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We prove oscillation theorems for the nonlinear delay differential equation " |y 0 (t)| �−2 y 0 (t) "0 + q(t)|y(�(t))| �−2 y(�(t)) = 0, tt� > 0, where � > 1, � > 1, q(t) � 0 and locally integrable on (t� ,1), �(t) is a continuous function satisfiying 0 < �(t) � t and limt!1 �(t) = 1. The results obtained essentially improve the known results in the literature and can be applied to linear and half-linear delay type differential equations.


Introduction
In the last decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions for different classes of second order differential equations with or without deviating arguments.For interested readers we refer to the papers [7,8,12,13,15] and the references quoted therein.
Before we continue with the description of the content of this paper, we present a short survey of the most basic results in the literature.
Let us consider the following linear differential equation where q(t) ≥ 0 is locally integrable on [t 0 , ∞).
In 1995, Kusano and Yoshida [9] generalized Theorems A and B as follows: then equation ( 11) is nonoscillatory.
Note that the equation (18) with τ (t) = t is referred to as a super-half-linear equation, a sub-half-linear equation and an Emden-Fowler type equation for β > α, β < α and β = α, respectively.We refer the readers to the introductory books by Agarwal et al. [2] and by Došlý and Ȓehák [4] for the equation (18) with τ (t) = t.
To present our results, we need the following lemma which is given by Erbe [5].
EJQTDE, 2011 No. 84, p. 4 First, we obtain two theorems which concern the oscillatory behaviour of equation (18) with β = α.Next, motivated by the ideas of Agarwal and Grace [1] and C ¸akmak [3], we present two other results for β = α.
Theorem 1 Suppose that for all sufficiently large t, q(t) satisfies , then all solutions of equation ( 18) with β = α are oscillatory.
Using the same argument as in the proof of Theorem 1, we can also prove the following result.
The proof is complete.
Remark 4 If the delayed argument is absent, i.e. τ (t) = t, then Theorems 1 and 2 reduce to Theorems L and N, respectively.Furthermore, Theorem 1 is an extension of Theorem J.
Remark 5 Let α = 2 and τ (t) = t.In this case, Theorem 1 is an extension of Theorem B.Moreover, Theorem 2 (or Theorem 2 with λ = 2) reduces to Theorem F (or Theorem D).
where the function f satisfies sf (t, s) ≥ q(t) |s| β for t ≥ t 0 and s ∈ R.
The proof of the following results are exactly as in that of above theorems and hence omitted.
Theorem 8 In addition to the conditions of Theorem 1 (or Theorem 2), if (34) with β = α holds, then all solutions of equation (33) are oscillatory.
Theorem 9 In addition to the conditions of Theorem 6 (or Theorem 7), if (34) with β > α holds, then all unbounded solutions of equation (33) are oscillatory.
Theorem 10 In addition to the conditions of Theorem 6 (or Theorem 7), if (34) with β < α holds, then all bounded solutions of equation ( 33) are oscillatory.