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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 10, Issue 6 (Dec 2012)

Issues

Invariant sets and Knaster-Tarski principle

Krzysztof Leśniak
  • Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Chopina 12/18, 87-100, Toruń, Poland
  • Email:
Published Online: 2012-10-12 | DOI: https://doi.org/10.2478/s11533-012-0109-4

Abstract

Our aim is to point out the applicability of the Knaster-Tarski fixed point principle to the problem of existence of invariant sets in discrete-time (multivalued) semi-dynamical systems, especially iterated function systems.

MSC: 54H25; 47H08; 54H20

Keywords: Invariant in closure set; Barnsley-Hutchinson operator; Fixed point; Monotone map; Measure of noncompactness; Strongly condensing multifunction

  • [1] Abian S., Brown A.B., A theorem on partially ordered sets, with applications to fixed point theorems, Canad. J. Math., 1961, 13, 78–82 http://dx.doi.org/10.4153/CJM-1961-007-5CrossrefGoogle Scholar

  • [2] Akhmerov R.R., Kamenskiĭ M.I., Potapov A.S., Rodkina A.E., Sadovskiĭ B.N., Measures of Noncompactness and Condensing Operators, Oper. Theory Adv. Appl., 55, Birkhäuser, Basel, 1992 Google Scholar

  • [3] Akin E., The General Topology of Dynamical Systems, Grad. Stud. Math., 1, American Mathematical Society, Providence, 1993 Google Scholar

  • [4] Andres J., Fišer J., Metric and topological multivalued fractals, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2004, 14(4), 1277–1289 http://dx.doi.org/10.1142/S021812740400979XCrossrefGoogle Scholar

  • [5] Andres J., Fišer J., Gabor G., Leśniak K., Multivalued fractals, Chaos Solitons Fractals, 2005, 24(3), 665–700 http://dx.doi.org/10.1016/j.chaos.2004.09.029CrossrefGoogle Scholar

  • [6] Andres J., Górniewicz L., Topological Fixed Point Principles for Boundary Value Problems, Topol. Fixed Point Theory Appl., 1, Kluwer, Dordrecht, 2003 Google Scholar

  • [7] Andres J., Väth M., Calculation of Lefschetz and Nielsen numbers in hyperspaces for fractals and dynamical systems, Proc. Amer. Math. Soc., 2007, 135(2), 479–487 http://dx.doi.org/10.1090/S0002-9939-06-08505-4CrossrefGoogle Scholar

  • [8] Ayerbe Toledano J.M., Domínguez Benavides T., López Acedo G., Measures of Noncompactness in Metric Fixed Point Theory, Oper. Theory Adv. Appl., 99, Birkhäuser, Basel, 1997 http://dx.doi.org/10.1007/978-3-0348-8920-9CrossrefGoogle Scholar

  • [9] Banaś J., Goebel K., Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math., 60, Marcel Dekker, New York, 1980 Google Scholar

  • [10] Bandt Ch., On the metric structure of hyperspaces with Hausdorff metric, Math. Nachr., 1986, 129, 175–183 http://dx.doi.org/10.1002/mana.19861290116CrossrefGoogle Scholar

  • [11] Barbashin E.A., On the theory of general dynamical systems, Učenye Zap. Moskov. Gos. Univ. Matematika, 1948, 135(2), 110–133 (in Russian) Google Scholar

  • [12] Barnsley M.F., Demko S., Iterated function systems and the global construction of fractals, Proc. Roy. Soc. London Ser. A, 1985, 399(1817), 243–275 http://dx.doi.org/10.1098/rspa.1985.0057CrossrefGoogle Scholar

  • [13] Barnsley M.F., Vince A., Real projective iterated function systems, J. Geom. Anal. (in press), DOI: 10.1007/s12220-011-9232-x CrossrefGoogle Scholar

  • [14] Beer G., Topologies on Closed and Closed Convex Sets, Math. Appl., 268, Kluwer, Dordrecht, 1993 Google Scholar

  • [15] Birkhoff G.D., Dynamical Systems, American Mathematical Society, New York, 1927 Google Scholar

