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• Published: 30 Baronial 2021

Some new oscillation criteria of fourth-lodge quasi-linear differential equations with neutral term

Advances in Difference Equations volume 2021, Article number:401 (2021) Cite this commodity

Abstract

In this commodity, nosotros are interested in studying the asymptotic behavior of quaternary-order neutral differential equations. Despite the growing interest in studying the oscillatory beliefs of delay differential equations of 2d-order, 4th-club equations have received less attention. We get more than ane benchmark to check the oscillation by the generalized Riccati method and the integral average technique. Our results are an extension and complement to some results published in the literature. Examples are given to prove the significance of new theorems.

Introduction

In this newspaper, we investigate the oscillation backdrop of solutions to the fourth-order neutral differential equations:

$$\brainstorm{aligned} \bigl( z ( x ) \varsigma _{r_{1}} \bigl( \delta ^{\prime \prime \prime } ( 10 ) \bigr) \bigr) ^{\prime number }+ \tilde{\omega } ( x ) \varsigma _{r_{2}} \bigl( \beta \bigl( \theta ( x ) \bigr) \bigr) =0, \end{aligned}$$

(i)

where \(\varsigma _{r_{i}}[southward]=|s|^{r_{i}-1}s\), \(\delta ( ten ) =\beta ( x ) +\tilde{y} ( x ) \beta ( \tilde{\theta } ( ten ) )\). Throughout this paper, we suppose that:

\(( S_{1} ) \):

\(r_{i} \) and \(r_{2} \) are quotients of odd positive integers,

\(( S_{two} ) \):

\(z,\tilde{y},\tilde{\omega }\in C[x_{0},\infty )\), \(z ( x ) >0\), \(z^{\prime } ( 10 ) \geq 0\), \(\tilde{\omega } ( 10 ) >0\), \(0\leq \tilde{y} ( 10 ) \leq \tilde{y}_{0}<i\), θ̃, \(\theta \in C[x_{0},\infty )\), \(\tilde{\theta } ( x ) \leq ten\), \(\lim_{x\rightarrow \infty }\tilde{\theta } ( x ) =\lim_{10 \rightarrow \infty }\theta ( x ) =\infty \),

and nether the assumption

$$\begin{aligned} \int _{x_{0}}^{\infty }\frac{i}{z^{i/r_{ane}} ( southward ) } \, \mathrm{d}southward=\infty. \stop{aligned}$$

(2)

Definition 1.one

([1])

Let

$$\begin{aligned} D=\bigl\{ ( x,s ) \in \mathbb{R} ^{two}:x\geq south\geq x_{0} \bigr\} \quad \text{and}\quad D_{0}=\bigl\{ ( x,s ) \in \mathbb{R} ^{two}:ten>southward\geq x_{0}\bigr\} . \stop{aligned}$$

The office \(G_{i}\in C ( D,\mathbb{R} ) \) fulfills the following conditions:

1. (i)

   \(G_{i} ( ten,southward ) =0\) for \(x\geq x_{0}, G_{i} ( 10,southward ) >0, ( 10,south ) \in D_{0}\);

2. (ii)

   The functions \(h,\upsilon \in C^{1} ( [ x_{0},\infty ), ( 0, \infty ) ) \) and \(g_{i}\in C ( D_{0},\mathbb{R} ) \) such that

   $$\begin{aligned} \frac{\fractional }{\partial south}G_{one} ( x,s ) + \frac{\alpha ^{\prime number } ( s ) }{\alpha ( south ) }Grand ( x,south ) =g_{1} ( x,s ) G_{1}^{r_{1}/ ( r_{1}+i ) } ( ten,s ) \terminate{aligned}$$

   (3)

   and

   $$\begin{aligned} \frac{\partial }{\partial s}G_{2} ( x,southward ) + \frac{h^{\prime } ( due south ) }{h ( s ) }G_{ii} ( x,south ) =g_{2} ( x,southward ) \sqrt{G_{2} ( 10,southward ) }. \end{aligned}$$

   (4)

Theory of oscillation of differential equations is a fertile study area and has attracted the attention of many authors recently. This is due to the existence of many important applications of this theory in neural networks, biology, social sciences, engineering science, etc., come across [2–10]. Very recently, a slap-up development was found in the study of oscillation of solutions to neutral differential equations, encounter [11–20]. In particular, quasilinear/Emden–Fowler differential equations have numerous applications in physics and engineering science (eastward.g., quasilinear/Emden–Fowler differential equations arise in the study of p-Laplace equations, porous medium issues, and so on); meet, e.one thousand., the papers [5, 21–24] for more details, the papers [5, 6, 25–28] for the oscillation of quasilinear/Emden–Fowler differential equations, and the papers [iv, 24, 29–35] for the oscillation and asymptotic behavior of quasilinear/Emden–Fowler differential equations with dissimilar neutral coefficients.

