A Necessary and Sufficient Condition for the Oscillation in a Class of Even Order Neutral Differential Equations

The even order neutral differential equation (1.1) d n dt n [x(t) + λx(t − τ)] + f (t, x(g(t))) = 0 is nondecreasing in u ∈ R for each fixed t ≥ t 0. It is shown that equation (1.1) is oscillatory if and only if the non-neutral differential equation x (n) (t) + 1 1 + λ f (t, x(g(t))) = 0 is oscillatory.


Introduction and main results
We shall be concerned with the oscillatory behavior of solutions of the even order neutral differential equation Throughout this paper, the following conditions are assumed to hold: n ≥ 2 is even; λ > 0; τ > 0; g ∈ C[t 0 , ∞), lim t→∞ g(t) = ∞; f ∈ C([t 0 , ∞) × R), uf (t, u) ≥ 0 for (t, u) ∈ [t 0 , ∞) × R, and f (t, u) is nondecreasing in u ∈ R for each fixed t ≥ t 0 .
By a solution of (1.1), we mean a function x(t) that is continuous and satisfies (1.1) on [t x , ∞) for some t x ≥ t 0 .Therefore, if x(t) is a solution of (1.1), then x(t) + λx(t − τ ) is n-times continuously differentiable on [t x , ∞).Note that, in general, x(t) itself is not continuously differentiable.
A solution is said to be oscillatory if it has arbitrarily large zeros; otherwise it is said to be nonoscillatory.This means that a solution x(t) is oscillatory if and only if there is a sequence {t i } ∞ i=1 such that t i → ∞ as i → ∞ and x(t i ) = 0 (i = 1, 2, . ..), and a solution x(t) is nonoscillatory if and only if x(t) is eventually positive or eventually negative.Equation (1.1) is said to be oscillatory if every solution of (1.1) is oscillatory.
There has been considerable investigation of the oscillations of even order neutral differential equations.For typical results we refer to the papers [1, 2, 4-8, 11, 12, 16, 18, 19, 21-25] and the monographs [3] and [9].Neutral differential equations find numerous applications in natural science and technology.For instance, they are frequently used for the study of distributed networks containing lossless transmission lines.See Hale [10].Now consider the linear equation and the nonlinear equation Here and hereafter we assume that σ ∈ R, γ > 0, γ = 1, p ∈ C[t 0 , ∞), p(t) > 0 for t ≥ t 0 .
In this paper we have the following oscillation theorem, which is able to narrow the above difference and gaps.
Theorem 1.1.Equation (1.1) is oscillatory if and only if The oscillatory behavior of solutions of non-neutral differential equations of the form has been intensively studied in the last three decades.We refer the reader to [9,14,15,17,20] and the references cited therein.In Section 2, using the known oscillation results for the equations we prove the following corollaries of Theorem 1.1.It is possible to obtain oscillation results for equations of the form (1.1).However, for simplicity, we have restricted our attention to equations (1.2) and (1.3).
We give an example illustrating Corollary 1.1.
Corollary 1.4.Suppose that λ ≤ λ, g(t) ≥ g(t) for t ≥ t 0 , and The proof of Corollary 1.4 is deferred to the next section.
In Section 3 we investigate the relation between functions u(t) and u(t) + λu(t − τ ).We show the "if" part and the "only if" part of Theorem 1.1 in Sections 4 and 5, respectively.
Such an approach as Theorem 1.1 has been conducted by Tang and Shen [23], and Zhang and Yang [25] for odd order neutral differential equations.

Proofs of Corollaries 1.1-1.4
In this section we prove Corollaries has an eventually positive solution, then the differential equation has an eventually positive solution.
Proof of Corollary 1.4.Assume that (1.15) has a nonoscillatory solution.Then Theorem 1.1 implies that has a nonoscillatory solution x(t).Without loss of generality, we may assume that x(t) > 0 for all large t.For the case where x(t) < 0 for all large t, y(t) ≡ −x(t) is an eventually positive solution of where f (t, u) = −f (t, −u), and hence the case x(t) < 0 can be treated similarly.From Lemma 4.1 below it follows that x(t) is eventually nondecreasing.
In view of the hypothesis of Corollary 1.4, we see that x(g(t)) ≥ x(g(t)) for all large t ≥ t 0 , and for all large t ≥ t 0 .By Lemma 2.2 and Theorem 1.1, equation (1.1) has a nonoscillatory solution.This completes the proof.

