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.
Similar content being viewed by others
References
Abian S., Brown A.B., A theorem on partially ordered sets, with applications to fixed point theorems, Canad. J. Math., 1961, 13, 78–82
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
Akin E., The General Topology of Dynamical Systems, Grad. Stud. Math., 1, American Mathematical Society, Providence, 1993
Andres J., Fišer J., Metric and topological multivalued fractals, Internat. J. Bifur. Chaos Appl. Sci. Engrg., 2004, 14(4), 1277–1289
Andres J., Fišer J., Gabor G., Leśniak K., Multivalued fractals, Chaos Solitons Fractals, 2005, 24(3), 665–700
Andres J., Górniewicz L., Topological Fixed Point Principles for Boundary Value Problems, Topol. Fixed Point Theory Appl., 1, Kluwer, Dordrecht, 2003
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
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
Banaś J., Goebel K., Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Appl. Math., 60, Marcel Dekker, New York, 1980
Bandt Ch., On the metric structure of hyperspaces with Hausdorff metric, Math. Nachr., 1986, 129, 175–183
Barbashin E.A., On the theory of general dynamical systems, Učenye Zap. Moskov. Gos. Univ. Matematika, 1948, 135(2), 110–133 (in Russian)
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
Barnsley M.F., Vince A., Real projective iterated function systems, J. Geom. Anal. (in press), DOI: 10.1007/s12220-011-9232-x
Beer G., Topologies on Closed and Closed Convex Sets, Math. Appl., 268, Kluwer, Dordrecht, 1993
Birkhoff G.D., Dynamical Systems, American Mathematical Society, New York, 1927
Bloom S.L., Ésik Z., The equational logic of fixed points, Theoret. Comput. Sci., 1997, 179(1–2), 1–60
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
Carl S., Heikkilä S., Fixed Point Theory in Ordered Sets and Applications, Springer, New York, 2011
Chueshov I.D., Introduction to the Theory of Infinite-Dimensional Dissipative Systems, Univ. Lektsii Sovrem. Mat., AKTA, Kharkov, 1999 (in Russian)
Conley C., Isolated Invariant Sets and the Morse Index, CBMS Regional Conf. Ser. in Math., 38, American Mathematical Society, Providence, 1978
Conley C., Easton R., Isolated invariant sets and isolating blocks, Trans. Amer. Math. Soc., 1971, 158, 35–61
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
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
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
Edalat A., Dynamical systems, measures and fractals via domain theory, Inform. and Comput., 1995, 120(1), 32–48
Edwards R.E., Functional Analysis. Theory and Applications, Holt, Rinehart and Winston, New York-Toronto-London, 1965
Elton J.H., An ergodic theorem for iterated maps, Ergodic Theory Dynam. Systems, 1987, 7(4), 481–488
Fečkan M., Topological Degree Approach to Bifurcation Problems, Topol. Fixed Point Theory Appl., 5, Springer, New Yok, 2008
Fleiner T., A fixed-point approach to stable matchings and some applications, Math. Oper. Res., 2003, 28(1), 103–126
Góebel K., Kirk W.A., Topics in Metric Fixed Point Theory, Cambridge Stud. Adv. Math., 28, Cambridge University Press, Cambridge, 1990
Górniewicz L., Topological Fixed Point Theory of Multivalued Mappings, 2nd ed., Topol. Fixed Point Theory Appl., 4, Springer, Dordrecht, 2006
Granas A., Dugundji J., Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003
Hale J.K., Asymptotic Behavior of Dissipative Systems, Math. Surveys Monogr., 25, American Mathematical Society, Providence, 1988
Hata M., On some properties of set-dynamical systems, Proc. Japan Acad. Ser. A Math. Sci., 1985, 61(4), 99–102
Hayashi S., Self-similar sets as Tarski’s fixed points, Publ. Res. Inst. Math. Sci., 1985, 21(5), 1059–1066
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
Heikkilä S., Fixed point results and their applications to Markov processes, Fixed Point Theory Appl., 2005, 3, 307–320
Hitzler P., Seda A.K., Generalized metrics and uniquely determined logic programs, Theoret. Comput. Sci., 2003, 305(1–3), 187–219
Hu S., Papageorgiou N.S., Handbook of Multivalued Analysis I, Math. Appl., 419, Kluwer, Dordrecht, 1997
Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J., 1981, 30(5), 713–747
