Neutral differential equations with noncanonical operator: Oscillation behavior of solutions

1.   Introduction

In this work, we consider the even-order neutral differential equation with several delays

where $l\geq l_{0}, \; \nu \left(l\right) = u\left(l\right) +p\left(l\right) u\left(\tau \left(l\right) \right)$, $n\geq 4$ is an even integer, $\alpha \in Q_{odd}^{+}: = \left\{ a/b:\text{ }a, b\in \mathbb{Z} ^{+}\text{ are odd}\right\}$ and the following conditions are fulfilled:

$(\textbf{i}) \; r$ is a differentiable real-valued function and $p, \; \tau, \; q_{i}$ are continuous real-valued functions on $\left[ l_{0}, \infty \right)$;

$(\textbf{ii}) \; r^{\prime }\left(l\right) \geq 0, \; p\left(l\right) \in \left[ 0, p_{0}\right], \; p_{0}$ is a constant, $\tau \left(l\right) \leq l,$ and $\lim_{l\rightarrow \infty }\tau \left(l\right) = \infty$;

$(\textbf{iii}) \; g_{i}\in C\left(\left[ l_{0}, \infty \right), \mathbb{R} \right), \; g_{i}\left(l\right) \leq l, \; g_{i}^{\prime }\left(l\right) > 0$ and $\lim_{l\rightarrow \infty }g_{i}\left(l\right) = \infty$;

$(\textbf{iv}) \; f\in C\left(\mathbb{R}, \mathbb{R} \right), \ f\left(u\right) \geq \varrho u^{\beta }$ for $u\neq 0$, $\ \varrho$ is a positive constant, $\beta$ is a ratio of odd positive integers; and

The function $u \; \in C\left([l_{u}, \infty)\right)$ with $l_{u}\geq l_{0},$ is said to be a solution of (1.1) if$\ u$ has the property $v\in C^{n-1}[l_{u}, \infty), \; r\left(\nu ^{\left(n-1\right) }\right) ^{\alpha }\in C^{1}[l_{u}, \infty), \ $and satisfies (1.1) on $[l_{u}, \infty)$. We consider only those solutions $u$ of (1.1) which satisfy $\sup \{\left\vert u\left(l\right) \right\vert :l\geq l\} > 0, \ $for all $l\geq l_{u}.$ As usual, a solution of (1.1) is called oscillatory if it has arbitrarily large zeros, otherwise, it is called nonoscillatory.

In numerical models of various physical, organic, and intrinsic phenomena, differential equations (even of the fourth order) are usually experienced. In particular, there are many applications of the delay differential equation, for example, in elasticity problems, structural deformation principles, or soil settlement; see ^[23,24].

The oscillation and nonoscillation of higher-order functional differential equations have concerned many authors, see ^{[2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33]}. A broad range of methods have been used to investigate the properties of solutions to various groups of equations. As a matter of fact, equation (1.1) (i.e., half-linear/Emden-Fowler differential equation) arises in a variety of real-world problems such as in the study of p-Laplace equations non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, and so forth; see the following papers for more details ^[5,6,7].

Agarwal et al. ^[2] and Zhang et al. ^[29] investigated the oscillatory behavior of a higher-order differential equation

and considered the both cases (1.2) and

In particular, assuming that $\tau \left(l\right) < l$, $\alpha \geq \beta$ and (1.2) holds, the results obtained by Zhang et al. ^[29] ensure that every solution $u$ of (1.3) is either oscillatory or satisfies $\lim_{l\rightarrow \infty }u\left(l\right) = 0.$

Meng and Xu ^[16] established oscillation criteria for even-order neutral differential equations

where $w\left(l\right) = \left(l\right) +p\left(l\right) u\left(l-\tau \right), \; a^{\prime }\left(l\right) \geq 0, \; f\left(u\right) /\left\vert u\right\vert ^{\alpha -1}u\geq k > 0, \; k$ is a constant and (1.4) holds.

Baculikova et al. ^[3] considered the equation

and proved this equation is oscillatory if the first-order equation

is oscillatory when (1.4) holds.

