Manojlivi`c; Oscillation criteria for second order superlinear neutral delay differential equations, Electron

New oscillation criteria for the second order nonlinear neutral delay dieren tial equation (y(t) +p(t)y(t )) 00 +q(t)f(y(g(t))) = 0, t t0 are given. The relevance of our theorems becomes clear due to a carefully selected example.

Our attention is restricted to those solutions of (E) that satisfies sup{|y(t)| : t ≥ T } > 0. We make a standing hypothesis that (E) does possess such solutions.By a solution of (E) we mean a function y(t) : [t 0 , ∞) → R, such that y(t) + p(t)y(t − τ ) ∈ C 2 (t ≥ t 0 ) and satisfies (E) on [t 0 , ∞).For further question concerning existence and uniqueness of solutions of neutral delay differential equations see Hale [18].A solution of Eq. (E) is said to be oscillatory if it is defined on some ray [T, ∞) and has an infinite sequence of zeros tending to infinity; otherwise it called nonoscillatory.An equation itself is called oscillatory if all its solutions are oscillatory.
In the last decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions of second order linear and nonlinear neutral delay differential equations.( See for example [5]- [17], [23], [24], [26] and the references quoted therein ).Most of these papers considered the equation (E) under the assumption that f (u) ≥ k > 0 for u = 0, which is not applicable for f (u) = |u| λ sgn uclassical Emden-Fowler case.Very recently, the results of Atkinson [3] and Belhorec [4] for Emden-Fowler differential equations have been extended to the equation (E) by Wong [26] under the assumption that the nonlinear function f (u) satisfies the sublinear condition < ∞ for all ε > 0 as well as the superlinear condtion The special case where f (u) = |u| γ sgn u, u ∈ R, (0 < γ = 1) is of particular interest.In this case, the differential equation (E) becomes (EF ) [y(t) + p(t)y(t − τ )] + q(t) y γ (g(t)) sgn y(g(t)) = 0, t ≥ t 0 .
Established oscillation criteria have been motivated by classical averaging criterion of Kamenev, for the linear differential equation x (t) + q(t)x(t) = 0.More recently, Philos [25] introduced the concept of general means and obtained further extensions of the Kamenev type oscillation criterion for the linear differential equation.The subject of extending oscillation criteria for the linear differential equation to that of the Emden-Fowler equation and the more general equation x (t) + q(t)f (x(t)) = 0 has been of considerable interest in the past 30 years.
The object of this paper is to prove oscillation criteria of Kamenev's and Philos's type for the equation (E).
For other oscillation results of various functional differential equation we refer the reader to the monographs [1,2,7,17,21].

Main Results
In this section we will establish some new oscillation criteria for oscillation of the superlinear equation (E) subject to the nonlinear condition where Considering the special case where f (u) = |u| γ sgn u, it is easy to see that (F 1 ) holds when γ > 1.
It will be convenient to make the following notations in the remainder of this paper.Let Φ(t, t 0 ) denotes the class of positive and locally integrable functions, but not integrable, which contains all the bounded functions for t ≥ t 0 .For arbitrary functions ∈ C 1 [[t 0 , ∞), R + ] and φ ∈ Φ(t, t 0 ), we define

Kamenev's Type Oscillation Criteria
Theorem 2.1 Assume that (1) holds and let the function f (u) satisfies the assumptions (F 1 ) and (F 2 ).Suppose that there exist φ ∈ Φ(t, t 0 ) and ∈ The superlinear equation (E) is oscilatory if Proof.Let y(t) be a nonoscillatory solution of Eq.(E).Without loss of generality, we assume that y(t) = 0 for t t 0 .Further, we suppose that there exists a t 1 ≥ t 0 such that y(t) > 0, y(t − τ ) > 0 and y(g(t)) > 0 for t t 1 , since the substitution u = −y transforms Eq. (E) into an equation of the same form subject to the assumption of Theorem.Let By (1) we see that x(t) ≥ y(t) > 0 for t t 1 , and from (E) it follows that EJQTDE, 2004 No. 10, p. 4 Therefore x (t) is decreasing function.Now, as x(t) > 0 and x (t) < 0 for t≥ t 1 , in view of Kiguradze Lemma [21], we have immediately that x (t) > 0, for t t 1 .Consequently, and there exists positive constant K 1 and T ≥ t 1 such that Now using (4) in (2), we have Using (F 2 ), we get Then, from (6) it follows that Define the function Then w(t) > 0. Differentiating (9) and using (8), we have Since f is nondecreasing function, g(t) ≤ t, taking into account (4), we have EJQTDE, 2004 No. 10, p. 5 Also, since Λ is nonincreasing function, from (5), we conclude that there exists positive constant K such that which together with (F 2 ), implies that Then from (10), we get Integrate ( 12) from T to t, so that we have where Using the fact that (t)/g (t) is positive, nonincreasing function, by Bonnet Theorem, there exists a ζ ∈ [T, t], so that Thus, for t ≥ T , we find from (13), that EJQTDE, 2004 No. 10, p. 6 where L = C + M .We multiply ( 14) by φ(t) and integrate from T to t, we get Using the condition (C 3 ), there exists a t 3 ≥ T such that L 1 − A ,φ (t, T ) ≤ 0 for t≥ t 3 .Then, for every t ≥ t 3 Since G is nonnegative, we have By Schwarz inequality, we obtain Now, From ( 15), ( 16) and ( 17), for all t ≥ t 3 and some µ, 0 < µ < 1, we get EJQTDE, 2004 No. 10, p. 7 Integrating (18) from t 3 to t, we obtain and this contradicts the assumption (C 2 ).Therefore, the equation (E) is oscillatory.
Since the differentiable function (t) ≡ 1 satisfied the condition (C 1 ), we have the following Corollary.
Corollary 2.1 Let the following condition holds The superlinear equation (E) is oscillatory if Consequently, all conditions of Theorem 2.1 are satisfied, and hence the equation (E 1 ) is oscillatory.
Theorem 2.2 Let the function satisfies the condition (C 1 ).If there exists a positive constant Ω and T ≥ t 0 such that then the superlinear equation (E) is oscillatory.
Proof.Let y(t) be a nonoscillatory solution of Eq. (E).Without loss of generality, we assume that y(t) =0 for t t 0 .Further, we suppose that there exists a t 1 ≥ t 0 such that y(t)>0, y(t − τ ) > 0 and y(g(t)) > 0 for t t 1 .Next consider the function w(t) defined with (9).Then, as in the proof of Theorem 2.1 we have that there exists a constant Ω > 0 and T ≥ t 1 , such that (12) is satisfied.From (12) we get or

