Necessary and Sufficient Conditions for Oscillation of a Mixed Neutral Equation

Consider the neutral differential equation with mixed arguments ddt[y(t) + py(t − τ)] + q1y(t − σ1) + q2y(t + σ2) = 0 (1) where p, q1, q2, τ, σ1, σ2 are positive real numbers. We prove that every solution of Eq. (1) oscillates if and only if the characteristic equation λ + λpe−λτ + q1e−λσ1 + q2eλσ2 = 0 has no real roots.