  • [16] Bloom S.L., Ésik Z., The equational logic of fixed points, Theoret. Comput. Sci., 1997, 179(1–2), 1–60 http://dx.doi.org/10.1016/S0304-3975(96)00248-4CrossrefGoogle Scholar

  • [17] Bogdewicz A., Herburt I., Moszyńska M., Quotient metrics with applications in convex geometry, Beitr. Algebra Geom. (in press), DOI: 10.1007/s13366-011-0082-2 CrossrefGoogle Scholar

  • [18] Carl S., Heikkilä S., Fixed Point Theory in Ordered Sets and Applications, Springer, New York, 2011 http://dx.doi.org/10.1007/978-1-4419-7585-0CrossrefGoogle Scholar

  • [19] Chueshov I.D., Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Univ. Lektsii Sovrem. Mat., AKTA, Kharkov, 1999 (in Russian) Google Scholar

  • [20] Conley C., Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math., 38, American Mathematical Society, Providence, 1978 Google Scholar

  • [21] Conley C., Easton R., Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 1971, 158, 35–61 http://dx.doi.org/10.1090/S0002-9947-1971-0279830-1CrossrefGoogle Scholar

  • [22] Conti G., Obukhovskiĭ V., Zecca P., On the topological structure of the solution set for a semilinear functionaldifferential inclusion in a Banach space, In: Topology in Nonlinear Analysis, Warsaw, September 5–18 and October 10–15, 1994, Banach Center Publ., 35, Polish Academy of Sciences, Institute of Mathematics, Warsaw, 1996, 159–169 Google Scholar

  • [23] De Blasi F.S., Georgiev P.Gr., Hukuhara’s topological degree for non compact valued multifunctions, Publ. Res. Inst. Math. Sci., 2003, 39(1), 183–203 http://dx.doi.org/10.2977/prims/1145476153CrossrefGoogle Scholar

  • [24] De Blasi F.S., Myjak J., A remark on the definition of topological degree for set-valued mappings, J. Math. Anal. Appl., 1983, 92(2), 445–451 http://dx.doi.org/10.1016/0022-247X(83)90261-5CrossrefGoogle Scholar

  • [25] Edalat A., Dynamical systems, measures and fractals via domain theory, Inform. and Comput., 1995, 120(1), 32–48 http://dx.doi.org/10.1006/inco.1995.1096CrossrefGoogle Scholar

  • [26] Edwards R.E., Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965 Google Scholar

  • [27] Elton J.H., An ergodic theorem for iterated maps, Ergodic Theory Dynam. Systems, 1987, 7(4), 481–488 http://dx.doi.org/10.1017/S0143385700004168CrossrefGoogle Scholar

  • [28] Fečkan M., Topological Degree Approach to Bifurcation Problems, Topol. Fixed Point Theory Appl., 5, Springer, New Yok, 2008 Google Scholar

  • [29] Fleiner T., A fixed-point approach to stable matchings and some applications, Math. Oper. Res., 2003, 28(1), 103–126 http://dx.doi.org/10.1287/moor.28.1.103.14256CrossrefGoogle Scholar

  • [30] Góebel K., Kirk W.A., Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., 28, Cambridge University Press, Cambridge, 1990 Google Scholar

  • [31] Górniewicz L., Topological Fixed Point Theory of Multivalued Mappings, 2nd ed., Topol. Fixed Point Theory Appl., 4, Springer, Dordrecht, 2006 Google Scholar

  • [32] Granas A., Dugundji J., Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003 Google Scholar

  • [33] Hale J.K., Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., 25, American Mathematical Society, Providence, 1988 Google Scholar

  • [34] Hata M., On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci., 1985, 61(4), 99–102 http://dx.doi.org/10.3792/pjaa.61.99CrossrefGoogle Scholar

  • [35] Hayashi S., Self-similar sets as Tarski’s fixed points, Publ. Res. Inst. Math. Sci., 1985, 21(5), 1059–1066 http://dx.doi.org/10.2977/prims/1195178796CrossrefGoogle Scholar

  • [36] Heikkilä S., On fixed points through a generalized iteration method with applications to differential and integral equations involving discontinuities, Nonlinear Anal., 1990, 14(5), 413–426 http://dx.doi.org/10.1016/0362-546X(90)90082-RCrossrefGoogle Scholar

