# Article

 Title: On condensing discrete dynamical systems (English) Author: Šeda, Valter Language: English Journal: Mathematica Bohemica ISSN: 0862-7959 (print) ISSN: 2464-7136 (online) Volume: 125 Issue: 3 Year: 2000 Pages: 275-306 Summary lang: English . Category: math . Summary: In the paper the fundamental properties of discrete dynamical systems generated by an $\alpha$-condensing mapping ($\alpha$ is the Kuratowski measure of noncompactness) are studied. The results extend and deepen those obtained by M. A. Krasnosel'skij and A. V. Lusnikov in \cite{21}. They are also applied to study a mathematical model for spreading of an infectious disease investigated by P. Takac in \cite{35}, \cite{36}. (English) Keyword: condensing discrete dynamical system Keyword: stability Keyword: singular interval Keyword: continuous branch connecting two points Keyword: continuous curve MSC: 34C25 MSC: 37B05 MSC: 47H07 MSC: 47H09 MSC: 47H10 MSC: 58F08 MSC: 58F22 idZBL: Zbl 0972.37009 idMR: MR1790121 DOI: 10.21136/MB.2000.126130 . Date available: 2009-09-24T21:43:36Z Last updated: 2020-07-29 Stable URL: http://hdl.handle.net/10338.dmlcz/126130 . Reference: [1] R. R. Achmerov M. I. Kamenskij A. S. Potapov, others: Measures of Noncompactness and Condensing Operators.Nauka, Novosibirsk, 1986. (In Russian.) Reference: [2] H. Amann: Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces.Siam Rev. 18 (1976), 620-709. Zbl 0345.47044, MR 0415432, 10.1137/1018114 Reference: [3] H. Amann: Gewöhnliche Differentialgleichungen.Walter de Gruyter, Berlin, 1983. MR 0713040 Reference: [4] J. P. Aubin A. Cellina: Differential Inclusions.Springer, Berlin, 1984. MR 0755330 Reference: [5] L. S. Block W. A. Coppel: Dynamics in One Dimension.Lecture Notes in Math., vol. 1513, Springer, Berlin, 1992. MR 1176513 Reference: [6] E. Čech: Point Sets.Academia, Praha, 1974. (In Czech.) Reference: [7] W. A. Coppel: Stability and Asymptotic Behavior of Differential Equations.D. C. Heath and Co., Boston, 1965. Zbl 0154.09301, MR 0190463 Reference: [8] J. L. Davy: Properties of the solutions of a generalized differential equation.Bull. Austral. Math. Soc. 6 (1972), 379-398. MR 0303023, 10.1017/S0004972700044646 Reference: [9] K. Deimling: Nonlinear Functional Analysis and Its Applications.Springer, Berlin, 1985. MR 0787404 Reference: [10] B. P. Demidovič: Lectures on Mathematical Theory of Stability.Nauka, Moskva, 1967. (In Russian.) MR 0226126 Reference: [11] R. Engelking: Outline of General Topology.North-Holland Publ. Co., Amsterdam, PWN-Polish Scientific Publishers, 1968. Zbl 0157.53001, MR 0230273 Reference: [12] M. Fukuhara: Sur une généralization d'un théorème de Kneser.Proc. Japan Acad. 29 (1953), 154-155. MR 0060084 Reference: [13] L. Górniewicz D. Rozploch-Nowakowska: On the Schauder fixed point theorem.Topology in Nonlinear Analysis. Banach Center Publications, vol. 35, Inst. Math., Polish Academy of Sciences, Warszawa, 1996. MR 1448438 Reference: [14] P. R. Halmos: Naive Set Theory.Springer, New York Inc., 1974. Zbl 0287.04001, MR 0453532 Reference: [15] A. Haščák: Fixed point theorems for multivalued mappings.Czechoslovak Math. J. 35 (1985), 533-542. MR 0809039 Reference: [16] P. Hess: Periodic-parabolic Boundary Value Problems and Positivity.Pitman Research Notes in Mathematics. Longman Sci and Tech., Burnt Mill, Harlow, 1991. Zbl 0731.35050, MR 1100011 Reference: [17] M. A. Krasnosel'skij A. I. Perov: On the existence of solutions of certain nonlinear operator equations.Dokl. Akad. Nauk SSSR 126 (1959), 15-18. (In Russian.) MR 0106421 Reference: [18] M. A. Krasnosel'skij G. M. Vajnikko P. P. Zabrejko, Ja. B. Rutickij V. Ja. Stecenko: Approximate Solutions of Operator Equations.Nauka, Moskva, 1969. (In Russian.) Reference: [19] M. A. Krasnosel'skij