Xing et al. [33] presented criteria for oscillation of the equation

$$\brainstorm{aligned} \bigl( z ( x ) \bigl( \delta ^{ ( north-i ) } ( ten ) \bigr) ^{r_{i}} \bigr) ^{\prime }+\tilde{\omega } ( x ) \beta ^{r_{ane}} \bigl( \theta ( x ) \bigr) =0 \end{aligned}$$

under the conditions

$$\begin{aligned} \bigl( \theta ^{-i} ( ten ) \bigr) ^{\prime }\geq \theta _{0}>0,\qquad \tilde{\theta }^{\prime } ( x ) \geq \tilde{\theta }_{0}>0,\qquad \tilde{\theta }^{-i} \bigl( \theta ( 10 ) \bigr) < x \end{aligned}$$

and

$$\begin{aligned} \liminf_{ten\rightarrow \infty } \int _{\tilde{\theta }^{-1} ( \theta ( ten ) ) }^{x} \frac{\widehat{\tilde{\omega }} ( s ) }{z ( s ) } \bigl( due south^{n-one} \bigr) ^{r_{1}}\,\mathrm{d}south> \biggl( \frac{1}{\theta _{0}}+ \frac{\tilde{y}_{0}^{r_{1}}}{\theta _{0}\tilde{\theta }_{0}} \biggr) > \frac{ ( ( n-1 ) ! ) ^{r_{1}}}{\mathrm{eastward}}, \cease{aligned}$$

where \(0\leq \tilde{y} ( x ) <\tilde{y}_{0}<\infty \) and \(\widehat{\tilde{\omega }} ( x ):=\min \{ \tilde{\omega } ( \theta ^{-1} ( x ) ),\tilde{\omega } ( \theta ^{-one} ( \tilde{\theta } ( x ) ) ) \} \).

Bazighifan et al. [xviii], Li and Rogovchenko [25], and Zhang et al. [26, 28] presented oscillation results for fourth-lodge equation

$$\begin{aligned} \bigl( z ( x ) \bigl( \delta ^{\prime \prime number \prime number } ( x ) \bigr) ^{r_{1}} \bigr) ^{\prime number }+\tilde{\omega } ( 10 ) \beta ^{r_{ane}} \bigl( \theta ( x ) \bigr) =0 \end{aligned}$$

under the status

$$\brainstorm{aligned} \int _{x_{0}}^{\infty }\frac{ane}{z^{1/r_{one}} ( southward ) } \, \mathrm{d}s< \infty, \end{aligned}$$

and they used the Riccati technique.

Zhang et al. [36] established oscillation criteria for the equation

$$\begin{aligned} \bigl( z ( ten ) \bigl( \delta ^{ ( n-ane ) } ( x ) \bigr) ^{r_{1}} \bigr) ^{\prime }+\tilde{\omega } ( x ) f \bigl( \beta \bigl( \theta ( x ) \bigr) \bigr) =0 \end{aligned}$$

and nether the condition

$$\begin{aligned} \int _{x_{0}}^{\infty } \biggl( g\rho ( ten ) E ( ten ) - \frac{1}{4\lambda } \biggl( \frac{\rho ^{\prime number } ( x ) }{\rho ( x ) } \biggr) ^{ii}\eta ( ten ) \biggr) \,\mathrm{d}z=\infty. \end{aligned}$$

Past using the Riccati transformation technique, Chatzarakis et al. [19] established asymptotic behavior for the neutral equation

$$\begin{aligned} \bigl( z ( ten ) \bigl( \delta ^{\prime number \prime number \prime } ( ten ) \bigr) ^{r_{1}} \bigr) ^{\prime }+ \int _{a}^{b} \tilde{\omega } ( 10,s ) f \bigl( \beta \bigl( \theta ( x,s ) \bigr) \bigr) \,\mathrm{d}s=0. \end{aligned}$$

In this piece of work, a new oscillation status is created for fourth-lodge differential equations with a canonical operator. We utilize the Riccati technique and the integral averaging technique to testify our results.

Here are the notations used for our written report:

$$\begin{aligned} &E_{1} ( 10 ) =\alpha ( x ) \tilde{\omega } ( x ) ( one- \tilde{y}_{0} ) ^{r_{2}}A_{one}^{r_{two}-r_{one}} \biggl( \frac{\theta ( x ) }{10} \biggr) ^{3r_{2}}, \\ &\Phi ( x ) = ( one-\tilde{y}_{0} ) ^{r_{2}/r_{1}}h ( x ) A_{2}^{r_{2}/r_{one}-one} ( 10 ) \int _{x}^{ \infty } \biggl( \frac{one}{z ( u ) } \int _{u}^{\infty }\tilde{\omega } ( s ) \frac{\theta ^{r_{2}} ( s ) }{due south^{r_{2}}}\,\mathrm{d}s \biggr) ^{i/r_{1}}\,\mathrm{d}u \stop{aligned}$$

and

$$\begin{aligned} \Theta ( x ) =r_{1}\mu _{1} \frac{x^{2}}{2z^{1/r_{1}} ( x ) \alpha ^{ane/r_{1}} ( ten ) }. \stop{aligned}$$

Oscillation criteria

We next present the lemmas needed for the proof of the original results.

