Oscillation of Second Order Delay Differential Equations with Nonlinear Nonpositive Neutral Term

Assume condition (1.2) holds. If x is a positive solution of equation (1.1) then the corresponding function z satisfies one of the following two cases:

• (I) z(t) > 0, z′(t) > 0, (a(t)z′(t))′ < 0;

• (II) z(t) < 0, z′(t) > 0, (a(t)z′(t))′ < 0,

Proof. Assume that x(t) > 0, x(τ(t)) > 0 and x(σ(t)) > 0 for all t ≥ t₁ for some t₁ ≥ t₀. From the definition of z(t) and equation (1.1), we have

Hence az′ is decreasing and of one sign for large t, that is, there exists a t₂ ≥ t₁ such that z′(t) > 0 or z′(t) < 0 for all t ≥ t₂.

If z′(t) < 0 for t ≥ t₂, then a(t)z′(t) ≤ –d < 0 for t ≥ t₂, where d = – a(t₂)z′(t₂) > 0. Thus, we have

In view of condition (1.2), we see that limt→∞z(t)=−∞. Now we consider the following two cases separately.

Case 1. If x is unbounded, then there exists a sequence {t_k} such that limk→∞tk=∞ and limk→∞x(tk)=∞ where x(t_k) = max{x(s) : t₀ ≤ s ≤ t_k}. Since limt→∞τ(t)=∞, τ(t_k) > t₀ for all sufficiently large k and τ(t) ≤ t, we have

Hence

as k → ∞ since 0 < α ≤ 1 and p(t) is bounded, which contradicts the fact that limt→∞z(t)=−∞.

Case 2. If x is bounded then z is also bounded, since p(t) is bounded, which contradicts limt→∞z(t)=−∞. Hence z satisfies one of the cases (I) and (II). This completes the proof. □

Assume condition (1.2) holds. Let x be a positive solution of equation (1.1) such that case (I) of Lemma 2.1holds. Then

Proof. From the definition of z and (C₂) we have x(t) > z(t) for all t ≥ t₁ ≥ t₀. In view of case (I), we obtain

Furthermore,

By (2.3), one obtains (z(t)R(t))′<0 for all t ≥ t₁. Thus, z(t)R(t) is strictly decreasing for all t ≥ t₁. This completes the proof. □

Let β < α, σ(t) < τ(t), and condition (1.2) hold. If

Proof. Assume that there is a nonoscillatory solution x of equation (1.1), say x(t) > 0, x(τ(t)) > 0 and x(σ(t)) > 0 for all t ≥ t₁ ≥ t₀. It follows from Lemma 2.1 that the corresponding function z satisfies either (I) or (II).

Let β = 1, and condition (1.2) hold. If

Proof. Assume that there exists a nonoscillatory solution x of equation (1.1), say, x(t) > 0, x(τ(t)) > 0, and x(σ(t)) > 0 for all t ≥ t₁ ≥ t₀ so that for z one of the cases (I) and (II) holds.

Let β > 1, and condition (1.2) hold. If there exists a positive nondecreasing function ρ(t) such that

Proof. Proceeding as in the proof of Theorem 2.1, we see that one of the cases of Lemma 2.1 holds for all t ≥ t₁.

Case (I) Proceeding as in the proof of Theorem 2.1 (Case (I)), we obtain

for all t ≥ t₁. Define

Then w(t) > 0, and from (2.15), we have

Since a(t)z′(t) is nonincreasing we have a(σ(t))z′(σ(t)) ≥ a(t)z′(t) and so

Using that z(t) is positive increasing and β > 1 there is a constant K > 0 such that z^β–1(t) ≥ K^β–1 > 0 for all t ≥ t₂ ≥ t₁ and the last inequality implies

Now using the completing the square, we have

Integrating the last inequality from t₂ to t, we obtain

Take lim sup as t → ∞ in the last inequality, we obtain a contradiction to (2.14).

Case (II) In this case z(t) < 0 and z′(t) > 0 for all t ≥ t₁. Then proceeding as in case (II) of Theorem 2.2, we obtain limt→∞x(t)=0. This completes the proof of the theorem. □

Consider a second order neutral differential equation

where p ∈ (0,1) is a constant. Here a(t) = 1, p(t) = p, q(t) = 8t, τ(t) = t/2, σ(t) = t/3 for t ≥ t₁ = 1, α = 1/3, β = 1/5 and R(t) = t – 1. Simple calculation shows that conditions (2.4) and (2.5) are satisfied. Therefore by Theorem 2.1 every solution of equation (3.1) is oscillatory.

Consider a second order neutral differential equation

where p ∈ (0,1) is a constant. Here a(t) = t, p(t) = p, q(t) = t, τ(t) = t/2, σ(t) = t/3 for t ≥ t₁ = 1, α = 1/3, β = 1 and R(t) = ln t. Now one can easily verify that all conditions of Theorem 2.2 are satisfied. Hence every solution of equation (3.2) is either oscillatory or tends to zero as t → ∞.

Consider a second order neutral differential equation

where p ∈ (0,1) is a constant. Here a(t) = t, p(t) = p, q(t)=1t, τ(t) = t – π/2, σ(t) = t – π or t ≥ t₁ = 1, α = 1/3, β = 3 and R(t) = ln t. By taking ρ(t) = 1, one can easily see that all conditions of Theorem 2.3 are satisfied, and hence every solution of equation (3.3) is either oscillatory or tends to zero as t → ∞.

We conclude this paper with the following remark.

Note that Theorem 2.1 guarantees that every solution of equation ((1.1) is oscillatory, and from other theorems we see that every solution is either oscillatory or tends to zero as t → ∞. Therefore it is interesting to improve Theorems 2.2, and 2.3 so that all solutions of equation (1.1) are oscillatory only. Further note that the results reported in [7, 11, 13, 17] cannot be applied to equations (3.1), 3.2) and (3.3) since the neutral term is not linear.