Oscillatory and asymptotic properties of higher-order quasilinear neutral differential equations

1.   Introduction

In this work, we consider the odd-order quasi-linear neutral differential equations of the form

where $n\geq 3$ is an odd integer, $\alpha$ is a ratio of positive odd integers and $m$ is a positive integer. Throughout this work, we assume the following:

$(\textbf{H1})$ $r$ $\in C^{1}\left[t_{0}, \infty \right)$ and $r^{\prime }\left(t\right) \geq 0,$ where

$(\textbf{H2})$ $p,$ $q_{\kappa }$ $\in C\left[t_{0}, \infty \right)$, $p\left(t\right) \in \left[0, p_{0}\right]$ such that $p_{0}$ is a constant and $q_{\kappa }\left(t\right) > 0$;

$(\textbf{H3})$ $\tau$, $\sigma _{\kappa }$ $\in C\left[t_{0}, \infty \right)$, $\tau \left(t\right) \leq t$, $\sigma _{\kappa }\left(t\right) \leq t$, $\lim_{t\rightarrow \infty }\tau \left(t\right) = \infty$ and $\lim_{t\rightarrow \infty }\sigma _{\kappa }\left(t\right) = \infty$ for all $\kappa = 1, 2, ..., m$.

By a solution of (1.1), we mean $x\in C\left([T_{x}, \infty), \mathbb{R} \right)$ with $T_{x}\geq t_{0}$, which satisfies the properties

and

and moreover satisfies (1.1) on $[T_{x}, \infty).$ We consider the nontrivial solutions of (1.1) existing on some half-line $[T_{x}, \infty)$ and satisfying the condition $\sup \{\left\vert x\left(t\right) \right\vert :t\geq t_{\ast }\} > 0\, \ $for any $t_{\ast }\geq T_{x}$. If there exists a $t_{1}\geq t_{0}$ such that either $x\left(t\right) > 0$ or $x\left(t\right) < 0$ for all $t\geq t_{1}$, then $x$ is said to be a nonoscillatory solution; otherwise, it is said to be an oscillatory solution.

Delay differential equations as a subclass of functional differential equations take into account the dependence on the systems past history where the theory of delay differential equations has enhanced our understanding of the qualitative behavior of their solutions and it has benefited significantly and wide from it, where many applications showed in various fields as mathematical biology and epidemiology (for instance, transport phenomena, distributed networks, interaction of species) and other related fields, etc., see ^[1,2,3].

Neutral delay differential equations are differential equations with delays, where the delays can appear in both the state variables and their time derivatives. There is considerable interest in studying of this type of equation because they are deemed to be adequate prescribing tool in modelling of the countless processes in all areas including problems concerning electric networks containing lossless transmission lines (as in high speed computers where such lines are used to interconnect switching circuits), in the study of vibrating masses attached to an elastic bar or in the solution of variational problems with time delays, or in the theory of automatic control and in neuro-mechanical systems in which inertia plays a major role, and in many areas of science as physical, biological and chemical, etc., see ^[4,5]. In addition, systems of delay differential equations were used to study stability properties of electrical power systems also, properties of delay differential equations were used in the study of singular fractional order differential equations, see ^[6,7] and the references cited therein.

As a matter of fact, quasilinear (i.e., half-linear) (neutral) differential equations with deviating arguments (delayed or advanced arguments or mixed arguments) have numerous applications in physics and engineering (e.g., quasilinear (i.e., half-linear) differential equations arise in a variety of real world problems such as in the study of $p$-Laplace equations, porous medium problems, chemotaxis models, and so forth), see ^{[8,9,10,11,12]}.

For several years, an increasing interest in obtaining sufficient conditions for oscillatory and nonoscillatory behavior of different classes of differential equations has been observed, see ^{[13,14,15,16,17,18]} for second-order equations. While the development of the study of the second-order equations was in turn reflected on the even-order equations in the works ^{[19,20,21,22,23,24,25,26,27,28]}. The development of the study of the odd-order equations can also be traced through works ^{[29,30,31,32,33,34]}, some of which are special cases of the studied equation.

