Some New Oscillation Results for Higher-Order Nonlinear Differential Equations with a Nonlinear Neutral Term

Abstract

The authors study the oscillatory behaviors of solutions of higher-order nonlinear differential equations with a nonlinear neutral term. The right hand side of their equation contains both an advanced and a delay term, and either (or both) of them can be sublinear or superlinear. The influence of these terms on the oscillatory and asymptotic behaviors of solutions is investigated by using a comparison to first-order advanced and delay differential equations. New oscillation criteria are presented that improve and extend many known oscillation criteria in the literature. An example is provided to illustrate the results.

1. Introduction

In this paper, we are concerned with the oscillatory and asymptotic behaviors of solutions to nonlinear higher-order differential equations with a nonlinear neutral term of the form:

$${\left(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }=q\left(t\right)x^{\gamma }\left(\tau \left(t\right)\right)+p\left(t\right)x^{\mu }\left(\omega \left(t\right)\right),$$

(1)

where $t\ge t_0>0$, $n\ge 2$ is a natural number, $y\left(t\right)=x\left(t\right)-p_1\left(t\right)x^{\beta }\left(\sigma \left(t\right)\right)$, and $\alpha$, $\beta$, $\gamma$, and $\mu$ are the ratios of positive odd integers with $\alpha \ge 1$. In the sequel, we assume that:

(i)

a, p, $p_1$, $q:[t_0,\infty )\to (0,\infty )$ are continuous functions with $a^{\prime }\left(t\right)\ge 0$;

(ii)

$\tau$, $\sigma$, $\omega :[t_0,\infty )\to \mathbb{R}$ are continuous functions such that $\sigma$ is strictly increasing, $\tau \left(t\right)\le t$, $\sigma \left(t\right)\le t$, $\omega \left(t\right)\ge t$, and ${lim}_{t\to \infty }\tau \left(t\right)={lim}_{t\to \infty }\sigma \left(t\right)=\infty$;

(iii)

$h\left(t\right):={\sigma }^{-1}\left(\tau \left(t\right)\right)\le t$ and ${lim}_{t\to \infty }h\left(t\right)=\infty$.

Note that here, the right hand side of Equation (1) contains both an advanced and a delay term, and either one (or both) can be sublinear or superlinear. We let

$$A(t,t_0)={\int }_{t_0}^t\frac{1}{a^{1/\alpha }\left(s\right)}ds,$$

and assume that

$$A(t,t_0)\to \infty \mathrm{as}t\to \infty .$$

(2)

By a solution of Equation (1), we mean a function $x\in C\left([t_x,\infty ),\mathbb{R}\right)$, for some $t_x\ge t_0$, such that $y\in C^{n-1}\left([t_x,\infty ),\mathbb{R}\right)$, $a{\left(y^{(n-1)}\right)}^{\alpha }\in C^1\left([t_x,\infty ),\mathbb{R}\right)$, and x satisfies Equation (1) on $[t_x,\infty )$. We consider only those solutions of (1) that exist on some half-line $[t_x,\infty )$ and that satisfy the condition $sup\left\{\left|x\left(t\right)\right|:T_1\le t<\infty \right\}>0\mathrm{for}\mathrm{any}{T}_1\ge t_x;$ such solutions are said to be continuable. We tacitly assume that Equation (1) possesses such solutions. A continuable solution $x\left(t\right)$ of (1) is said to be oscillatory if it has infinitely many zeros; otherwise, it is called nonoscillatory. Equation (1) is said to be oscillatory if all its solutions are oscillatory.

Due to their many applications in engineering and the natural sciences, oscillation theory for functional differential equations has received a great deal of attention in the last several decades. In particular, Equation (1) can be viewed as a generalization of the higher-order linear delay differential equation

$$x^{\left(n\right)}\left(t\right)=q\left(t\right)x\left(\tau \left(t\right)\right),$$

(3)

the higher-order linear advanced equation

$$x^{\left(n\right)}\left(t\right)=p\left(t\right)x\left(\omega \left(t\right)\right),$$

(4)

or the higher-order linear differential equation involving both advanced and delay arguments

$$x^{\left(n\right)}\left(t\right)=q\left(t\right)x\left(\tau \left(t\right)\right)+p\left(t\right)x\left(\omega \left(t\right)\right).$$

(5)

While Equations (3) and (4) with $q\left(t\right)$ and $p\left(t\right)$ negative, and a number of their generalizations, have been widely studied in the literature (see as examples [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28] and the references cited therein), there have been few works for Equation (5), especially with $q\left(t\right)$ and $p\left(t\right)$ positive. We refer the reader to [29,30] for some oscillation results in the even-order case.

