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

An n -th order delayed differential equation y (n) (t) = f (t , y t , y t ′ , ... , y t (n − 1) ) is considered, where y t (θ) = y (t + θ) , θ ∈ [ − τ , 0 ] , τ > 0 , if t → ∞. 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 φ (i − 1) (t) , i = 1 , ... , n where φ is an n -times continuously differentiable function such that 0 < (− 1) i φ (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. [ABSTRACT FROM AUTHOR]