Nonlinear Differential Equations with Distributed Delay: Some New Oscillatory Solutions

Abstract

The oscillation of a class of fourth-order nonlinear damped delay differential equations with distributed deviating arguments is the subject of this research. We propose a new explanation of the fourth-order equation oscillation in terms of the oscillation of a similar well-studied second-order linear differential equation without damping. The extended Riccati transformation, integral averaging approach, and comparison principles are used to provide some additional oscillatory criteria. An example demonstrates the efficacy of the acquired criteria.

1. Introduction

In our current study, we take into consideration the following fourth-order nonlinear damped delay differential equations with distributed deviating arguments:

$$\begin{array}{c}\hfill {\left(x_2\left(t\right){\left(x_1\left(t\right){\left(u^{″}\left(t\right)\right)}^{\alpha }\right)}^{\prime }\right)}^{\prime }+p\left(t\right){\left(u^{″}\left(\delta \left(t\right)\right)\right)}^{\alpha }+{\int }_c^{d}q(t,\varrho )f(t,u\left(g(t,\varrho )\right))d\varrho =0,\end{array}$$

(1)

where $\alpha \ge 1$ is a ratio of odd non-negative natural numbers and $c<d$. We consider the below assertions all through this article:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{x}_1,x_2,p,\delta \in C(I,[0,\infty ))\mathrm{and}{x}_1,x_2>0,\mathrm{where}I=[t_0,+\infty );\hfill \\ q,g\in C[I\times [c,d],[0,\infty )),\delta \left(t\right)\le t,\underset{t\to +\infty }{lim}\delta \left(t\right)=\infty ,g(t,\varrho )\mathrm{is}\mathrm{an}\mathrm{non-decreasing}\hfill \\ \mathrm{function}\mathrm{for}\varrho \in [c,d]\mathrm{satisfying}g(t,\varrho )\le t\mathrm{and}\underset{t\to +\infty }{lim}g(t,\varrho )=\infty ;\hfill \\ f\in C(\mathbb{R},\mathbb{R}),\mathrm{there}\mathrm{is}\mathrm{a}\mathrm{constant}{k}_1>0\mathrm{such}\mathrm{that}f(t,u\left(t\right))/u^{\beta }\ge k_1.\hfill \end{array}\right.\end{array}$$

We define the operators,

$$\begin{array}{c}\hfill L^{\left[0\right]}u=u,L^{\left[1\right]}u=u^{\prime },L^{\left[2\right]}u=x_1{\left({\left(L^{\left[0\right]}u\right)}^{″}\right)}^{\alpha },L^{\left[3\right]}u=x_2{\left(L^{\left[2\right]}u\right)}^{\prime }\mathrm{as}\mathrm{well}\mathrm{as}\\ \hfill L^{\left[4\right]}u={\left(L^{\left[3\right]}u\right)}^{\prime }.\end{array}$$

The meaning of having a solution to Equation (1) is the function $u\left(t\right)$ in $C^2[T_u,\infty )$, for which $L^{\left[2\right]}u$, $L^{\left[4\right]}u$ is in $C^1[T_u,\infty )$, and Equation (1) holds on $[T_u,\infty )$, such that $T_u\ge t_0$. We only take into consideration the solutions $u\left(t\right)$ when $sup\left\{\right|u\left(t\right)|:t\ge T\}>0$ for every $T\ge T_u$. On one hand, such a solution to Equation (1) is termed oscillatory when this solution is not eventually negative and, at the same time, not eventually positive on the interval $[T_u,\infty )$. On the other hand, the same solution is termed non-oscillatory if it is eventually negative or eventually positive. Finally, when every solution is oscillating, the equation is said to be oscillatory.

We define

$$\begin{array}{ccc}& & A_1(t_1,t)={\int }_{t_1}^t{x}_1^{-1/\alpha }\left(s\right)ds,\hfill \\ & & A_2(t_1,t)={\int }_{t_1}^t{x}_2^{-1}\left(s\right)ds,\hfill \\ & & A_3(t_1,t)={\int }_{t_1}^t{\left({\left(x_1\left(s\right)\right)}^{-1}{A}_2(t_1,s)\right)}^{1/\alpha }ds,\hfill \\ & & A_4(t_1,t)={\int }_{t_1}^t{\int }_{t_1}^u{\left({\left(x_1\left(s\right)\right)}^{-1}{A}_2(t_1,s)\right)}^{1/\alpha }dsdu,\hfill \end{array}$$

for $t_0\le t_1\le t<\infty$ and assume that

$$\begin{array}{c}\hfill A_1(t_1,t)\to \infty ,A_2(t_1,t)\to \infty \mathrm{as}t\to \infty .\end{array}$$

(2)

Fourth-order differential equations are often used in mathematical models of a wide range of physical, chemical, and biological processes [1,2,3,4]. Problems with elasticity, structural deformation, and soil settling are examples of applications of this type of equation. In addition, in mechanical and engineering fields, questions about the presence of oscillatory and non-oscillatory solutions are mostly arising, and the solutions require the presence of the same mentioned equation [5]. Many researchers have intensively studied the topic of oscillation of fourth or higher order differential equations in depth, and many strategies for establishing oscillatory criteria for fourth or higher order differential equations have been developed. Several works, see [6,7,8,9,10,11,12,13,14,15,16,17,18], contain extremely interesting results linked to oscillatory features of solutions of neutral differential equations and damped delay differential equations with or without distributed deviating arguments.

