Dynamics of smooth essentially strongly order-preserving semiflows with application to delay differential equations

Abstract In this paper, we introduce a class of smooth essentially strongly order-preserving semiflows and improve the limit set dichotomy for essentially strongly order-preserving semiflows. Generic convergence and stability properties of this class of smooth essentially strongly order-preserving semiflows are then developed. We also establish the generalized Krein–Rutman Theorem for a compact and eventually essentially strongly positive linear operator. By applying the main results of this paper to essentially cooperative and irreducible systems of delay differential equations, we obtain some results on generic convergence and stability, the linearized stability of an equilibrium and the existence of the most unstable manifold in these systems. The obtained results improve some corresponding ones already known.

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

[2]  P. Polácik,et al.  Convergence in smooth strongly monotone flows defined by semilinear parabolic equations , 1989 .

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

[4]  Jianhong Wu.  Theory and Applications of Partial Functional Differential Equations , 1996 .

[5]  W. D. Evans,et al.  PARTIAL DIFFERENTIAL EQUATIONS , 1941 .

[6]  K. Deimling.  Nonlinear functional analysis , 1985 .

[7]  Hal L. Smith.  Invariant curves for mappings , 1986 .

[8]  Hal L. Smith.  Systems of ordinary differential equations which generate an order preserving flow. A survey of results , 1988 .

[9]  H. H. Schaefer,et al.  Topological Vector Spaces , 1967 .

[10]  Jifa Jiang,et al.  On the ω-limit Set Dichotomy of Cooperating Kolmogorov Systems , 2003 .

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

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

[13]  Angus E. Taylor.  Introduction to functional analysis , 1959 .

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

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

[16]  Lihong Huang,et al.  Convergence and stability for essentially strongly order-preserving semiflows , 2006 .

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