Criteria for the Nonexistence of Kneser Solutions of DDEs and Their Applications in Oscillation Theory

Abstract

The objective of this study was to improve existing oscillation criteria for delay differential equations (DDEs) of the fourth order by establishing new criteria for the nonexistence of so-called Kneser solutions. The new criteria are characterized by taking into account the effect of delay argument. All previous relevant results have neglected the effect of the delay argument, so our results substantially improve the well-known results reported in the literature. The effectiveness of our new criteria is illustrated via an example.

1. Introduction

The issue of studying the oscillatory behavior of delay differential equations (DDEs) is one of the most important branches of qualitative theory. The oscillation theory of DDEs has captured the attention of many researchers for several decades. Recently, an active research movement has emerged to improve, complement and simplify the criteria for oscillations of many classes of differential equations of different orders; for second-order, see [1,2,3,4,5,6,7,8,9]; for third-order, see [10,11,12,13]; for fourth-order & higher-order, see [14,15,16,17,18,19,20,21,22,23,24,25]; and for special cases, see [26,27,28,29,30,31,32,33,34,35,36,37,38]. Fourth-order differential equations appear in models related to physical, biological and chemical phenomena, for example, elasticity problems, soil leveling and the deformation of structures; see, for example, [7,23,32]. It is also worth mentioning the oscillatory muscle movement model represented by a fourth-order delay differential equation, which can arise due to the interaction of a muscle with its inertial load [37].

In this paper we are concerned with the study of the asymptotic behavior of the fourth-order delay differential equation:

$${\left(a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }\right)}^{\prime }+f(l,x\left(\tau \left(l\right)\right))=0,l\ge l_0.$$

(1)

Throughout the paper, we assume $\alpha \in {\mathbb{Q}}_{odd}^{+}:=\left\{\beta /\gamma :\beta ,\gamma \in {\mathbb{Z}}^{+}\mathrm{are}\mathrm{odd}\right\}$, $a\in C^1\left(I_0,{\mathbb{R}}^{+}\right),$ $a^{\prime }\left(l\right)\ge 0,$ ${\int }^{\infty }{a}^{-1/\alpha }\left(\varrho \right)\mathrm{d}\varrho <\infty ,$ $\tau \in C\left(I_0,{\mathbb{R}}^{+}\right),$ $\tau \left(l\right)<l,$ ${lim}_{l\to \infty }\tau \left(l\right)=\infty ,$ $I_{\varrho }:=\left[l_{\varrho },\infty \right),$ $f\in C\left(I_0\times \mathbb{R},\mathbb{R}\right),$ $xf\left(l,x\right)>0$ for all $x\ne 0$ and there exists a function $q\in C\left(I_0,\left[0,\infty \right)\right)$ such that $f\left(l,x\right)\ge q\left(l\right)x^{\alpha }$.

If there exists a $l_x\ge l_0$ such that the real-valued function $x$is continuous, $a{\left(x^{‴}\right)}^{\alpha }$ is continuously differentiable and satisfies (1), for all $l\in I_x$, then x is said to be a solution of (1). We take into account these solutions x of (1) such that $sup\left\{\left|x\left(s\right)\right|:s\ge l_x\right\}>0$ for every $l_x$ in $I_{\ast }$. A solution x of (1) is said to be a Kneser solution if $x\left(l\right)x^{\prime }\left(l\right)<0$ for all $l\ge l_{\ast }$, where $l_{\ast }$ is large enough. The set of all eventually positive Kneser solutions of Equation (1) is denoted by ℜ. A solution x of (1) is said to be non-oscillatory if it is positive or negative, ultimately; otherwise, it is said to be oscillatory. The equation itself is said to be oscillatory if all its solutions oscillate.

Below, we mention specifically some related works that were the motivation for this paper.

Zhang et al. [25] studied the oscillatory behavior of (1) when $f\left(l,x\right):=q\left(l\right)x^{\beta }$. Results in [25] used an approach that leads to two independent conditions in comparison with first-order delay differential equations and a condition in a traditional form ($limsup\left(·\right)=+\infty$). However, to use (Lemma 2.2.3, [27]), they conditioned ${lim}_{l\to \infty }x\left(l\right)\ne 0$. Thus, under the conditions of (Theorem 1, [25]), Equation (1) still has a non-oscillatory solution that tends to zero. To surmount this problem, Zhang, et al. [38] considered—by using (Lemma 2.2.1, [27])—three possible cases for the derivatives of the solutions, and they followed the same approach as in (Theorem 1, [25]). However, in the case where $x^{\prime }>0$, they ensured that ${lim}_{l\to \infty }x\left(l\right)\ne 0$, so they ensured that every solution of (1) is oscillatory.

