New Criteria on Oscillatory Behavior of Third Order Half-Linear Functional Differential Equations

This paper deals with some new criteria for the oscillation of third order half-linear differential equations. The purpose of the present paper is the linearization of equation 1.1 in the sense that we would deduce oscillation of studied equation from that of the linear form and to provide new oscillation criteria via comparison with first order equations whose oscillatory behavior are known. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature for equation 1.1. The results are illustrated by some examples.


Introduction
This paper is concerned with oscillatory behavior of all solutions of the halflinear third order differential equations of the form a (t) (x (t)) α = q (t) x α (τ (t)) + p (t) x α (ω (t)) . (1.1) We assume that A solution of (1.1) is said to be oscillatory if it has arbitrarily large zeros, and otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if al l its solutions are oscillatory.
In recent years, the oscillation theory of functional differential equations has received much attention since it has a great number of applications in engineering and natural sciences. For some related contributions on the oscillatory behavior of various classes of functional differential equations, we refer the reader to  and the references cited therein.
The problem of the oscillation of solutions of differential equations has been widely studied by many authors and by many techniques since the pioneering work of Sturm on second order linear differential equations. In the past 30 years, the oscillation theory for second order neutral delay differential equations and third-order retarded delay differential equations have been well developed; see, for example, the monographs [4,7] and papers [3,5,6,[8][9][10][11][12][13] as well as the references cites therein. Compared to second order delay differential equations, it seems that not much work has been done concerning with the oscillation of third order differential equations [5,12]. However, to the best of our knowledge, there are no results for third order differential equations. The aim of the present paper is the linearization of Eq. (1.1) in the sense that we would deduce oscillation of studied equation from that of the linear form and to provide new oscillation criteria (taking the linear form of Eq. (1.1) into account) via comparison with first order equations whose oscillatory behavior are known. The obtained results are new, improve and correlate many of the known oscillation criteria appeared in the literature for Eq. (1.1).

Main Results
In this section, we study some oscillation criteria for equation (1.1).
To obtain our result, we need the following lemmas: The first order delay differential inequality (i.e., g (t) ≤t) has an eventually positive solution, so does the delay equation (II) The first order advanced differential inequality (i.e., g (t) ≥ t) has an eventually positive solution, so does the delay equation This Lemma is an extension of known results in [18,19] and [21,Corollary 1] and the proof is immediate. (1.2) hold and assume that and there exists a nondecreasing functions ϕ (t) and If the advanced equation is oscillatory and the first order delay differential equation Hence a (t) (x (t)) α is of one sign. We shall distinguish the following two cases: Differentiating the above equality, we have and so, we get ) . (2.5) First, we consider the Case (I) and the inequality In this case, we see that x (i) (t) > 0 for i = 0, 1, 2.  Consequently, Once again, we integrate this inequality from ϕ ( t) to t yields (A(s, ϕ(s))) ds a 1 α (ϕ (ϕ (ω (t)))) x (ϕ (ϕ (ω (t)))) . (2.7) Using (2.7) in (2.6), we have Since a(t) is a nondecreasing function and α > 1 we have (A(s, ϕ(s)))ds α W (ρ (t)) .
It follows from Lemma 2.1(II) that the corresponding differential equation (2.3) also has a positive solution, which is a contradiction. This completes the proof.
Next, we consider Case (II). It is easy to see that there exists a constant θ ∈ (0, 1) such that (2.8) Using this inequality in Eq. (2.5) with p(t) = 0, we have Using (2.8) in (2.9), we obtain or, Now. For v ≥ u ≥t 1 , we see that Setting u = τ (t) and v = ξ(t), we have (2.12) Using this inequality in (2.10), we have It follows from Lemma 2.1(I) that the corresponding differential equation (2.4) also has a positive solution, which is a contradiction. This completes the proof.
The following corollary is immediate.