P. P. Zabrejko: Geometric Methods of Nonlinear Analysis.Nauka, Moskva, 1975. (In Russian.) Reference: [20] M. A. Krasnosel'skij E.A. Lifšitc A. V. Sobolev: Positive Linear Systems: Method of Positive Operators.Nauka, Moskva, 1985. (In Russian.) Reference: [21] M. A. Krasnosel'skij A. V. Lusnikov: Fixed points with special properties.Dokl. Akad. Nauk 345 (1995), 303-305. (In Russian.) MR 1372832 Reference: [22] Z. Kubáček: A generalization of N. Arouszajn's theorem on connectedness of the fixed point set of a compact mapping.Czechoslovak Math. J. 37 (1987), 415 423. MR 0904769 Reference: [23] Z. Kubáček: On the structure of fixed point sets of some compact maps in the Fréchet space.Math. Bohem. 118 (1993), 343-358. MR 1251881 Reference: [24] C. Kuratowski: Topologie. Vol. II.Pol. Tow. Mat., Warszawa, 1952. Zbl 0049.39704, MR 0054232 Reference: [25] A. Pelczar: Introduction to Theory of Differential Equations. Part 2. Elements of the Qualitative Theory of Differential Equations.PWN. Warszawa 1989. (In Polish.) Reference: [26] V. A. Pliss: Nonlocal Problems of Oscillation Theory.Nauka, Moskva, 1904. (In Russian). Reference: [27] P. Poláčik I. Tereščák: Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems.Arch. Rational Mech. Anal. 116 (1991), 339-361. MR 1132766, 10.1007/BF00375672 Reference: [28] N. Rouche P. Habets M. Laloy: Stability Theory by Liapunov's Direct Method.Springer, New York, 1977. MR 0450715 Reference: [29] N. Rouche J. Mawhin: Équations Différentielles Ordinaires, Tome II, Stabilité et Solutions Périodiques.Masson et Cie, Paris. 1973. MR 0481182 Reference: [30] B. Rudolf: Existence theorems for nonlinear operator equation $Lu + Nu = f$ and some properties of the set of its solutions.Math. Slovaca 42 (1992). 55-63. MR 1159491 Reference: [31] B. Rudolf: A periodic boundary value problem in Hilbert space.Math. Bohem. 119 (1994), 347-358. Zbl 0815.34059, MR 1316586 Reference: [32] B. Rudolf: Monotone iterative technique and connectedness of solutions.Preprint. To appear. Reference: [33] B. Rudolf Z. Kubáček: Remarks on J. Nieto's paper: Nonlinear second-order periodic boundary value problems.J. Math. Anal. Appl. 46 (1990), 203-206. 10.1016/0022-247X(90)90341-C Reference: [34] W. Sobieszek P. Kowalski: On the different deffinitions of the lower semicontinuity, upper semicontinuity, upper scmicompacity. closity and continuity of the point-to-set maps.Demonstratio Math. 11 (1978), 1059-1003. MR 0529647 Reference: [35] P. Takáč: Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications to biology.Nonlinear Anal. 14 (1990), 35-42. MR 1028245, 10.1016/0362-546X(90)90133-2 Reference: [36] P. Takáč: Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups.J. Math. Anal. Appl. 148 (1990), 223-244. MR 1052057, 10.1016/0022-247X(90)90040-M Reference: [37] V. Šeda J. J. Nieto M. Gera: Periodic boundary value problems for nonlinear higher order ordinary differential equations.Appl. Math. Comp. 48 (1992). 71-82. MR 1147728, 10.1016/0096-3003(92)90019-W Reference: [38] V. Šeda Z. Kubáček: On the connectedness of the set of fixed points of a compact operator in the Fréchet space $C^m ([b,\infty), R^n)$.Czechoslovak Math. J. 42 (1992), 577-588. MR 1182189 Reference: [39] V. Šeda: Fredholm mappings and the generalized boundary value problem.Differential Integral Equations 8 (1995), 19-40. MR 1296108 Reference: [40] T. Yoshizawa: Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions.Springer, New York. 1975. Zbl 0304.34051, MR 0466797 Reference: [41] K. Yosida: Functional Analysis.Springer, Berlin, 1980. Zbl 0435.46002, MR 0617913 Reference: [42] E. Zeidler: Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems.Springer, New York Inc., 1986. Zbl 0583.47050, MR 0816732 .

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