Neutral differential equations, i.e., equations in which the highest-order derivative of the unknown function appears both with and without a deviation, arise in many areas of applied mathematics, and have important applications in the natural sciences and technology. Readers interested in the application of equations of this kind can refer to the monograph by Hale [31], among the most cited sources. Consequently, oscillation theory for second- and third-order neutral differential equations has been well developed in recent decades; see, for example, the monographs [4,32] and the papers [2,8,9,10,11,12,16,20], as well as the references cited therein. However, not much work has been done on higher-order neutral equations (see [3,15] for general higher-order equations with a linear neutral term; [17,18] for even-order equations with a sublinear ($\beta \in [0,1)$) neutral term; or [13] for odd-order equations with a negative linear neutral term). To the best of our knowledge, there appear to be no results for higher-order equations with a general nonlinear neutral term of the form in Equation (1), even in the case of a single delay or advanced argument (i.e., if either $p\left(t\right)=0$ or $q\left(t\right)=0$).

The aim of the present paper is to initiate such a study of the oscillatory properties of Equation (1) in case all the above mentioned factors (higher-order, sublinear, superlinear, negative neutral term, and advanced and delayed nonlinear terms) jointly contribute to the behavior of solutions. We investigate their influence on the oscillatory and asymptotic behavior of solutions of (1) by comparisons to first-order advanced and delay differential equations whose behavior is known. As a result, we are able to deduce the oscillation of solutions of Equation (1) from that of one suitable advanced and four delay first-order differential equations. The results obtained in this work are new, and they improve and contain many known oscillation criteria in the literature, even for important particular cases of Equation (1), such as if $p_1\left(t\right)=0$ (i.e., the non-neutral case), $p\left(t\right)=0$ (i.e., the delay case), or $q\left(t\right)=0$ (i.e., the advanced case).

In the sequel, all functional inequalities are to hold eventually, i.e., they are satisfied for all large t. Without loss of generality, we deal only with positive solutions of (1), since if $x\left(t\right)$ is a solution of (1), then $-x\left(t\right)$ is also a solution.

2. Auxiliary Lemmas

For convenience, we introduce the following notation. Let

$${\eta }_1\left(t\right)=\eta \left(t\right),{\eta }_{i+1}\left(t\right)={\eta }_i\left(\eta \left(t\right)\right),I_1\left(t\right)=\eta \left(t\right)-t,\mathrm{and}{I}_{i+1}\left(t\right)={\int }_t^{\eta \left(t\right)}{I}_i\left(s\right)ds,$$

as well as

$${\phi }_1\left(t\right)=\phi \left(t\right),{\phi }_{i+1}\left(t\right)={\phi }_i\left(\phi \left(t\right)\right),J_1\left(t\right)=t-\phi \left(t\right),\mathrm{and}{J}_{i+1}\left(t\right)={\int }_{\phi \left(t\right)}^t{J}_i\left(s\right)ds,$$

where $\eta$, $\phi \in C([t_0,\infty ),\mathbb{R})$ and $i=1,2,\dots ,n-1$. We also let

$$Q\left(t\right)=\frac{q\left(t\right)}{{\left(p_1\left(h\left(t\right)\right)\right)}^{\gamma /\beta }}.$$

Next, we present some preliminary lemmas that will help to simplify the proofs of our main results. It is clear from Equation (1) that for $y\left(t\right)>0$, we have that $(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha })$ is increasing and eventually of one sign, which means that $y^{(n-1)}\left(t\right)$ is eventually of one sign. In an effort to bring some clarity to our analysis, we let $m=n-1$ and apply the well known lemma of Kiguradze (see [33] or ([34], Lemma 1.1)) to $y^{\left(m\right)}\left(t\right)$.

First suppose that $y^{\left(m\right)}\left(t\right)<0$. Then, there exist $t_1\ge t_0$ and $\ell \in \{0,1,\dots ,m\}$, such that $m+\ell$ is odd and for $t\ge t_1$,

$$\left\{\begin{array}{cc}{y}^{\left(i\right)}\left(t\right)>0,\hfill & i=0,1,\dots ,\ell -1,\hfill \\ {(-1)}^{i+\ell }{y}^{\left(i\right)}\left(t\right)>0,\hfill & i=\ell ,\ell +1,\dots ,m-1.\hfill \end{array}\right.$$

That is, a positive solution y will be said to belong to the class ${\mathcal{N}}_{\ell }$ if its first ℓ derivatives are positive and then their signs alternate. Hence, if $m=n-1$ is even so that n is odd, the set ${\mathcal{N}}_o$ of all positive solutions has the following decomposition:

$${\mathcal{N}}_o={\mathcal{N}}_1\cup {\mathcal{N}}_3\cup \dots \cup {\mathcal{N}}_{n-2},$$

and if $m=n-1$ is odd (so n is even), the set ${\mathcal{N}}_e$ of all positive solutions has the following decomposition:

$${\mathcal{N}}_e={\mathcal{N}}_0\cup {\mathcal{N}}_2\cup \dots \cup {\mathcal{N}}_{n-2}.$$

Notice that if $y^{(n-1)}\left(t\right)=y^{\left(m\right)}\left(t\right)<0$, then in order to have

$$\begin{array}{cc}\hfill 0\le {(a(t){(y^{(n-1)}(t))}^{\alpha })}^{\prime }& =a^{\prime }(t){(y^{(n-1)}(t))}^{\alpha }+\alpha a(t){(y^{(n-1)}(t))}^{\alpha -1}{y}^{(n)}(t)\hfill \\ \hfill & ={(y^{(n-1)}\left(t\right))}^{\alpha -1}[a^{\prime }\left(t\right)y^{(n-1)}\left(t\right)+\alpha a\left(t\right)y^{\left(n\right)}\left(t\right)],\hfill \end{array}$$

we must have $y^{\left(n\right)}\left(t\right)>0$.