In fact, for the following equation, Bazighifan et al. [19] have developed some oscillation criteria

$${\left(r\left(t\right){\left(N_x^{‴}\left(t\right)\right)}^{\beta }\right)}^{\prime }+{\int }_a^{b}q(t,\varrho )x^{\beta }\left(g(t,\varrho )\right)d\varrho =0.$$

Moreover, Dzurina et al. [20] introduced some oscillation findings of the below fourth-order equation

$${\left(r_3\left(t\right){\left(r_2\left(t\right)\left(r_1\left(t\right)y^{\prime }\left(t\right)\right)\right)}^{\prime }\right)}^{\prime }+p\left(t\right)y^{\prime }\left(t\right)+q\left(t\right)y\left(\tau \left(t\right)\right)=0.$$

More specifically, there are no requirements for the oscillation of Equation (1) in the previous studies.

By the motivations above, our contribution would be giving certain adequate conditions that ensure that every solution to Equation (1) oscillates, utilizing proper Riccati-type transformation, integral averaging condition, and comparison technique, when the following second-order equation

$$\begin{array}{c}\hfill {\left(x_2\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }+\frac{p\left(t\right)}{x_1\left(\delta \left(t\right)\right)}z\left(t\right)=0,\end{array}$$

(3)

is oscillatory or non-oscillatory.

2. Basic Lemmas

We state in the current section several Lemmas along with their proofs, which are mostly needed in the rest of this study.

Lemma 1

([8]). Assume that Equation (3) is non-oscillatory. If Equation (1) has a non-oscillatory solution $u\left(t\right)$ on I, $t_1\ge t_0$, then there is a $t_2\in I$ in a way that $u\left(t\right)L^{\left[2\right]}u\left(t\right)>0$ or $u\left(t\right)L^{\left[2\right]}u\left(t\right)<0$ for $t\ge t_2$.

Lemma 2.

If the Equation (1) has a non-oscillatory solution $u\left(t\right)$ that satisfies $u\left(t\right)L^{\left[2\right]}u\left(t\right)>0$ in Lemma 1 for $t\ge t_1\ge t_0$, then

$$\begin{array}{ccc}& & L^{\left[2\right]}u\left(t\right)>A_2(t_1,t)L^{\left[3\right]}u\left(t\right),t\ge t_1,\hfill \end{array}$$

(4)

$$\begin{array}{ccc}& & L^{\left[1\right]}u\left(t\right)>A_3(t_1,t){\left(L^{\left[3\right]}u\left(t\right)\right)}^{1/\alpha },t\ge t_1,\hfill \end{array}$$

(5)

and

$$\begin{array}{c}\hfill u\left(t\right)>A_4(t_1,t){\left(L^{\left[3\right]}u\left(t\right)\right)}^{1/\alpha },t\ge t_1.\end{array}$$

(6)

Proof. 

We suppose that there is a $t_1\ge t_0$ in a way that $u\left(t\right)>0$ and $u\left(g\right(t,\varrho \left)\right)>0$ for $t\ge t_1$. From Equation (1), we have

$$\begin{array}{c}\hfill L^{\left[4\right]}u\left(t\right)=-\left(\frac{p\left(t\right)}{x_1\left(\delta \left(t\right)\right)}\right)L^{\left[2\right]}u\left(\delta \left(t\right)\right)-k_1{\int }_c^{d}q(t,\varrho )u^{\beta }\left(g(t,\varrho )\right)d\varrho \le 0,\end{array}$$

and $L^{\left[3\right]}u\left(t\right)$ is non increasing on I, we obtain

$$\begin{array}{c}\hfill L^{\left[2\right]}u\left(t\right)\ge {\int }_{t_1}^t{\left(L^{\left[2\right]}u\left(s\right)\right)}^{\prime }ds={\int }_{t_1}^t{\left(x_2\left(s\right)\right)}^{-1}{L}^{\left[3\right]}u\left(s\right)ds\ge A_2(t_1,t)L^{\left[3\right]}u\left(t\right),\end{array}$$

which implies that

$$\begin{array}{c}\hfill u^{’’}\left(t\right)\ge {\left(L^{\left[3\right]}u\left(t\right)\right)}^{1/\alpha }{\left({\left(x_1\left(t\right)\right)}^{-1}{A}_2(t_1,t)\right)}^{1/\alpha }.\end{array}$$

Now, twice integrating above from $t_1$ to t and using $L^{\left[3\right]}u\left(t\right)\le 0$, we find

$$\begin{array}{c}\hfill u^{\prime }\left(t\right)\ge {\left(L^{\left[3\right]}u\left(t\right)\right)}^{1/\alpha }{\int }_{t_1}^t{\left({\left(x_1\left(s\right)\right)}^{-1}{A}_2(t_1,s)\right)}^{1/\alpha }ds\end{array}$$

and

$$\begin{array}{c}\hfill u\left(t\right)\ge {\left(L^{\left[3\right]}u\left(t\right)\right)}^{1/\alpha }{\int }_{t_1}^t{\int }_{t_1}^u{\left({\left(x_1\left(s\right)\right)}^{-1}{A}_2(t_1,s)\right)}^{1/\alpha }dsdu\mathrm{for}t\le t_1.\end{array}$$