Lemma 2.1

([37])

If \(\beta ^{(i)} ( x ) >0\), \(i=0,1,\ldots,n\), and \(\beta ^{ ( north+1 ) } ( x ) <0\), then

$$\brainstorm{aligned} n!\frac{\beta ( 10 ) }{x^{north}}\geq ( due north-1 ) ! \frac{\beta ^{\prime } ( ten ) }{x^{n-i}}. \end{aligned}$$

Lemma 2.ii

([20])

Let \(\beta \in C^{n} ( [ x_{0},\infty ), ( 0, \infty ) ) \). Assume that \(\beta ^{ ( due north ) } ( x ) \) is of a stock-still sign and not identically nil on \([ x_{0},\infty ) \) and that at that place exists \(x_{one}\geq x_{0}\) such that \(\beta ^{ ( northward-one ) } ( x ) \beta ^{ ( n ) } ( x ) \leq 0\) for all \(10\geq x_{1}\). If \(\lim_{x\rightarrow \infty }\beta ( x ) \neq 0\), then for every \(\mu \in ( 0,1 ) \) there exists \(x_{\mu }\geq x_{one}\) such that

$$\begin{aligned} \beta ( 10 ) \geq \frac{\mu }{ ( north-1 ) !}x^{north-1} \bigl\vert \beta ^{ ( n-1 ) } ( 10 ) \bigr\vert \quad\textit{for }x\geq x_{\mu }. \stop{aligned}$$

Lemma ii.iii

([27])

Permit \(a\geq 0\). So

$$\begin{aligned} X\beta -Y\beta ^{ ( a+ane ) /a}\leq a^{a}(a+one)^{- ( a+1 ) }Y^{-a}X^{a+1}, \end{aligned}$$

where \(Y>0\) and Ten are constants.

Lemma 2.4

([38])

$$\begin{aligned} \textit{Assume that }\beta \textit{is an eventually positive solution of (one).} \end{aligned}$$

(5)

Then

$$\begin{aligned} &\textit{Case } ( \mathbf{N}_{i} ): \quad \delta ^{ ( j ) } ( x ) >0 \quad \textit{for } j=0,1,2,iii, \\ &\textit{Case } ( \mathbf{North}_{two} ): \quad \delta ^{ ( j ) } ( 10 ) >0 \quad \textit{for } j=0,1,3\textit{ and }\delta ^{\prime \prime } ( 10 ) < 0, \end{aligned}$$

for \(ten\geq x_{1}\), where \(x_{i}\geq x_{0}\) is sufficiently large.

Lemma 2.5

Let (v) hold. So

$$\begin{aligned} \bigl( z ( x ) \bigl( \delta ^{\prime number \prime \prime } ( ten ) \bigr) ^{r_{1}} \bigr) ^{\prime number }\leq -M ( x ) \bigl( \delta ^{\prime \prime \prime } \bigl( \theta ( x ) \bigr) \bigr) ^{r_{2}}, \end{aligned}$$

(six)

where

$$\begin{aligned} Thousand ( x ) =\tilde{\omega } ( x ) ( ane- \tilde{y}_{0} ) ^{r_{2}} \biggl( \frac{\mu }{6}\theta ^{3} ( x ) \biggr) ^{r_{two}}. \end{aligned}$$

Proof

Allow (5) agree. From the definition of δ, we get

$$\begin{aligned} \beta ( x ) &\geq \delta ( x ) -\tilde{y}_{0}\beta \bigl( \tilde{\theta } ( 10 ) \bigr) \\ &\geq \delta ( x ) -\tilde{y}_{0}\delta \bigl( \tilde{\theta } ( x ) \bigr) \\ &\geq ( 1-\tilde{y}_{0} ) \delta ( x ), \end{aligned}$$

which with (1) gives

$$\begin{aligned} \bigl( z ( ten ) \bigl( \delta ^{\prime \prime \prime } ( x ) \bigr) ^{r_{1}} \bigr) ^{\prime }+\tilde{\omega } ( ten ) ( 1-\tilde{y}_{0} ) ^{r_{2}}\delta ^{r_{2}} \bigl( \theta ( ten ) \bigr) \leq 0. \end{aligned}$$

(7)

Using Lemma 2.ii, nosotros see that

$$\brainstorm{aligned} \delta ( x ) \geq \frac{\mu }{6}x^{3}\delta ^{\prime \prime \prime number } ( x ). \cease{aligned}$$

(8)

Combining (seven) and (8), we detect

$$\begin{aligned} \bigl( z ( x ) \bigl( \delta ^{\prime \prime number \prime number } ( x ) \bigr) ^{r_{ane}} \bigr) ^{\prime }+\tilde{\omega } ( ten ) ( i-\tilde{y}_{0} ) ^{r_{ii}} \biggl( \frac{\mu }{6}\theta ^{3} ( x ) \biggr) ^{r_{2}} \bigl( \delta ^{\prime \prime number \prime } \bigl( \theta ( x ) \bigr) \bigr) ^{r_{2}}\leq 0. \stop{aligned}$$

