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

Abstract

An n-th order delayed differential equation y(n)(t)=f(t,yt,yt′,…,yt(n−1)) is considered, where yt(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−1y(i−1)(t)<(−1)i−1φ(i−1)(t), i=1,…,n where φ is an n-times continuously differentiable function such that 0<(−1)iφ(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.