Moaaz et al. ^[21] investigated the oscillatory behavior of the equation

where $\ 0\leq p\left(l\right) \leq p_{0} < \infty, \ \left\vert f\left(l, u\right) \right\vert \geq q\left(l\right) \left\vert u\right\vert ^{\alpha }$ and under the condition (1.4).

In this work, based on the Riccati substitution technique and comparison with delay equations of first-order, we obtain new sufficient conditions for oscillation of (1.1). Unlike most of the previous related works, we are interested in studying (1.1) in the noncanonical case (1.2). Examples illustrating our new results are also given.

The following lemmas are needed in the proofs of our results:

Lemma 1.1. ^[1] Let $\psi \in C^{n}(\left[ l_{0}, \infty \right), \mathbb{R} ^{+}), \; \psi ^{\left(n\right) }$ be of fixed sign and not identically zero on a subray of $\left[ l_{0}, \infty \right)$, and $\psi ^{\left(n-1\right) }\psi ^{\left(n\right) }\leq 0\ $for $l\geq l_{1}\in \left[ l_{0}, \infty \right)$. If $\lim_{l\rightarrow \infty }\psi \left(l\right) \neq 0, \ $then

for every $\lambda \in (0, 1)$ and $l\geq l_{\lambda }\in \left[ l_{1}, \infty \right)$.

Lemma 1.2. ^[20] Assume that $s\geq 0, \; B\geq 0$ and $A > 0.$ Then

Lemma 1.3. [11,Lemma 1.1] Assume that $f\in C^{m}\left(\left[ l_{0}, \infty \right), \left(0, \infty \right) \right)$ and $f^{\left(m\right) }$ is eventually of one sign for all large $l$. Then, there exists a nonnegative integer $h\leq m,$ with $m+h$ even for $f^{\left(m\right) }\geq 0$, or $m+h$ odd for $f^{\left(m\right) }\leq 0$ such that

and

eventually.

Lemma 1.4. ^[11] If $\varphi \in C^{m}\left(\left[ l_{0}, \infty \right), \left(0, \infty \right) \right), \; \varphi ^{\left(k\right) }\left(l\right) > 0, \; k = 0, 1, ..., m$ and $\varphi ^{\left(m+1\right) }\left(l\right) < 0.$ Then,

for every $\lambda \in \left(0, 1\right)$ eventually.

2.   Main results

Let us define the following:

and

where $c_{1}, \; c_{2}, \; c_{3}$ and $c_{4}\ $are any positive constants. It is recognized that the identification of the signs of the solution derivatives is required and, before studying the oscillation of the delay differential equations, causes a significant effect.

Lemma 2.1. Assume that $u\in C\left(\left[ l_{0}, \infty \right), \left(0, \infty \right) \right)$ is a solution of (1.1). Then, $\left(r\left(l\right) \left(\nu ^{\left(n-1\right) }\left(l\right) \right) ^{\alpha }\right) ^{\prime }\leq 0$, and one of the next cases holds, for $l$ large enough

Proof. Assume that $u$ is an eventually positive solution of (1.1). Then, there exists $l_{1}\geq l_{0}$ such that $u\left(l\right), \; u\left(\tau \left(l\right) \right)$ and $u\left(g\left(l\right) \right)$ are positive for all $l\geq l_{1}$. Hence, by the definition of $\nu$, we have that $\nu \left(l\right) > 0$ for $l\geq l_{1}$. It follows from (1.1) that $\left(r\left(l\right) \left(\nu ^{\left(n-1\right) }\left(l\right) \right) ^{\alpha }\right) ^{\prime }\leq 0$. Next, using Lemma 1.3 and considering that $n$ is even, we directly get the cases $\left(\bf{A}\right) -\left(\bf{C}\right)$.