Many authors as Ladde and Zhang in ^[22,28] established a criterion for oscillatory behavior of the higher-order differential equation

Grace ^[20] extended some new results to the equation

under the assumptions that $\alpha$ is even,

Agarwal et al. ^[19] studied Eq (1.3) under conditions

Karpuz et al. ^[21] investigated the oscillatory behavior of linear neutral differential equations

where $n$ is an odd integer and $0\leq p\left(t\right) < 1.$

Li and Thandapani ^[31] established some oscillation criteria for certain higher-order neutral differential equation

with $0\leq p\left(t\right) \leq p_{0} < \infty.$

Yildiz et al. ^[27] examined the oscillation of odd-order neutral differential equation

where $0\leq p\left(t\right) \leq p_{1} < 1.$

In the present paper, we aim to improve the results in previous studies and present some new sufficient conditions which ensure that every solution of (1.1) oscillates or tends to zero.

2.   Auxiliary lemmas

Here are some lemmas that we need during the next results.

Lemma 2.1. [18, Lemma (2.3)] Let $g\left(v\right) = Cv-Dv^{\alpha +1/\alpha }$ where $C, D > 0$. Then $g$ attains its maximum value on $\mathbb{R}$ at $v^{\ast } = \left(\alpha C/\left(\alpha +1\right) D\right) ^{\alpha }$ and

Lemma 2.2. ^[34] Assume that $c_{1}, c_{2}\in \lbrack 0, \infty)$ and $\gamma > 0$. Then

where

Lemma 2.3. ^[35] Let $f\in C^{n}\left(\left[t_{0}, \infty \right), \left(0, \infty \right) \right).$ Assume that $f^{\left(n\right) }\left(t\right)$ is of fixed sign and not identically zero on $\left[t_{0}, \infty \right)$ and that there exists a $t_{1}\geq t_{0}$ such that $f^{\left(n-1\right) }\left(t\right) f^{\left(n\right) }\left(t\right) \leq 0$ for all $t\geq t_{1}$. If $\lim_{t\rightarrow \infty }f\left(t\right) \neq 0,$ then for every $\mu \in \left(0, 1\right)$ there exists $t_{\mu }\geq t_{1}$ such that

3.   Main results

Through the rest of this paper, we will use the following definitions:

and

Lemma 3.1. Let $x$ be a positive solution of (1.1). Then $z\left(t\right) > 0,$ $\left(r\left(\left(z\right) ^{\left(n-1\right) }\right) ^{\alpha }\right) ^{\prime }\leq 0$ and there are two possible cases for derivatives of $z:$

Proof. Assume that $x$ is a positive solution of (1.1) on $[t_{0}, \infty).$ Then, there exists $t_{1}\geq t_{0}$ such that $x\left(t\right) > 0,$ $x\left(\sigma _{\kappa }\left(t\right) \right) > 0$ and $x\left(\tau \left(t\right) \right) > 0,$ for $t\geq t_{1}.$ By the definition of $z$, it is easy to see that $z\left(t\right) \geq x\left(t\right) > 0.$ Furthermore, from (1.1), we have $\left(r\left(\left(z\right) ^{\left(n-1\right) }\right) ^{\alpha }\right) ^{\prime }\leq 0.$ The rest of the proof is similar to proof of Lemma in ^[29]. Thus, the proof is complete.

Lemma 3.2. Let $x\left(t\right)$ be a positive solution of (1.1) and $z\left(t\right)$ satisfy $(\bf{II)}$. If

then $\lim_{t\rightarrow \infty }x\left(t\right) = \lim_{t\rightarrow \infty }z\left(t\right) = 0,$ where

