Nielsen number, impulsive differential equations and problem of Jean Leray

Jan Andres


We will show that, unlike to usual (i.e.\ non-impulsive) differential equations, the Nielsen theory results for single-valued as well as multivalued maps on tori can be effectively applied to impulsive differential equations and inclusions. With this respect, two main aims will be focused, namely: (i) multiplicity results for harmonic periodic solutions, (ii) the coexistence of subharmonic periodic solutions with various periods. In both cases, we will try to contribute at least partly to the problem posed already in 1950 by Jean Leray. A dynamic complexity of the related maps, measured in terms of entropy, will be also examined.


Nielsen number; admissible maps; periodic orbits; impulsive differential equations and inclusions; harmonic and subharmonic solutions; multiplicity results; coexistence of subharmonics; entropy

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L. Alsedà, S. Baldwin, J. Llibre, R. Swanson and W. Szlenk, Minimal sets of periods for torus maps via Nielsen numbers, Pacific J. Math. 169 (1995), no. 1, 1–32.

J. Andres, Nielsen number and differential equations, Fixed Point Theory Appl. 2005 (2005), (268678), 1–31.

J. Andres, On the notion of random chaos, Proc. Amer. Math. Soc. 145 (2017), no. 8, 3223–3435.

J. Andres, On the coexistence of irreducible orbits of coincidences for multivalued admissible maps on the circle via Nielsen theory, Topology Appl. 221 (2017), 596–609.

J. Andres, Coexistence of periodic solutions with various periods of impulsive differential equations and inclusions on tori via Poincaré operators, Topology Appl. 255 (2019), 126–140.

J. Andres, Randomized Sharkovsky-type theorems and their application to random impulsive differential equations and inclusions on tori, stochastics and dynamics 19 (2019), no. 1, 1–30.

J. Andres and J. Fišer, Sharkovsky-type theorems on S 1 applicable to differential equations, Internar. J. Bifur. Chaos Appl. sci. Engrg. 27 (2017), no. 3, 1–21.

J. Andres and L. Górniewicz, Topological Fixed Point Principles for Boundary Value Problems, Kluwer, Dordrecht, 2003.

J. Andres, L. Górniewicz and J. Jezierski, A generalized Nielsen number and multiplicity results for differential inclusions, Topology Appl. 100 (2000), no. 2–3, 193–209.

J. Andres, L. Górniewicz and J. Jezierski, Relative versions of the multivalued Lefschetz and Nielsen theorems and their application to admissible semi-flows, Topol. Methods Nonlinear Anal. 16 (2001), no. 1, 73–92.

J. Andres, L. Górniewicz and J. Jezierski, Periodic points of multivalued mappings with applications to differential inclusions on tori, Topology Appl. 127 (2003), no. 3, 447–472.

C. Bernhardt, Periodic orbits of continuous mappings of the circle without fixed points, Ergodic Theory Dynam. Systems 1 (1981), no. 4, 413–417.

L. Block, Periods of periodic points of maps of the circle which have a fixed point, Proc. Amer. Math. Soc. 82 (1981), no. 3, 481–486.

L. Block, J. Guckenheimer, M. Misiurewicz and L.-S. Young, Periodic points and topological entropy of one-dimensional maps, Global Theory of Dynamical Systems (Z. Nitecki, C. Robinson, eds.) Lecture Notes in Math., vol. 819, Springer, Berlin, 1980, pp. 18–34.

R.F. Brown, The Lefschetz Fixed Point Theorem, Scott, Foresman and Co., Glenview, Illinois, 1971.

R.F. Brown, Nielsen fixed point theory and parametrized differential equations, Fixed Point Theory and Its Applications (R.F. Brown, ed.), Contemp. Math. 72, Amer. Math. Soc., Providence, R.I., 1989, pp. 33–46.

R.F. Brown, Topological identification of multiple solutions to parametrized nonlinear equations, Pacific J. Math. 131 (1988), no. 1, 51–69.

R.F. Brown, Fixed points of n-valued multimaps of the circle, Bull. Pol. Acad. Sci. Math. 54 (2006), 153–162.