By comparing with one or a couple of first-order delay differential equations, Baculikova et al. [14] studied the oscillatory behavior of (1) under the conditions

$$f^{\prime }\left(x\right)\ge 0\mathrm{and}-f\left(-xy\right)\ge f\left(xy\right)\ge f\left(x\right)f\left(y\right),\mathrm{for}xy>0.$$

In this study, we first create new criteria for the nonexistence of Kneser solutions of nonlinear fourth-order differential Equations (1). By using these new criteria, we introduce sufficient conditions for oscillation which take into account the effect of delay argument $\tau \left(l\right)$. All previous relevant results have neglected the effect of the delay argument, so our results substantially improve the well-known results reported in the literature. The effectiveness of our new criteria is illustrated via an example.

2. Main Results

Firstly, for simplicity’s sake, we assume ${\delta }_0\left(l\right):={\int }_l^{\infty }{a}^{-1/\alpha }\left(\varrho \right)\mathrm{d}\varrho$ and ${\delta }_m\left(l\right):=$ ${\int }_l^{\infty }{\delta }_{m-1}\left(\varrho \right)\mathrm{d}\varrho ,$ for $m=1,2$. Moreover, we let

$$\left(\mathrm{H}\right)\mathrm{there}\mathrm{is}\mathrm{a}\mathrm{constant}h>1\mathrm{such}\mathrm{that}\frac{{\delta }_2\left(\tau \left(l\right)\right)}{{\delta }_2\left(l\right)}\ge h\mathrm{for}l\ge l_0.$$

When checking the behavior of positive solutions of DDE (1), we have—by using (Lemma 2.2.1, [27])—three cases:

$$\begin{array}{ll}\mathrm{Case} \left(1\right):& x^{\prime }\left(l\right)>0,x^{‴}\left(l\right)>0 \mathrm{and} x^{\left(4\right)}\left(l\right)<0;\\ \mathrm{Case} \left(2\right):& x^{\prime }\left(l\right)>0,x^{″}\left(l\right)>0 \mathrm{and} x^{‴}\left(l\right)<0;\\ \mathrm{Case} \left(3\right):& x^{\prime }\left(l\right)<0,x^{″}\left(l\right)>0 \mathrm{and} x^{‴}\left(l\right)<0.\end{array}$$

Moreover, from (1), we have that ${\left(a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }\right)}^{\prime }\le 0$, for $l\in I_1$. We note that if $x\in \Re$, then x satisfies Case $\left(3\right)$.

Lemma 1.

Assume that $x\in \Re$. If

$${\int }_{l_0}^{\infty }{\left(\frac{1}{a\left(\nu \right)}{\int }_{l_0}^{\nu }q\left(\varrho \right)\mathrm{d}\varrho \right)}^{1/\alpha }\mathrm{d}\nu =\infty ,$$

(2)

then

$$\underset{l\to \infty }{lim}x\left(l\right)=0.$$

(3)

Lemma 2.

Assume that $x\in \Re$ and (2) hold. Then

$$\eta :=\underset{l\to \infty }{limsup}{\delta }_2{\left({\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho \right)}^{1/\alpha }\le 1.$$

(4)

Proof. 

Suppose $x\in \Re$. Integrating (1) from $l_1$ to l and using that fact that $x^{\prime }\left(l\right)<0$, we get

$$\begin{array}{lll}-a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }& \ge & -a\left(l_1\right){\left(x^{‴}\left(l_1\right)\right)}^{\alpha }+{\int }_{l_1}^{l}q\left(\varrho \right)x^{\alpha }\left(\tau \left(\varrho \right)\right)\mathrm{d}\varrho \\ & \ge & -a\left(l_1\right){\left(x^{‴}\left(l_1\right)\right)}^{\alpha }+x^{\alpha }\left(\tau \left(l\right)\right){\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho -x^{\alpha }\left(\tau \left(l\right)\right){\int }_{l_0}^{l_1}q\left(\varrho \right)\mathrm{d}\varrho ,\end{array}$$

(5)

for all $l\in I_1$. In view of (3), there is a $l_2\in I_1$ such that

$$a\left(l_1\right){\left(x^{‴}\left(l_1\right)\right)}^{\alpha }+x^{\alpha }\left(\tau \left(l\right)\right){\int }_{l_0}^{l_1}q\left(\varrho \right)\mathrm{d}\varrho <0,$$

for $l\in I_2$. Thus, (5) becomes

$$-a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }\ge x^{\alpha }\left(\tau \left(l\right)\right){\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho \ge x^{\alpha }\left(l\right){\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho .$$

(6)

Now, by using the monotonicity of $a^{1/\alpha }\left(l\right)x^{‴}\left(l\right)$, we have

$$x^{″}\left(\tau \left(l\right)\right)\ge x^{″}\left(l\right)\ge {\int }_l^{\infty }\frac{1}{a^{1/\alpha }\left(\varrho \right)}\left(-a^{1/\alpha }\left(\varrho \right)x^{‴}\left(\varrho \right)\right)\mathrm{d}\varrho \ge -a^{1/\alpha }\left(l\right)x^{‴}\left(l\right){\delta }_0\left(l\right).$$

(7)

Integrating (7) twice from l to ∞ and using ${\left(a^{1/\alpha }\left(l\right)x^{‴}\left(l\right)\right)}^{\prime }\le 0$, we get

$$-x^{\prime }\left(l\right)\ge -a^{1/\alpha }\left(l\right)x^{‴}\left(l\right){\delta }_1\left(l\right)$$

(8)

and

$$x\left(l\right)\ge -a^{1/\alpha }\left(l\right)x^{‴}\left(l\right){\delta }_2\left(l\right).$$

(9)

From (9) and (6), we see that

$$-a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }\ge -a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }{\delta }_2^{\alpha }\left(l\right){\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho ,$$

and so

$$1\ge {\delta }_2^{\alpha }\left(l\right){\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho .$$

Taking the limsup on both sides of the inequality, we arrive at (4). The proof is complete. □

Lemma 3.