Now, consider the situation where $y^{\left(m\right)}\left(t\right)=y^{(n-1)}\left(t\right)>0$. Then, $m+\ell$ is even. In view of condition (2), $y^{(n-2)}\left(t\right)=y^{(m-1)}\left(t\right)>0$ eventually, so all derivatives of $y\left(t\right)$ up to $y^{(n-2)}\left(t\right)$ must be positive as well. Here we do not know the sign of $y^{\left(n\right)}\left(t\right)$, which in fact may be oscillatory (see [35,36] for related problems).

To summarize, we then have the following possible cases for the eventual behavior of the positive solutions of the equations

$${\left(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }=q\left(t\right)y^{\gamma }\left(\tau \left(t\right)\right)$$

(6)

and

$${\left(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }=p\left(t\right)y^{\mu }\left(\omega \left(t\right)\right).$$

(7)

Lemma 1.

Let conditions (i)–(ii) and (2) hold. Then, the equations (6) and (7) can eventually have positive solutions satisfying:

(I)

$y\left(t\right)>0$, $y^{\prime }\left(t\right)>0$, $y^{(n-1)}\left(t\right)<0$, $y^{\left(n\right)}\left(t\right)>0$,

(II)

${(-1)}^i{y}^{\left(i\right)}\left(t\right)>0$, $i=0,1,2,\cdots ,n-1$, if n is even,

(III)

$y^{\left(i\right)}\left(t\right)>0$ for $i=0,1,2,\cdots ,n-1$.

In each of the following lemmas, we show how the existence of positive solutions belonging to one of these three classes can be eliminated from consideration.

Lemma 2.

Let conditions (i)–(iii) and (2) hold, and assume that there exists a nondecreasing function $\xi \in C([t_0,\infty ),\mathbb{R})$ such that $\tau \left(t\right)\le \xi \left(t\right)\le t$ for $t\ge t_0$. If there is a constant $\theta \in (0,1)$ such that the first-order delay differential equation

$$Y^{\prime }\left(t\right)+\frac{{\theta }^{\gamma }}{{((n-2)!)}^{\gamma }}q\left(t\right)a^{-\gamma /\alpha }\left(\xi \left(t\right)\right){\left({\tau }^{n-2}\left(t\right)[\xi \left(t\right)-\tau \left(t\right)]\right)}^{\gamma }{Y}^{\gamma /\alpha }\left(\xi \left(t\right)\right)=0$$

(8)

is oscillatory, then Equation (6) has no eventually positive solution satisfying Case (I).

Proof. 

Let $y\left(t\right)$ be a nonoscillatory solution of Equation (6) satisfying Case (I); say, $y\left(t\right)>0$ and $y\left(\tau \right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$. Since $y\left(t\right)>0$, $y^{\prime }\left(t\right)>0$ and $y^{(n-1)}\left(t\right)<0$ for $t\ge t_2$ for some $t_2\ge t_1$, by ([4], Lemma 2.2.3), for every $\theta \in (0,1)$, there exists a $t_{\theta }\ge t_2$, such that

$$y\left(t\right)\ge \frac{\theta }{(n-2)!}{t}^{n-2}{y}^{(n-2)}\left(t\right)\mathrm{for}t\ge t_{\theta }.$$

Since ${lim}_{t\to \infty }\tau \left(t\right)=\infty$, we have

$$y\left(\tau \left(t\right)\right)\ge \frac{\theta }{(n-2)!}{\left(\tau \left(t\right)\right)}^{n-2}{y}^{(n-2)}\left(\tau \left(t\right)\right)$$

(9)

for $t\ge t_3$ for some $t_3\ge t_{\theta }$. Using (9) in (6) gives

$${\left(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\ge \frac{{\theta }^{\gamma }}{{((n-2)!)}^{\gamma }}q\left(t\right){\left(\tau \left(t\right)\right)}^{\gamma (n-2)}{\left(y^{(n-2)}\left(\tau \left(t\right)\right)\right)}^{\gamma },$$

(10)

Letting $Z\left(t\right)=y^{(n-2)}\left(t\right)$, it follows from (10) that

$${\left(a\left(t\right){\left(Z^{\prime }\left(t\right)\right)}^{\alpha }\right)}^{\prime }\ge \frac{{\theta }^{\gamma }}{{((n-2)!)}^{\gamma }}q\left(t\right){\left(\tau \left(t\right)\right)}^{\gamma (n-2)}{Z}^{\gamma }\left(\tau \left(t\right)\right).$$

(11)

