OSCILLATION RESULTS FOR SECOND ORDER QUASILINEAR NEUTRAL DELAY DIFFERENTIAL EQUATIONS WITH “MAXIMA”

In this paper, some new oscillation criteria are obtained for the second order quasilinear neutral delay differential equation

In the last few years, the qualitative theory of differential equations with "maxima" received very little attention even though such equations often arise in the problem of automatic regulation of various real system, see for example [1,10,12].The oscillatory behavior of solutions of differential equations with "maxima" are discussed in [1-6, 11, 15], and the references cited therein.In [2], the authors have established some oscillation criteria for equation (1.1) when α = 1 and a(t) ≡ 1.Also in [14], the authors discussed oscillatory behavior of equation (1.1) when α = 1 and ∞ t 0 dt a(t) = ∞.Motivated by these observations, in this paper, we analyse the oscillatory and asymptotic behavior of solutions of equation (1.1) under the condition In Section 2, we establish sufficient conditions for the almost oscillation of all solutions of equation (1.1).In Section 3, we present sufficient conditions for the existence of nonoscillatory solutions for the equation (1.1) using contraction mapping principle.In Section 4, we present some examples to illustrate the main results.

Oscillation Results
In this section, we will derive some new sufficient condition for the almost oscillation of equation (1.1).Define and . Lemma 2.1.Let x(t) be an eventually positive solution of equation (1.1).Then one of the following holds: Proof.Let x(t) be an eventually positive solution of equation (1.1).Then we may assume that x(σ(τ (t))) > 0, x(τ (t)) > 0 for all t ≥ T .Then inview of (C 2 ), we have z(t) > 0 for t ≥ T .From the equation (1.1) we obtain Hence, a(t) (z ′ (t)) α and z(t) are of eventually of one sign.This completes the proof.
Lemma 2.2.Let x(t) be an eventually negative solution of equation (1.1).Then one of the following holds: Proof.The assertion of Lemma 2.3 can be verified easly.Proof.Let x(t) be a positive solution of equation (1.1).Then by Lemma 2.1(II), we have z(t) > 0 and z ′ (t) < 0 for all t ≥ T .Therefore z(t) → L ≥ 0 as t → ∞.If L > 0, then for ǫ = L(1−p) 2p > 0, there exists T ≥ t 0 such that L < z(t) < L + ǫ for t ≥ T .Then for t ≥ T , we have From equation (1.1), we have Integrating from T to ∞ and using the fact that a(t) (z ′ (t)) α is positive and decreasing we obtain Divide the last inequality by a(t) and then integrating the resulting inequality we obtain where m ≥ 1 is an integer, then every solution of equation (1.1) is almost oscillatory.
Proof.Let x(t) be a nonoscillatory solution of equation (1.1).Then either x(t) > 0 eventually or x(t) < 0 eventually.We shall consider the case when x(t) > 0. Since the other case can be investigated analogously.
Let x(σ(t)) > 0, x(τ (t)) > 0 for all t ≥ T , where T is chosen so large enough that the conclusions of Lemma 2.1 hold for all t ≥ T .
First we assume Lemma 2.1(I) holds.Then and max x(s) ≥ max (1 − p(t)). (2.4) From the equation (1.1) and (2.4) we have Then from (2.5) and (2.6) we have Multiply by (t − s) m and then integrating from T to t, we obtain Dividing the last inequality by t m and then taking limit supremum, we obtain lim Next assume that Lemma 2.1(II) holds.Then by Lemma 2.4 we obtain that lim t→∞ x(t) = 0.This completes the proof.
hold then every solution of equation (1.1) is almost oscillatory.
Proof.Assume that there exists a nonoscillatory solution x(t) of equation (1.1) such that x(σ(t)) > 0, x(τ (t)) > 0 for all t ≤ T where T is chosen large enough that the conclusion of Lemma 2.1 hold for all t ≥ T .Integrating (1.1) from T to t yields, x β (u)ds = 0. (2.8) Letting t → ∞, we have x β (u)ds < ∞. (2.9) In this case z(t) is increasing, so there exists a positive number C such that z(t) > C for t ≥ T .This, together with (2.3) yields.
Next assume that Lemma 2.1(II) holds.Then by Lemma 2.4, we obtain that lim t→∞ x(t) = 0.This completes the proof.
hold then every solution of equation (1.1) is almost oscillatory.
Proof.Proceeding as in the proof of Theorem 2.2, we have that Lemma 2.1 holds.For Case(I), we have (2.9) and (2.10).For large t, we have A(t) ≤ 1 and This contradicts (2.10).Next assume that Lemma 2.1(II) holds.Then by Lemma 2.4, we obtain that lim t→∞ x(t) = 0.This completes the proof.

Existence of Nonoscillatory Solutions
In this section, we provide sufficient conditions for the existence of nonoscillatory solutions of equation (1.1) in case α > β > 1 or β < α < 1.Note that in this section we do not require p(t) ≡ p.
then the equation (1.1) has a bounded nonoscillatory solution.
Proof.Choose T ≥ t 0 sufficiently large so that and Let ψ be the set of all bounded continuous function on [t 0 , ∞) with norm and let Define the operator T : S → ψ by Clearly, T is continuous, now for every x ∈ S and t ≥ T Also, Thus we have that T S ⊂ S. Since S is bounded closed and convex subset of ψ.
We only need to show that T is contraction mapping on S, in order to apply contraction principle.For x, y ∈ S and t ≥ T , we have |dt By the Mean Value Theorem for derivative applied to the function V (u) = u β , α > β > 1, we see that for any x, y ∈ S, we have Thus, and we see that T is a contraction on S. Hence, T has a unique fixed point which is clearly a positive solution of equation (1.1).This completes the proof.
Proof.Choose T ≥ t 0 sufficiently large so that, and Let, ψ be the set of all bounded continuous function on [t 0 , ∞) with norm and let Define the operator T : S → ψ by Clearly, T is continuous.Now for every x ∈ S and t ≥ T . Also, Thus, we have that T S ⊂ S. Since S is bounded closed and convex subset of ψ.We only need to show that T is contraction mapping on S in order to apply contraction principle.For x, y ∈ S and t ≥ T we have By the Mean Value Theorem of derivatives applied to the function V (u) = u β , β < α < 1.We see that for any x, y ∈ S, we have Thus, T is a contraction mapping.So T has a unique fixed point x, that is clearly a positive solution of equation (1.1).This completes the proof.

Examples
In this section we present some examples to illustrate the main results.x 1/6 (s) = 0, t ≥ 1. (4.5) It is easy to see that all conditions of Theorem 3.2 are satisfies and hence every bounded nonoscillatory solution of equation (4.5) tends to zero as t → ∞.Infact x(t) = e −t is one such solution of equation (4.5).
We can easily check that all conditions of Theorem 2.6 are satisfying and hence every solution of equation (1.1) is almost oscillatory.It is easy to prove that all conditions of Theorem 2.7 are satisfies and hence every solution of equation (1.1) is almost oscillatory.It is easily verified that all conditions of Theorem 3.1 are satisfies and hence every nonoscillatory solution of equation (4.4) tends to zero as t → ∞.Infact x(t) = e −t is one such solution of equation (4.4). [t−1,t]