□

Lemma 3

([10]). Let $\xi \in C^1(I,{\mathbb{R}}^{+})$, $\xi \left(t\right)\le t$, ${\xi }^{\prime }\left(t\right)\ge 0$ and $G\left(t\right)\in C(I,{\mathbb{R}}^{+})$ for $t\ge t_0$. Assume that $y\left(t\right)$ is a bounded solution of a second-order delay differential equation:

$$\begin{array}{c}\hfill (x_2\left(t\right)y^{\prime }{\left(t\right))}^{\prime }-\mathsf{\Theta }\left(t\right)y\left(\xi \left(t\right)\right)=0.\end{array}$$

(7)

If

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}{\int }_{\xi \left(t\right)}^t\mathsf{\Theta }\left(s\right)A_2(\xi \left(t\right),\xi \left(s\right))ds>1\end{array}$$

(8)

or

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}{\int }_{\xi \left(t\right)}^t\left({\left(x_2\left(t\right)\right)}^{-1}{\int }_u^t\mathsf{\Theta }\left(s\right)ds\right)du>1,\end{array}$$

(9)

where $x_2\left(t\right)$ is as in Equation (1); thus, the solutions of Equation (7) are oscillatory.

3. Oscillation—Comparison Principle Method

In this section, we shall establish some oscillation criteria for Equation (1). For convenience, we denote

$$\begin{array}{cc}\hfill Q\left(t\right)& =\left(\frac{p\left(t\right)}{x_1\left(\delta \left(t\right)\right)}\right)A_2(t_1,\delta \left(t\right)),\psi \left(t\right)=exp\left({\int }_{t_1}^{t}Q\left(s\right)ds\right),\hfill \\ \hfill \tilde{q}(t,\varrho )& ={\int }_c^{d}q(t,\varrho )d\varrho ,{\mathsf{\Theta }}^{*}\left(t\right)=k_1\tilde{q}(t,\varrho ){\left(A_4(t_1,g(t,d))\right)}^{\beta }.\hfill \end{array}$$

Theorem 1.

Assume that $\alpha \ge \beta$ and the conditions in Equation (2) hold, and Equation (3) is non-oscillatory. Suppose there exists a $\xi \in C^1(I,\mathbb{R})$ such that

$$g(t,\varrho )\le \xi \left(t\right)\le \delta \left(t\right)\le t,{\xi }^{\prime }\left(t\right)\ge 0\mathrm{for}t\ge t_1,$$

and Equations (8) or (9) holds with

$$\begin{array}{c}\hfill \mathsf{\Theta }\left(t\right)={\ell }_{*}{k}_1\tilde{q}(t,\varrho )g^{\beta }(t,d){\left(A_1(\xi \left(t\right),g(t,d))\right)}^{\beta }-\frac{p\left(t\right)}{x_1\left(\delta \left(t\right)\right)}\ge 0,t\ge t_1,\end{array}$$

for constant ${\ell }_{*}$$>0$. Moreover, suppose that every solution of the first-order delay equation

$$\begin{array}{c}\hfill z^{\prime }\left(t\right)+{\psi }^{1-\frac{\beta }{\alpha }}\left(g(t,d)\right){\mathsf{\Theta }}^{*}\left(t\right)z^{\frac{\beta }{\alpha }}\left(g(t,d)\right)=0.\end{array}$$

(10)

Then, every solution of Equation (1) is oscillatory.

Proof. 

Let Equation (1) have a non-oscillatory solution $u\left(t\right)$. Assume there exists a $t\ge t_1$ such that $u\left(t\right)>0$ and $u\left(g\right(t,\varrho \left)\right)>0$ for some $t_{}\ge t_0$. From Lemma 1, $u\left(t\right)$ has the conditions either $L^{\left[2\right]}u\left(t\right)>0$ or $L^{\left[2\right]}u\left(t\right)<0$ for $t\ge t_1$.

Assume that $u\left(t\right)$ has the condition $L^{\left[2\right]}u\left(t\right)>0$ for $t\ge t_1$, then one can easily see that $L^{\left[3\right]}u\left(t\right)>0$ for $t\ge t_1$. We can choose $t_2\ge t_1$ such that $g(t,\varrho )\ge t_1$ for $t\ge t_2$, $g(t,\varrho )$→∞ as $t\to \infty$, and we have Equation (6),

$$\begin{array}{c}\hfill u\left(g(t,d)\right)>A_4(t_1,g(t,d)){\left(L^{\left[3\right]}u\left(g(t,d)\right)\right)}^{1/\alpha },t\ge t_2.\end{array}$$

(11)

By substituting Equations (4) and (11) into Equation (1) and when $L^{\left[3\right]}u\left(t\right)$ is decreasing,

$$\begin{array}{ccc}\hfill & & {\left(L^{\left[3\right]}u\left(t\right)\right)}^{\prime }+\left(\frac{p\left(t\right)}{x_1\left(\delta \left(t\right)\right)}\right)L^{\left[3\right]}u\left(t\right)A_2(t_1,\delta \left(t\right))\hfill \\ & & +k_1\tilde{q}(t,\varrho ){\left(A_4(t_1,g(t,d))\right)}^{\beta }{\left(L^{\left[3\right]}u\left(g(t,d)\right)\right)}^{\beta /\alpha }\le 0.\hfill \end{array}$$