Lemma 2.2. Assume that $u\in C\left(\left[ l_{0}, \infty \right), \left(0, \infty \right) \right)$ is a solution of (1.1) whose $\nu$ satisfies $\left(\bf{B}\right)$. Then $\left(\nu ^{\left(n-2\right) }\left(l\right) \right) ^{\beta -\alpha }\geq \eta \left(l\right)$, eventually$.$

Proof. The proof for this lemma is analogous to the proof of Lemma 2.1 in ^[18]. Hence, we omit it here.

Lemma 2.3. Assume that $u\in C\left(\left[ l_{0}, \infty \right), \left(0, \infty \right) \right)$ is a solution of (1.1) whose $\nu$ satisfies $\left(\bf{C}\right)$. Then $\nu ^{\beta -\alpha }\left(l\right) \geq \mu \left(l\right)$, eventually$.$

Proof. The proof for this lemma is similar to the proof of Lemma 2.2 in ^[18]. Hence, we omit it here.

Lemma 2.4. Assume that $u\in C\left(\left[ l_{0}, \infty \right), \left(0, \infty \right) \right)$ is a solution of (1.1) whose $\nu$ satisfies $\left(\bf{C}\right)$. If

then $\lim_{l\rightarrow \infty }u\left(l\right) = 0$.

Proof. Let $u\in C\left(\left[ l_{0}, \infty \right), \left(0, \infty \right) \right)$ be a solution of (1.1) and case $\left(\bf{C}\right)$ holds. Then, $\lim_{l\rightarrow \infty }\nu \left(l\right) = D$. We claim that $D = 0$. Indeed, for the sake of a contradiction, suppose that $D > 0$, there exists a $l_{1}\geq l_{0}$ such that $u\left(g\left(l\right) \right) \geq D$ for $l\geq l_{1}.$ Integrating (1.1) on $\left[ l_{1}, l\right]$, we have

that is,

Integrating (2.3) twice on $\left[ l, \infty \right)$, we have

and

Similarly, by integrating the above inequality ($n-4$) times on $\left[ l, \infty \right)$, we get

Integrating this inequality on $\left[ l_{1}, \infty \right)$, we find

which is a contradiction with (2.2). Thus, $D = 0$; moreover the inequality $u\leq \nu$ implies $\lim_{l\rightarrow \infty }u\left(l\right) = 0$. The proof of this lemma is complete.

Theorem 2.1. Assume that (2.2) holds and $p_{0} < 1.$ If the first-order delay differential equation

is oscillatory for some and

holds for some $\ \lambda, \lambda _{0}, \lambda _{1}\in \left(0, 1\right)$, then every solution of (1.1) is either oscillatory or converges to zero as $l\rightarrow \infty$.

Proof. Assume the contrary that there is a nonoscillatory solution $u$ of (1.1). Then, we can assume $u\left(l\right), \; u\left(\tau \left(l\right) \right)$ and $u\left(g\left(l\right) \right)$ are positive for $l\geq l_{1}\geq l_{0}$. It follows from Lemma 2.1 that the behavior of $\nu$ and its derivatives is possible in three cases. First, suppose that case $\left(\bf{A}\right)$ holds. Based on the definition of $\nu$, we see that

and so

from (ⅳ) and (2.7), we have

which with (1.1) gives

From Lemma 1.1, we have

for every $\lambda \in \left(0, 1\right).$ From (2.9) and (2.8), we obtain

Let $y\left(l\right) = r\left(l\right) \left(\nu ^{\left(n-1\right) }\left(l\right) \right) ^{\alpha }$. Clearly, $y$ is a positive solution of the first-order delay differential inequality

It follows from [22,Theorem 1] that the corresponding differential equation (2.4) also has a positive solution for all $\lambda _{0}\in \left(0, 1\right)$, which is a contradiction.