  • [37] Heikkilä S., Fixed point results and their applications to Markov processes, Fixed Point Theory Appl., 2005, 3, 307–320 Google Scholar

  • [38] Hitzler P., Seda A.K., Generalized metrics and uniquely determined logic programs, Theoret. Comput. Sci., 2003, 305(1–3), 187–219 http://dx.doi.org/10.1016/S0304-3975(02)00709-0CrossrefGoogle Scholar

  • [39] Hu S., Papageorgiou N.S., Handbook of Multivalued Analysis I, Math. Appl., 419, Kluwer, Dordrecht, 1997 Google Scholar

  • [40] Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747 http://dx.doi.org/10.1512/iumj.1981.30.30055CrossrefGoogle Scholar

  • [41] Illanes A., Nadler S.B. Jr., Hyperspaces, Monogr. Textbooks Pure Appl. Math., 216, Marcel Dekker, New York, 1999 Google Scholar

  • [42] Iosifescu M., Iterated function systems. A critical survey, Math. Rep. (Bucur.), 2009, 11(61)(3), 181–229 Google Scholar

  • [43] Jachymski J., Order-theoretic aspects of metric fixed point theory, In: Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, 2001, 613–641 Google Scholar

  • [44] Jachymski J., Gajek L., Pokarowski P., The Tarski-Kantorovitch principle and the theory of iterated function systems, Bull. Austral. Math. Soc., 2000, 61(2), 247–261 http://dx.doi.org/10.1017/S0004972700022243CrossrefGoogle Scholar

  • [45] Joseph J.E., Multifunctions and graphs, Pacific J. Math., 1978, 79(2), 509–529 Google Scholar

  • [46] Kantorovitch L., The method of successive approximation for functional equations, Acta Math., 1939, 71, 63–97 http://dx.doi.org/10.1007/BF02547750CrossrefGoogle Scholar

  • [47] Kieninger B., Iterated Function Systems on Compact Hausdorff Spaces, PhD thesis, Universität Augsburg, 2002 Google Scholar

  • [48] Knaster B., Un théorème sur les fonctions d’ensembles, Annales de la Société Polonaise de Mathématique, 1928, 6, 133–134 Google Scholar

  • [49] Krasnosel’skii M.A., Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964 Google Scholar

  • [50] Kuratowski C., Topologie I, Monogr. Mat., 20, PWN, Warsaw, 1958 Google Scholar

  • [51] Lasota A., Myjak J., Attractors of multifunctions, Bull. Polish Acad. Sci. Math., 2000, 48(3), 319–334 Google Scholar

  • [52] Leśniak K., Extremal sets as fractals, Nonlinear Anal. Forum, 2002, 7(2), 199–208 Google Scholar

  • [53] Leśniak K., Stability and invariance of multivalued iterated function systems, Math. Slovaca, 2003, 53(4), 393–405 Google Scholar

  • [54] Leśniak K., Infinite iterated function systems: a multivalued approach, Bull. Pol. Acad. Sci. Math., 2004, 52(1), 1–8 http://dx.doi.org/10.4064/ba52-1-1CrossrefGoogle Scholar

  • [55] Leśniak K., Fixed points of the Barnsley-Hutchinson operators induced by hyper-condensing maps, Matematiche (Catania), 2005, 60(1), 67–80 Google Scholar

  • [56] Leśniak K., On the Lifshits constant for hyperspaces, Bull. Pol. Acad. Sci. Math., 2007, 55(2), 155–160 http://dx.doi.org/10.4064/ba55-2-6CrossrefGoogle Scholar

  • [57] Leśniak K., Note on the Kuratowski theorem for abstract measures of noncompactness, preprint available at www-users.mat.umk.pl/_much/works/kuratow.ps Google Scholar

  • [58] Marchaud A., Sur les champs de demi-droites et les équations différentielles du premier ordre, Bull. Soc. Math. France, 1934, 62, 1–38 Google Scholar

  • [59] Mauldin R.D., Urbański M., Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 1996, 73(1), 105–154 http://dx.doi.org/10.1112/plms/s3-73.1.105CrossrefGoogle Scholar

