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Cesarano, C. Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order. Encyclopedia. Available online: https://encyclopedia.pub/entry/17396 (accessed on 20 April 2024).

Cesarano C. Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order. Encyclopedia. Available at: https://encyclopedia.pub/entry/17396. Accessed April 20, 2024.

Cesarano, Clemente. "Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order" *Encyclopedia*, https://encyclopedia.pub/entry/17396 (accessed April 20, 2024).

Cesarano, C. (2021, December 21). Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order. In *Encyclopedia*. https://encyclopedia.pub/entry/17396

Cesarano, Clemente. "Oscillatory Properties of Noncanonical Neutral DDEs of Second-Order." *Encyclopedia*. Web. 21 December, 2021.

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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

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

(**1**)

where $t\in \left[{t}_{0},\infty \right)$ and $v\left(t\right):=u\left(t\right)+{a}_{1}\left(t\right)u\left({g}_{0}\left(t\right)\right)$ . In this entry, we obtain new sufficient criteria for the oscillation of solutions of (1) under the following hypotheses:

(A1) $\beta \ge 1$ is a ratio of odd integers;

(A2) ${a}_{i}\in C\left(\left[{t}_{0},\infty \right),\left[0,\infty \right)\right)$ for $i=0,1,2,$ ${a}_{0}\left(t\right)>0$ ${a}_{1}\le {c}_{0}$ a constant (this constant plays an important role in the results), and ${a}_{2}$ does not vanish identically on any half-line $\left[{t}_{*},\infty \right)$ with ${t}_{*}\in \left[{t}_{0},\infty \right);$

(A3) ${g}_{j}\in C\left(\left[{t}_{0},\infty \right),\mathbb{R}\right)$ ${g}_{j}\left(t\right)\le t$ ${g}_{0}^{\prime}\left(t\right)\ge {g}_{0}^{*}>0,$ ${g}_{0}\circ {g}_{1}={g}_{1}\circ {g}_{0}$ and ${lim}_{t\to \infty}{g}_{j}\left(t\right)=\infty $ for $j=0,1$

By a proper solution of (1), we mean a $u\in {C}^{1}\left(\left[{t}_{0},\infty \right)\right)$ with ${a}_{0}\xb7{\left({v}^{\prime}\right)}^{\beta}\in {C}^{1}\left(\left[{t}_{0},\infty \right)\right)$ and $sup\{\left|u\left(t\right)\right|:t\ge {t}_{*}\}>0,$ for ${t}_{*}\in \left[{t}_{0},\infty \right),$ and u satisfies (1) on $\left[{t}_{0},\infty \right)$ . 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

We begin with the following notations: ${U}^{+}$ is the set of all eventually positive solutions of (1), $V\left(t\right):={a}_{0}^{1/\beta}\left(t\right){v}^{\prime}\left(t\right)$ ,

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

(**3**)

Then, *v* eventually satisfies:

and:

**Proof.** Let $u\in {U}^{+}$ . Then, we have that $u\left({g}_{0}\left(t\right)\right)$ , and $u\left({g}_{1}\left(t\right)\right)$ are positive for $t\ge {t}_{1}$ , for some ${t}_{1}\ge {t}_{0}$ . 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 ${\left({V}^{\beta}\left(t\right)\right)}^{\prime}\le 0$ , we find:

(**5**)

$\left({\mathbf{C}}_{1}\right)$ Assume the contrary, that ${v}^{\prime}\left(t\right)>0$ for $t\ge {t}_{1}$ . Thus, from (5), we have:

This, from (3), implies:

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

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

or

and so,

(**6**)

Using the fact that ${\eta}^{\prime}\left(t\right)<0$ , we obtain that $\eta \left(t\right)<{\eta}^{\prime}\left({t}_{2}\right)<{\eta}^{\prime}\left({t}_{1}\right)$ for all $t\ge {t}_{2}\ge {t}_{1}$ . Hence, by integrating (6) from ${t}_{1}$ to *t*, we obtain:

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

$\left({\mathbf{C}}_{2}\right)$ Since $V\left(t\right)$ is decreasing, we obtain:

and:

(**7**)

Then, ${\left(v/\eta \right)}^{\prime}\ge 0$ .

