Strongly order preserving semiflows generated by functional differential equations

Abstract

The aim of this paper is to apply some recently developed improvements in the theory of monotone dynamical systems, due to the authors [ 12, 13 J, to certain systems of functional differential equations (FDE’s) which do not enjoy the quasimonotone property considered by one of us in [9]. The present work extends our previous results in this direction for scalar equations in [ 111 to systems. In the fundamental paper [S], M. W. Hirsch establishes that most orbits of a strongly monotone scmiflow on a strongly ordered space X tend to the set E of equilibria. Employing ideas of Hirsch and of H. Matano [7], the authors have extended the theory to obtain some improvements in the results of Hirsch and Matano under Matano’s weaker assumption that the semiflow is strongly order preserving. Recently, in [13], we gave sufficient conditions for most orbits of a strongly order preserving semiflow to converge to an equilibrium. These conditions require additional smoothness of the semiflow and a certain strong monotonicity condition for the linearized flow determined by the variational equation about each equilibrium. P. PoliCik [S] has also given sufficient conditions for most orbits to converge for semilinear parabolic evolution equations possessing certain strong monotonicity properties. In this paper, WC apply the convergence results developed in [ 131 to nonquasimonotone FDE’s. In this Introduction, WC briefly review the application of monotone dynamical systems theory to FDE’s enjoying the quasimonotone property. This research appeared in [9, 133. Then we state one of the main results of this paper which applies when the quasimonotone property fails but when another property (M,) holds. Some simple examples are given which