Assume that $x\in \Re$ and (2) hold. Then there exists a $l_{\epsilon }\ge l_1$ such that

$$\frac{\mathrm{d}}{\mathrm{d}l}\left(\frac{x\left(l\right)}{{\delta }_2^{\eta -\epsilon }\left(l\right)}\right)\le 0,$$

for any $\epsilon >0$ and $l\in I_{\epsilon }$. Moreover, if $\left(H\right)$ holds, then

$$x\left(\tau \left(l\right)\right)\ge h^{\eta -\epsilon }x\left(l\right)forl\in I_{\epsilon }.$$

(10)

Proof. 

Suppose $x\in \Re$. Then, there is a $l_1\in I_0$ such that $x\left(\tau \left(l\right)\right)>0$. Proceeding as in the proof of Lemma 2, we arrive at (6) and (8). Thus, for $l\ge l_2$, where $l_2\in I_1$is large enough, we have

$$-a{\left(l\right)}^{1/\alpha }{x}^{‴}\left(l\right)\ge x\left(l\right){\left({\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho \right)}^{1/\alpha }.$$

From the definition of $\eta$, for every $\epsilon >0$, there exists a $l_3\ge l_2$ such that

$${\delta }_2\left(l\right){\left({\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho \right)}^{1/\alpha }>{\eta }_{\ast }:=\eta -\epsilon ,$$

for $l\in I_3$. Hence, from (8), we have

$$\frac{\mathrm{d}}{\mathrm{d}l}\left(\frac{x\left(l\right)}{{\delta }_2^{{\eta }_{\ast }}\left(l\right)}\right)\le \frac{{\delta }_2^{{\eta }_{\ast }}\left(l\right)a^{1/\alpha }\left(l\right)x^{‴}\left(l\right){\delta }_1\left(l\right)+{\eta }_{\ast }x\left(l\right){\delta }_2^{{\eta }_{\ast }-1}\left(l\right){\delta }_1\left(l\right)}{{\delta }_2^{2{\eta }_{\ast }}\left(l\right)},$$

which with (6) gives

$$\begin{array}{lll}\frac{\mathrm{d}}{\mathrm{d}l}\left(\frac{x\left(l\right)}{{\delta }_2^{{\eta }_{\ast }}\left(l\right)}\right)& \le & \frac{1}{{\delta }_2^{2{\eta }_{\ast }}\left(l\right)}\left(-x\left(l\right){\delta }_2^{{\eta }_{\ast }}\left(l\right){\delta }_1\left(l\right){\left({\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho \right)}^{1/\alpha }+{\eta }_{\ast }x\left(l\right){\delta }_2^{{\eta }_{\ast }-1}\left(l\right){\delta }_1\left(l\right)\right)\\ & \le & \frac{1}{{\delta }_2^{2{\eta }_{\ast }}\left(l\right)}x\left(l\right){\delta }_2^{{\eta }_{\ast }-1}\left(l\right){\delta }_1\left(l\right)\left({\eta }_{\ast }-{\delta }_2\left(l\right){\left({\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho \right)}^{1/\alpha }\right)\le 0.\end{array}$$

Using this fact, one can easily see that

$$x\left(\tau \left(l\right)\right)\ge x\left(l\right){\left(\frac{{\delta }_2\left(\tau \left(l\right)\right)}{{\delta }_2\left(l\right)}\right)}^{{\eta }_{\ast }}\ge h^{{\eta }_{\ast }}x\left(l\right).$$

The proof is complete. □

Lemma 4.

Assume that $x\in \Re$ and $\left(H\right)$, (2) hold. Then

$$h^{\eta }\eta \le 1.$$

(11)

Proof. 

Suppose $x\in \Re$. Using Lemma 3, we get that (10) holds. As in the proof of Lemma 2, we have that (6) holds. From (6) and (10), we have

$$-a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }\ge x^{\alpha }\left(l\right)h^{\alpha {\eta }_{\ast }}{\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho ,$$

(12)

which implies

$$-a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }\ge -a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }{\delta }_2^{\alpha }\left(l\right)h^{\alpha {\eta }_{\ast }}{\int }_{l_0}^{l}q\left(\varrho \right)\mathrm{d}\varrho .$$

Taking the limsup on both sides of the latter inequality, we obtain $h^{{\eta }_{\ast }}\eta \le 1$. Since $\epsilon$ is arbitrary, we obtain that (11) holds. The proof is complete. □

Lemma 5.