Proof. Let $x$ be a positive solution of (1.1) on $[t_{0}, \infty).$ Then, there exists $t_{1}\geq t_{0}$ such that $x\left(t\right) > 0,$ $x\left(\sigma _{\kappa }\left(t\right) \right) > 0$ and $x\left(\tau \left(t\right) \right) > 0,$ for $t\geq t_{1}$. Since the corresponding function $z\left(t\right) > 0$ and $z^{\prime }\left(t\right) < 0,$ then there exists a finite limit $\lim_{t\rightarrow \infty }z\left(t\right) = c\geq 0.$ Let $c > 0$. Then for any $\epsilon > 0$, we have $\epsilon +c$ $> z\left(t\right) > c,$ eventually. It is easy to see that

thus,

This implies that

where $\varrho = c-p_{0}\left(\epsilon +c\right) /\epsilon +c > 0,$ that is,

Using (3.3) in (1.1), we obtain

By (3.1) and $\sigma \left(t\right) < t$, we see that

Integrating last inequality from $t$ to $\infty$, we get

By $\lim_{t\rightarrow \infty }z\left(\sigma \left(t\right) \right) > c$, it follows that

Integrating (3.4) twice from $t$ to $\infty$, we have

Repeating this procedure, we arrive at

Now, integrating from $t_{1}$ to $\infty$, we see that

This contradicts (3.2). Then we have $\lim_{t\rightarrow \infty }z\left(t\right) = 0.$

In the following lemma, we will use the notions

and

Lemma 3.3. If $x\left(t\right)$ is a positive solution of (1.1) and $z\left(t\right)$ satisfy $(\bf{I)}$, (3.5) and $\sigma _{\kappa }\circ \tau = \tau \circ \sigma _{\kappa }$ hold, then

Moreover, if (3.6) and $\left(\sigma _{\kappa }\circ \sigma ^{-1}\right) \circ \tau = \tau \circ \left(\sigma _{\kappa }\circ \sigma ^{-1}\right)$ hold, then

Proof. Let $x$ be a positive solution of (1.1). Then, there exists $t_{1}\geq t_{0}$ such that $x\left(t\right) > 0,$ $x\left(\sigma _{\kappa }\left(t\right) \right) > 0$ and $x\left(\tau \left(t\right) \right) > 0$ for $t\geq t_{1}$. By Lemma 2, we see that

From (1.1), (3.5) and property $\ \sigma _{\kappa }\circ \tau = \tau \circ \sigma _{\kappa }$, we get

Using (1.1) with above inequality and taking (3.9) into account, we have

By (3.1), we see that

Using (3.1) and (3.6) in (1.1), we are led to

Also, using (3.1) and (3.5) in (1.1), we obtain

Combining (3.10) with (3.11) and taking into account (3.9), one can see that

That is,

By the fact $z^{\prime } > 0,$ we note that $z\left(\sigma _{\kappa }\left(\sigma ^{-1}\left(t\right) \right) \right) > z\left(t\right)$ which implies that

The proof of lemma is complete.

Theorem 3.1. Assume that (3.2), (3.5), $\sigma \left(t\right) \leq \tau \left(t\right)$, $\sigma ^{\prime }\left(t\right) > 0$ and $\sigma _{\kappa }\circ \tau = \tau \circ \sigma _{\kappa }$ hold. If there exists a function $\delta \in C^{1}\left(\left[t_{0}, \infty \right), \left(0, \infty \right) \right)$, such that

then every solution of (1.1) is oscillatory or tends to zero.

Proof. Let $x$ be a positive solution of (1.1). Then, there exists $t_{1}\geq t_{0}$ such that $x\left(t\right) > 0,$ $x\left(\sigma _{\kappa }\left(t\right) \right) > 0$ and $x\left(\tau \left(t\right) \right) > 0$ for $t\geq t_{1}$. Let $z$ $\ $satisfying case $(\bf{I).}$ Define the positive function $\omega \left(t\right)$ by

Hence, by differentiating (3.13), we get

Since $z^{^{\prime }} > 0,$ $z^{^{\prime \prime }} > 0, \,$we see that $\lim_{t\rightarrow \infty }z^{^{\prime }}\neq 0,$ using Lemma 2.3 with $f = z^{^{\prime }}$, we see that

for every $\mu \in (0, 1).$ By $z^{n}\left(t\right) \leq 0,$ we get

Substituting (3.13) and (3.15) into (3.14) implies

that is,

Now, define another positive function $v\left(t\right)$ by

By differentiating (3.17), we get

From (3.15), $\sigma \left(t\right) \leq \tau \left(t\right)$ and $z^{n}\left(t\right) \leq 0$, we have

Substituting (3.19) and (3.17) into (3.18), implies

By $r^{^{\prime }}\left(t\right) > 0,$ we get

Now, using inequalities (3.16) and (3.20), we get

By (3.7), we obtain

Applying the following inequality inequality (2.1) with

we get

Integrating the last inequality from $t_{2}$ to $t,$ we obtain

The proof is complete.

