ¼ fX g and if ðX n Þn2N 2 P cp ðXÞ is such that HðX n ; T F ðX n ÞÞ ! 0, then X n ! X as n ! þ1. Since F i : X ! P cp ðXÞ is a multivalued Meir–Keeler operator, for each i 2 f1; . . . ; mg then, we have that T F is a (single-valued) Meir–Keeler type operator having (from Petrusßel [33]) a unique fixed point X 2 P cp ðXÞ. Next, we have HðX n ; X Þ 6 HðX n ; T F ðX n ÞÞ þ HðT F ðX n Þ; T F ðX ÞÞ < HðX n ; T F ðX n ÞÞ þ uðHðX n ; X ÞÞ, where u is the L-function given in Lim’s characterization theorem. Hence HðX n ; X Þ uðHðX n ; X ÞÞ < HðX n ; T F ðX n ÞÞ and so HðX n ; X Þ uðHðX n ; X ÞÞ ! 0 as n ! þ1. This proves that HðX n ; X Þ ! 0 as n ! þ1. h Acknowledgement The authors are thankful to anonymous reviewer whose suggestions improved the quality of the paper in a significant way. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49]
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