Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order

Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order: History

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Subjects: Mathematics

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A DDE is a single-variable differential equation, usually called time, in which the derivative of the solution at a certain time is given in terms of the values of the solution at earlier times. Moreover, if the highest-order derivative of the solution appears both with and without delay, then the DDE is called of the neutral type. The neutral DDEs have many interesting applications in various branches of applied science, as these equations appear in the modeling of many technological phenomena. The problem of studying the oscillatory and nonoscillatory properties of DDEs has been a very active area of research in the past few decades.

delay differential equation
neutral
oscillation
noncanonical case

1. Introduction

Consider the 2nd-order delay differential equation (DDE) of the neutral type:

(1)

where

t \in [t_{0}, \infty)

and

v (t) : = u (t) + a_{1} (t) u (g_{0} (t))

. In this entry, we obtain new sufficient criteria for the oscillation of solutions of (1) under the following hypotheses:

(A1)

β \geq 1

is a ratio of odd integers;

(A2)

a_{i} \in C ([t_{0}, \infty), [0, \infty))

for

i = 0, 1, 2,

a_{0} (t) > 0

a_{1} \leq c_{0}

a constant (this constant plays an important role in the results), and

a_{2}

does not vanish identically on any half-line

[t_{*}, \infty)

with

t_{*} \in [t_{0}, \infty);

(A3)

g_{j} \in C ([t_{0}, \infty), R)

g_{j} (t) \leq t

g_{0}^{'} (t) \geq g_{0}^{*} > 0,

g_{0} \circ g_{1} = g_{1} \circ g_{0}

and

{lim}_{t \to \infty} g_{j} (t) = \infty

for

j = 0, 1

By a proper solution of (1), we mean a

u \in C^{1} ([t_{0}, \infty))

with

a_{0} \cdot {(v^{'})}^{β} \in C^{1} ([t_{0}, \infty))

and

sup {|u (t)| : t \geq t_{*}} > 0,

for

t_{*} \in [t_{0}, \infty),

and u satisfies (1) on

[t_{0}, \infty)

. A solution u of (1) is called nonoscillatory if it is eventually positive or eventually negative; otherwise, it is called oscillatory.

The oscillatory properties of solutions of second-order neutral DDE (1) in the noncanonical case, that is:

(2)

where

2. Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order

We begin with the following notations: $U^{+}$ is the set of all eventually positive solutions of (1), $V (t) : = a_{0}^{1 / β} (t) v^{'} (t)$ ,

Lemma 1. Assume that $v \in U^{+}$ and there exists a $δ_{0} \in (0, 1)$ such that:

(3)

Then, v eventually satisfies:

and:

Proof. Let $u \in U^{+}$ . Then, we have that $u (g_{0} (t))$ , and $u (g_{1} (t))$ are positive for $t \geq t_{1}$ , for some $t_{1} \geq t_{0}$ . Therefore, it follows from (1) that:

Using (1) and Lemma 1 in ^[1], we see that:

and so:

(4)

Integrating this inequality from $t_{1}$ to t and using the fact ${(V^{β} (t))}^{'} \leq 0$ , we find: ${(V^{β} (t))}^{'} \leq 0$

(5)

$(C_{1})$ Assume the contrary, that $v^{'} (t) > 0$ for $t \geq t_{1}$ . Thus, from (5), we have:

This, from (3), implies:

Letting $t \to \infty$ and taking the fact that $η (t) \to 0$ as $t \to \infty$ , we obtain $V^{β} (t) \to - \infty$ $V^{β} (t) \to - \infty$ , which contradicts the positivity of $V (t)$ . $V (t)$

Next, since v is positive decreasing, we have that ${lim}_{t \to \infty} v (t) = v_{0} \geq 0$ . Assume the contrary, that $v_{0} > 0$ . Then, $v (t) \geq v_{0}$ for all $t \geq t_{2}$ , for some $t_{2} \geq t_{1}$ $t_{2} \geq t_{1}$ . Thus, from (3) and (5), we have:

and so,

(6)

Using the fact that $η^{'} (t) < 0$ , we obtain that $η (t) < η^{'} (t_{2}) < η^{'} (t_{1})$ for all $t \geq t_{2} \geq t_{1}$ . Hence, by integrating (6) from $t_{1}$ to t, we obtain:

Letting $t \to \infty$ and taking the fact that $η (t) \to 0$ as $t \to \infty$ , we obtain $v (t) \to - \infty$ , which contradicts the positivity of $v (t)$ . Therefore, $v_{0} = 0$ . $v (t)$

$(C_{2})$ Since $V (t)$ is decreasing, we obtain:

and:

(7)

Then, ${(v / η)}^{'} \geq 0$ .

$(C_{3})$ From (7), we obtain:

Thus, from (4) and the fact $V^{'} (t) \leq 0$ , we obtain:

and then:

The proof is complete.

Lemma 2. Assume that $u \in U^{+}$ $u \in U^{+}$ and there exists a $δ_{0} \in (0, 1)$ such that (3) holds. Then:

Proof. Let $u \in U^{+}$ . From Lemma 1, we have that $(C_{1})$ – $(C_{3})$ hold for $t \geq t_{1}$ . $u \in U^{+}$

Integrating $(C_{3})$ from $t_{1}$ to t, we arrive at:

From (3), we obtain:

and:

(8)

Using $(C_{1})$ , we eventually have:

Hence, (8) becomes:

This implies that $v / η^{γ_{0} δ_{0}}$ is a decreasing function.
The proof is complete.