Assume that $x\in \Re$ and $\left(H\right)$, (2) hold. Then

$$\tilde{\eta }:=\underset{l\to \infty }{liminf}\frac{h^{\eta }}{{\delta }_2\left(l\right)}{\int }_l^{\infty }{\delta }_2^{\alpha +1}\left(\varrho \right)q\left(\varrho \right)\mathrm{d}\varrho \le {\left(\frac{\alpha }{\alpha +1}\right)}^{\alpha +1}.$$

(13)

Proof. 

Suppose $x\in \Re$. Proceeding as in the proof of Lemma 2, we obtain (8) and (9). Define the function $\omega \left(l\right)\in C^1\left(\left[l_0,\infty \right),R\right),$such that

$$\omega \left(l\right)=\frac{a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }}{x^{\alpha }\left(l\right)}.$$

Differentiating $\omega \left(l\right)$ and using (1), (8), (10) and the fact that $x^{\prime }\left(l\right)<0$, we have

$${\omega }^{\prime }\left(l\right)=\frac{{\left(a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }\right)}^{\prime }}{x^{\alpha }\left(l\right)}-\frac{\alpha a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }{x}^{\prime }}{x^{\alpha +1}\left(l\right)}\le -h^{{\eta }_{\ast }}q\left(l\right)-\alpha {\delta }_1\left(l\right){\omega }^{\left(\alpha +1\right)/\alpha }\left(l\right).$$

(14)

Multiplying (14) by ${\delta }_2^{\alpha }$ and integrating the resulting inequality from $l_1$ to l, we obtain

$$\begin{array}{lll}{\delta }_2^{\alpha }\left(l\right)\omega \left(l\right)-{\delta }_2^{\alpha }\left(l_1\right)\omega \left(l_1\right)+{\int }_{l_1}^l{h}^{{\eta }_{\ast }}q\left(\varrho \right){\delta }_2^{\alpha }\left(\varrho \right)\mathrm{d}\varrho & \le & -{\int }_{l_1}^l\alpha {\delta }_2^{\alpha -1}\left(\varrho \right){\delta }_1\left(\varrho \right)\omega \left(\varrho \right)\mathrm{d}\varrho \\ & & -{\int }_{l_1}^l\alpha {\delta }_1\left(\varrho \right){\delta }_2^{\alpha }\left(\varrho \right){\omega }^{\left(\alpha +1\right)/\alpha }\left(\varrho \right)\mathrm{d}\varrho .\end{array}$$

Using the inequality

$$-By+A{y}^{\left(\alpha +1\right)/\alpha }\ge -\frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{B^{\alpha +1}}{A^{\alpha }},A,B>0,$$

with $A={\delta }_1\left(\varrho \right){\delta }_2^{\alpha }\left(\varrho \right),B={\delta }_1\left(\varrho \right){\delta }_2^{\alpha -1}\left(\varrho \right)$and $y=-\omega \left(\varrho \right)$, we conclude that

$${\int }_{l_1}^l\left(h^{{\eta }_{\ast }}q\left(\varrho \right){\delta }_2^{\alpha }\left(\varrho \right)-\frac{{\alpha }^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\delta }_1\left(\varrho \right)}{{\delta }_2\left(\varrho \right)}\right)\mathrm{d}\varrho \le {\delta }_2^{\alpha }\left(l_1\right)\omega \left(l_1\right)-{\delta }_2^{\alpha }\left(l\right)\omega \left(l\right).$$

(15)

From (9), one can easily see that $-1\le \omega \left(l\right){\delta }_2^{\alpha }\left(l\right)<0$, which with (15) gives

$${\int }_{l_1}^l\left(h^{{\eta }_{\ast }}q\left(\varrho \right){\delta }_2^{\alpha }\left(\varrho \right)-\frac{{\alpha }^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\delta }_1\left(\varrho \right)}{{\delta }_2\left(\varrho \right)}\right)\mathrm{d}\varrho <\infty .$$

Hence, there is a $l_{ϵ}\ge l_1$ such that

$${\int }_l^{\infty }\left(h^{{\eta }_{\ast }}q\left(\varrho \right){\delta }_2^{\alpha }\left(\varrho \right)-\frac{{\alpha }^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}}\frac{{\delta }_1\left(\varrho \right)}{{\delta }_2\left(\varrho \right)}\right)\mathrm{d}\varrho <ϵ,$$

for any $ϵ>0$ and $l\in I_{ϵ}$. Since ${\delta }_2$ is decreasing, we get

$$\begin{array}{lll}ϵ& >& \frac{1}{{\delta }_2\left(l\right)}{\int }_l^{\infty }\left(h^{{\eta }_{\ast }}q\left(\varrho \right){\delta }_2^{\alpha +1}\left(\varrho \right)-\frac{{\alpha }^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}}{\delta }_1\left(\varrho \right)\right)\mathrm{d}\varrho \\ & >& \frac{h^{{\eta }_{\ast }}}{{\delta }_2\left(l\right)}{\int }_l^{\infty }q\left(\varrho \right){\delta }_2^{\alpha +1}\left(\varrho \right)\mathrm{d}\varrho -\frac{{\alpha }^{\alpha +1}}{{\left(\alpha +1\right)}^{\alpha +1}}.\end{array}$$

Taking the limsup on both sides of the inequality, we arrive at (13). The proof is complete. □

From the previous results, the following theory can be inferred.