For $v\ge u\ge t_3$, we see that

$$Z\left(u\right)-Z\left(v\right)\ge (v-u)(-Z^{\prime }\left(v\right)).$$

Since $Z\left(t\right)>0$, setting $u=\tau \left(t\right)$ and $v=\xi \left(t\right)$, we have

$$Z\left(\tau \left(t\right)\right)\ge (\xi \left(t\right)-\tau \left(t\right))(-Z^{\prime }\left(\xi \left(t\right)\right)).$$

Using this inequality in (11) and letting $Y\left(t\right)=-a\left(t\right){\left(Z^{\prime }\left(t\right)\right)}^{\alpha }>0$, we see that $Y\left(t\right)$ is an eventually (for $t\ge t_{\theta }$, where $t_{\theta }$ depends on the choice of $\theta$) positive solution of the delay differential inequality

$$Y^{\prime }\left(t\right)+\frac{{\theta }^{\gamma }}{{((n-2)!)}^{\gamma }}q\left(t\right)a^{-\gamma /\alpha }\left(\xi \left(t\right)\right){\left({\tau }^{n-2}\left(t\right)[\xi \left(t\right)-\tau \left(t\right)]\right)}^{\gamma }{Y}^{\gamma /\alpha }\left(\xi \left(t\right)\right)\le 0$$

(12)

for every $\theta \in (0,1)$. Therefore, by virtue of ([37], Theorem 1), the associated delay differential Equation (8) also has a positive solution, which is a contradiction. This completes the proof. □

Lemma 3.

Let conditions (i)–(iii) and (2) hold, and assume that there exists a nondecreasing function $\eta \in C([t_0,\infty ),\mathbb{R})$ such that

$$\eta \left(t\right)>t\mathrm{and}\rho \left(t\right):={\eta }_{n-1}\left(\tau \left(t\right)\right)<t\mathrm{for}t\ge t_0.$$

(13)

If the first-order delay differential equation

$$X^{\prime }\left(t\right)+I_{n-1}^{\gamma }\left(\tau \left(t\right)\right)q\left(t\right)a^{-\gamma /\alpha }\left(\rho \left(t\right)\right)X^{\gamma /\alpha }\left(\rho \left(t\right)\right)=0$$

(14)

is oscillatory, then Equation (6) has no eventually positive solution satisfying Case (II).

Proof. 

Let $y\left(t\right)$ be a nonoscillatory solution of Equation (6), such that $y\left(t\right)>0$ and $y\left(\tau \right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$, and for which $y^{\prime }\left(t\right)<0$ and $y^{(n-1)}\left(t\right)<0$ for $t\ge t_2$ for some $t_2\ge t_1$; i.e., we are in Case (II). Integrating the function $y^{(n-1)}\left(t\right)$ from t to $\eta \left(t\right)$ gives

$$\begin{array}{cc}\hfill y^{(n-2)}\left(t\right)& \ge -y^{(n-2)}\left(\eta \left(t\right)\right)+y^{(n-2)}\left(t\right)={\int }_t^{\eta \left(t\right)}-y^{(n-1)}\left(s\right)ds\hfill \\ \hfill & \ge -(\eta \left(t\right)-t)y^{(n-1)}\left(\eta \left(t\right)\right)=-I_1\left(t\right)y^{(n-1)}\left(\eta \left(t\right)\right).\hfill \end{array}$$

Repeated integrations of this inequality from t to $\eta \left(t\right)$ yield

$$y\left(t\right)\ge -I_{n-1}\left(t\right)y^{(n-1)}\left({\eta }_{n-1}\left(t\right)\right),$$

or

$$y\left(\tau \left(t\right)\right)\ge -I_{n-1}\left(\tau \left(t\right)\right)y^{(n-1)}\left(\rho \left(t\right)\right).$$

Using this in (6) and letting $X\left(t\right)=-a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }>0$, we obtain

$$-X^{\prime }\left(t\right)\ge I_{n-1}^{\gamma }\left(\tau \left(t\right)\right)q\left(t\right)a^{-\gamma /\alpha }\left(\rho \left(t\right)\right)X^{\gamma /\alpha }\left(\rho \left(t\right)\right).$$

The remainder of the proof is similar to that of Lemma 2 and hence, is omitted. □

Lemma 4.

Let conditions (i)–(iii) and (2) hold, and assume that there exist nondecreasing functions φ, $\xi \in C([t_0,\infty ),\mathbb{R})$, such that

$$\phi \left(t\right)<t,\delta \left(t\right)={\phi }_{n-1}\left(\omega \left(t\right)\right)>t,andh\left(t\right)\le \xi \left(t\right)\le \phi \left(t\right)fort\ge t_0.$$

(15)

If the first-order advanced differential equation

$$W^{\prime }\left(t\right)-p\left(t\right)J_{n-1}^{\mu }\left(\omega \left(t\right)\right)a^{-\mu /\alpha }\left(\omega \left(t\right)\right)W^{\mu /\alpha }\left(\delta \left(t\right)\right)=0$$

(16)

is oscillatory, then Equation (7) has no eventually positive solution satisfying Case (III).