(12)

Taking $\varphi =L^{\left[3\right]}u$, we have

$$\begin{array}{c}\hfill {\varphi }^{\prime }\left(t\right)+Q\left(t\right)\varphi \left(t\right)+{\mathsf{\Theta }}^{*}\left(t\right){\varphi }^{\frac{\beta }{\alpha }}\left(g(t,d)\right)\le 0\end{array}$$

(13)

or

$$\begin{array}{c}\hfill {\left(\psi \left(t\right)\varphi \left(t\right)\right)}^{\prime }+\psi \left(t\right){\mathsf{\Theta }}^{*}\left(t\right){\varphi }^{\frac{\beta }{\alpha }}\left(g(t,d)\right)\le 0,\mathrm{for}t\ge t_2.\end{array}$$

(14)

Next, setting $z=\psi \varphi >0$ and $\psi \left(g\right(t,d\left)\right)\le \varphi \left(t\right)$, we have

$$\begin{array}{c}\hfill z^{\prime }\left(t\right)+{\psi }^{1-\frac{\beta }{\alpha }}\left(g(t,d)\right){\mathsf{\Theta }}^{*}\left(t\right)z^{\frac{\beta }{\alpha }}\left(g(t,d)\right)\le 0.\end{array}$$

(15)

This means Equation (15) is positive for this inequality. Furthermore, by ([21], Corollary 2.3.5), it can be seen that Equation (1) has a positive solution, a contradiction.

Next, assume $u\left(t\right)$ has the condition $L^{\left[2\right]}u\left(t\right)<0$, for $t\ge t_1$, then one can easily see that $L^{\left[1\right]}u\left(t\right)\ge 0$, $L^{\left[3\right]}u\left(t\right)>0$ for $t\ge t_3(\ge t_2)$. Using the monotonicity of $u^{\prime }\left(t\right)$ and mean value property of differentiation, there exists a $\theta \in (0,1)$ such that

$$\begin{array}{c}\hfill u\left(t\right)\ge \theta t{u}^{\prime }\left(t\right),\mathrm{for}t\ge t_3.\end{array}$$

(16)

Set $w\left(t\right)=L^{\left[1\right]}u\left(t\right)$, then $w^{\prime }\left(t\right)=u^{″}\left(t\right)<0$. Using Equation (16) in Equation (1), we obtain

$$\begin{array}{c}\hfill {\left(x_2\left(t\right){\left(x_1\left(t\right){\left[w^{\prime }\left(t\right)\right]}^{\alpha }\right)}^{\prime }\right)}^{\prime }+p\left(t\right){\left(w^{\prime }\left(\delta \left(t\right)\right)\right)}^{\alpha }+k_1{\left(t\theta \right)}^{\beta }\tilde{q}(t,\varrho )w^{\beta }\left(g(t,d)\right)\le 0,\end{array}$$

and so $\left(x_1\left(t\right){\left[w^{\prime }\left(t\right)\right]}^{\alpha }\right)<0$, we have ${\left(x_1\left(t\right){\left[w^{\prime }\left(t\right)\right]}^{\alpha }\right)}^{\prime }>0$ for $t\ge t_3$. Now, for $v\ge u\ge t_3$, we obtain

$$\begin{array}{ccc}\hfill w\left(u\right)>w\left(u\right)-w\left(v\right)& =& -{\int }_u^v-x_1^{-1/\alpha }\left(\tau \right){\left(x_1\left(\tau \right){\left(w^{\prime }\left(\tau \right)\right)}^{\alpha }\right)}^{1/\alpha }d\tau \hfill \\ & \ge & x_1^{1/\alpha }\left(v\right)(-w^{\prime }\left(v\right)))\left({\int }_u^v{x}_1^{-1/\alpha }\left(\tau \right)d\tau \right)\hfill \\ & =& x_1^{1/\alpha }\left(v\right)(-w^{\prime }\left(v\right))A_1(u,v).\hfill \end{array}$$

Taking $u=\xi \left(t\right)$ and $v=g(t,d)$, we obtain

$$\begin{array}{c}\hfill w\left(g(t,d)\right)>A_1(g(t,d),\xi \left(t\right))\left(x_1^{1/\alpha }\left(\xi \left(t\right)\right)(-w^{\prime }\left(\xi \left(t\right)\right))\right)=A_1(g(t,d),\xi \left(t\right))y\left(\xi \left(t\right)\right),\end{array}$$

where $y\left(t\right)=x_1^{1/\alpha }\left(\xi \left(t\right)\right)(-w^{\prime }\left(\xi \left(t\right)\right))>0$ for $t\ge t_3$. From Equation (1), we have that $y\left(t\right)$ is decreasing and $g(t,d)\le \xi \left(t\right)\le \delta \left(t\right)\le t$; thus, we obtain

$$\begin{array}{c}\hfill {\left(x_2\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }+\frac{p\left(t\right)}{x_1\left(\delta \left(t\right)\right)}z\left(\delta \left(t\right)\right)\ge k_1{\left(\theta g(t,d)\right)}^{\beta }\tilde{q}(t,\varrho )A_1(g(t,d),\xi \left(t\right))z^{\frac{\beta }{\alpha }-1}\left(\xi \left(t\right)\right)z\left(\xi \left(t\right)\right).\end{array}$$