Theorem 1.

Assume that (2) holds. If one of the following conditions holds:

$$\begin{array}{l}\left({\mathbf{C}}_1\right)\eta >1;\\ \left({\mathbf{C}}_2\right)h^{\eta }\eta >1 and \left(H\right);\\ \left({\mathbf{C}}_3\right)\tilde{\eta }>{\left(\alpha /\left(\alpha +1\right)\right)}^{\alpha +1} and \left(H\right),\end{array}$$

then the set ℜ is empty.

Proof. 

Suppose $x\in \Re$. Using Lemmas 2, 4 and 5, we have that (4), (11) and (13) hold. Then we obtain a contradiction with $\left({\mathbf{C}}_1\right)-\left({\mathbf{C}}_3\right)$ respectively. The proof is complete. □

Lemma 6.

Assume that $M>0,$ L and N are constants $\psi \left(\vartheta \right)=L\vartheta -M{\left(\vartheta -N\right)}^{\left(\alpha +1\right)/\alpha }.$ Then,

$$\psi \left(\vartheta \right)=LN+\frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{L^{\alpha +1}}{M^{\alpha }}.$$

Proof. 

It is easy to see that the maximum value of $\psi$ on $R$at ${\vartheta }^{\ast }=N+{\left(\alpha L/\left(\left(\alpha +1\right)M\right)\right)}^{\alpha }$ is

$$\underset{\vartheta \in R}{max}\psi \left(\vartheta \right)=\psi \left({\vartheta }^{\ast }\right)=LN+\frac{{\alpha }^{\alpha }}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{L^{\alpha +1}}{M^{\alpha }}.$$

(16)

Then, the proof is complete. □

Theorem 2.

Assume $\left(H\right)$ and (2) hold. If

$$\underset{l\to \infty }{limsup}\frac{{\delta }_2^{\alpha }\left(l\right)}{\rho \left(l\right)}{\int }_{l_1}^l\left(\rho \left(\zeta \right)h^{\alpha \eta }q\left(\zeta \right)-\frac{1}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\left({\rho }^{\prime }\left(\zeta \right)\right)}^{\alpha +1}}{{\rho }^{\alpha }\left(\zeta \right){\delta }_1^{\alpha }\left(\zeta \right)}\right)\mathrm{d}\zeta >1,$$

(17)

then the set ℜ is empty.

Proof. 

Suppose $x\in \Re$. As in the proof of Lemma 2, we have that (8) and (9) hold. From (9), we obtain

$$\frac{a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }}{x^{\alpha }\left(l\right)}\ge -\frac{1}{{\delta }_2^{\alpha }\left(l\right)}.$$

(18)

Thus, if we define the a generalized Riccati substitution as

$$w\left(l\right):=\rho \left(l\right)\left(\frac{a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }}{x^{\alpha }\left(l\right)}+\frac{1}{{\delta }_2^{\alpha }\left(l\right)}\right),$$

(19)

where $\omega \left(l\right)\in C^1\left(\left[l_0,\infty \right),R\right),$ then $w\left(l\right)>0$ for all $l\ge l_1$. Differentiating $\omega \left(l\right)$, we have

$$\begin{array}{lll}{w}^{\prime }\left(l\right)& =& \frac{{\rho }^{\prime }\left(l\right)}{\rho \left(l\right)}w\left(l\right)+\rho \left(l\right)\frac{{\left(a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }\right)}^{\prime }}{x^{\alpha }\left(l\right)}-\alpha \rho \left(l\right)\frac{a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }}{x^{\alpha +1}\left(l\right)}{x}^{\prime }\left(l\right)\\ & & -\frac{\alpha {\delta }_2^{\prime }\left(l\right)}{{\delta }_2^{\alpha +1}\left(l\right)}.\end{array}$$

(20)

From (1), we see that

$$\begin{array}{lll}{\left(a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }\right)}^{\prime }& =& -f(l,x(\tau \left(l\right)\left)\right)\\ & \le & -q\left(l\right)x^{\alpha }\left(\tau \left(l\right)\right).\end{array}$$

(21)

Using (8) and (21), (20) becomes

$$\begin{array}{lll}{w}^{\prime }\left(l\right)& \le & \frac{{\rho }^{\prime }\left(l\right)}{\rho \left(l\right)}w\left(l\right)-\rho \left(l\right)q\left(l\right){\left(\frac{x\left(\tau \left(l\right)\right)}{x\left(l\right)}\right)}^{\alpha }\\ & & -\alpha \rho \left(l\right)a\left(l\right){\left(\frac{x^{‴}\left(l\right)}{x\left(l\right)}\right)}^{\alpha +1}{a}^{1/\alpha }\left(l\right){\delta }_1\left(l\right)+\frac{\alpha {\delta }_1\left(l\right)}{{\delta }_2^{\alpha +1}\left(l\right)}.\end{array}$$