Proof. 

Let $y\left(t\right)$ be a nonoscillatory solution of Equation (7), such that $y\left(t\right)>0$ and $y\left(\omega \right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$, and which satisfies Case (III) for $t\ge t_2$ for some $t_2\ge t_1$. Using the monotonicity of $a^{1/\alpha }\left(t\right)y^{(n-1)}\left(t\right)$ and the assumption that $a^{\prime }\left(t\right)\ge 0$, we obtain

$$\begin{array}{cc}\hfill y^{(n-2)}\left(t\right)-y^{(n-2)}\left(\phi \left(t\right)\right)& ={\int }_{\phi \left(t\right)}^t{a}^{-1/\alpha }\left(s\right)a^{1/\alpha }\left(s\right)y^{(n-1)}\left(s\right)ds\hfill \\ \hfill & \ge \left({\int }_{\phi \left(t\right)}^{t}ds\right)a^{-1/\alpha }\left(t\right)a^{1/\alpha }\left(\phi \left(t\right)\right)y^{(n-1)}\left(\phi \left(t\right)\right)\hfill \\ \hfill & \ge J_1\left(t\right)a^{-1/\alpha }\left(t\right)\left(a^{1/\alpha }{y}^{(n-1)}\right)\left(\phi \left(t\right)\right).\hfill \end{array}$$

(17)

Repeated integrations of (17) from $\phi \left(t\right)$ to t give

$$y\left(t\right)\ge J_{n-1}\left(t\right)a^{-1/\alpha }\left(t\right)\left(a^{1/\alpha }{y}^{n-1}\right)\left({\phi }_{n-1}\left(t\right)\right).$$

Using this inequality in (7) and letting $W\left(t\right)=a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }$, we obtain

$$W^{\prime }\left(t\right)\ge J_{n-1}^{\mu }\left(\omega \left(t\right)\right)p\left(t\right)a^{-\mu /\alpha }\left(\omega \left(t\right)\right)W^{\mu /\alpha }\left(\delta \left(t\right)\right).$$

(18)

We then see from ([8], Lemma 2.3) that the associated advanced differential Equation (16) also has a positive solution. This contradiction completes the proof of the lemma. □

Analogous to Lemma 1, a similar analysis can be made for the equation

$${\left(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }+Q\left(t\right)y^{\gamma /\beta }\left(h\left(t\right)\right)=0.$$

(19)

Then, as before, we can easily conclude that $y^{(n-1)}\left(t\right)$ is eventually of one sign. First, we claim that $y^{(n-1)}\left(t\right)>0$. If we suppose, to the contrary, that there exists $t_1\ge t_0$ such that $y^{(n-1)}\left(t\right)<0$ for $t\ge t_1$, then

$$\begin{array}{cc}\hfill y^{(n-2)}\left(t\right)-y^{(n-2)}\left(t_1\right)& ={\int }_{t_1}^t{a}^{-1/\alpha }\left(s\right)a^{1/\alpha }\left(s\right)y^{(n-1)}\left(s\right)ds\hfill \\ & \le a^{1/\alpha }\left(t_1\right)y^{(n-1)}\left(t_1\right)A(t,t_1)\to -\infty \mathrm{as}t\to \infty .\hfill \end{array}$$

That is, $y^{(n-2)}\left(t\right)<0$ eventually, which leads to a contradiction to the positivity of $y\left(t\right)$. Hence, $y^{(n-1)}\left(t\right)>0$ for $t\ge t_1$. Now, notice that in order to have

$$\begin{array}{cc}\hfill 0\ge {\left(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }& =a^{\prime }\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }+\alpha a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha -1}{y}^{\left(n\right)}\left(t\right)\hfill \\ \hfill & ={\left(y^{(n-1)}\left(t\right)\right)}^{\alpha -1}[a^{\prime }\left(t\right)y^{(n-1)}\left(t\right)+\alpha a\left(t\right)y^{\left(n\right)}\left(t\right)],\hfill \end{array}$$

we must have $y^{\left(n\right)}\left(t\right)<0$. Then, it follows from the lemma of Kiguradze that if n is odd, the set ${\mathcal{N}}_o$ of all positive solutions has the following decomposition

$$\mathcal{N}o={\mathcal{N}}_0\cup {\mathcal{N}}_2\cup \dots \cup {\mathcal{N}}_{n-1},$$

and if n is even, then the set ${\mathcal{N}}_e$ of all positive solutions has the decomposition

$${\mathcal{N}}_e={\mathcal{N}}_1\cup {\mathcal{N}}_3\cup \dots \cup {\mathcal{N}}_{n-1}.$$

To summarize, we then have the following possible cases for the eventual behavior of the positive solutions of (19).

Lemma 5.

