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We prove that a compact family of bounded condensing multifunctions has bounded condensing set-theoretic union. Compactness is understood in the sense of the Chebyshev uniform semimetric induced by the Hausdorff distance and condensity is taken w.r.t. the Hausdorff measure of noncompactness. As a tool, we present an estimate for the measure of an infinite union. Then we apply our result to infinite iterated function systems.
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Tom
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1--8
Opis fizyczny
Bibliogr. 19 poz.
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autor
- Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Toruń, Poland, much@mat.uni.torun.pl
Bibliografia
- [AKPRS] R. R. Akhmerov, M. I. Kamenskiĭ, A. S. Potapov, A. E. Rodkina and B. N. Sadovskiĭ, Measures of Noncompactness and Condensing Operators, Nauka, Novosibirsk, 1986 (in Russian).
- [AF] J. Andres and J. Fišer, Metric and Topological Multivalued Fractals, Internat. J. Bifur. Chaos 14 (2004), to appear.
- [AG] J. Andres and L. Górniewicz, On the Banach contraction principle for multivalued mappings, in: Approximation, Optimization and Mathematical Economics, M. Lassonde, (ed.), Physica-Verlag and Springer, 2001, 1-23.
- [CV] C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Math. 580, Springer, Berlin, 1977.
- [CoV] C. Costantini and P. Vitolo, Decomposition of topologies on lattices and hyperspaces, Dissertationes Math. 381 (1999).
- [D] K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, 1985.
- [E] A. Edalat, Dynamical systems, measures and fractals via domain theory, Inform. and Comput. 120 (1995), 32-48.
- [H] M. Hata, On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci. 61 (1985), 99-102.
- [Ha] S. Hayashi, Self-similar sets as Tarski's fixed points, Publ. RIMS Kyoto Univ. 21 (1985), 1059-1066.
- [HP] S. Hu and N. S. Papageorgiou, Handbook of Multivalued Analysis, Volume I : Theory, Math. Appl., Kluwer, Dordrecht, 1997.
- [Hu] J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
- [JGP] J. Jachymski, L. Gajek and P. Pokarowski, The Tarski-Kantorovitch principle and the theory of iterated function systems, Bull. Austral. Math. Soc. 61 (2000), 247-261.
- [K] B. Kieninger, Iterated Function Systems on Compact Hausdorff Spaces, Berichte Math., Shaker-Verlag, Aachen, 2002.
- [LM] A. Lasota and J. Myjak, Attractors of multifunctions, Bull. Polish Acad. Sci. Math. 48 (2000), 319-334.
- [L1] K. Leśniak, Extremal sets as fractals, Nonlinear Anal. Forum 7 (2002), 199-208.
- [L2] -, Stability and invariance of multivalued iterated function systems, Math. Slovaca 53 (2003), 393-405.
- [S] V. Šeda, On condensing discrete dynamical systems, Math. Bohem. 125 (2000), 275-306.
- [SV] J. Soto-Andrade and F. J. Varela, Self-reference and fixed points: A discussion and an extension of Lawvere's theorem, Acta Appl. Math. 2 (1984), 1-19.
- [W] K. R. Wicks, Fractals and Hyperspaces, Lecture Notes in Math. 1492, Springer, Berlin, 1991.
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Bibliografia
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bwmeta1.element.baztech-article-BAT5-0004-0001