Next, we assume that the case $\left(\bf{B}\right)$ holds. We define the function $\Phi$ by

Then $\Phi \left(l\right) < 0$ for $l\geq l_{1}$. Since $r\left(l\right) \left(\nu ^{\left(n-1\right) }\left(l\right) \right) ^{\alpha }$ is decreasing, we get

Multiplying (2.12) by $r^{-1/\alpha }\left(s\right)$ and integrating it on $\left[ l, \infty \right)$, we obtain

that is,

From (2.11), we see that

Differentiating (2.11), we have

which, in view of (1.1) and (2.11), becomes

Since $\nu ^{\prime }\left(l\right) > 0$, we get that (2.8) holds. Hence, (2.14) becomes

From Lemma 1.1, we find

for all sufficiently large $l$ and for every $\lambda \in \left(0, 1\right)$. Then, (2.15) become

Since $l\geq g\left(l\right)$ and $\nu ^{\left(n-2\right) }\left(l\right)$ is decreasing, we have

Multiplying (2.16) by $\delta _{0}^{\alpha }\left(l\right)$ and integrating it on $\left[ l_{1}, l\right]$, we get

Setting $A = \delta _{0}^{\alpha }\left(s\right) /r^{1/\alpha }\left(s\right)$, $B = \delta _{0}^{\alpha -1}\left(s\right) /r^{1/\alpha }\left(s\right)$ and $\vartheta = -\Phi \left(s\right)$, and using Lemma 1.2, we get

due to (2.13), that contradicts (2.5).

Finally, suppose that $\left(\bf{C}\right)$ holds. From Lemma 2.4, one can see that $\lim_{l\rightarrow \infty }u\left(l\right) = 0,$ which is a contradiction.

This complete the proof.

Theorem 2.2. Suppose that the first-order delay differential equation (2.4) is oscillatory for some $\lambda _{0}\in \left(0, 1\right)$ and (2.5) holds for some $\lambda _{1}\in \left(0, 1\right)$. If

and

then every solution of (1.1) is oscillatory, where $\Omega _{i}\left(l\right) = \min \left\{ q_{i}\left(l\right), q_{i}\left(\tau \left(l\right) \right) \right\},$

and $\kappa = 1$ if $\beta \in \left(0, 1\right]$; otherwise, $\kappa = 2^{\beta -1}$.

Proof. Assume that there is a nonoscillatory solution $u$ of (1.1). Then, we can assume $u\left(l\right), \; u\left(\tau \left(l\right) \right)$ and $u\left(g\left(l\right) \right)$ are positive for $l\geq l_{1}\geq l_{0}$. It follows from Lemma 2.1 that the behavior of $\nu$ and its derivatives is possible in three cases.

The proof of the case where $\left(\bf{A}\right)$ or $\left(\bf{B} \right)$ holds is the same as that of Theorem 2.1.

Suppose that $\left(\bf{C}\right)$ holds. Since $\left(r\left(l\right) \left(\nu ^{\left(n-1\right) }\left(l\right) \right) ^{\alpha }\right) ^{\prime }\leq 0$, we have that

or

Integrating this inequality from $l$ to $\infty$ and making use of the fact that $\nu ^{\left(n-2\right) }$ is a positive decreasing function, we arrive at

Taking into account the behavior of derivatives of $\nu$ and integrating (2.18) ($n-2$) times from $l$ to $\infty$, we see that

for $k = 0, 1, ..., n-3$. From (1.1), we get

and

that is,

From (2.20) and (2.21), we find

By integrating this inequality from $l_{1}$ to $l$, we get

Since $\left(r\left(l\right) \nu ^{\left(n-1\right) }\left(l\right) \right) ^{\prime }\leq 0$, we arrive at

which, with Lemma 2.3, gives

Combining [(2.19), $k = 0$] and (2.22), we have that

which is a contradicts with (2.17). This completes the proof.

In the following theorem, we set new conditions for testing the oscillation of (1.1) when $n = 4$, which apply in the ordinary case.

