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Invariant Compact Sets of Nonexpansive Iterated Function Systems

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Invariant Compact Sets of Nonexpansive Iterated Function Systems Asian Research Journal of Mathematics 3(4): 1-10, 2017; Article no.ARJOM.32634 ISSN: 2456-477X SCIENCEDOMAIN international www.sciencedomain.org Invariant Compact Sets of Nonexpansive Iterated Function Systems ∗ Francisco J. Solis1 and Ezequiel Ojeda-Gomez1 1Center of Research in Mathematics, CIMAT, Guanajuato 36000, M´exico. Authors' contributions This work was carried out in collaboration between both authors. Author FJS designed and managed the analyses of the study. Author EOG wrote the first draft of the manuscript and managed literature searches. Both authors read and approved the final manuscript. Article Information DOI: 10.9734/ARJOM/2017/32634 Editor(s): (1) Firdous Ahmad Shah, Department of Mathematics, University of Kashmir, South Campus, Anantnag, Jammu and Kashmir, India. Reviewers: (1) William H. Sulis, University of Waterloo, Waterloo, Canada and McMaster University, Hamilton, Canada. (2) Andrzej Wisnicki, Rzeszow University of Technology, Rzeszow, Poland. (3) Srijanani Anurag Prasad, The Northcap University, Gurgaon, India. (4) Sergei Soldatenko, Institute for Informatics and Automation of the Russian Academy of Sciences, Russia. Complete Peer review History: http://www.sciencedomain.org/review-history/18676 Received: 6th March 2017 Accepted: 11th April 2017 Original Research Article Published: 17th April 2017 Abstract In this work we prove the existence of invariant compact sets under the action of nonexpansive iterated functions systems on normed spaces. We generalized the previous result to convex metric spaces and we also show that uniqueness of invariant compact sets is not guaranteed. Finally, in the last section we prove a result that provides a method for calculating an invariant set under nonexpansive iterated functions systems which is maximal with respect to inclusion. Keywords: Nonexpansive; iterated function systems; compact sets. 2010 Mathematics Subject Classification: 47H09, 47H10. *Corresponding author: E-mail: [email protected]; Solis and Ojeda-Gomez; ARJOM, 3(4), 1-10, 2017; Article no.ARJOM.32634 1 Introduction In the decade 1980-1990, John E. Hutchinson introduced a new way to describe fractal sets by treating them as fixed points of hyperbolic iterated function systems (IFS) acting on complete metric spaces using the Banach fixed-point theorem [1]. Later, M. Barnsley gave important results and found many applications for their use, see for example [2, 3, 4, 5]. Subsequently, many authors have studied variations of Hutchinson's results by modifying the space where the iterative function systems act, by choosing diverse contractionary IFS applications or by modifying the properties of such applications, see for example [6, 7, 8, 9, 10, 11]. It is important to remark that different analytical techniques have been used in all these studies. Our goal in this work is to prove the existence and potential uniqueness of fixed points, invariant compact sets, of iterate functions systems generated by nonexpansive applications. Here we follow Hutchinson's philosophy of using fixed point theorems. The importance of this work resides in the fact that there is only basic comprehension on the existence and/or uniqueness of fixed points of the Hutchinson-Barnsley operator induced by nonexpansive IFS. Moreover, uniqueness can not be guaranteed unlike the hyperbolic IFS, and nonexpansive applications are one of the main fundamental pieces of study of Fixed point theory and they are especially important as new computational techniques for solving mathematical problems. The structure of the paper is as follows: In the first section we give an alternative proof of a basic known result using Edelstein´sfixed point theorem in order to show its use in the theory of nonhyperbolic systems of iterated functions. In the second section we prove a theorem that guarantees the existence of invariant compact sets under the action of nonexpansive IFS on normed spaces by using a result from [12]. In Section 3 we generalized the previous result to convex metric spaces along with a simple example to show that uniqueness is not guaranteed. Finally, in the last section we prove a result that provides a method for calculating an invariant set under a nonexpansive IFS which is maximal with respect to inclusion. 