Illanes A., Nadler S.B. Jr., Hyperspaces, Monogr. Textbooks Pure Appl. Math., 216, Marcel Dekker, New York, 1999
Iosifescu M., Iterated function systems. A critical survey, Math. Rep. (Bucur.), 2009, 11(61)(3), 181–229
Jachymski J., Order-theoretic aspects of metric fixed point theory, In: Handbook of Metric Fixed Point Theory, Kluwer, Dordrecht, 2001, 613–641
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
Joseph J.E., Multifunctions and graphs, Pacific J. Math., 1978, 79(2), 509–529
Kantorovitch L., The method of successive approximation for functional equations, Acta Math., 1939, 71, 63–97
Kieninger B., Iterated Function Systems on Compact Hausdorff Spaces, PhD thesis, Universität Augsburg, 2002
Knaster B., Un théorème sur les fonctions d’ensembles, Annales de la Société Polonaise de Mathématique, 1928, 6, 133–134
Krasnosel’skii M.A., Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964
Kuratowski C., Topologie I, Monogr. Mat., 20, PWN, Warsaw, 1958
Lasota A., Myjak J., Attractors of multifunctions, Bull. Polish Acad. Sci. Math., 2000, 48(3), 319–334
Leśniak K., Extremal sets as fractals, Nonlinear Anal. Forum, 2002, 7(2), 199–208
Leśniak K., Stability and invariance of multivalued iterated function systems, Math. Slovaca, 2003, 53(4), 393–405
Leśniak K., Infinite iterated function systems: a multivalued approach, Bull. Pol. Acad. Sci. Math., 2004, 52(1), 1–8
Leśniak K., Fixed points of the Barnsley-Hutchinson operators induced by hyper-condensing maps, Matematiche (Catania), 2005, 60(1), 67–80
Leśniak K., On the Lifshits constant for hyperspaces, Bull. Pol. Acad. Sci. Math., 2007, 55(2), 155–160
Leśniak K., Note on the Kuratowski theorem for abstract measures of noncompactness, preprint available at www-users.mat.umk.pl/_much/works/kuratow.ps
Marchaud A., Sur les champs de demi-droites et les équations différentielles du premier ordre, Bull. Soc. Math. France, 1934, 62, 1–38
Mauldin R.D., Urbański M., Dimensions and measures in infinite iterated function systems, Proc. London Math. Soc. (3), 1996, 73(1), 105–154
McGehee R., Attractors for closed relations on compact Hausdorff spaces, Indiana Univ. Math. J., 1992, 41(4), 1165–1209
Melnik V.S., Valero J., On attractors of multivalued semi-flows and differential inclusions, Set-Valued Anal., 1998, 6(1), 83–111
Ok E.A., Fixed set theory for closed correspondences with applications to self-similarity and games, Nonlinear Anal., 2004, 56(3), 309–330
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
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
Ponomarev V.I., On common fixed set for two continuous multivalued selfmappings of bicompactum, Colloq. Math. 1963, 10, 227–231 (in Russian)
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
Schröder B.S.W., Algorithms for the fixed point property, Theoret. Comput. Sci., 1999, 217(2), 301–358
Šeda V., On condensing discrete dynamical systems, Math. Bohem., 2000, 125(3), 275–306
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
Stenflo Ö., A survey of average contractive iterated function systems, J. Difference Equ. Appl., 2012, 18(8), 1355–1380
Strother W., Fixed points, fixed sets, and M-retracts, Duke Math. J., 1955, 22(4), 551–556
Tarafdar E.U., Chowdhury M.S.R., Topological Methods for Set-Valued Nonlinear Analysis, World Scientific, Hackensack, 2008
Tarski A., A lattice-theoretical fixpoint theorem and its applications, Pacific J. Math., 1955, 5(2), 285–309
de Vries J., Elements of Topological Dynamics, Math. Appl., 257, Kluwer, Dordrecht, 1993
Waszkiewicz P., Kostanek M., Reconciliation of elementary order and metric fixpoint theorems (manuscript)
Wicks K.R., Fractals and Hyperspaces, Lecture Notes in Math., 1492, Springer, Berlin, 1991
Yamaguchi M., Hata M., Kigami J., Transl. Math. Monogr., 167, Mathematics of Fractals, American Mathematical Society, Providence, 1997
Zaremba S.C., Sur les équations au paratingent, Bull. Sci. Math., 1936, 60, 139–160
Author information
Authors and Affiliations
Corresponding author
About this article
Cite this article
Leśniak, K. Invariant sets and Knaster-Tarski principle. centr.eur.j.math. 10, 2077–2087 (2012). https://doi.org/10.2478/s11533-012-0109-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.2478/s11533-012-0109-4