Some remarks on a boundedness assumption for monotone dynamical systems
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Abstract:
We study the consequences of the assumption that all forward orbits are bounded for monotone dynamical systems. In particular, it turns out that this assumption has more implications than is immediately apparent.References
- Y. A. Abramovich and A. W. Wickstead, A compact regular operator without modulus, Proc. Amer. Math. Soc. 116 (1992), no. 3, 721–726. MR 1098395, DOI 10.1090/S0002-9939-1992-1098395-7
- E. N. Dancer, Upper and lower stability and index theory for positive mappings and applications, Nonlinear Anal. 17 (1991), no. 3, 205–217. MR 1120974, DOI 10.1016/0362-546X(91)90048-6
- E. N. Dancer, Global solution branches for positive mappings, Arch. Rational Mech. Anal. 52 (1973), 181–192. MR 353077, DOI 10.1007/BF00282326
- E. N. Dancer, Multiple fixed points of positive mappings, J. Reine Angew. Math. 371 (1986), 46–66. MR 859319, DOI 10.1515/crll.1986.371.46
- Michele Matzeu and Alfonso Vignoli (eds.), Topological nonlinear analysis, Progress in Nonlinear Differential Equations and their Applications, vol. 15, Birkhäuser Boston, Inc., Boston, MA, 1995. Degree, singularity, and variations. MR 1322322, DOI 10.1007/978-1-4612-2570-6
- E. N. Dancer and P. Hess, Stability of fixed points for order-preserving discrete-time dynamical systems, J. Reine Angew. Math. 419 (1991), 125–139. MR 1116922
- Morris W. Hirsch, Fixed points of monotone maps, J. Differential Equations 123 (1995), no. 1, 171–179. MR 1359916, DOI 10.1006/jdeq.1995.1161
- M. A. Krasnosel′skiĭ, Positive solutions of operator equations, P. Noordhoff Ltd., Groningen, 1964. Translated from the Russian by Richard E. Flaherty; edited by Leo F. Boron. MR 0181881
- Jiang Ji-Fa, On the global stability of cooperative systems, Bull London Math Soc 26 (1994), 455-458.
- P. Poláčik and I. Tereščák, Convergence to cycles as a typical asymptotic behavior in smooth strongly monotone discrete-time dynamical systems, Arch. Rational Mech. Anal. 116 (1992), no. 4, 339–360. MR 1132766, DOI 10.1007/BF00375672
- Helmut H. Schaefer, Topological vector spaces, Graduate Texts in Mathematics, Vol. 3, Springer-Verlag, New York-Berlin, 1971. Third printing corrected. MR 0342978, DOI 10.1007/978-1-4684-9928-5
- Hal L. Smith and Horst R. Thieme, Convergence for strongly order-preserving semiflows, SIAM J. Math. Anal. 22 (1991), no. 4, 1081–1101. MR 1112067, DOI 10.1137/0522070
- Hal L. Smith, Monotone dynamical systems, Mathematical Surveys and Monographs, vol. 41, American Mathematical Society, Providence, RI, 1995. An introduction to the theory of competitive and cooperative systems. MR 1319817
- A. Vanderbauwhede, Invariant manifolds in infinite dimensions, Dynamics of infinite-dimensional systems (Lisbon, 1986) NATO Adv. Sci. Inst. Ser. F: Comput. Systems Sci., vol. 37, Springer, Berlin, 1987, pp. 409–420. MR 921925
Additional Information
- E. N. Dancer
- Affiliation: School of Mathematics and Statistics, University of Sydney, N.S.W. 2006, Australia
- Email: dancer_n@maths.su.oz.au
- Received by editor(s): September 5, 1996
- Communicated by: Hal L. Smith
- © Copyright 1998 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 126 (1998), 801-807
- MSC (1991): Primary 47H15
- DOI: https://doi.org/10.1090/S0002-9939-98-04276-2
- MathSciNet review: 1443378