Thus, (6) holds. This completes the proof. □

Lemma 2.6

Let (5) concur. If δ satisfies \(( \mathbf{Due north}_{1} )\), and so

$$\begin{aligned} B^{\prime } ( 10 ) \leq \frac{\alpha ^{\prime number } ( 10 ) }{\alpha ( x ) }B ( x ) -E_{1} ( 10 ) -r_{one} \mu _{1}\frac{x^{2}}{2z^{1/r_{1}} ( ten ) \alpha ^{1/r_{one}} ( x ) }B^{\frac{r_{1}+i}{r_{1}}} ( x ), \end{aligned}$$

(9)

if δ satisfies \(( \mathbf{Due north}_{2} )\), then

$$\begin{aligned} A^{\prime } ( x ) \leq -\Phi ( x ) + \frac{h^{\prime number } ( x ) }{h ( x ) }A ( 10 ) - \frac{1}{h ( x ) }A^{2} ( ten ), \end{aligned}$$

(10)

where

$$\begin{aligned} B ( x ):=\alpha ( x ) \frac{z ( 10 ) ( \delta ^{\prime number \prime number \prime number } ( ten ) ) ^{r_{1}}}{\delta ^{r_{1}} ( x ) }>0 \end{aligned}$$

(11)

and

$$\begin{aligned} A ( x ):=h ( ten ) \frac{\delta ^{\prime } ( ten ) }{\delta ( x ) }, \quad x\geq x_{1}. \cease{aligned}$$

(12)

Proof

Let (5) and \(( \mathbf{Due north}_{1} ) \) hold. From (eleven) and (7), we detect

$$\begin{aligned} B^{\prime } ( x ) \leq \frac{\alpha ^{\prime } ( x ) }{\alpha ( x ) }B ( x ) -\blastoff ( x ) \tilde{\omega } ( 10 ) ( 1-\tilde{y}_{0} ) ^{r_{2}} \frac{\delta ^{r_{ii}} ( \theta ( x ) ) }{\delta ^{r_{ane}} ( x ) }-r_{ane}\blastoff ( ten ) \frac{z ( x ) ( \delta ^{\prime number \prime \prime } ( x ) ) ^{r_{one}}}{\delta ^{r_{1}+one} ( 10 ) }\delta ^{\prime } ( x ) . \end{aligned}$$

(thirteen)

Using Lemma 2.1, we find

$$\brainstorm{aligned} \delta ( x ) \geq \frac{x}{three}\delta ^{\prime } ( x ), \end{aligned}$$

and hence

$$\begin{aligned} \frac{\delta ( \theta ( x ) ) }{\delta ( 10 ) }\geq \frac{\theta ^{3} ( ten ) }{10^{three}}. \end{aligned}$$

(14)

It follows from Lemma 2.ii that

$$\brainstorm{aligned} \delta ^{\prime } ( x ) \geq \frac{\mu _{1}}{2}x^{2} \delta ^{\prime \prime number \prime number } ( x ) \end{aligned}$$

(15)

for all \(\mu _{one}\in ( 0,1 ) \) and every sufficiently large ten. Thus, by (xiii), (14), and (xv), we get

$$\begin{aligned} B^{\prime number } ( x )\leq {}&\frac{\alpha ^{\prime number } ( x ) }{\alpha ( ten ) }B ( x ) -\alpha ( ten ) \tilde{ \omega } ( x ) ( 1-\tilde{y}_{0} ) ^{r_{two}}\delta ^{r_{ii}-r_{ane}} ( ten ) \biggl( \frac{\theta ( x ) }{x} \biggr) ^{3r_{two}} \\ &{}-r_{1}\mu _{1} \frac{x^{2}}{2z^{1/r_{i}} ( ten ) \blastoff ^{1/r_{ane}} ( x ) }B^{ \frac{r_{1}+1}{r_{1}}} ( x ). \cease{aligned}$$

Since \(\delta ^{\prime } ( ten ) >0\), at that place be \(x_{2}\geq x_{i}\) and \(A_{1}>0\) such that

$$\begin{aligned} \delta ( x ) >A_{1}. \stop{aligned}$$

(16)

Thus, we obtain

$$\begin{aligned} B^{\prime } ( 10 ) \leq{} &\frac{\blastoff ^{\prime } ( x ) }{\alpha ( x ) }B ( ten ) -\alpha ( x ) \tilde{ \omega } ( x ) ( 1-\tilde{y}_{0} ) ^{r_{two}}A^{r_{2}-r_{1}} \biggl( \frac{\theta ( ten ) }{x} \biggr) ^{3r_{2}} \\ &{}-r_{1}\mu _{1} \frac{x^{two}}{2z^{1/r_{1}} ( x ) \alpha ^{one/r_{1}} ( 10 ) }B^{ \frac{r_{one}+1}{r_{1}}} ( 10 ), \finish{aligned}$$

which yields

$$\begin{aligned} B^{\prime } ( x ) \leq \frac{\alpha ^{\prime } ( x ) }{\alpha ( 10 ) }B ( x ) -E_{one} ( 10 ) -r_{ane} \mu _{1}\frac{10^{two}}{2z^{1/r_{1}} ( x ) \alpha ^{one/r_{one}} ( x ) }B^{\frac{r_{1}+1}{r_{1}}} ( x ). \end{aligned}$$

Thus, (9) holds.