$\left({\mathbf{C}}_{3}\right)$ From (7), we obtain:

Thus, from (4) and the fact ${V}^{\prime}\left(t\right)\le 0$ , we obtain:

and then:

The proof is complete.

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

**Proof.** Let $u\in {U}^{+}$ . From Lemma 1, we have that $\left({\mathbf{C}}_{1}\right)$ – $\left({\mathbf{C}}_{3}\right)$ hold for $t\ge {t}_{1}$ .

Integrating $\left({\mathbf{C}}_{3}\right)$ from ${t}_{1}$ to t, we arrive at:

From (3), we obtain:

and:

(**8**)

Using $\left({\mathbf{C}}_{1}\right)$ , we eventually have:

Hence, (8) becomes:

This implies that $v/{\eta}^{{\gamma}_{0}{\delta}_{0}}$ is a decreasing function.

The proof is complete.

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).

(**9**)

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

Next, we define:

From $\left({\mathbf{C}}_{1}\right)$ , $w\left(t\right)>0$ for $t\ge {t}_{1}$ . Thus,

Thus, it follows from $\left({\mathbf{C}}_{3}\right)$ that:

(**10**)

Using $\left({\mathbf{C}}_{4}\right)$ , we obtain that:

which with (10) gives:

(**11**)

Now, we set:

Then, $W\left(t\right)\le {\tilde{c}}_{0}w\left({g}_{0}\left(t\right)\right)$ , 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.

(**12**)

then every solution of (1) is oscillatory.

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

(**13**)

then every solution of (1) is oscillatory.

Now, we define the functions:

$${\Theta}_{1}:=\frac{V}{v},$$

and:

$${\Theta}_{2}:=\frac{V\circ {g}_{0}}{v\circ {g}_{0}}.$$

Then, ${\Theta}_{1}$ and ${\Theta}_{2}$ are negative for $t\ge {t}_{1}$ . From $\left({\mathbf{C}}_{4}\right)$ , we obtain:

Hence,

and:

Then:

(**14**)

and:

(**15**)

Integrating this inequality from ${t}_{1}$ to *t*, we have:

From $\left({\mathbf{C}}_{2}\right),$ we obtain $\eta \left(t\right){\Theta}_{1}\left(t\right)\ge -1$ . Therefore,

where:

Since ${\eta}^{\prime}\left(t\right)<0$ and ${a}^{\prime}\left(t\right)\ge 0$ , we find:

Taking $lim{sup}_{t\to \infty}$ and using (13), we arrive at a contradiction.

The proof is complete.

**Remark** **1.** It is easy to see that the previous works that dealt with the noncanonical case required either ${a}_{1}\left(t\right)<1$ or ${a}_{1}\left(t\right)<\eta \left(t\right)/\eta \left({g}_{0}\left(t\right)\right)$ . Since η is decreasing and ${g}_{0}\left(t\right)\le t$ , we have that $\eta \left({g}_{0}\left(t\right)\right)\ge \eta \left(t\right)$ . Then, the results of these works only apply when ${a}_{1}\left(t\right)\in \left(0,1\right)$ .

**Example** **1.** Consider the DDE:

(**16**)

where $t\ge 1,$ ${a}_{1}^{*}>0,$ and $\kappa <\lambda \in \left(0,1\right)$ . By choosing ${\delta}_{0}={a}_{2}^{*},$ the condition (12) becomes:

(**17**)

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

(**18**)

or:

(**19**)

or:

(**20**)

Consider the following most specific special case:

(**21**)

which ensures the oscillation of (21).

- 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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