2.2. Oscillation Theorems

In the next theorem, by using the principle of comparison with an equation of the first-order, we obtain a new criterion for the oscillation of (1).

Theorem 6. Assume that

g_{1} (t) \leq g_{0} (t)

and there exists a

δ_{0} \in (0, 1)

such that (3) holds. If the delay differential equation:

(9)

is oscillatory, then every solution of (1) is oscillatory.

Proof. Assume the contrary, that (1) has a solution

u \in U^{+}

. Then, we have that

u (t),

and

u (g_{1} (t))

are positive for

t \geq t_{1}

t \geq t_{1}

, for some

t_{1} \geq t_{0}

t_{1} \geq t_{0}

. From Lemmas 1 and 2, we have that

(C_{1})

–

(C_{4})

hold for

t \geq t_{1}

Next, we define:

From

(C_{1})

w (t) > 0

for

t \geq t_{1}

t \geq t_{1}

. Thus,

Thus, it follows from

(C_{3})

that:

(10)

Using

(C_{4})

, we obtain that:

which with (10) gives:

(11)

Now, we set:

Then,

W (t) \leq {\tilde{c}}_{0} w (g_{0} (t))

W (t) \leq {\tilde{c}}_{0} w (g_{0} (t))

, and so, (11) becomes:

which has a positive solution. In view of ^[2] (Theorem 1), (9) also has a positive solution, which is a contradiction.

The proof is complete.

Corollary 1. Assume that

g_{1} (t) \leq g_{0} (t)

and there exists a

δ_{0} \in (0, 1)

such that (3) holds. If:

(12)

then every solution of (1) is oscillatory.

Proof. It follows from Theorem 2 in ^[3] that the condition (12) implies the oscillation of (9).

Next, by introducing two Riccati substitution, we obtain a new oscillation criterion for (1).

Theorem 7. Assume that

g_{1} (t) \leq g_{0} (t)

and there exists a

δ_{0} \in (0, 1)

such that (3) holds. If:

(13)

then every solution of (1) is oscillatory.

Proof. Assume the contrary, that (1) has a solution

u \in U^{+}

. Then, we have that

u (t),

and

u (g_{1} (t))

are positive for

t \geq t_{1}

, for some

t_{1} \geq t_{0}

t_{1} \geq t_{0}

. From Lemmas 1 and 2, we have that

(C_{1})

–

(C_{4})

hold for

t \geq t_{1}

Now, we define the functions:

Θ_{1} : = \frac{V}{v},

and:

Θ_{2} : = \frac{V \circ g_{0}}{v \circ g_{0}} .

Then,

Θ_{1}

and

Θ_{2}

are negative for

t \geq t_{1}

. From

(C_{4})

, we obtain:

t \geq t_{1}

Hence,

and:

Then:

(14)

and:

(15)

Combining (14) and (15), we obtain:

Integrating this inequality from

t_{1}

t_{1}

to t, we have:

From

(C_{2}),

we obtain

η (t) Θ_{1} (t) \geq - 1

η (t) Θ_{1} (t) \geq - 1

. Therefore,

where:

Since

η^{'} (t) < 0

and

a^{'} (t) \geq 0

, we find:

η^{'} (t) < 0

Taking

lim {sup}_{t \to \infty}

lim {sup}_{t \to \infty}

and using (13), we arrive at a contradiction.

The proof is complete.

2.3. Applications and Discussion

Remark 1. It is easy to see that the previous works that dealt with the noncanonical case required either $a_{1} (t) < 1$ or $a_{1} (t) < η (t) / η (g_{0} (t))$ . Since η is decreasing and $g_{0} (t) \leq t$ , we have that $η (g_{0} (t)) \geq η (t)$ . Then, the results of these works only apply when $a_{1} (t) \in (0, 1)$ .

Example 1. Consider the DDE:

(16)

where

t \geq 1,

a_{1}^{*} > 0,

and

κ < λ \in (0, 1)

. By choosing

δ_{0} = a_{2}^{*},

the condition (12) becomes:

(17)

Using Corollary 1, Equation (16) is oscillatory if (17) holds.

Remark 2. To apply Theorems 3 and 4 on (16), we must stipulate that

a_{1}^{*} < 1

. Let a special case of (16), namely,

A simple computation shows that (16) is oscillatory if:

(18)

or:

(19)

or:

(20)

Consider the following most specific special case:

(21)

Note that (18)–(20) fail to apply. However, (17) reduces to:

which ensures the oscillation of (21).

This entry is adapted from the peer-reviewed paper 10.3390/math9172026

References

Baculikova, B.; Dzurina, J. Oscillation theorems for second-order nonlinear neutral differential equations. Comput. Math. Appl. 2011, 62, 4472–4478.
Philos, C. On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delays. Arch. Math. (Basel) 1981, 36, 168–178.
Kitamura, Y.; Kusano, T. Oscillation of first-order nonlinear differential equations with deviating arguments. Proc. Amer. Math. Soc. 1980, 78, 64–68.

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