(22)

Using Lemma 3, we have that (10) holds. Thus, (22) yields

$$\begin{array}{lll}{w}^{\prime }\left(l\right)& \le & \frac{{\rho }^{\prime }\left(l\right)}{\rho \left(l\right)}w\left(l\right)-\rho \left(l\right)h^{\alpha \left(\eta -\epsilon \right)}q\left(l\right)-\alpha \rho \left(l\right)a\left(l\right){\left(\frac{x^{‴}\left(l\right)}{x\left(l\right)}\right)}^{\alpha +1}{a}^{1/\alpha }\left(l\right){\delta }_1\left(l\right)\\ & & +\frac{\alpha {\delta }_1\left(l\right)}{{\delta }_2^{\alpha +1}\left(l\right)}.\end{array}$$

Hence, from the definition of w, we obtain

$$\begin{array}{lll}{w}^{\prime }\left(l\right)& \le & -\rho \left(l\right)h^{\alpha \left(\eta -\epsilon \right)}q\left(l\right)+\frac{{\rho }^{\prime }\left(l\right)}{\rho \left(l\right)}w\left(l\right)-\alpha \frac{1}{{\rho }^{1/\alpha }\left(l\right)}{\left(w\left(l\right)-\frac{\rho \left(l\right)}{{\delta }_2^{\alpha }\left(l\right)}\right)}^{1+1/\alpha }{\delta }_1\left(l\right)\\ & & +\frac{\alpha {\delta }_1\left(l\right)}{{\delta }_2^{\alpha +1}\left(l\right)}.\end{array}$$

(23)

Using inequality (16) with

$$L:=\frac{{\rho }^{\prime }\left(l\right)}{\rho \left(l\right)},M:=\alpha \frac{{\delta }_1\left(l\right)}{{\rho }^{1/\alpha }\left(l\right)},N:=\frac{\rho \left(l\right)}{{\delta }_2^{\alpha }\left(l\right)}$$

and $\vartheta :=w$, we obtain

$$\begin{array}{lll}\frac{{\rho }^{\prime }\left(l\right)}{\rho \left(l\right)}w\left(l\right)& \le & \alpha \frac{1}{{\rho }^{1/\alpha }\left(l\right)}{\left(w\left(l\right)-\frac{\rho \left(l\right)}{{\delta }_2^{\alpha }\left(l\right)}\right)}^{1+1/\alpha }+\frac{1}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\left({\rho }^{\prime }\left(l\right)\right)}^{\alpha +1}}{{\rho }^{\alpha }\left(l\right){\delta }_1^{\alpha }\left(l\right)}\\ & & +\frac{{\rho }^{\prime }\left(l\right)}{{\delta }_2^{\alpha }\left(l\right)},\end{array}$$

which, with (23), gives

$$w^{\prime }\left(l\right)\le -\rho \left(l\right)h^{\alpha \left(\eta -\epsilon \right)}q\left(l\right)+\frac{1}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\left({\rho }^{\prime }\left(l\right)\right)}^{\alpha +1}}{{\rho }^{\alpha }\left(l\right){\delta }_1^{\alpha }\left(l\right)}+\frac{{\rho }^{\prime }\left(l\right)}{{\delta }_2^{\alpha }\left(l\right)}+\frac{\alpha {\delta }_1\left(l\right)}{{\delta }_2^{\alpha +1}\left(l\right)}$$

or

$$w^{\prime }\left(l\right)\le -\rho \left(l\right)h^{\alpha \left(\eta -\epsilon \right)}q\left(l\right)+\frac{1}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\left({\rho }^{\prime }\left(l\right)\right)}^{\alpha +1}}{{\rho }^{\alpha }\left(l\right){\delta }_1^{\alpha }\left(l\right)}+\frac{\mathrm{d}}{\mathrm{d}l}\left(\frac{\rho \left(l\right)}{{\delta }_2^{\alpha }\left(l\right)}\right).$$

(24)

By integrating (24) from $l_1$ to l, we obtain

$$\begin{array}{lll}w\left(l\right)-w\left(l_1\right)& \le & -{\int }_{l_1}^l\left(\rho \left(\zeta \right)h^{\alpha \left(\eta -\epsilon \right)}q\left(\zeta \right)-\frac{1}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\left({\rho }^{\prime }\left(\zeta \right)\right)}^{\alpha +1}}{{\rho }^{\alpha }\left(\zeta \right){\delta }_1^{\alpha }\left(\zeta \right)}\right)\mathrm{d}\zeta \\ & & +\frac{\rho \left(l\right)}{{\delta }_2^{\alpha }\left(l\right)}-\frac{\rho \left(l_1\right)}{{\delta }_2^{\alpha }\left(l_1\right)}.\end{array}$$