Since z is decreasing and $\alpha \ge \beta$, there exists a constant ℓ such that $z^{\frac{\beta }{\alpha }-1}\left(t\right)\ge \ell$ for $t\ge t_3$. Thus, we obtain

$$\begin{array}{c}\hfill {\left(x_2\left(t\right)z^{\prime }\left(t\right)\right)}^{\prime }\ge \left(\ell k_1{\left(\theta g(t,d)\right)}^{\beta }\tilde{q}(t,\varrho )A_1(g(t,d),\xi \left(t\right))-\frac{p\left(t\right)}{x_1\left(\delta \left(t\right)\right)}\right)z\left(\xi \left(t\right)\right).\end{array}$$

Proceeding the rest of the proof in Lemma (3), we arrive at the required conclusion, and so it is omitted. □

4. Oscillation—Riccati Method

This section deals with some oscillation criteria for Equation (1) using the Ricatti Method.

Theorem 2.

Assume $\alpha \ge \beta$ and the conditions in Equation (2) hold, Equation (3) is non-oscillatory. Suppose there exists η, $\xi \in C^1(I,\mathbb{R})$ such that $g(t,\varrho )\le \xi \left(t\right)\le \delta \left(t\right)\le t$, ${\xi }^{\prime }\left(t\right)\ge 0$ and $\eta >0$ for $t\ge t_1$ with

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}{\int }_{t_5}^t\left(k_1\eta \left(s\right)\tilde{q}(s,\varrho )-\frac{A^2\left(s\right)}{4B\left(s\right)}\right)ds=\infty forall{t}_1\in I,\end{array}$$

(17)

where, for $t\ge t_1$,

$$\begin{array}{c}\hfill A\left(t\right)=\frac{{\eta }^{\prime }\left(t\right)}{\eta \left(t\right)}-\frac{p\left(t\right)}{x_1\left(\delta \left(t\right)\right)}{A}_2(t_1,\delta \left(t\right))\end{array}$$

(18)

and

$$\begin{array}{c}\hfill B\left(t\right)=\frac{\beta {\ell }_2^{\beta -\alpha }{g}^{\prime }(t,d)}{\eta \left(t\right)}{\left(A_4(t_1,g(t,d))\right)}^{\beta -1}{\left(A_3(t_1,g(t,d))\right)}^{1/\alpha },\end{array}$$

(19)

also Equations (8) or (9) hold with $\mathsf{\Theta }\left(t\right)$ as in Theorem 1. Then every solution of Equation (1) is oscillatory.

Proof. 

Suppose that Equation (1) has a non-oscillatory solution $u\left(t\right)$. Assume that, there exists a $t\ge t_1$ such that $u\left(t\right)>0$ and $u\left(g\right(t,\varrho \left)\right)>0$ for some $t_{}\ge t_0$. From Lemma 1, $u\left(t\right)$ has the conditions either $L^{\left[2\right]}u\left(t\right)>0$ or $L^{\left[2\right]}u\left(t\right)<0$ for $t\ge t_1$. If condition $L^{\left[2\right]}u\left(t\right)<0$ holds, the proof follows from Theorem 1.

Next, if condition $L^{\left[2\right]}u\left(t\right)>0$ holds, define

$$\begin{array}{c}\hfill \omega \left(t\right)=\eta \left(t\right)\frac{L^{\left[3\right]}u\left(t\right)}{u^{\beta }\left(g(t,d)\right)},t\in I,\end{array}$$

(20)

then $\omega \left(t\right)>0$ for $t\ge t_1$. From Equation (6) and $L^{\left[4\right]}u\left(t\right)<0$, we have

$$\begin{array}{c}\hfill \omega \left(t\right)=\eta \left(t\right)\frac{L^{\left[3\right]}u\left(t\right)}{u^{\beta }\left(g(t,d)\right)}\le \eta \left(t\right)\frac{L^{\left[3\right]}u\left(g(t,d)\right)}{u^{\beta }\left(g(t,d)\right)}\le \eta \left(t\right){\left(A_4(t_1,g(t,d))\right)}^{-\alpha }{u}^{\alpha -\beta }\left(g(t,d)\right),\end{array}$$

(21)

for $t\ge t_1$. From Equation (5) and definition $L^{\left[2\right]}u\left(t\right)$, we find

$$\begin{array}{c}\hfill u^{\prime }\left(g(t,d)\right)=L^{\left[1\right]}u\left(g(t,d)\right)\ge A_3(t_1,g(t,d)){\left(L^{\left[3\right]}u\left(\delta \left(t\right)\right)\right)}^{1/\alpha }\ge A_3(t_1,g(t,d)){\left(L^{\left[3\right]}u\left(g(t,d)\right)\right)}^{1/\alpha }.\end{array}$$

