Existence of solutions in cones to delayed higher-order differential equations

An n-th order delayed differential equation y((n))(t) = f (t, yt, y(t)', ... , y(t)((n-1))) is considered, where y(t)(theta) = y(t + theta), theta is an element of [-tau,0], tau > 0, if t -> infinity. A criterion is formulated guaranteeing the existence of a solution y = y(t) in a cone 0 < (-1)(i-1)y((i-1))(t) < (-1)i-1 phi((i-1))(t), i = 1,..., n where phi is an n-times continuously differentiable function such that 0 < (-1)(i)phi((i))(t), i = 0,..., n. The proof is based on a similar result proved first for a system of delayed differential equations equivalent in a sense. Particular linear cases are considered and an open problem is formulated as well. (c) 2022 Elsevier Ltd. All rights reserved.