Convergence of solutions for an equation with state-dependent delay

A result of Smith and Thieme shows that if a semiflow is strongly order preserving, then a typical orbit converges to the set of equilibria. For the equation with state-dependent delay (x)over dot(t) = - mux(t) + f(x(t - r(x(t))), where mu > 0 and f and r are smooth real functions with f(0) = 0 and f' > 0, we construct a semiflow which is monotone but not strongly order preserving. We prove a convergence result under a monotonicity condition different from the strong order preserving property, and apply it to the above equation to obtain generic convergence. (C) 2001 Academic Press.