Theorem 3.2. Assume that (3.2), (3.5), (3.6), $\sigma \left(t\right) \leq \tau \left(t\right)$ and $\sigma _{\kappa }\circ \sigma ^{-1}\circ \tau = \tau \circ \sigma _{\kappa }\circ \sigma ^{-1}$ hold. If there exists a function $\delta \in C^{1}\left(\left[t_{0}, \infty \right), \left(0, \infty \right) \right)$, such that

then every solution of (1.1) is oscillatory or tends to zero.

Proof. Let $x$ be a positive solution of (1.1). Then, there exist $t_{1}\geq t_{0}$ such that $x\left(t\right) > 0,$ $x\left(\sigma _{\kappa }\left(t\right) \right) > 0$ and $x\left(\tau \left(t\right) \right) > 0$ for $t\geq t_{1}$. Let $z$ $\ $satisfying case $(\bf{I).}$ Define the positive function by

Hence, by differentiating (3.13), we get

Since $z^{^{\prime }} > 0,$ $z^{^{\prime \prime }} > 0, \,$we see that $\lim_{t\rightarrow \infty }z^{^{\prime }}\neq 0,$ using Lemma 2.3 with $f = z^{^{\prime }}$, we obtain

for every $\mu \in (0, 1).$ Thus, by $\sigma ^{-1}\left(t\right) > t$ and $z^{n}\left(t\right) \leq 0,$ we get

Substituting (3.23) and (3.26) into (3.24) implies

that is,

Now, define another positive function $v\left(t\right)$ by

By differentiating (3.28), we get

From (3.25), $\sigma ^{-1}\left(\tau \left(t\right) \right) \geq t$ and $z^{n}\left(t\right) \leq 0$, we have

Substituting (3.30) and (3.28) into (3.29), implies

By $r^{^{\prime }}\left(t\right) > 0,$ we get

Now, using inequalities (3.27) and (3.31), we get

By (3.9), we obtain

Applying the following inequality (2.1) with

we get

Integrating last the inequality from $t_{2}$ to $t,$ we obtain

The proof is complete.

Example 3.1. Consider the odd order neutral delay differential equation

we note that

It is easy to see that the conditions (3.5), (3.6) and (3.2) hold.

Applying Theorem 3.2, we have that every solution of (3.32) is oscillatory or tends to zero as $t\rightarrow \infty$ when

Remark 3.1. If we consider the special case $\left(x\left(t\right) +\frac{17}{18} x\left(t/2\right) \right) ^{\left(3\right) }+\frac{q_{0}}{t^{3}}x\left(t/2^{2}\right) = 0,$ then every solution is oscillatory or tends to zero if $q_{0} > 46. 22$, while by using the result in ^[21], we have that every solution is oscillatory or tends to zero if $q_{0} > 144$. Consequently, our results apply to the equation $\left(x\left(t\right) + \frac{17}{18}x\left(t/2\right) \right) ^{\left(3\right) }+\frac{70}{t^{3}} x\left(t/2^{2}\right) = 0$, while the other results fail to study this equation.

4.   Conclusions

In this study, oscillatory properties of a class of odd-order quasi-linear neutral differential equations are established. By introducing some Riccati substitution, we obtained new conditions that guarantee that all nonoscillatory solutions of (1.1) converge to zero. Our results extend and complement the previous results in the literature. An interesting issue is obtaining new criteria that ensure that all solutions of (1.1) oscillate.

Acknowledgments

The authors is grateful to the editors and two anonymous referees for a very thorough reading of the manuscript and for some valuable notes that improved the manuscript.

Conflict of interest

There are no competing interests