In this paper, we are concerned with the oscillation of third order nonlinear delay differential equations of the formBy using a generalized Riccati transformation and integral averaging technique, we establish some new sufficient conditions which insure that any solution of this equation oscillates or converges to zero. In particular, several examples are given to illustrate the importance of our results.

a b s t r a c tIn this work, we are concerned with oscillation of third-order nonlinear functional differential equations of the formBy using a Riccati type transformation and integral averaging technique, we establish some new sufficient conditions under which every solution y(t) either oscillates or converges to zero as t → ∞.Unlike ones from the known works in the literature, our results are applicable to nonlinear functional differential equations of the above form when f (u) = |u| α−1 u, α > 0.

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In this paper, by using elementary analysis, we establish some new Lyapunov-type inequalities for nonlinear systems of differential equations, special cases of which contain the well-known equations such as Emden-Fowler-type and half-linear equations. The inequalities obtained here can be used as handy tools in the study of qualitative behaviour of solutions of the associated equations.

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