Skip to main content
×
Home
    • Aa
    • Aa

Sublinear discrete-time order-preserving dynamical systems

  • J. F. Jiang (a1)
Abstract
Abstract

Suppose that the continuous mapping is order-preserving and sublinear. If every positive semi-orbit has compact closure, then every positive semi-orbit converges to a fixed point. This result does not require that the order be strongly preserved.

Copyright
References
Hide All
[1]Hirsch M. W.. Systems of differential equations that are competitive or cooperative. II: Convergence almost everywhere. SIAM J. Math. Anal. 16 (1985), 432439.
[2]Hirsch M. W.. Systems of differential equations that are competitive or cooperative. V: Convergence in 3-dimensional systems. J. Diff. Eqns. 80 (1989), 94106.
[3]Hirsch M. W.. Stability and convergence in strongly monotone dynamical systems. J. reine angew. Math. 383 (1988), 153.
[4]Hirsch M. W.. The dynamical systems approach to differential equations. Bull. Amer. Math. Soc. 11 (1984), 164.
[5]Matano H.. Strong comparison principle in nonlinear parabolic equations: in Nonlinear parabolic equations: qualitative properties of solutions (Boccardo L. and Tesei A., eds.), Pitman Res. Notes in Math. 149 (Longman Scientific and Technical, 1987), 148155.
[6]Smith H. L. and Thieme H. R.. Quasiconvergence and stability for strongly order-preserving semiflows. SIAM J. Math. Anal. 21 (1990), 673692.
[7]Smith H. L. and Thieme H. R.. Convergence for strongly order-preserving semiflows. SIAM J.Math. Anal. 22 (1991), 10811101.
[8]Smith H. L.. Cooperative systems of differential equations with concave nonlinearities. J. Nonlinear Anal. 10 (1986), 10371052.
[9]Polaĉik P.. Convergence in smooth strongly monotone flows defined by semilinear parabolic equations. J. Diff. Eqns. 79 (1989), 89110.
[10]Alikakos N. D. and Hess P.. On stabilization of discrete monotone dynamical systems. Israel J.Math. 59 (1987), 185194.
[11]Alikakos N. D., Hess P. and Matano H.. Discrete order preserving semigroups and stability for periodic parabolic differential equations. J. Diff. Eqns. 82 (1989), 322341.
[12]Takáĉ P.. Convergence to equilibrium on invariant d-hypersurfaces for strongly increasing discrete-time semigroups. J. Math. Anal. Appl. 148 (1990), 223244.
[13]Takáĉc P.. Asymptotic behavior of discrete-time semigroups of sublinear, strongly increasing mappings with applications to biology. J. Nonlinear Anal. 14 (1990). 3542.
[14]Dancer E. N. and Hess P.. Stability of fixed points for order-preserving discrete-time dynamical systems. J. reine angew. Math. 419 (1991), 125139.
[15]Jiang J. F.. Convergence to trap almost everywhere for flows generated by cooperative and irreducible vector fields. Chin. Ann. of Math. Series B 14 (1993), 165174.
[16]Jiang J. F.. A Liapunov function for three-dimensional feedback systems. Proc. Amer. Math. Soc. 114 (1992), 10091013.
[17]Jiang J. F.. A Liapunov function for 4-dimensional positive feedback systems. Quarterly Appl. Math. LII (1994), 601614.
[18]Jiang J. F.. The algebraic criteria for the asymptotic behavior of cooperative systems with concave nonlinearities. J. Systems Sci. Math. Sci. 6 (1993), 193208.
[19]Jiang J. F.. A note on a global stability theorem of M. W. Hirsch. Proc. Amer.Math. Soc. 112 (1991), 803806.
[20]Jiang J. F.. On the global stability of cooperative systems. Bull. LondonMath. Soc. 26 (1994), 455458.
[21]Jiang J. F.. Three- and four-dimensional cooperative systems with every equilibrium stable. J. Math. Anal. Appl. 188 (1994), 92100.
[22]Jiang J. F.. On the analytic order-preserving discrete-time dynamical systems in Rn with every fixed point stable. To appear in J. LondonMath. Soc.
[23]Coppel W. A.. Stability and asymptotic behavior of differential equations (Heath, 1965).
[24]Smith H. L.. Periodic solutions of periodic competitive and cooperative systems. SIAM J. Math. Anal. 17 (1986), 12891318.
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 6 *
Loading metrics...

Abstract views

Total abstract views: 40 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 22nd October 2017. This data will be updated every 24 hours.