From (19), we are led to

$$\begin{array}{l}{\int }_{l_1}^l\left(\rho \left(\zeta \right)h^{\alpha \left(\eta -\epsilon \right)}q\left(\zeta \right)-\frac{1}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\left({\rho }^{\prime }\left(\zeta \right)\right)}^{\alpha +1}}{{\rho }^{\alpha }\left(\zeta \right){\delta }_1^{\alpha }\left(\zeta \right)}\right)\mathrm{d}\zeta \\ \le -\rho \left(l\right)\frac{a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }}{x^{\alpha }\left(l\right)}+\rho \left(l_1\right)\frac{a\left(l_1\right){\left(x^{‴}\left(l_1\right)\right)}^{\alpha }}{x^{\alpha }\left(l_1\right)}\\ \le -\rho \left(l\right)\frac{a\left(l\right){\left(x^{‴}\left(l\right)\right)}^{\alpha }}{x^{\alpha }\left(l\right)}.\end{array}$$

In view of (18), we get

$${\int }_{l_1}^l\left(\rho \left(\zeta \right)h^{\alpha \left(\eta -\epsilon \right)}q\left(\zeta \right)-\frac{1}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\left({\rho }^{\prime }\left(\zeta \right)\right)}^{\alpha +1}}{{\rho }^{\alpha }\left(\zeta \right){\delta }_1^{\alpha }\left(\zeta \right)}\right)\mathrm{d}\zeta \le \frac{\rho \left(l\right)}{{\delta }_2^{\alpha }\left(l\right)}$$

or

$$\frac{{\delta }_2^{\alpha }\left(l\right)}{\rho \left(l\right)}{\int }_{l_1}^l\left(\rho \left(\zeta \right)h^{\alpha \left(\eta -\epsilon \right)}q\left(\zeta \right)-\frac{1}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\left({\rho }^{\prime }\left(\zeta \right)\right)}^{\alpha +1}}{{\rho }^{\alpha }\left(\zeta \right){\delta }_1^{\alpha }\left(\zeta \right)}\right)\mathrm{d}\zeta \le 1.$$

Taking limit supremum, we obtain a contradiction with (17). This completes the proof. □

Corollary 1.

Assume $\left(H\right)$ and (2) hold. If one of the following conditions holds:

$$\underset{l\to \infty }{limsup}{\delta }_2^{\alpha }\left(l\right){\int }_{l_1}^l{h}^{\alpha \eta }q\left(\zeta \right)\mathrm{d}\zeta >1$$

(25)

or

$$\underset{l\to \infty }{limsup}{\delta }_2^{\alpha -1}\left(l\right){\int }_{l_1}^l\left(h^{\alpha \eta }q\left(\zeta \right){\delta }_2\left(\zeta \right)-\frac{1}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\delta }_1\left(\zeta \right)}{{\delta }_2^{\alpha }\left(\zeta \right)}\right)\mathrm{d}\zeta >1$$

(26)

or

$$\underset{l\to \infty }{limsup}{\int }_{l_1}^l\left(h^{\alpha \eta }q\left(\zeta \right){\delta }_2^{\alpha }\left(\zeta \right)-\frac{{\alpha }^{\alpha +1}}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{{\delta }_1\left(l\right)}{{\delta }_2\left(l\right)}\right)\mathrm{d}\zeta >1,$$

(27)

then the set ℜ is empty.

Proof. 

By choosing $\rho \left(l\right)=1$, $\rho \left(l\right)={\delta }_2\left(l\right)$ or $\rho \left(l\right)={\delta }_2^{\alpha }\left(l\right)$, the condition (17) in Theorem 2 becomes as (25), (26) or (27), respectively. □

3. Discussion and Applications

Depending on the new criteria for the nonexistence of Kneser solutions, we introduced new criteria for oscillation of (1). When checking the behavior of positive solutions of DDE (1), we have three Cases $\left(1\right)–\left(3\right)$. In order to illustrate the importance of the results obtained for Case $\left(3\right)$, we recall an existing criterion for a particular case of (1) with $\alpha =\beta$:

Theorem 3

(Theorem 2.1 with $n=4$, [25]). Assume that $\alpha =\beta$,

$$\underset{l\to \infty }{liminf}{\int }_{\tau \left(l\right)}^{l}q\left(\varrho \right)\frac{{\tau }^{3\alpha }\left(\varrho \right)}{a\left(\tau \left(\varrho \right)\right)}\mathrm{d}\varrho >\frac{6^{\alpha }}{\mathrm{e}}$$

(28)

and there exists a $\rho \in C^1\left(I_0,{\mathbb{R}}^{+}\right)$ such that

$$\underset{l\to \infty }{limsup}{\int }_{l_0}^l\left({\delta }_0^{\alpha }\left(\varrho \right)q\left(\varrho \right){\left(\frac{\lambda }{2!}{\tau }^2\left(\varrho \right)\right)}^{\alpha }-\frac{{\alpha }^{\alpha +1}}{{\left(\alpha +1\right)}^{\left(\alpha +1\right)}}\frac{1}{\delta \left(\varrho \right)a^{1/\alpha }\left(\varrho \right)}\right)\mathrm{d}\varrho =\infty ,$$

(29)

for some $\lambda \in \left(0,1\right)$. Then every solution of (1) is oscillatory or tends to zero.