2 Preliminaries and Strict Contractions Let (X; d) be a complete metric space. A hyperbolic iterated function system (IFS) is a finite set N of contractive maps fwigi=1 over (X; d). We denote by H(X) to the set of all non-empty compact subsets of (X; d). Such space can be metrizable with Hausdorf's metric h defined by h(A; B) = d(A; B) _ d(B; A); where x _ y denotes the maximum between x and y, d(A; B) = max d(a; B) and d(a; B) = min d(a; b): a2A b2B Hutchinson proved, in [1], that when (X; d) is complete then (H(X); h) is also complete. The metric h has several properties (see [2]), in particular given A; B; C and D in H(X), h(A [ B; C [ D) ≤ h(A; C) _ h(B; D): (2.1) Each IFS induces a transformation W : H(X) ! H(X) such that for each element C of H(X), [N W (C) = wi(C): i=1 Such map is known as Hutchinson-Barnsley's operator which is a contraction on H(X). It is known that (see [1]) there is a unique element A in H(X) that is a fixed point of W , that is, [N A = wi(A): i=1 2 Solis and Ojeda-Gomez; ARJOM, 3(4), 1-10, 2017; Article no.ARJOM.32634 As a first step we prove theorem 2.2 from [6], using Edelstein's fixed point theorem in order to show its diverse uses in the theory of nonhyperbolic systems of iterated functions. Recall that an application f that acts over a metric space (X; d) is a strict contraction if d(fx; fy) < d(x; y) 8x; y 2 X: Theorem 2.1 (Edelstein). Let f be a strict contraction over a compact metric space (X; d). Then n f has a unique fixed point x0 and for all x on X, f x ! x0 as n ! 1. It is important to recall that if X is compact then H(X) is also compact (see [2]). The theorem to prove is the following: Theorem 2.2 ([8]). Let (X; d) be a compact metric space, N 2 N and wi : X ! X with i 2 f1;:::;Ng strict contractions. Then W : H(X) ! H(X) has a unique fixed point, that is, there exists a unique compact subset A of X which is invariant with respect to the IFS. Moreover, for each C 2 H(X), W n(C) ! A when n ! 1. Proof. Let w be a strict contraction on X. Let B; C elements of H(X) and x 2 B. Since C is compact and d(x; ·) is continuous, there exists y∗ in C such that ∗ ∗ d(wx; w(C)) ≤ d(wx; wy ) < d(x; y ) = d(x; C): Thus, d(wx; w(C)) < d(x; C). Analogously, by being w(B) compact and d(·; w(C)) continuous, there exists x∗ 2 B such that ∗ ∗ d(w(B); w(C)) = d(wx ; w(C)) < d(x ;C) ≤ d(B; C); that is, d(w(B); w(C)) < d(B; C). In the same way it can be proven that d(w(C); w(B)) < d(C; B). Which implies that h(w(B); w(C)) < h(B; C). By induction and using (2.1), it follows that the N operator W defined by fwigi=1 is a strict contraction on H(X). Finally, since H(X) is compact and W is a strict contraction, the result follows from theorem 2.1. So far, theorem 2.2 is just another example of using fixed point theorems in the theory of IFS. 3 Nonexpansive IFS Over Normed Spaces We begin by recalling that a map f : X ! X is nonexpansive if for all x; y 2 X, d(fx; fy) ≤ d(x; y). N Proposition 3.1. Let(X; d) be a complete metric space. If fwigi=1 are nonexpansive maps on (X; d), then the induced operator W is a nonexpansive map on (H(X); h). Proof. See [2]. Let us recall that a convex set is a set of points such that, given any two points x; y in that set, the line xy joining them lies entirely within that set. Definition 3.1. A subset S of a linear space is called star domain or convex star if there exists a point x0 2 S such that for any other point x 2 S, the segment that joins x with x0 is contained in S. Note that any convex set is star convex. Our first theorem is as follows: Theorem 3.1. Let S ⊆ X be a compact start domain on a normed linear space X. Then any nonexpansive map on H(S) has a fixed point. In particular, the Hutchinson-Barnsley's operator defined by a nonexpansive IFS on S has a fixed point, that is, there exists a compact subset A of S such that W (A) = A: 3 Solis and Ojeda-Gomez; ARJOM, 3(4), 1-10, 2017; Article no.ARJOM.32634 Proof. Let I = [0; 1] and since S is a star domain, there exists x0 in S such that for any other element x of S, f(1 − t)x0 + txgt2I ⊆ S: We define then f : I × H(S) ! H(S) (t; A) 7! f(t; A); where f(t; A) = fz 2 X j z = (1 − t)x0 + ta; for some a 2 Ag: First, let us prove that f(t; A) 2 H(S). Given t 2 I and A 2 H(S), let z 2 f(t; A).
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