Let conditions (i)–(ii) and (2) hold. Then, Equation (19) can have eventually positive solutions satisfying:

(J)

$y\left(t\right)>0$, $y^{\prime }\left(t\right)>0$, $y^{(n-1)}\left(t\right)>0$, $y^{\left(n\right)}<0$,

(JJ)

${(-1)}^i{y}^{\left(i\right)}\left(t\right)>0$, $i=0,1,2,\dots ,n$, if n is odd.

Lemma 6.

Let conditions (i)–(iii) and (2) hold. If there exists a constant $\theta \in (0,1)$ such that the first-order delay differential equation

$$W^{\prime }\left(t\right)+\frac{{\theta }^{\gamma /\beta }}{{((n-1)!)}^{\gamma /\beta }}{a}^{-\gamma /\alpha \beta }\left(h\left(t\right)\right){\left(h^{n-1}\left(t\right)\right)}^{\gamma /\beta }Q\left(t\right)W^{\gamma /\alpha \beta }\left(h\left(t\right)\right)=0$$

(20)

is oscillatory, then Equation (19) has no eventually positive solution satisfying Case (J).

Proof. 

Let $y\left(t\right)$ be a nonoscillatory solution of Equation (19), such that $y\left(t\right)>0$ and $y\left(h\right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$ and for which $y^{\prime }\left(t\right)>0$ and $y^{(n-1)}\left(t\right)>0$ for $t\ge t_2$ for some $t_2\ge t_1$. By ([4], Lemma 2.2.3), for every $\theta \in (0,1)$, there exists a $t_{\theta }\ge t_2$, such that

$$y\left(t\right)\ge \frac{\theta }{(n-1)!}{t}^{n-1}{y}^{(n-1)}\left(t\right)\mathrm{for}t\ge t_{\theta },$$

and so

$$y\left(h\left(t\right)\right)\ge \frac{\theta }{(n-1)!}{h}^{n-1}\left(t\right)y^{(n-1)}\left(h\left(t\right)\right)$$

for $t\ge t_3$ for some $t_3\ge t_{\theta }$. Using this inequality in (19) and letting $W\left(t\right)=a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }$, we obtain

$$W^{\prime }\left(t\right)\le -\frac{{\theta }^{\gamma /\beta }}{{((n-1)!)}^{\gamma /\beta }}{a}^{-\gamma /\alpha \beta }\left(h\left(t\right)\right){\left(h^{n-1}\left(t\right)\right)}^{\gamma /\beta }Q\left(t\right)W^{\gamma /\alpha \beta }\left(h\left(t\right)\right).$$

(21)

The rest of the proof is similar to that of Lemma 2 and is omitted. □

Lemma 7.

Let conditions (i)–(iii) and (2) hold, and assume that there exists a nondecreasing function $\eta \in C([t_0,\infty ),\mathbb{R})$, such that

$$\eta \left(t\right)>t\mathrm{and}\pi \left(t\right):={\eta }_{n-1}\left(h\left(t\right)\right)<t\mathrm{for}t\ge t_0.$$

(22)

If the first-order delay differential equation

$$X^{\prime }\left(t\right)+Q\left(t\right)I_{n-1}^{\gamma /\beta }\left(h\left(t\right)\right)a^{-\gamma /\alpha \beta }\left(\pi \left(t\right)\right)X^{\gamma /\alpha \beta }\left(\pi \left(t\right)\right)=0$$

(23)

is oscillatory, then Equation (19) has no eventually positive solution satisfying Case (JJ).

Proof. 

Let $y\left(t\right)$ be a nonoscillatory solution of Equation (19), such that $y\left(t\right)>0$ and $y\left(h\right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$ and satisfying Case (JJ) for $t\ge t_2$ for some $t_2\ge t_1$. Integrating the function $y^{(n-1)}\left(t\right)$ from $t\ge t_2$ to $\eta \left(t\right)$, we see that

$$\begin{array}{cc}\hfill -y^{(n-2)}\left(t\right)& \ge y^{(n-2)}\left(\eta \left(t\right)\right)-y^{(n-2)}\left(t\right)={\int }_t^{\eta \left(t\right)}{y}^{(n-1)}\left(s\right)ds\hfill \\ \hfill & \ge (\eta \left(t\right)-t)y^{(n-1)}\left(\eta \left(t\right)\right)=I_1\left(t\right)y^{(n-1)}\left(\eta \left(t\right)\right).\hfill \end{array}$$

Repeated integrations from t to $\eta \left(t\right)$ yield

$$y\left(t\right)\ge I_{n-1}\left(t\right)y^{(n-1)}\left({\eta }_{n-1}\left(t\right)\right),$$

or

$$y\left(h\left(t\right)\right)\ge I_{n-1}\left(h\left(t\right)\right)y^{(n-1)}\left(\pi \left(t\right)\right).$$

Using this inequality in (19) and letting $X\left(t\right)=a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }>0$, we obtain

$$X^{\prime }\left(t\right)\le -Q\left(t\right)I_{n-1}^{\gamma /\beta }\left(h\left(t\right)\right)a^{-\gamma /\alpha \beta }\left(\pi \left(t\right)\right)X^{\gamma /\alpha \beta }\left(\pi \left(t\right)\right).$$

The remainder of the proof is similar to that of Lemma 2 and is omitted. □

3. Main Results

In this section, we present our main oscillation results.

