On the oscillation of the solutions to delay and difference equations

Consider the first-order linear delay differential equation xʹ(t) + p(t)x(τ(t)) = 0, t≥ t₀, (1) where p, τ ∈ C ([t₀,∞, ℝ⁺, τ(t) is nondecreasing, τ(t) < t for t ≥ t⁰ and lim_t→∞ τ(t) = ∞, and the (discrete analogue) difference equation Δx(n) + p(n)x(τ(n)) = 0, n= 0, 1, 2,…, (1)ʹ where Δx(n) = x(n + 1) − x(n), p(n) is a sequence of nonnegative real numbers and τ(n) is a nondecreasing sequence of integers such that τ(n) ≤ n − 1 for all n ≥ 0 and lim_n→∞ τ(n) = ∞. Optimal conditions for the oscillation of all solutions to the above equations are presented.