Philos-Type Oscillation Results for Third-Order Differential Equation with Mixed Neutral Terms

: The motivation for this paper is to create new Philos-type oscillation criteria that are established for third-order mixed neutral differential equations with distributed deviating arguments. The key idea of our approach is to use the triple of the Riccati transformation techniques and the integral averaging technique. The established criteria improve, simplify and complement results that have been published recently in the literature. An example is also given to demonstrate the applicability of the obtained conditions.


introduction
It is prudent to say that neutral differential equations have drawn obvious regard because of their wide uses and applications in science and technology, including physical sciences, gas and fluid mechanics, signal processing, robotics and traffic systems, engineering, population dynamics, medicine and the like. Of late, the theory of oscillation of differential equations of the third order has become an important topic, and therefore the oscillatory properties of this type of equation have already been obtained [1][2][3][4][5][6]. In particular, it is a necessary and invaluable issue, either theoretically or practically, to probe into neutral differential equations with distributed deviating arguments. Hence, a scientific study of the qualitative properties of solutions of these equations is proposed for applications, see for example the book [7,8] and the papers [9][10][11][12][13][14][15].
Cuimei et al. [2] established an important extension of the Kamenev oscillation criterion for a third-order equation with a middle term. Ganesan et al. [3] studied the oscillatory properties of a third-order equation with a neutral type. Kumar et al. [6] extended the oscillation results of a third-order equation with distributed deviating arguments.
By defining and for all µ ≥ µ 1 , respectively, and the proof of Theorem 1, we obtain which contradicts (32).

Remark 2.
In this paper, we suggest a new Philos-type oscillation criterion for a third-order mixed neutral differential Equation (1) by using the triple Riccati transformation technique and inequalities technique.

Conclusions
In this paper, we used the triple of the Riccati transformation techniques to establish Philos-type oscillation theorems for (1) in the case of (2). Our result improves and complements results in the cited papers. It would also be of interest to find another method to study (1) in the case where R[µ, µ 0 ] < ∞ as µ → ∞. These results easily extend to the corresponding dynamic equations on time scales. The details are left to the reader.  Acknowledgments: The authors thank the reviewers for their useful comments, which led to the improvement of the content of the paper.

Conflicts of Interest:
The authors declare no conflict of interest.