On proper oscillatory and vanishing-at-infinity solutions of differential equations with a deviating argument

Sufficient conditions are found for the existence of multiparametric families of proper oscillatory and vanishing-at-infinity solutions of the differential equation $$u^{(n)} (t) = g\left( {t, u(\tau _0 (t)), \ldots ,u^{(m - 1)} (\tau _{m - 1} (t))} \right)$$ , wheren≥4,m is the integer part of π/2,g:R ₊×R ^m →R is a function satisfying the local Carathéodory conditions, and τ _i :R ₊→R(i=0,...,m?1) are measurable functions such that τ _i (t) →+∞ fort→+∞(i=0,...,m?1).