Let \(( \mathbf{North}_{2} ) \) hold. Integrating (vii) from x to u, nosotros find

$$\brainstorm{aligned} z ( u ) \bigl( \delta ^{\prime \prime number \prime number } ( u ) \bigr) ^{r_{1}}-z ( x ) \bigl( \delta ^{ \prime \prime number \prime } ( x ) \bigr) ^{r_{ane}}\leq - \int _{10}^{u} \tilde{\omega } ( s ) ( ane- \tilde{y}_{0} ) ^{r_{2}} \delta ^{r_{2}} \bigl( \theta ( s ) \bigr) \,\mathrm{d}southward. \end{aligned}$$

(17)

From Lemma 2.1, nosotros obtain

$$\begin{aligned} \delta ( x ) \geq ten\delta ^{\prime } ( x ), \end{aligned}$$

and hence

$$\begin{aligned} \delta \bigl( \theta ( ten ) \bigr) \geq \frac{\theta ( ten ) }{x}\delta ( x ). \end{aligned}$$

(18)

For (17), letting \(u\rightarrow \infty \) and using (18), we get

$$\begin{aligned} z ( ten ) \bigl( \delta ^{\prime \prime \prime } ( x ) \bigr) ^{r_{ane}}\geq ( 1- \tilde{y}_{0} ) ^{r_{2}} \delta ^{r_{two}} ( 10 ) \int _{ten}^{\infty }\tilde{\omega } ( s ) \frac{\theta ^{r_{ii}} ( s ) }{s^{r_{two}}} \,\mathrm{d}southward. \end{aligned}$$

(xix)

Integrating (xix) from 10 to ∞, we find

$$\begin{aligned} \delta ^{\prime \prime } ( x ) \leq - ( ane-\tilde{y}_{0} ) ^{r_{2}/r_{1}}\delta ^{r_{2}/r_{i}} ( ten ) \int _{x}^{\infty } \biggl( \frac{1}{z ( u ) } \int _{u}^{ \infty }\tilde{\omega } ( s ) \frac{\theta ^{r_{2}} ( due south ) }{s^{r_{ii}}}\,\mathrm{d}s \biggr) ^{i/r_{ane}}\,\mathrm{d}u. \stop{aligned}$$

(20)

From the definition of \(A ( 10 ) \), we encounter that \(A ( x ) >0 \) for \(ten\geq x_{1}\), and using (16) and (twenty), nosotros find

$$\begin{aligned} A^{\prime number } ( ten ) ={}&\frac{h^{\prime } ( x ) }{h ( x ) }A ( x ) +h ( x ) \frac{\delta ^{\prime \prime } ( 10 ) }{\delta ( x ) }-h ( 10 ) \biggl( \frac{\delta ^{\prime number } ( 10 ) }{\delta ( x ) } \biggr) ^{two} \\ \leq{} &\frac{h^{\prime } ( ten ) }{h ( x ) }A ( x ) -\frac{1}{h ( x ) }A^{2} ( x ) \\ &{}- ( 1-\tilde{y}_{0} ) ^{r_{2}/r_{1}}h ( x ) \delta ^{r_{2}/r_{i}-i} ( 10 ) \int _{ten}^{\infty } \biggl( \frac{1}{z ( u ) } \int _{u}^{\infty }\tilde{\omega } ( s ) \frac{\theta ^{r_{ii}} ( s ) }{s^{r_{ii}}}\,\mathrm{d}s \biggr) ^{1/r_{ane}}\,\mathrm{d}u. \end{aligned}$$

Since \(\delta ^{\prime } ( x ) >0\), there exist \(x_{2}\geq x_{one}\) and \(A_{2}>0\) such that

$$\begin{aligned} \delta ( ten ) >A_{2}. \end{aligned}$$

Thus, we obtain

$$\begin{aligned} A^{\prime number } ( x ) \leq -\Phi ( x ) + \frac{h^{\prime } ( x ) }{h ( x ) }A ( ten ) - \frac{ane}{h ( 10 ) }A^{2} ( x ). \end{aligned}$$

Thus, (10) holds. The proof of the theorem is completed. □

Now, we present some Philos-type oscillation criteria for (i).

Theorem 2.7

Permit (25) agree. If \(\alpha,h\in C^{1} ( [ x_{0},\infty ),\mathbb{R} ) \) such that

$$\begin{aligned} \underset{x\rightarrow \infty }{\lim \sup } \frac{1}{G ( 10,x_{1} ) }\int _{x_{1}}^{x}G ( 10,s ) E_{1} ( s ) - \frac{g_{ane}^{r_{1}+i} ( x,south ) G_{1}^{r_{1}} ( 10,south ) }{ ( r_{ane}+1 ) ^{r_{1}+1}} \frac{2^{r_{1}}z ( s ) \alpha ( south ) }{ ( \mu _{1}s^{2} ) ^{r_{1}}}\,\mathrm{d}s= \infty \end{aligned}$$

(21)

for all \(\mu _{2}\in ( 0,1 ) \), and

$$\brainstorm{aligned} \underset{10\rightarrow \infty }{\lim \sup } \frac{1}{G_{2} ( 10,x_{1} ) } \int _{x_{1}}^{10} \biggl( G_{ii} ( x,s ) \Phi ( due south ) - \frac{h ( s ) g_{ii}^{two} ( ten,s ) }{4} \biggr) \,\mathrm{d}s=\infty, \end{aligned}$$

(22)

then (1) is oscillatory.