Then,

$$\begin{array}{ccc}\hfill \frac{u^{\prime }\left(g(t,d)\right)}{u\left(g\right(t,d\left)\right)}& \ge & {\left(\frac{A_3(t_1,g(t,d))}{\eta \left(\delta \right(t\left)\right)}\right)}^{1/\alpha }\frac{{\eta }^{1/\alpha }\left(\delta \left(t\right)\right){\left(L^{\left[3\right]}u\left(t\right)\right)}^{1/\alpha }}{u^{\beta /\alpha }\left(g(\delta \left(t\right),d)\right)}{u}^{\beta /\alpha -1}\left(g(\delta \left(t\right),d)\right)\hfill \\ & =& {\left(\frac{A_3(t_1,g(t,d))}{\eta \left(t\right)}\right)}^{1/\alpha }{\omega }^{1/\alpha }\left(t\right)u^{\beta /\alpha -1}\left(g(\delta \left(t\right),d)\right).\hfill \end{array}$$

(22)

Furthermore, since there exists a constant ${\ell }_1$ and $t_2\ge t_1$ such that for $L^{\left[3\right]}u\left(t\right)$ ≤ $L^{\left[3\right]}u\left(t_2\right)=$ ${\ell }_1$. Therefore,

$$\begin{array}{c}\hfill L^{\left[2\right]}u\left(t\right)=L^{\left[2\right]}u\left(t_2\right)+{\int }_{t_2}^t{\left(L^{\left[2\right]}u\left(s\right)\right)}^{\prime }ds\le L^{\left[2\right]}u\left(t_2\right)+{\ell }_1{\int }_{t_2}^t\frac{ds}{x_2\left(s\right)}\\ \hfill =L^{\left[2\right]}u\left(t_2\right)+{\ell }_1{A}_2(t_2,t)=\left[\frac{L^{\left[2\right]}u\left(t_2\right)}{A_2(t_2,t)}+{\ell }_1\right]A_2(t_2,t)\\ \hfill \le \left[\frac{L^{\left[2\right]}u\left(t_2\right)}{A_2(t_2,t_3)}+{\ell }_1\right]A_2(t_2,t)={\ell }_1^{*}{A}_2(t_2,t),\end{array}$$

(23)

holds for all $t\ge t_2$, where ${\ell }_1^{*}={\ell }_1+\frac{L^{\left[2\right]}u\left(t_1\right)}{A_2(t_2,t_3)}$, which implies that

$$\begin{array}{c}\hfill u^{\prime }\left(t\right)=u^{\prime }\left(t_3\right)+{\int }_{t_3}^t{u}^{″}\left(s\right)ds\le u^{\prime }\left(t_3\right)+{\int }_{t_3}^t{\left(\frac{{\ell }_1^{*}{A}_2(t_2,s)}{x_1\left(s\right)}\right)}^{1/\alpha }ds\\ \hfill =u\left(t_3\right)+{\left({\ell }_1^{*}\right)}^{1/\alpha }{A}_3(t_3,t)={\ell }_2{A}_3(t_3,t),\end{array}$$

holds for all $t\ge t_3(\ge t_2)$, where ${\ell }_2=\frac{u\left(t_2\right)}{A_3(t_3,t_4)}+{\left({\ell }_1^{*}\right)}^{1/\alpha }$. Then,

$$\begin{array}{c}\hfill u\left(t\right)=u\left(t_4\right)+{\int }_{t_4}^t{u}^{\prime }\left(s\right)ds\le u\left(t_4\right)+{\int }_{t_4}^t\left({\ell }_2{A}_3(t_3,s)\right)ds\\ \hfill =u\left(t_4\right)+{\ell }_2{A}_4(t_4,t)={\ell }_2^{*}{A}_4(t_4,t),\end{array}$$

(24)

holds for all $t\ge t_4(\ge t_3)$, where ${\ell }_2^{*}=\frac{u\left(t_4\right)}{A_4(t_4,t_1)}+{\ell }_2$. Further,

$$\begin{array}{c}\hfill u^{\beta /\alpha -1}\left(g(t,d)\right)\ge {\left({\ell }_2^{*}\right)}^{\beta /\alpha -1}{\left(A_4(t_4,g(t,d))\right)}^{\beta /\alpha -1},t\ge t_4.\end{array}$$

(25)

By using Equation (24) in Equation (21), we obtain

$$\begin{array}{c}\hfill \omega \left(t\right)\le {\left({\ell }_2^{*}\right)}^{\alpha -\beta }\eta \left(t\right){\left(A_4(t_1,g(t,d))\right)}^{-\beta },\end{array}$$

(26)

and hence

$$\begin{array}{c}\hfill {\omega }^{\frac{1}{\alpha }-1}\left(t\right)\le {\left({\ell }_2^{*}\right)}^{(\alpha -\beta )\left(\frac{1}{\alpha }-1\right)}{\eta }^{\frac{1}{\alpha }-1}\left(t\right){\left(A_4(t_1,g(t,d))\right)}^{-\beta \left(\frac{1}{\alpha }-1\right)}.\end{array}$$

(27)

Now differentiating Equation (20), we obtain

$$\begin{array}{c}\hfill {\omega }^{\prime }\left(t\right)=\frac{{\eta }^{\prime }\left(t\right)}{\eta \left(t\right)}\omega \left(t\right)+\frac{L^{\left[4\right]}u\left(t\right)}{L^{\left[3\right]}u\left(t\right)}\omega \left(t\right)-\beta g^{\prime }(t,d)\frac{u^{\prime }\left(g(t,d)\right)}{u\left(g\right(t,d\left)\right)}\omega \left(t\right).\end{array}$$

(28)