R.F. Brown, M. Furi, L. Górniewicz and B. Jiang (eds.), Handbook of Topological Fixed Point Theory, Springer, Berlin, 2005.

R.F. Brown, and D.L. Gonçalves, On the topology of n-valued maps, Adv. Fixed Point Theory 8 (2) (2018) 205–220.

E.A. Coddington and N. Levinson, Theory of Differential Equations, McGraw–Hill, New York, 1955.

Z. Dzedzej, Fixed Point Index Theory for a Class of Nonacyclic Multivalued Mappings, Dissertationes Mathematicae, vol. 253, PAN, Warsaw, 1985.

L.S. Efremova, Periodic orbits and a degree of a continuous map of a circle, Differential Integral Equationsns (Gor’kiı̆) 2 (1978) 109–115. (in Russian)

M. Fečkan, Multiple solutions of nonlinear equations via Nielsen fixed-point theory: a survey, Nonlinear Analysis in Geometry and Topology (T.M. Rassias, ed.), Hadronic Press, Palm Harbor, FL, 2000, pp. 77–97.

A. Fel’shtyn, Dynamical Zeta Functions, Nielsen Theory, and Reidemeister Torsion, Mem. Amer. Math. Soc., vol. 147, no. 699, Amer. Math. Soc., Providence, RI, 2000.

B. Hasselblatt and A. Katok, A First Course in Dynamics: With a Panorama of Recent Developments, Cambridge Univ. Press, Cambridge, 2003.

N.V. Ivanov, Entropy and Nielsen numbers, Soviet Math. Dokl. 26 (1982), 63–66; Dokl. Akad. Nauk SSSR 265 (1982), no. 2, 284–286. (in Russian)

J. Jezierski and W. Marzantowicz, Homotopy Methods in Topological Fixed and Periodic Points Theory, Springer, Berlin, 2006.

B. Jiang, Lectures on the Nielsen Fixed Point Theory, Contemp. Math., vol. 14, Amer. Math. Soc., Providence, RI, 1983.

B. Jiang, Nielsen theory for periodic orbits and applications to dynamical systems, Nielsen Theory and Dynamical Systems (Ch. McCord, ed.), Contemp. Math., vol. 152, Amer. Math. Soc., Providence, RI, 1993, pp. 183–202.

B. Jiang, Estimation of the number of periodic orbits, Pacific J. Math. 172 (1996), no. 1, 151–185.

B. Jiang, Applications of Nielsen theory to dynamics, Nielsen Theory and Reidemeister Torsion (J. Jezierski, ed.), Banach Center Publications, vol. 49, Institute of Mathematics, Polish Acad. Sci.’, Warsaw, 1999, pp. 203–221.

B. Jiang and J. Llibre, Minimal sets of periods for torus maps, Discrete Cont. Dynam. Syst. 4 (1998), no. 2, 301–320.

T.-H. Kiang, The Theory of Fixed Point Classes, Springer’, Berlin, 1989.

D. Kwietniak and P. Oprocha, Topological entropy and chaos for maps induced on hyperspaces, Chaos Solitons Fractals 33 (2007), no. 1, 76–86.

J. Leray, La theorie des points fixes et ses applications en analyse, Proc. International Congress of Math. 1950, vol. 2, Amer. Math. Soc., 1952.

M. Misiurewicz, Periodic orbits of maps of degree one of a circle, Ergodic Theory Dynam. Systems 2 (1982), no. 221–227.

N.A. Perestyuk, V.A. Plotnikov, A.M. Samoilenko and N.V. Skripnik, Differential Equations with Impulse Effects. Multivalued Right-hand Sides with Discontinuities, De Gruyter Studies in Mathematics, vol. 40, De Gruyter, Berlin, 2011.

H.W. Siegberg, Chaotic mappings on S 1 , periods one, two, three imply chaos on S 1 , Proceedings of the Conference: Numerical Solutions of Nonlinear Equations (Bremen, 1980), Lecture Notes in Math., vol. 878, Springer, Berlin, 1981, pp.,351–370.

X. Zhao, Periodic orbits with least period three on the circle, Fixed Point Theory Appl. 2008 (2008), (194875), 1–8.


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