Proof

Let β exist a nonoscillatory solution of (1), nosotros see that \(\beta >0\). Assume that \(( \mathbf{North}_{ane} ) \) holds. Multiplying (9) by \(G ( x,s )\) and integrating the resulting inequality from \(x_{1}\) to ten; we obtain

$$\begin{aligned} \int _{x_{ane}}^{x}K ( x,s ) E_{ane} ( due south ) \, \mathrm{d}due south \leq {}&B ( x_{one} ) 1000 ( ten,x_{1} ) + \int _{x_{1}}^{x} \biggl( \frac{\partial }{\partial due south}Chiliad ( x,s ) + \frac{\blastoff ^{\prime number } ( s ) }{\alpha ( s ) }One thousand ( 10,s ) \biggr) B ( south ) \,\mathrm{d}s \\ &{}- \int _{x_{1}}^{x}\Theta ( southward ) G ( x,due south ) B^{ \frac{r_{1}+1}{r_{i}}} ( s ) \,\mathrm{d}s. \end{aligned}$$

From (3), nosotros get

$$\begin{aligned} \int _{x_{ane}}^{x}G ( x,s ) E_{ane} ( s ) \, \mathrm{d}southward \leq {}&B ( x_{one} ) G ( x,x_{1} ) + \int _{x_{i}}^{x}g_{1} ( x,s ) G_{1}^{r_{1}/ ( r_{i}+i ) } ( ten,s ) B ( s ) \,\mathrm{d}s \\ &{}- \int _{x_{1}}^{x}\Theta ( southward ) G ( ten,southward ) B^{ \frac{r_{1}+one}{r_{i}}} ( south ) \,\mathrm{d}south. \terminate{aligned}$$

(23)

Using Lemma 2.iii with \(Five=\Theta ( s ) G ( ten,s ), U=g_{ane} ( x,southward ) G_{ane}^{r_{one}/ ( r_{1}+1 ) } ( 10,south ) \), and \(\beta =B ( due south ) \), we get

$$\begin{aligned} &g_{1} ( x,s ) G_{1}^{r_{1}/ ( r_{ane}+one ) } ( x,s ) B ( s ) -\Theta ( s ) 1000 ( x,s ) B^{\frac{r_{1}+1}{r_{ane}}} ( s ) \\ &\quad\leq\frac{g_{ane}^{r_{1}+one} ( x,due south ) G_{1}^{r_{1}} ( 10,s ) }{ ( r_{1}+1 ) ^{r_{1}+ane}} \frac{2^{r_{1}}z ( ten ) \alpha ( ten ) }{ ( \mu _{1}x^{two} ) ^{r_{1}}}, \cease{aligned}$$

which with (23) gives

$$\begin{aligned} \frac{1}{Grand ( x,x_{1} ) } \int _{x_{ane}}^{x} \biggl( G ( ten,s ) E_{1} ( due south ) - \frac{g_{1}^{r_{1}+1} ( x,s ) G_{1}^{r_{1}} ( x,due south ) }{ ( r_{ane}+1 ) ^{r_{one}+one}} \frac{ii^{r_{1}}z ( due south ) \alpha ( s ) }{ ( \mu _{1}s^{ii} ) ^{r_{1}}} \biggr) \,\mathrm{d}s\leq B ( x_{1} ), \end{aligned}$$

which contradicts (21).

Assume that \(( \mathbf{N}_{2} ) \) holds. Multiplying (10) by \(G_{two} ( ten,southward )\) and integrating the resulting inequality from \(x_{1}\) to x, we notice

$$\begin{aligned} \int _{x_{1}}^{x}G_{2} ( ten,south ) \Phi ( s ) \, \mathrm{d}s \leq {}&A ( x_{i} ) G_{2} ( x,x_{i} ) \\ &{}+ \int _{x_{one}}^{x} \biggl( \frac{\fractional }{\fractional southward}G_{2} ( ten,s ) +\frac{h^{\prime } ( south ) }{h ( s ) }G_{2} ( x,s ) \biggr) A ( due south ) \,\mathrm{d}southward \\ &{}- \int _{x_{one}}^{x}\frac{1}{h ( south ) }G_{2} ( ten,s ) A^{ii} ( southward ) \,\mathrm{d}s. \stop{aligned}$$