Using Equations (1) and (4) in Equation (28), we have

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& \le & \left[\frac{{\eta }^{\prime }\left(t\right)}{\eta \left(t\right)}-\frac{p\left(t\right)}{x_1\left(g(t,d)\right)}{A}_2(t_4,g(t,d))\right]\omega \left(t\right)-k_1\eta \left(t\right)\tilde{q}(t,\varrho )-\beta g^{\prime }\left(t\right)\frac{u^{\prime }\left(g(t,d)\right)}{u\left(g\right(t,d\left)\right)}\omega \left(t\right)\hfill \\ & \le & A\left(t\right)\omega \left(t\right)-k_1\eta \left(t\right)\tilde{q}(t,\varrho )-\beta g^{\prime }\left(t\right)\frac{u^{\prime }\left(g(t,d)\right)}{u\left(g\right(t,d\left)\right)}\omega \left(t\right).\hfill \end{array}$$

(29)

By using Equations (22), (25) and (28) in Equation (29), we have

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& \le & A\left(t\right)\omega \left(t\right)-k_1\eta \left(t\right)\tilde{q}(t,\varrho )-\frac{\beta {\ell }_2^{\beta -\alpha }{g}^{\prime }\left(t\right)}{\eta \left(t\right)}{\left(A_4(t_1,g(t,d))\right)}^{\beta -1}{\left(A_3(t_1,g(t,d))\right)}^{1/\alpha }{\omega }^2\left(t\right)\hfill \\ & =& A\left(t\right)\omega \left(t\right)-k_1\eta \left(t\right)\tilde{q}(t,\varrho )+B\left(t\right){\omega }^2\left(t\right)\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& =& -k_1\eta \left(t\right)\tilde{q}(t,\varrho )+{\left[\sqrt{B\left(t\right)}\omega \left(t\right)-\frac{1}{2}\frac{A\left(t\right)}{\sqrt{B\left(t\right)}}\right]}^2+\frac{1}{4}\frac{A^2\left(t\right)}{B\left(t\right)}\hfill \\ & \le & -k_1\eta \left(t\right)\tilde{q}(t,\varrho )+\frac{1}{4}\frac{A^2\left(t\right)}{B\left(t\right)}.\hfill \end{array}$$

(31)

Integrating Equation (31) from $t_5(>t_4)$ to t gives

$$\begin{array}{c}\hfill {\int }_{t_5}^t\left(k_1\eta \left(s\right)\tilde{q}(s,\varrho )-\frac{1}{4}\frac{A^2\left(s\right)}{B\left(s\right)}\right)ds\le \omega \left(t_5\right),\end{array}$$

(32)

which contradicts Equation (17). □

Corollary 1.

Assume $\alpha \ge \beta$ and the conditions in Equation (2) hold, Equation (3) is non-oscillatory. Suppose there exists η, $\xi \in C^1(I,\mathbb{R})$ such that $g(t,\varrho )\le \xi \left(t\right)\le \delta \left(t\right)\le t$, ${\xi }^{\prime }\left(t\right)\ge 0$ and $\eta >0$ for $t\ge t_1$ such that the function $A\left(t\right)\le 0$,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}{\int }_{t_5}^t\left(\eta \left(s\right)\tilde{q}(s,\varrho )\right)ds=\infty forall{t}_1\in I,\end{array}$$

(33)

where $A\left(t\right)$ is defined in Equation (18), and Equations (8) or (9) holds with $\mathsf{\Theta }\left(t\right)$ as in Theorem 1. Then, every solution of Equation (1) is oscillatory.

Next, we examine the oscillation results of solutions to Equation (1) by Philos-type. Let ${\mathbb{D}}_0=\{(t,s):a\le s<t<+\infty \}$, $\mathbb{D}=\left\{\right(t,s):a\le s\le t<+\infty \}$, the continuous function $H(t,s)$, $H:\mathbb{D}\to \mathbb{R}$ belongs to the class function $\mathbb{R}$:

(i)

$H(t,t)=0$ for $t\ge t_0$ and $H(t,s)>0$ for $(t,s)\in {\mathbb{D}}_0$;

(ii)

H has a continuous and non-positive partial derivative on ${\mathbb{D}}_0$ with respect to the second variable such that

$$-\frac{\partial H(t,s)}{\partial s}=h(t,s){\left[H(t,s)\right]}^{1/2},$$

for all $(t,s)\in {\mathbb{D}}_0$.

Theorem 3.

Assume $\alpha \ge 1$ and the conditions in Equation (2) hold, and Equation (3) is non-oscillatory. Suppose there exists η, $\xi \in C^1(I,\mathbb{R})$ such that $g(t,\varrho )\le \xi \left(t\right)\le \delta \left(t\right)\le t$, ${\xi }^{\prime }\left(t\right)\ge 0$, $\eta >0$ and $H(t,s)$∈$\mathbb{R}$ for $t\ge t_1$ with

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}\frac{1}{H(t,t_5)}{\int }_{t_5}^t\left(k_1\eta \left(s\right)\tilde{q}(s,\varrho )H(t,s)-\frac{P^2(t,s)}{4B\left(s\right)}\right)ds=\infty forall{t}_1\in I,\end{array}$$

(34)

where $P(t,s)=h(t,s)-A\left(s\right)\sqrt{H(t,s)}$ and $A\left(t\right)$, $B\left(t\right)$ are defined in Theorem 2, and Equations (8) or (9) holds with $\mathsf{\Theta }\left(t\right)$ as in Theorem 1. Then, every solution of Equation (1) is oscillatory.