Philos's Type Oscillation Criteria
Next, we present some new oscillation results for Eq.(E), by using integral averages condition of Philos-type.Following Philos [25], we introduce a class of functions .Let where .
Theorem 2.4 Let H belongs to the class and assume that where G(t, s) and η(t) are defined as in Theorem 2.3.The superlinear equation (E) is oscillatory if there exist a continuous function ψ on [t 0 , ∞), such that for every T ≥ t 0 and for every Ω > 0 EJQTDE, 2004 No. 10, p. 13 and where ψ + (t) = max{ψ(t), 0}.
Proof.We suppose that there exists a solution y(t) of the equation (E), such that y(t) > 0 on [T 0 , +∞) for some T 0 ≥ t 0 .Defining the function w(t) by ( 9) in the same way as in the proof of Theorem 2.3, we obtain the inequality (20).By (20), we have for t > T ≥ T 0

H(t, s) G(t, s) w(s) ds
Hence, for T≥ T 0 we get By the condition (C 7 ) and the previous inequality, we see that Define the functions α(t) and β(t) as follows EJQTDE, 2004 No. 10, p. 14 Then, (21) In order to show that we suppose on the contrary, that (24) fails, i.e. there exists a T 1 > T 0 such that where µ is an arbitrary positive number and ζ is a positive constant satisfying Using integration by parts and (25), we have for all t ≥ T 1 In view of ( 23), there exists a constant µ 2 such that for all sufficiently large n.It follows from (27) that and (28) implies Then, by ( 28) and (30), for n large enough we derive Thus On the other hand by Schwarz inequality, we have Then, by (26), for large enough n we get Because of (31), we have which gives But the latter contradicts the assumption (C 5 ).Thus, (24) holds.Finally, by (22), we obtain EJQTDE, 2004 No. 10, p. 17 which contradicts the assumption (C 7 ).This completes the proof.
Theorem 2.5 Let H belongs to the class satisfying the condition (III) and ∈ The superlinear equation (E) is oscillatory if there exist a continuous function ψ on [t 0 , ∞), such that for every T ≥ t 0 and for every Ω > 0, (C 7 ) and are satisfied.
Proof.For the nonoscillatory solution y(t) of the equation (E), such that y(t) > 0 on [T 0 , ∞) for some T 0 ≥ t 0 , as in the proof of Theorem 2.4, ( 21) is satisfied.We conclude by ( 21) that for From the condition (C 10 ) it follows that  28) is verified for all n ≥ N .Proceeding as in the proof of Theorem 2.4.we obtain (32), which contradicts (37).Therefore, (24) holds.Using (22), we conclude by (24) and using the procedure of the proof of , we conclude that (34) is satisfied, which contradicts the assumption (C 7 ).Hence, the superlinear equation (E) is oscillatory.
Remark.With the appropriate choice of functions H and h, it is possible to derive a number of oscillation criteria for Eq.(E).Defining, for example, for some integer n >1, the function H(t,s) by H(t, s) = (t − s) n , (t, s) ∈ D. ( we can easily check that H∈ as well as that it satisfies the condition (III).Therefore, as a consequence of Theorems 2.3.-2.5.we can obtain a number of oscillation criteria.
Of course, we are not limited only to choice of functions H defined by (38), which has become standard and goes back to the well known Kamenevtype conditions.With a different choice of these functions it is possible to derive from Theorems 2.3.-2.5.other sets of oscillation criteria.In fact, another possibilities are to choose the function H as follows:

Corollary 2 . 2
the former inequality from T to t, we are lead to g (s) ds < w(T ) − w(t) < w(T ) < ∞ and this contradicts (C 4 ).Then every solution of Eq. (E) oscillates.By choosing (t)≡ 1, we get the following Corollary.The superlinear equation (E) is oscillatory if lim sup t→∞ t T Q(s)ds = ∞ .