Thus,

$$\brainstorm{aligned} \int _{x_{1}}^{x}G_{2} ( x,south ) \Phi ( s ) \, \mathrm{d}s \leq {}&A ( x_{1} ) G_{2} ( x,x_{one} ) + \int _{x_{1}}^{10}g_{ii} ( x,s ) \sqrt{G_{2} ( x,s ) }A ( s ) \,\mathrm{d}s \\ &{}- \int _{x_{1}}^{ten}\frac{one}{h ( s ) }G_{2} ( x,southward ) A^{2} ( s ) \,\mathrm{d}south \\ \leq {}&A ( x_{i} ) G_{ii} ( x,x_{1} ) + \int _{x_{1}}^{x}\frac{h ( southward ) g_{2}^{2} ( ten,south ) }{iv}\, \mathrm{d}s, \end{aligned}$$

and so

$$\begin{aligned} \frac{1}{G_{ii} ( x,x_{1} ) } \int _{x_{i}}^{x} \biggl( G_{two} ( ten,s ) \Phi ( s ) - \frac{h ( s ) g_{2}^{2} ( ten,due south ) }{4} \biggr) \,\mathrm{d}due south\leq A ( x_{ane} ), \finish{aligned}$$

which contradicts (22). The proof of the theorem completed. □

Corollary ii.eight

Allow (25) hold. If \(\alpha,h\in C^{ane} ( [ x_{0},\infty ),\mathbb{R} ) \) such that

$$\begin{aligned} \int _{x_{0}}^{\infty } \biggl( E_{1} ( south ) - \frac{2^{r_{i}}}{ ( r_{one}+1 ) ^{r_{1}+ane}} \frac{z ( southward ) ( \blastoff ^{\prime } ( due south ) ) ^{r_{1}+1}}{\mu _{1}^{r_{one}}s^{2r_{ane}}\alpha ^{r_{1}} ( s ) } \biggr) \,\mathrm{d}s=\infty \end{aligned}$$

(24)

and

$$\brainstorm{aligned} \int _{x_{0}}^{\infty } \biggl( \Phi ( s ) - \frac{ ( h^{\prime } ( s ) ) ^{2}}{4h ( s ) } \biggr) \,\mathrm{d}s=\infty \terminate{aligned}$$

(25)

for some \(\mu _{1}\in ( 0,1 ) \) and every \(A_{1},A_{ii}>0\), then (i) is oscillatory.

Example 2.9

Consider the equation

$$\begin{aligned} \biggl( \beta +\frac{1}{two}\beta \biggl( \frac{1}{3}ten \biggr) \biggr) ^{ ( iv ) }+\frac{\tilde{\omega }_{0}}{x^{4}}\beta \biggl( \frac{1}{2}x \biggr) =0, \quad x\geq 1,\tilde{\omega }_{0}>0. \end{aligned}$$

(26)

Permit \(r_{1}=r_{2}=1\), \(z ( ten ) =ane\), \(\tilde{y} ( x ) =1/two \), \(\tilde{\theta } ( x ) =10/three\), \(\theta ( ten ) =ten/two\), and \(\tilde{\omega } ( x ) =\tilde{\omega }_{0}/x^{4}\). Hence, it is easy to see that

$$\begin{aligned} \int _{x_{0}}^{\infty }\frac{ane}{z^{1/r_{1}} ( s ) } \, \mathrm{d}due south=\infty,\qquad E_{1} ( x ) =\frac{\tilde{\omega }_{0}}{16s} \end{aligned}$$

and

$$\begin{aligned} \Phi ( x ):=\frac{\tilde{\omega }_{0}}{24}. \end{aligned}$$

If nosotros put \(\alpha ( due south ):=ten^{3}\) and \(h ( x ):=ten^{2}\), so we observe

$$\brainstorm{aligned} & \int _{x_{0}}^{\infty } \biggl( E_{1} ( s ) - \frac{ii^{r_{1}}}{ ( r_{1}+i ) ^{r_{i}+1}} \frac{z ( due south ) ( \blastoff ^{\prime } ( due south ) ) ^{r_{1}+one}}{\mu _{ane}^{r_{1}}s^{2r_{1}}\alpha ^{r_{one}} ( s ) } \biggr) \,\mathrm{d}s \\ &\quad = \int _{x_{0}}^{\infty } \biggl( \frac{\tilde{\omega }_{0}}{16s}- \frac{9}{2\mu _{one}due south} \biggr) \,\mathrm{d}s \cease{aligned}$$

and

$$\begin{aligned} & \int _{x_{0}}^{\infty } \biggl( \Phi ( due south ) - \frac{ ( h^{\prime } ( s ) ) ^{2}}{4h ( s ) } \biggr) \,\mathrm{d}s \\ &\quad= \int _{x_{0}}^{\infty } \biggl( \frac{\tilde{\omega }_{0}}{24}-i \biggr) \,\mathrm{d}southward. \finish{aligned}$$

Thus,

$$\begin{aligned} \tilde{\omega }_{0}>72 \stop{aligned}$$

(27)

and

$$\brainstorm{aligned} \tilde{\omega }_{0}>24. \end{aligned}$$

(28)

From Corollary ii.eight, equation (26) is oscillatory if \(\tilde{\omega }_{0}>72\).