Proof. 

Suppose that Equation (1) has a non-oscillatory solution $u\left(t\right)$. Assume that there exists a $t\ge t_1$ such that $u\left(t\right)>0$ and $u\left(g\right(t,\varrho \left)\right)>0$ for some $t_{}\ge t_0$. Proceeding as in the proof of Theorem 2, we obtain the inequality from Equation (30), i.e.,

$$\begin{array}{ccc}\hfill {\omega }^{\prime }\left(t\right)& \le & A\left(t\right)\omega \left(t\right)-k_1\eta \left(t\right)\tilde{q}(t,\varrho )+B\left(t\right){\omega }^2\left(t\right),\hfill \end{array}$$

and so,

$$\begin{array}{ccc}\hfill {\int }_{t_5}^{t}H(t,s)\eta \left(s\right)\tilde{q}(s,\varrho )ds& \le & {\int }_{t_5}^{t}H(t,s)[-{\omega }^{\prime }\left(s\right)+A\left(s\right)\omega \left(s\right)-B\left(s\right){\omega }^2\left(s\right)]ds\hfill \\ & =& -H(t,s){\left[\omega \left(s\right)\right]}_{t_5}^t+{\int }_{t_5}^t[\frac{\partial H(t,s)}{\partial s}\omega \left(s\right)\hfill \\ & & +H(t,s)\left[A\left(s\right)\omega \left(s\right)-B\left(s\right){\omega }^2\left(s\right)\right]]ds\hfill \\ & =& H(t,t_5)\omega \left(t_5\right)-{\int }_{t_5}^t[{\omega }^2\left(s\right)B\left(s\right)H(t,s)\hfill \\ & & +\omega \left(s\right)\left(h(t,s)\sqrt{H(t,s)}-H(t,s)A\left(s\right)\right)]ds\hfill \\ & \le & H(t,t_5)\omega \left(t_5\right)+{\int }_{t_5}^t\frac{P^2(t,s)}{4B\left(s\right)}ds,\hfill \end{array}$$

which contradicts Equation (34). The rest of the proof is similar to that of Theorem 2 and hence is omitted. □

5. Examples

Below, we present an example to show the application of the main results. This example is given to demonstrate Theorem 2.

Example 1.

For $t\ge 1$, consider the fourth-order differential equation

$$\begin{array}{c}\hfill {\left(1/2t{\left(9e^{-t}\left(t\right)\left(u^{″}\left(t\right)\right)\right)}^{\prime }\right)}^{\prime }+36e^{-s/2}{u}^{″}(t/2)+{\int }_1^2\frac{t}{3}u(\varrho ,36e^{t/3})d\varrho =0.\end{array}$$

(35)

Here, $x_1=9e^{-t}$, $x_2=1/2t$, $\alpha =\beta =1$, $p\left(t\right)=36e^{-s/2}$, $q(t,\varrho )=t/3$ and $\delta \left(t\right)=t/2$, $g(t,\varrho )=t/3$. Now, pick $\eta \left(t\right)=36e^{t/3}$, so we obtain

$$\begin{array}{ccc}& & A_1(t_1,t)={\int }_1^t{\left(9e^s\right)}^{-1}ds=9(e^t-e),\hfill \\ & & A_2(t_1,t)={\int }_1^{t}2sds=t^2-1=(t+1)(t-1),\hfill \\ & & A_3(t_1,t/3)={\int }_1^{t/3}{\left(9e^s\right)}^{-1}(s^2-1)ds=e^{t/3}{(t-3)}^2,\hfill \\ & & \tilde{q}(s,\varrho )=\frac{s}{3}{\int }_1^{2}d\varrho =s/3,\hfill \end{array}$$

$A^2\left(s\right)=\frac{{(3t^2-5)}^2}{9}$ and $B\left(s\right)=\frac{{(s-3)}^2}{36}.$ Now,

$$\begin{array}{c}\hfill \underset{t\to \infty }{lim\; sup}{\int }_2^t\left(k_1\eta \left(s\right)\tilde{q}(s,\varrho )-\frac{A^2\left(s\right)}{4B\left(s\right)}\right)ds=\underset{t\to \infty }{lim\; sup}{\int }_2^t\left(12k_{1}s{e}^{s/3}-{\left(\frac{3s^2-5}{s-3}\right)}^2\right)ds\to \infty as t\to \infty ,\end{array}$$

and all hypotheses of Theorem 2 are satisfied, so every solution of Equation (35) is oscillatory.

6. Conclusions

The form in Equation (1) is clearly more generic than all of the problems covered in the literature. In this paper, we provided some oscillatory properties using the appropriate Riccati-type transformation, integral averaging condition, and comparison method, ensuring that any solution of Equation (1) oscillates under the assumption of $A_1(t_1,t)\to \infty$, $A_2(t_1,t)\to \infty$ as $t\to \infty$. Furthermore, based on the condition of $A_1(t_1,t)<\infty$, $A_2(t_1,t)<\infty$ as $t\to \infty$, it would be desirable to expand the oscillation criteria of Equation (1).