Topological Methods in Nonlinear Analysis

Relative versions of the multivalued Lefschetz and Nielsen theorems and their application to admissible semi-flows

Jan Andres, Lech Górniewicz, and Jerzy Jezierski

Full-text: Open access

Abstract

The relative Lefschetz and Nielsen fixed-point theorems are generalized for compact absorbing contractions on ANR-spaces and nilmanifolds. The nontrivial Lefschetz number implies the existence of a fixed-point in the closure of the complementary domain. The relative Nielsen numbers improve the lower estimate of the number of coincidences on the total space or indicate the location of fixed-points on the complement. Nontrivial applications of these topological invariants (under homotopy) are given to admissible semi-flows and differential inclusions.

Article information

Source
Topol. Methods Nonlinear Anal., Volume 16, Number 1 (2000), 73-92.

Dates
First available in Project Euclid: 22 August 2016

Permanent link to this document
https://projecteuclid.org/euclid.tmna/1471875423

Mathematical Reviews number (MathSciNet)
MR1805040

Zentralblatt MATH identifier
0991.47040

Citation

Andres, Jan; Górniewicz, Lech; Jezierski, Jerzy. Relative versions of the multivalued Lefschetz and Nielsen theorems and their application to admissible semi-flows. Topol. Methods Nonlinear Anal. 16 (2000), no. 1, 73--92. https://projecteuclid.org/euclid.tmna/1471875423


Export citation

References

  • J. Andres, On the multivalued Poincaré operators , Topol. Methods Nonlinear Anal., 10 (1997), 171–182 \ref\key 2 ––––, A target problem for differential inclusions with state-space constraints , Demonstratio Math., 30 (1997), 783–790 \ref\key 3 ––––, Multiple bounded solutions of differential inclusions: the Nielsen theory approach , J. Differential Equations, 155 (1999), 285–310 \ref\key 4
  • K. Boucher, M. Brown and E. E. Slaminka, A Nielsen-type theorems for area-preserving homeomorphisms of the two disc , Continuum Theory and Dynamical Systems (T. West, ed.), LNPAM, 149 , M. Dekker Inc., New York, 43–50 \ref\key 7
  • C. Bowszyc, Fixed point theorems for the pairs of spaces , Bull. Polish Acad. Sci. Math., 16 (1968), 845–850 \ref\key 8
  • R. F. Brown and R. E. Green, An interior fixed point property of the disc , Amer. Math. Monthly, 101 (1994), 39–47 \ref\key 9
  • R. F. Brown, R. E. Green and H. Schirmer, Fixed points of map extensions , Topological Fixed Point Theory and Applications, Lecture Notes in Math., 141 , Springer, Berlin (1989), 24–45 \ref\key 10
  • R. F. Brown and H. Schirmer, Nielsen theory of roots of maps of pairs , Topology Appl., 92 (1999), 247–274 \ref\key 11
  • F. S. P. Cardona, Reidemeister theory for maps of pairs , Far East J. Math. Sci. Special Volume, Part I, Geometry and Topology (1999), 109–136 \ref\key 12
  • F. S. P. Cardona and P. N.-S. Wong, On the computation of the relative Nielsen number , Topology Appl., to appear \ref\key 13
  • M. L. Fernandes and F. Zanolin, Remarks on strongly flow-invariant sets , J. Math. Anal. Appl., 128 (1987), 176–188 \ref\key 14
  • J. Jezierski, A modification of the relative Nielsen number of H. Schirmer , Topology Appl., 62 (1995), 45–63 \ref\key 18
  • W. Kryszewski, The fixed point index for the class of compositions of acyclic set-valued maps of \romANRs, Bull. Soc. Math. France, 120 (1996), 129–151 \ref\key 19
  • G. S. Ladde and V. Lakshmikantham, On flow-invariant sets , J. Math. Anal. Appl., 128 (1987), 176–188 \ref\key 20
  • C. K. McCord, Computing Nielsen numbers , Nielsen Theory and Dynamical Systems, Contemp. Math. (C. K. McCord, ed.), 152 , Amer. Math. Soc., Providence, R. I. (1993), 249–267 \ref\key 21
  • M. Mrozek, A cohomological index of Conley type for multi-valued admissible flows , J. Differential Equations, 84 (1990), 15–51 \ref\key 22
  • B. Norton-Odenthal and P. Wong, On the computation of the relative Nielsen number , Topology Appl., 56 (1994), 141–157 \ref\key 23
  • R. M. Redheffer and W. Walter, Flow-invariant sets and differential inequalities in normed spaces , Appl. Anal., 5 (1975), 149–161 \ref\key 24
  • K. P. Rybakowski, The Homotopy Index and Partial Differential Equations, Springer, Berlin (1987) \ref\key 25
  • H. Schirmer, A relative Nielsen number , Pacific J. Math., 112 (1986), 459-473 \ref\key 26 ––––, On the location of fixed points on a pair of spaces , Topology Appl., 30 (1988), 253–266 \ref\key 27 ––––, A survey of relative Nielsen fixed point theory , in Nielsen Theory and Dynamical Systems, Contemp. Math. (C. K. McCord, ed.), 152 , Amer. Math. Soc. Colloq. Publ. Providence R.I. (1993), 291–309 \ref\key 28
  • R. Srzednicki, Periodic and constant solutions via topological principle of Wa$\dot{z}$ewski , Acta. Math. Univ. Iag., 26 (1987), 183–190 \ref\key 29 ––––, Periodic and bounded solutions in blocks for time-periodic nonautonomous ordinary differential equations , Nonlinear Anal., 22 (1994), 707–737 \ref\key 30 ––––, Generalized Lefschetz theorem and fixed point index formula , Topology Appl., 81 (1997), 207–224 \ref\key 31 ––––, A generalization of the Lefschetz fixed point theorem and detection of chaos , Proc. Amer. Math. Soc., to appear \ref\key 32
  • P. Wong, A note on the local and the extension Nielsen numbers , Topology Appl., 78 (1992), 207–213 \ref\key 33 ––––, Fixed points on pairs of nilmanifolds , Topology Appl., 62 (1995), 173–179 \ref\key 34
  • W. X. Zhao, A relative Nielsen number for the complement , Topological Fixed Point Theory and Applications, Lecture Notes in Math., 1411 , Springer, Berlin (1989), 189–199 \ref\key 35 ––––, Estimation of the number of fixed points on the complement , Topology Appl., 37 (1990), 257–265 \ref\key 36 ––––, Basic relative Nielsen numbers , Topology, Hawaii, World Scientific, Singapore (1992), 215–222