Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order

Lemma 1. Consider the NDDE

where  τ ≥ 0,    q_i > 0,   and σ_i ≥ 0    for all  i = 1,2, …, n.

Let x(t)   be a positive solution of (4). Set

If p ≥ −1, then z(t) is a positive and decreasing solution of (4); that is,

Lemma 2. Let p and τ  be positive constants. Let x(t)   be an eventually positive solution of the delay differential inequality

Then for t sufficiently large,

where

Theorem 3. Consider NDDE (3). Assume that

• (i)

  q₂ ∈ [0, ∞), σ ≥ τ, $$ and

• (ii)

  $$,

where $$ is the unique real root of the equation

Then all solutions of (3) are oscillatory.

Proof. Assume, for the sake of a contradiction, that (3) has a nonoscillatory solution  x(t). Without loss of generality, assume that  x(t) > 0    ∀ t ≥ t₀ > 0. Let

So that y(t) is also a positive solution of (3).

That is,

where

Set for t ≥ t₀ + 2τ

Thus it follows from Lemma 1 that z(t)   is a positive and decreasing solution of

and in particular (as σ > τ implies that t − τ_i ≤ t − τ,   i = 1,2.), it follows that

But we have

This implies that

Applying Lemma 2 with (18) we get

Then  w(t)  is bounded.

Dividing (16) by  z(t) > 0 and integrating from  t − τ  to  t, we get

Let m = lim _t→∞inf  w(t).

Then, it follows from (20) that for ε > 0 and sufficiently small,

As ε is arbitrary, so we have

Let

Then

Let $$ be the unique real root of the equation

Then

Hence

This contradicts condition (ii) and then completes the proof.

Example 4. Consider the NDDE

We note that

Then we have

•  (i)  

  $$,

•  (ii)  
  $$ ()

where  $$  is the unique real root of the equation Then all the hypotheses of Theorem 3 are satisfied, and therefore every solution of (28) oscillates. (Indeed  x(t) = sint  is such a solution.)

Theorem 5. Consider the NDDE (1). Assume that

•  (iii)

    $$,  and   q₂(t) ∈ C[[t₀, ∞), (0, ∞)]  is periodic with period  τ,

•  (iv)

    $$,

where  $$  is defined as in Theorem 3. Then all solutions of (1) are oscillatory.

Proof. Assume, for the sake of contradiction, that (1) has a nonoscillatory solution x(t). Without loss of generality, assume that (t) > 0    ∀ t ≥ t₀ > 0. Let

which is oscillation invariant transformation. Then y(t) is a positive solution of the equation where  $$   is periodic with period τ.

Let

Then  z(t) is decreasing positive solution of the equation

Set

This implies that  w(t) ≥ 1, since  z(t − τ) ≥ z(t).

Dividing both sides of (33) by z(t) and then integrating from t − τ to  t, we obtain that

Hence

Since q(t) is periodic with period  τ, then we obtain

Substituting in (38) we find, for all t ≥ t₀,

Now, we want to prove that w(t)  is bounded.

Applying the assumption (iv), we can find t^* ∈ (t − τ, t) such that

where $$ is similar as in the proof of Theorem 3.

Integrating (33) from t^* to t we obtain

Using Bonnet’s Theorem and in particular (as z^′(t − τ) < o), we get

Integrating (33) from t − τ to t^*, we get

Using Bonnet’s Theorem and in particular (as z^′(t − τ) < o), we get

Combining (43) and (45), we conclude

or

Then  w(t)  is bounded.

Now, let

But we have proved that w(t)  is bounded; that is, m  is finite.

From (40), we obtain

Therefore, we get

Hence

This contradicts our assumption (iv) and then completes the proof.

Example 6. Consider the NDDE

where

Then we have

• (1)

  $$;

• (2)

  q₂(t) = 1 + cos 2t ∈ C[[0, ∞), (0, ∞)] is periodic with period π and satisfies

  $$ ()

where $$ is the unique real root of the equation

Therefore (52) satisfies all the hypotheses of Theorem 5. Hence every solution of this equation is oscillatory.

Theorem 7. Suppose that condition (iii) holds. If

•  (v)

    $$,

then every solution of (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 5, we get (49) which implies that

Hence

But this is a contradiction of assumption (v), and then the proof is complete.

Example 8. Consider the NDDE

Here we have

Note that q₂(t) = e + sin4t  is positive and periodic with period π/2, and also

• (1)

  $$,

• (2)
  $$ ()

Then (58) satisfies hypotheses of Theorem 7, and so all its solutions are oscillatory.