On the Lifshits constant for hyperspaces

• Autorzy: Leśniak K.
• Języki publikacji: EN

Abstrakty

EN

The Lifshits theorem states that any k-uniformly Lipschitz map with a bounded orbit on a complete metric space X has a fixed point provided k < x(X] where x(X) is the so-called Lifshits constant of X. For many spaces we have x(X) > 1. It is interesting whether we can use the Lifshits theorem in the theory of iterated function systems. Therefore we investigate the value of the Lifshits constant for several classes of hyperspaces.

Słowa kluczowe

EN

Lifshits constant; hyperspace; Hausdorff metric; Pompeiu metric; lp-product; ψ-product

• Wydawca: Instytut Matematyczny PAN
• Czasopismo: Bulletin of the Polish Academy of Sciences. Mathematics
• Rocznik: 2007
• Tom: Vol. 55, no 2
• Strony: 155--160
• Opis fizyczny: Bibliogr. 160 poz.

Twórcy

autor

Leśniak K.

• Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100 Torun, Poland,

Bibliografia

• [AG] J. Andres and L. Gorniewicz, On the Banach contraction principle for multivalued mappings, in: Approximation, Optimization and Mathematical Economics, M. Lassonde (ed.), Physica-Verlag, 2001, 1-23.
• [AT] J. M. Ayerbe Toledano, T. Dominguez Benavides and G. Lopez Acedo, Measures of Noncompactness in Metric Fixed Point Theory, Birkhauser, Basel, 1997.
• [Bl] K. Borsuk, On some metrizations of the hyperspace of compact sets, Fund. Math. 41 (1955), 168-202.
• [B2] —, On a metrization of the hyperspace of a metric space, ibid. 94 (1977), 191-207.
• [E] A. Edalat, Dynamical systems, measures and fractals via domain theory, Inform. Comput. 120 (1995), 32-48.
• [GK] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge Univ. Press, 1990.
• [G] L. Gorniewicz, Present state of the Brouwer fixed point theorem for multivalued mappings, Ann. Sci. Math. Quebec 22 (1998), 169-179.
• [H] M. Hata, On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A 61 (1985), 99-102.
• [Ha] S. Hayashi, Self-similar sets as Tarski's fixed points, Publ. RIMS Kyoto Univ. 21 (1985), 1059-1066.
• [Hu] J. E. Hutchinson, Fractals and self similarity, Indiana Univ. Math. J. 30 (1981), 713-747.
• [IN] A. Illanes and S. B. Nadler Jr., Hyperspaces. Fundamentals and Recent Advances, Dekker, New York, 1999.
• [KST] M. Kato, K.-S. Saito and T. Tamura, Uniform non-squareness of φ-direct sums of Banach spaces X [direct sum]φ Y, Math. Inequal. Appl. 7 (2004), 429-437.
• [LM] A. Lasota and J. Myjak, Attractors of multifunction, Bull. Polish Acad. Sci. Math. 48 (2000), 319-334.
• [LI] K. Leśniak, Extremal sets as fractals, Nonlinear Anal. Forum 7 (2002), 199-208.
• [L2] —, Towards computing Lifshits constant for hyperspaces, Fixed Point Theory 4 (2003), 159-163.
• [LF] E. Llorens Fuster, Moduli and constants.,. what a show!, http://www.uv.es/ llorens, 2006; extended version of: Some moduli and constants related to metric fixed point theory, in: Handbook of Metric Fixed Point Theory, W. A. Kirk and B. Sims (eds.), Kluwer, Dordrecht, 2001, 133-175.
• [M] D. Miklaszewski, The Role of Various Kinds of Continuity in The Fixed Point Theory of Set-Valued Mappings, Lecture Notes in Nonlinear Anal. 7, Juliusz Schauder Center for Nonlinear Studies, Nicolaus Copernicus Univ., Toruń, 2005.
• [O] E. A. Ok, Fixed set theory for closed correspondences with applications to self-similarity and games, Nonlinear Anal. 56 (2004), 309-330.
• [WW] J. Wiśnicki and J. Wośko, On relative Hausdorff mesures of noncompactness and relative Chebyshev radii in Banach spaces, Proc. Amer. Math. Soc. 124 (1996), 2465-2474.