Relation between u(t) and u(t) + λu(t − τ )
In this section we study the relation between functions u(t) and u(t) + λu(t − τ ).
Proof of Lemma 3.1 for the case λ > 1. Assume that λ > 1, and that lim t→∞ λ −t/τ u(t) = 0. Let t ≥ T be fixed.Since we find that From (3.10)-(3.13) it follows that for m ∈ N. By lim t→∞ λ −t/τ u(t) = 0, we find that for each fixed t ≥ T .In a similar fashion as in the proof of Lemma 3.1 for the case 0 < λ < 1, we conclude that A(t) = o(t l ) (t → ∞).This completes the proof.
is nondecreasing and concave on [T, ∞), then there exists a constant α such that is nondecreasing and convex on [T, ∞), then there exists a constant α such that Proof of Lemma 3.2.Since (∆u)(t) is concave, we find that for t ≥ T + τ , and Combining (3.15) and (3.16) with (3.17), we have EJQTDE, 2000 No. 4, p. 12 and since (∆u)(t) is nondecreasing.In the same way, we see that We have This completes the proof.
Proof of Lemma 3.3.We see that for t ≥ T + τ .By using (3.17) and the same arguments as in the proof of Lemma 3.2, we conclude that and and we have for t ∈ [T + (2m − 1)τ, T + (2m + 1)τ ], m = 1, 2, . . . .In view of the nondecreasing nature of (∆u)(t), we obtain It is easy to see that The proof is complete.
From Lemmas 3.2 and 3.3, we obtain the next result.

Proof of the "if " part of Theorem 1.1
In this section we prove the "if" part of Theorem 1.1.We make use of the following well-known lemma of Kiguradze [13].
for all large t ≥ T .
The proof is complete.

Proof of the "only if " part of Theorem 1.1
In this section we give the proof of the "only if" part of Theorem 1.1.To this end, we require the following result concerning an "inverse" of the operator ∆.Then there exists a mapping Φ on Y which has the following properties (i)-(v): Here and hereafter, C[T * , ∞) is regarded as the Fréchet space of all continuous functions on [T * , ∞) with the topology of uniform convergence on every compact subinterval of [T * , ∞).
We divide the proof of Lemma 5.1 into the two cases 0 < λ ≤ 1 and λ > 1.
(ii) Take an arbitrary compact subinterval I of [T − τ, ∞).Let ε > 0. There is an integer q ≥ 1 such that Let {y j } ∞ j=1 be a sequence in Y converging to y ∈ Y uniformly on every compact subinterval of [T * , ∞).There exists an integer j 0 ≥ 1 such that It follows from (5.1) and (5.2) that which implies that Φy j converges Φy uniformly on I.We see that Φy The proof for the case λ > 1 is complete.
Proof.We may assume that w(t) > 0 for all large t.Recall that w(t) satisfies one of (4.In exactly the same way, we have lim t→∞ w(t + ρ)/w(t) = 1 for the case (4.5).Assume that (4.4) holds.By the mean value theorem, for each large fixed t ≥ T , there is a number η(t) such that w(t + ρ) − w(t) = ρw (η(t)) and t < η(t) < t + ρ.
Thus we obtain .
Now we prove the "only if" part of Theorem 1.1.
Proof of the "only if " part of Theorem 1.1.We show that if equation (1.10) has a nonoscillatory solution, then equation (1.1) has a nonoscillatory solution.Let z(t) be a nonoscillatory solution of (1.10).Without loss of generality, we may assume that z(t) is eventually positive.Set w(t) = (1 + λ)z(t).Then w(t) is an eventually positive solution of (5.3) Lemma 4.1 implies that w(t) is a function of Kiguradze degree k for some k ∈ {1, 3, . . ., n − 1}, and one of the cases (4.3)-(4.5)holds.Hence, lim t→∞ w(t)/t k = const ≥ 0. From Lemma 5.2 it follows that We can take a sufficiently large number T ≥ T 1 such that w (i) (t) > 0 (i = 0, 1, 2, . . ., k − 1), w(g(t)) > 0 for t ≥ T , and Recall (4.2).Integrating (5.3), we have for t ≥ T , where For each y ∈ Y , we define the mapping where  By applying the Schauder-Tychonoff fixed point theorem to the operator F , there exists a y ∈ Y such that y = F y.
Consider the set Y of functions y ∈ C[T * , ∞) which satisfiesy(t) = 0 for t ∈ [T * , T ] and 0 ≤ y(t) ≤ w(t) − P (t) for t ≥ T.Then Y is closed and convex.Note that there is a constant M > 0 such that |y(t)| ≤ M t k on [T, ∞) for y ∈ Y , by lim t→∞ w(t)/t k = const ≥ 0.
we see that F is well defined on Y and maps Y into itself.Since Φ is continuous on Y , by the Lebesgue dominated convergence theorem, we can show that F is continuous on Y as a routine computation.Now we claim that F (Y ) is relatively compact.We note that F (Y ) is uniformly bounded on every compact subinterval of [T * , ∞), because of F (Y ) ⊂ Y .By the Ascoli-Arzelà theorem, it suffices to verify that F (Y ) is EJQTDE, 2000 No. 4, p. 23 equicontinuous on every compact subinterval of [T * , ∞).Let I be an arbitrary T and y ∈ Y .Thus we see that {(F y) (t) : y ∈ Y } is uniformly bounded on I.The mean value theorem implies that F (Y ) is equicontinuous on I. Since |(F y)(t 1 ) − (F y)(t 2 )| = 0 for t 1 , t 2 ∈ [T * , T ], we conclude that F (Y ) is equicontinuous on every compact subinterval of [T * , ∞).
, we conclude that