Convergence to equilibria in scalar nonquasimonotone functional differential equations. (Q1406505)

This paper studies a class of scalar autonomous functional-differential equations (FDEs). The basic assumption is that the generated semiflow is strongly order-preserving with respect to the exponential ordering introduced by \textit{H. L. Smith} and \textit{H. R. Thieme} [J. Math. Anal. Appl. 150, No. 2, 289--306 (1990; Zbl 0719.34123)]. It is shown that the boundedness of all solutions of the FDE is the same as for the associated ODE obtained by ignoring the delays. This fact combined with a generic convergence result of Smith and Thieme, explicitly characterizes the convergence of solutions starting from an open and dense subset of the phase space. Under certain conditions, it is also shown that the stability of an equilibrium of the FDE is equivalent to the stability of the corresponding ODE without delay. The most significant result of the paper gives conditions for the convergence of all solutions, not only for the generic ones. The basic idea is that under certain conditions, the convergence of all solutions of the FDE is equivalent to the boundedness of the solutions of the associated ODE without delay. Although, the applicability of this result is quite restrictive, the method is very interesting in the theory of monotone dynamical systems.