Oscillation criteria for a certain second-order nonlinear differential equations with deviating arguments

In this paper, by using the generalized Riccati technique and the integral averaging technique, some new oscillation criteria for certain second order retarded differential equation of the form " r (t) ˛ ˛ u ′ (t) ˛ ˛ α−1 u ′ (t) " ′ + p (t) f (u (τ (t))) = 0 are established. The results obtained essentially improve known results in the literature and can be applied to the well known half-linear and Emden-Fowler type equations.

The oscillatory behavior of functional differential equations with deviating arguments has been the subject of intensive study in the last three decades, see for example the monographs Agarwal et al [2], Dosly and Rehak [9], Gyori and Ladas [13], and Ladde et al [20].
The study of oscillation of second order differential equations is of great interest.Many criteria have been found which involve the behavior of the integral of a EJQTDE, 2009 No. 61, p. 1 combination of the coefficients.Recently Agarwal et al [4], Chern et al [8], Elbert [11], Kusano et al [16,17,18,19], and Mirzov [23,24] have observed some similar properties between the half-linear equation and the corresponding linear equation 3) Some of the above results are improved by Dzurina and Stavroulakis [10].On the other hand Ladde et al [20] presented the following oscillatory criteria for Eq.(1.3) In this paper, we shall continue in this direction the study of oscillatory properties of (1.1).By using the generalized Riccati technique and the integral averaging technique, we shall establish some new oscillatory criteria.The first our purpose is to improve the above mentioned results.The second aim is to show that many others known criteria are included in the our obtained results.The third intention of paper, is to apply obtained results for investigation of oscillation of the generalized Emden-Fowler equation Note that, in this direction, although there is an extensive literature on the oscillatory behavior of Eq. (1.2) and (1.3), there is not much done for Eq.(1.5).We refer to the reader, see [1,2,3,4,26,30] in delay case and [15,29] in ordinary case.

Main Results
In this section we prove our main result.
Theorem 1.Let there exist a constant k > 0 such that If and there exists a differentiable function ρ where µ := α α + 1 Proof.Assume the theorem false.Let u (t) be a nonoscillatory solution of (1.1).
Without loss of generality we may assume that u (t) > 0. This implies that Hence the function r (t) |u ′ (t)| α−1 u ′ (t) is decreasing and therefore there are two cases u ′ (t) > 0 and u ′ (t) < 0. The case u ′ (t) < 0, by the hypothesis (H 2 ) , is impossible and we see that u ′ (t) > 0 on [t, ∞) for some t 1 ≥ t 0 .Define Then w (t) > 0. Differentiating w (t) and using Eq.(1.1), since r (t) u ′ α (t) is decreasing, we have Riccati type inequality Let us assume that u (t) is bounded.Then there exist some positive constants c 1 and c 2 such that for all t ≥ t 0 Integrating Eq. (1.1) from t to ∞, we obtain Since r (t) u ′ α (t) is positive and decreasing, we have Integrating this inequality again from t 0 to t, we have (2.6) Letting t → ∞ the last inequality contradicts to (2.2).Therefore, we conclude (2.7) By using the inequality (2.9) Integrating this inequality from t 1 to t, we get Letting lim sup t→∞ , we get in view of (2.3) that w (t) → −∞, which contradicts w (t) > 0 and the proof is complete.
The conclusions of Theorem 1 leads to the following.
Corollary 1.Let the condition (2.3) in Theorem 1 be replaced by then the conclusion of Theorem 1 holds.
Next we state the following result.
Proof.It is enough to show that (2.12) and (2.13) together implies (2.3).From (2.13), it follows that there exist ǫ > 0 such that for all large t This means that  that is for all ǫ > 0 there exists a t 1 such that for all t ≥ t 1
For this purpose, we first define the sets We introduce a general class of parameter functions H : D → R which have continuous partial derivative on D with respect to the second variable and satisfy Note that, by choosing specific functions H, it is possible to derive several oscillation criteria for a wide range of differential equations, see [5,6,21,22,27].More general types of such functions have been constructed in [28,31] then Eq. (1.1) is oscillatory.
Proof.Using the function w (t) defined in (2.4) and proceeding similarly as in the proof of Theorem 1, we have inequality (2.9).
Multiplying this inequality by H (t, s) , t > s, and next integrating from t 1 to t after simple computation we have This gives which contradicts (3.1).This completes the proof of the theorem.
r (τ (t)) Note that, by choosing specific functions ρ and H, it is possible to derive several oscillation criteria for Eq.(1.1) and its special cases Half-linear Eq. (1.2) and generalized Emden-Fowler Eq. (1.5) with β ≥ α.
and we can find always a positive constant k large enough such that f ′ (x) / |f (x)| 1−1/α ≥ k for all x ∈ R D .Hence, all above oscillation criteria are valid for the generalized Emden-Fowler Eq. (1.5).We note that these conclusions, for Eq.(1.5) with β ≥ α, complement Theorem 3.1 in [4] and they do not appear to follow from the known oscillation criteria in the literature.
Note that criteria reported in the references do not apply to Eq. (3.6).