Theorem 1.

Let conditions (i)–(iii) and (2) hold, and assume that there exist nondecreasing functions ξ, η, $\phi \in C([t_0,\infty ),\mathbb{R})$, such that (13), (15), and (22) hold. If the advanced Equation (16) and the delay equations (8), (14), (20), and (23) are oscillatory, then Equation (1) is oscillatory.

Proof. 

Let $x\left(t\right)$ be a nonoscillatory solution of Equation (1), such that $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, $x\left(\sigma \right(t\left)\right)>0$, and $x\left(\omega \right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$. It follows from (1) that

$${\left(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }=q\left(t\right)x^{\gamma }\left(\tau \left(t\right)\right)+p\left(t\right)x^{\mu }\left(\omega \left(t\right)\right)\ge 0,$$

(24)

and so $a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }$ is nondecreasing and eventually of one sign. That is, there exists a $t_2\ge t_1$ such that $y^{(n-1)}\left(t\right)>0$ or $y^{(n-1)}\left(t\right)<0$ for $t\ge t_2$. From the definition of y, we see that

$$y\left(t\right)=x\left(t\right)-p_1\left(t\right)x^{\beta }\left(\sigma \left(t\right)\right)\le x\left(t\right).$$

Now, if $y\left(t\right)$ is positive, we have the inequalities

$${\left(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\ge q\left(t\right)y^{\gamma }\left(\tau \left(t\right)\right)\ge 0$$

and

$${\left(a\left(t\right){\left(y^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\ge p\left(t\right)y^{\mu }\left(\omega \left(t\right)\right)\ge 0.$$

We shall distinguish the Cases (I)–(III) of Lemma 1. For for Case (I) we apply Lemma 2, for Case (II) we apply Lemma 3, and for Case (III) we apply Lemma 4.

Next, we consider the situation where $y\left(t\right)<0$ for $t\ge t_2$. Let

$$z\left(t\right)=-y\left(t\right)=-x\left(t\right)+p_1\left(t\right)x^{\beta }\left(\sigma \left(t\right)\right)\le p_1\left(t\right)x^{\beta }\left(\sigma \left(t\right)\right).$$

Then,

$$x\left(t\right)\ge {\left(\frac{z\left({\sigma }^{-1}\left(t\right)\right)}{p_1\left({\sigma }^{-1}\left(t\right)\right)}\right)}^{1/\beta }$$

Using this in (24) yields

$${\left(a\left(t\right){\left(z^{(n-1)}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\le -q\left(t\right)x^{\gamma }\left(\tau \left(t\right)\right)\le -Q\left(t\right)z^{\gamma /\beta }\left(h\left(t\right)\right).$$

Finally, we consider the two cases in Lemma 5, i.e., the Cases (J) and (JJ). By applying Lemmas 6 and 7 we obtain the desired conclusions. □

In the following theorem, we remove some of the assumptions in Theorem 1 and show that a solution either oscillates or converges to zero.

Theorem 2.

If, in Theorem 1, we exclude the Equations (14) and (23), then any solution $x\left(t\right)$ of Equation (1) is either oscillatory or converges to zero as $t\to \infty$.

Proof. 

Let $x\left(t\right)$ be a nonoscillatory solution of Equation (1) with ${lim}_{x\to \infty }x\left(t\right)\ne 0$, $x\left(t\right)>0$, $x\left(\tau \right(t\left)\right)>0$, $x\left(\sigma \right(t\left)\right)>0$, and $x\left(\omega \right(t\left)\right)>0$ for $t\ge t_1$ for some $t_1\ge t_0$. We then exclude Cases (II) and (JJ). The remainder proof is similar to that of Theorem 1 and we omit the details. □

The following corollary is immediate.

Corollary 1.

Let conditions (i)–(iii) and (2) hold, and assume that there exist nondecreasing functions ξ, $\phi \in C([t_0,\infty ),\mathbb{R})$, such that (15) holds. If

$${\int }_{t_0}^{\infty }q\left(s\right)a^{-\gamma /\alpha }\left(\xi \left(s\right)\right){\left({\tau }^{n-2}\left(s\right)[\xi \left(s\right)-\tau \left(s\right)]\right)}^{\gamma }ds=\infty \mathrm{for}\gamma /\alpha <1,$$

(25)

$${\int }_{t_0}^{\infty }p\left(s\right)J_{n-1}^{\mu }\left(\omega \left(s\right)\right)a^{-\mu /\alpha }\left(\omega \left(s\right)\right)ds=\infty \mathrm{for}\mu /\alpha >1,$$

(26)

and

$${\int }_{t_0}^{\infty }{a}^{-\gamma /\alpha \beta }\left(h\left(s\right)\right){\left(h^{n-1}\left(s\right)\right)}^{\gamma /\beta }Q\left(s\right)ds=\infty \mathrm{for}\gamma /\alpha \beta <1,$$

(27)

then a solution of Equation (1) is either oscillatory or converges to zero.

