Generic Quasi-convergence for Strongly Order Preserving Semiflows: A New Approach

The principal result of the theory of monotone semiflows says that for an open and dense set of initial data, the trajectory converges to the set of equilibria. We show that strong compactness assumptions on the semiflow, required for the proof, can be replaced by the assumption that limit sets have infima and suprema in the space. This assumption is automatically satisfied in nice subsets of the space of continuous functions on a compact set and in euclidean space with respect to typical orderings.

[1]  M. Hirsch.  Systems of differential equations that are competitive or cooperative. VI: A local Cr Closing Lemma for 3-dimensional systems , 1985, Ergodic Theory and Dynamical Systems.

[2]  P. Takáč.  Domains of attraction of generic ω-limit sets for strongly monotone discrete-time semigroups. , 1992 .

[3]  M. Hirsch.  Stability and convergence in strongly monotone dynamical systems. , 1988 .

[4]  Horst R. Thieme,et al.  Quasi convergence and stability for strongly order-preserving semiflows , 1990 .

[5]  H. H. Schaefer.  Banach Lattices and Positive Operators , 1975 .

[6]  Horst R. Thieme,et al.  Convergence for strongly order-preserving semiflows , 1991 .

[7]  Hiroshi Matano,et al.  Existence of nontrivial unstable sets for equilibriums of strongly order-preserving systems , 1984 .

[8]  M. Hirsch.  Systems of di erential equations which are competitive or cooperative I: limit sets , 1982 .

[9]  M. Hirsch,et al.  4. Monotone Dynamical Systems , 2005 .

[10]  Hal L. Smith,et al.  Monotone Dynamical Systems: An Introduction To The Theory Of Competitive And Cooperative Systems (Mathematical Surveys And Monographs) By Hal L. Smith , 1995 .

[11]  H. Smith,et al.  Dynamics of competition , 1999 .

[12]  M. Hirsch.  Systems of Differential Equations that are Competitive or Cooperative II: Convergence Almost Everywhere , 1985 .

[13]  Hal L. Smith.  Monotone semiflows generated by functional differential equations , 1987 .

[14]  Morris W. Hirsch,et al.  System of differential equations that are competitive or cooperative. IV: structural stability in three-dimensional systems , 1990 .