Analytical and numerical treatment of oscillatory mixed differential equations with differentiable delays and advances

In this work, we study the oscillatory behaviour of the differential equation of mixed type x ′ ( t ) = ∫ − 1 0 x ( t − r ( θ ) ) d ν ( θ ) + ∫ − 1 0 x ( t + τ ( θ ) ) d η ( θ ) with delays r ( θ ) and advances τ ( θ ) , both differentiable. Some analytical and numerical criteria are obtained in order to guarantee that all solutions are oscillatory.