  • [60] McGehee R., Attractors for closed relations on compact Hausdorff spaces, Indiana Univ. Math. J., 1992, 41(4), 1165–1209 http://dx.doi.org/10.1512/iumj.1992.41.41058CrossrefGoogle Scholar

  • [61] Melnik V.S., Valero J., On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 1998, 6(1), 83–111 http://dx.doi.org/10.1023/A:1008608431399CrossrefGoogle Scholar

  • [62] Ok E.A., Fixed set theory for closed correspondences with applications to self-similarity and games, Nonlinear Anal., 2004, 56(3), 309–330 http://dx.doi.org/10.1016/j.na.2003.08.001CrossrefGoogle Scholar

  • [63] d’Orey V., Fixed point theorems for correspondences with values in a partially ordered set and extended supermodular games, J. Math. Econom., 1996, 25(3), 345–354 http://dx.doi.org/10.1016/0304-4068(95)00728-8CrossrefGoogle Scholar

  • [64] Petruşel A., Rus I.A., Dynamics on (P cp(X), H d) generated by a finite family of multi-valued operators on (X, d), Math. Morav., 2001, 5, 103–110 Google Scholar

  • [65] Ponomarev V.I., On common fixed set for two continuous multivalued selfmappings of bicompactum, Colloq. Math. 1963, 10, 227–231 (in Russian) Google Scholar

  • [66] Ran A.C.M., Reurings M.C.B., A fixed point theorem in partially ordered sets and some applications to matrix equations, Proc. Amer. Math. Soc., 2004, 132(5), 1435–1443 http://dx.doi.org/10.1090/S0002-9939-03-07220-4CrossrefGoogle Scholar

  • [67] Schröder B.S.W., Algorithms for the fixed point property, Theoret. Comput. Sci., 1999, 217(2), 301–358 http://dx.doi.org/10.1016/S0304-3975(98)00273-4CrossrefGoogle Scholar

  • [68] Šeda V., On condensing discrete dynamical systems, Math. Bohem., 2000, 125(3), 275–306 Google Scholar

  • [69] Soto-Andrade J., Varela F.J., Self-reference and fixed points: a discussion and an extension of Lawvere’s theorem, Acta Appl. Math., 1984, 2(1), 1–19 http://dx.doi.org/10.1007/BF01405490CrossrefGoogle Scholar

  • [70] Stenflo Ö., A survey of average contractive iterated function systems, J. Difference Equ. Appl., 2012, 18(8), 1355–1380 http://dx.doi.org/10.1080/10236198.2011.610793CrossrefGoogle Scholar

  • [71] Strother W., Fixed points, fixed sets, and M-retracts, Duke Math. J., 1955, 22(4), 551–556 http://dx.doi.org/10.1215/S0012-7094-55-02261-4CrossrefGoogle Scholar

  • [72] Tarafdar E.U., Chowdhury M.S.R., Topological Methods for Set-Valued Nonlinear Analysis, World Scientific, Hackensack, 2008 http://dx.doi.org/10.1142/6347CrossrefGoogle Scholar

  • [73] Tarski A., A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math., 1955, 5(2), 285–309 Google Scholar

  • [74] de Vries J., Elements of Topological Dynamics, Math. Appl., 257, Kluwer, Dordrecht, 1993 Google Scholar

  • [75] Waszkiewicz P., Kostanek M., Reconciliation of elementary order and metric fixpoint theorems (manuscript) Google Scholar

  • [76] Wicks K.R., Fractals and Hyperspaces, Lecture Notes in Math., 1492, Springer, Berlin, 1991 Google Scholar

  • [77] Yamaguchi M., Hata M., Kigami J., Transl. Math. Monogr., 167, Mathematics of Fractals, American Mathematical Society, Providence, 1997 Google Scholar

  • [78] Zaremba S.C., Sur les équations au paratingent, Bull. Sci. Math., 1936, 60, 139–160 Google Scholar

About the article

Published Online: 2012-10-12

Published in Print: 2012-12-01


Citation Information: Open Mathematics, ISSN (Online) 2391-5455, DOI: https://doi.org/10.2478/s11533-012-0109-4.

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