Theorem 2.3. Assume that $n = 4, \; \alpha = \beta = 1, \; p_{0} < 1$ and (2.2) hold. Suppose also that

for some constant $\lambda _{1}\in \left(0, 1\right)$. Assume further that there exist two positive functions $\rho \left(l\right), \vartheta \left(l\right) \in C^{1}\left[ l_{0}, \infty \right)$, such that

and

where $\lambda _{2}\in \left(0, 1\right).$ Then every solution of (1.1) is oscillatory or tends to zero as $l\rightarrow \infty.$

Proof. Assume that Eq (1.1) has a positive solution $u\left(l\right)$. It follows from (1.1) and Lemma 2.1 that there exist four possible cases for the behavior of $\nu$ and its derivatives:

Let $\left(i\right)$ hold. Now, we define

Then clearly $\phi \left(l\right)$ is positive for $l\geq l_{1}$ and satisfies

From (1.1) and (2.26), we have

by using (2.8) and (2.27), we get

Now, it follows from Lemmas 1.1 and 1.4 that

and

respectively. Substituting (2.29) and (2.30) into (2.28), we get

from the definition of $\phi \left(l\right)$, we obtain

Set $A = \lambda _{2}l^{2}/2\rho \left(l\right) r\left(l\right)$, $B = \rho ^{\prime }\left(l\right) /\rho \left(l\right)$ and $s = \phi \left(s\right)$. Using Lemma 1.2, we have

integrating (2.31) from $l_{1}$ to $l$, we have

which contradicts (2.24).

Assume that case $\left(ii\right)$ holds. Define the function $\varphi \left(l\right)$ by

Then clearly $\varphi \left(l\right)$ is positive for $l\geq l_{1}$ and satisfies

from the definition of $\varphi \left(l\right)$, we obtain

Now integrating (1.1) from $l$ to $\infty,$ we have

by using (2.8) and (2.33), we get

From Lemma 1.4, we get

that is,

Combining (2.35) and (2.34), we get

that is

integrating the above inequality from $l$ to $\infty,$ we have

Combining above inequality with (2.32), we obtain

Thus, we have

integrating (2.36) from $l_{1}$ to $l$, we have

which contradicts (2.25).

The proof of the case where $\left(iii\right)$ or $\left(iv\right)$ holds is the same as that of Theorem 2.2 and Theorem 2.1 respectively.

This completes the proof.

Example 2.1. Consider the NDDE

where $a, b\in \left(0, 1\right)$ and $g_{1} > g_{2}.$ Then, we note that

Therefore, it is easy to verify that

Next, to apply Theorem 2.1. We first check the condition (2.2), (2.4) and (2.5). By substitution and a simple computation, (2.4) becomes

By applying a well known criterion [12,Theorem 2.1.1] for first-order delay differential equation (2.38) to be oscillatory, the criterion is immediately obtained.

that is,

Now, we note that (2.5) reduces to

which satisfies if

thus, if condition (2.39) and (2.40) hold, then every solution of (2.37) is oscillatory or tends to zero.

On the other hand, by Theorem 2.2, we see that (2.17) becomes

and so

thus, if (2.39), (2.40) and (2.41) hold, then every solution of (2.37) is oscillatory.

Example 2.2. Consider the NDDE

where $l\geq 1, 0 < a < b$ and $b > c.$ Then, we can clearly note that $\alpha = \beta = 3, n = 4.$

Therefore, it is easy to verify that

By substitution and a simple computation, (2.4) becomes

Applying a well-known criterion [12,Theorem 2.1.1], we see that (2.38) is oscillatory. Moreover, (2.5) reduces to

which satisfies if $\left(q_{1}+q_{2}\right) > 81/32.$ Thus, every solution of (2.42) is oscillatory or tends to zero if $\left(q_{1}+q_{2}\right) > 81/32.$

To apply Theorem 2.2, we see that (2.17) becomes

that is, $\left(q_{1}+q_{2}\right) > 24e^{3a}.$ Then, every solution of (2.42) is oscillatory if $\left(q_{1}+q_{2}\right) > \max \left\{ 24e^{3a}, 81/32\right\}.$

Acknowledgments

The authors present their sincere thanks to the two anonymous referees. (Taher A. Nofal) Taif University Researchers Supporting Project number (TURSP-2020/031), Taif University, Taif, Saudi Arabia.

Conflict of interest

There are no competing interests