Invariant cones of positive initial functions for delay differential equations

Consider the scalar delay differential equation x(t) + ∑ a_kx(t-r_k) = 0 (1) where a_k and r_k ≥ 0, (0 ≤k ≤ m), are real numbers.In this paper we show that there exists an invariant cone of the positive initial functions if and only if the characteristic equation of Eq. (1) has a real root. We also give the constraction of the maximal invariant cone among the positive initial functions with respect to Eq. (1). At the end of the paper we show the generalizations of these results for systems