Proof. 

The proof follows from $\xi \left(t\right)\le t$ and (12), $\delta \left(t\right)>t$ and (18), and $h\left(t\right)\le t$ and (21), respectively. The details of the proof are left to the reader. □

The following corollary is a consequence of ([38], Theorem 1) (also see ([4], Lemma 2.2.9)) for the delay cases, and ([4], Lemma 2.2.10) for the advanced case.

Corollary 2.

Let conditions (i)–(iii) and (2) hold, and assume that there exists nondecreasing functions ξ, $\phi \in C([t_0,\infty ),\mathbb{R})$, such that (15) holds. If

$$\underset{t\to \infty }{lim\; inf}{\int }_{\xi \left(t\right)}^{t}q\left(s\right)a^{-\gamma /\alpha }\left(\xi \left(s\right)\right){\left({\tau }^{n-2}\left(s\right)[\xi \left(s\right)-\tau \left(s\right)]\right)}^{\gamma }ds\ge \frac{{((n-2)!)}^{\gamma }}{e},\mathrm{for}\gamma =\alpha ,$$

(28)

$$\underset{t\to \infty }{lim\; inf}{\int }_t^{\delta \left(t\right)}p\left(s\right)J_{n-1}^{\mu }\left(\omega \left(s\right)\right)a^{-\mu /\alpha }\left(\omega \left(s\right)\right)ds>\frac{1}{e},\mathrm{for}\mu =\alpha ,$$

(29)

and

$$\underset{t\to \infty }{lim\; inf}{\int }_{h\left(t\right)}^t{a}^{-\gamma /\alpha \beta }\left(h\left(s\right)\right){\left(h^{n-1}\left(s\right)\right)}^{\gamma /\beta }Q\left(s\right)ds>\frac{{((n-1)!)}^{\gamma /\beta }}{e},\mathrm{for}\gamma =\alpha \beta ,$$

(30)

then any solution of Equation (1) is either oscillatory or converges to zero.

Remark 1.

It is important to notice that any positive solution from Case (III) of Lemma 4 tends to infinity eventually. This is easily seen from the fact that $y^{\left(i\right)}\left(t\right)>0$, $i=0,1,2$ for $n\ge 3$ or from $y^{\left(i\right)}\left(t\right)>0$, $i=0,1$, the fact that ${\left(a^{1/\alpha }\left(t\right)y^{\prime }\left(t\right)\right)}^{\prime }>0$, and condition (2) with $n=2$. Clearly, if in Corollary 1 (or 2) we remove conditions (26) and (29) eliminating solutions satisfying Case (III), then the conclusions of these corollaries become “then a solution of (1) is either oscillatory, converges to zero, or diverges to infinity."

We conclude this paper with an example to illustrate our results.

Example 1.

Consider the equation

$${\left(t\left({\left(x\left(t\right)-\frac{1}{2}{x}^{1/3}\left(\frac{t}{2}\right)\right)}^{(n-1)}\right)\right)}^{\prime }=t{x}^{1/5}\left(\frac{t}{8}\right)+(1+t^2)x^3\left(2^{n}t\right),t\ge 1.$$

(31)

Here, we have $\alpha =1$, $\beta =1/3$, $\gamma =1/5$, $\mu =3$, $p_1\left(t\right)=1/2$, $a\left(t\right)=t$, $q\left(t\right)=t$, $p\left(t\right)=1+t^2$, $\tau \left(t\right)=t/8$, $\sigma \left(t\right)=t/2$, $\omega \left(t\right)=2^{n}t$, and $n\ge 2$ is a natural number. Letting $\xi \left(t\right)=t/3$ and $\phi \left(t\right)=t/2$, we see that $h\left(t\right)=t/4\le \xi \left(t\right)\le \phi \left(t\right)$ and $\delta \left(t\right)={\phi }_{n-1}\left(\omega \left(t\right)\right)=2t$; i.e., condition (15) holds. Since $A(t,t_0)=A(t,1)=lnt\to \infty$ as $t\to \infty$, condition (2) holds. Since

$$J_1\left(t\right)=\frac{t}{2},J_2\left(t\right)=\frac{3t^2}{16},\cdots ,J_{n-1}\left(t\right)=d{t}^{n-1}\mathrm{with}d>0\mathrm{is}\mathrm{a}\mathrm{constant},$$

it is easy to see that condition (26) holds. It is also easy to see that conditions (25) and (27) hold. Therefore, by Corollary 1, a solution of Equation (31) is either oscillatory or converges to zero.

Remark 2.

Interesting problems for further research would be to obtain the oscillation criteria for (1) without requiring $a^{\prime }\left(t\right)\ge 0$ or that $\alpha \ge 1$. Also, notice that there are 144 possible combinations of $p>0$, $q>0$, $p<0$, $q<0$, sublinear, superlinear, delay, advanced, and ordinary for the right hand side of Equation (1). All of these would be fertile ground for additional research.