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Necessary and sufficient conditions for oscillation of second-order differential equations with nonpositive neutral coefficients

In this work, we present necessary and sufficient conditions for oscillation of all solutions of a second-order functional differential equation of type (r(t)(z(t))) + m ∑ i=1 qi(t)x (σi(t)) = 0, t > t0, where z(t) = x(t) + p(t)x(τ(t)). Under the assumption ∫ ∞ (r(η)) dη =∞, we consider two cases when γ > αi and γ < αi. Our main tool is Lebesgue’s dominated convergence theorem. Finally, we provide examples illustrating our results and state an open problem.