From the previous Theorems, we conclude under the assumptions of the Theorem that every positive solution x of (1) tends to zero, and hence x satisfies Case $\left(3\right)$. Therefore, conditions (28) and (29) ensure (3) without verifying the extra condition (2). In view of Theorems 1 and 3, we obtain the following:

Corollary 2.

Assume that (28) and (29) hold for some $\lambda \in \left(0,1\right)$. If $\left({\mathbf{C}}_1\right),$ $\left({\mathbf{C}}_2\right)$ or $\left({\mathbf{C}}_3\right)$ holds, then (1) is oscillatory.

Proof. 

Suppose that x is a nonoscillatory solution of (1). Thus, we have three cases. From Theorem 3, we find (28) and (29) contradicts Case (1) and Case (2) respectively.

For Case (3), using Theorem 1, if one of the conditions $\left({\mathbf{C}}_1\right)–\left({\mathbf{C}}_3\right)$ holds, then we obtain a contradiction. The proof is complete. □

Corollary 3.

Assume that (28) and (29) hold for some $\lambda \in \left(0,1\right)$. If (25), (26) or (27) holds, then (1) is oscillatory.

Proof. 

Suppose the x is a nonoscillatory solution of (1). Thus, we have three cases. From Theorem 3, we find (28) and (29) contradict Case (1) and Case (2) respectively.

For Case (3), using Corollary 1, if one of the conditions (25)–(27) holds, then we obtain a contradiction. The proof is complete. □

We state now an Example:

Example 1.

We have the fourth-order DDE

$${\left({\mathrm{e}}^{\alpha l}{x}^{‴}\left(l\right)\right)}^{\prime }+q_0{\mathrm{e}}^{\alpha l}{x}^{\alpha }\left({\tau }_{0}l\right)=0,$$

(30)

where $l\ge 1,{\tau }_0\in \left(0,1-1/\mathrm{e}\right)$and $q_0>0$. Note that $a\left(l\right)={\mathrm{e}}^{\alpha l},q\left(l\right)=q_0{\mathrm{e}}^{\alpha l},\tau \left(l\right)={\tau }_{0}l$. It is easy to conclude that${\delta }_m\left(l\right)={\mathrm{e}}^{-l}$for $m=0,1,2$. Then, we see that (28) and (29) are satisfied for all $q_0>0$.

For condition $\left({\mathbf{C}}_1\right)$, we have

$$\eta ={\left(\frac{q_0}{\alpha }\right)}^{1/\alpha }>1.$$

(31)

By using the fact that ${\mathrm{e}}^y>\mathrm{e}y$ for $y>0$, we get

$$\frac{{\delta }_2\left(\tau \left(l\right)\right)}{{\delta }_2\left(l\right)}={\mathrm{e}}^{\left(1-{\tau }_0\right)l}>\mathrm{e}\left(1-{\tau }_0\right)l\ge \mathrm{e}\left(1-{\tau }_0\right):=h>1,$$

for $l\ge 1$. Hence, Conditions $\left({\mathbf{C}}_2\right)$ or $\left({\mathbf{C}}_3\right)$ reduce to

$${\left(\frac{q_0}{\alpha }\right)}^{1/\alpha }{\left(\mathrm{e}\left(1-{\tau }_0\right)\right)}^{{\left(q_0/\alpha \right)}^{1/\alpha }}>1$$

(32)

and

$$q_0{\left(\mathrm{e}\left(1-{\tau }_0\right)\right)}^{{\left(q_0/\alpha \right)}^{1/\alpha }}>{\left(\frac{\alpha }{\alpha +1}\right)}^{\alpha +1},$$

(33)

respectively.

Thus, by Corollary, if (31), (32) or (33) holds, then (30) is oscillatory.

Remark 1.

To the best of our knowledge, the known related sharp criterion for (30) based on (Theorem 2.1, [38]) gives

$$q_0>{\left(\frac{\alpha }{\alpha +1}\right)}^{\alpha +1}.$$

(34)

Note firstly that our criteria (32) and (33) essentially take into account the influence of delay argument $\tau \left(l\right)$, which has been neglected in all previous results of fourth-order equations.Secondly, in the case where $\alpha =1$ and ${\tau }_0=1/2$, we have

Condition	(31)	(32)	(33)	(34)
Criterion	q0 > 1.00	q0 > 0.786	q0 > 0.233	q0 > 0.250.

Condition (33) supports the most efficient and sharp criterion for oscillation of Equation (30).

4. Conclusions

We worked on extending and improving existing oscillation criteria for DDEs of the fourth order for the nonexistence of Kneser solutions. The new criteria that we proved are characterized by taking into account the effect of the delay argument.