Example ii.10

Consider the equation

$$\brainstorm{aligned} \bigl( x \bigl( \beta +\tilde{y}_{0}\beta ( \gamma x ) \bigr) ^{\prime \prime \prime number } \bigr) ^{\prime }+ \frac{\tilde{\omega }_{0}}{x^{3}}\beta ( \eta x ) =0, \quad ten\geq 1, \terminate{aligned}$$

(29)

where \(\tilde{y}_{0}\in [ 0,one ), \gamma,\eta \in ( 0,1 ) \), and \(\tilde{\omega }_{0}>0\). Let \(r_{1}=r_{2}=1\), \(z ( 10 ) =ten\), \(\tilde{y} ( ten ) =\tilde{y}_{0}\), \(\tilde{\theta } ( ten ) =\gamma x\), \(\theta ( x ) =\eta x\), and \(\tilde{\omega } ( x ) =\tilde{\omega }_{0}/10^{3}\). Hence, if we set \(\alpha ( south ):=x^{2}\) and \(h ( x ):=x\), then nosotros go

$$\brainstorm{aligned} E_{1} ( 10 ) = \frac{\tilde{\omega }_{0} ( ane-\tilde{y}_{0} ) \eta ^{3}}{ten}, \qquad \Phi ( x ) = \frac{\tilde{\omega }_{0} ( one-\tilde{y}_{0} ) \eta }{4x}. \end{aligned}$$

Thus, (24) and (25) become

$$\begin{aligned} & \int _{x_{0}}^{\infty } \biggl( E_{ane} ( south ) - \frac{2^{r_{1}}}{ ( r_{ane}+ane ) ^{r_{1}+one}} \frac{z ( s ) ( \blastoff ^{\prime } ( s ) ) ^{r_{1}+one}}{\mu _{1}^{r_{1}}s^{2r_{i}}\alpha ^{r_{1}} ( due south ) } \biggr) \,\mathrm{d}s \\ &\quad = \int _{x_{0}}^{\infty } \biggl( \frac{\tilde{\omega }_{0} ( one-\tilde{y}_{0} ) \eta ^{3}}{s}- \frac{2}{\mu _{i}s} \biggr) \,\mathrm{d}southward \stop{aligned}$$

and

$$\begin{aligned} & \int _{x_{0}}^{\infty } \biggl( \Phi ( s ) - \frac{ ( h^{\prime } ( s ) ) ^{2}}{4h ( s ) } \biggr) \,\mathrm{d}s \\ &\quad= \int _{x_{0}}^{\infty } \biggl( \frac{\tilde{\omega }_{0} ( 1-\tilde{y}_{0} ) \eta }{4s}- \frac{1}{4s} \biggr) \,\mathrm{d}s. \terminate{aligned}$$

And so,

$$\begin{aligned} \tilde{\omega }_{0}> \frac{ii}{ ( 1-\tilde{y}_{0} ) \eta ^{3}} \terminate{aligned}$$

(thirty)

and

$$\brainstorm{aligned} \tilde{\omega }_{0}>\frac{1}{ ( i-\tilde{y}_{0} ) \eta }. \end{aligned}$$

From Corollary 2.8, equation (26) is oscillatory if (30) holds.

Conclusion

In this work, we proved some new oscillation theorems for (1). New oscillation results are established that complement related contributions to the subject. We used the Riccati technique and the integral averages technique to go some new results to oscillation of equation (1) nether the condition \(\int _{x_{0}}^{\infty }\frac{i}{z^{1/r_{1}} ( s ) }\,\mathrm{d}s=\infty \). We may say that, in future work, we volition study this type of equation under the condition

$$\begin{aligned} \int _{x_{0}}^{\infty }\frac{ane}{z^{1/r_{one}} ( s ) } \, \mathrm{d}south< \infty. \end{aligned}$$

Too nosotros will try to innovate some important oscillation criteria of differential equations of 4th-society and under

$$\begin{aligned} \delta ( x ) =\beta ( x ) +\tilde{y} ( x ) \sum_{i=i}^{j} \beta _{i} \bigl( \tilde{\theta } ( x ) \bigr). \end{aligned}$$

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Acknowledgements

The authors give thanks the editors and the reviewers for their useful comments.

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Affiliations

1. Department of Mathematics and Statistics, College of Science, Taif Academy, P.O. Box 11099, Taif, 21944, Saudi arabia

   Saeed Althubiti

2. Department of Mathematics, College of Science and Homo Studies, Hotat Sudair, Majmaah University, Majmaah, 11952, Kingdom of saudi arabia

   Fahad Alsharari

3. Mathematics Departments, Higher of Science and Arts in Gurayat, Jouf University, Sakaka, 77455, Saudi Arabia

   Fahad Alsharari

4. Department of Mathematics, Kinesthesia of Science, Hadhramout University, Hadhramout, 50512, Republic of yemen

   Omar Bazighifan

5. Department of Electrical and Electronic Engineering Educators, School of Pedagogical and Technological Education (ASPETE), Marousi, 15122, Athens, Greece

   George E. Chatzarakis

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Althubiti, South., Alsharari, F., Bazighifan, O. et al. Some new oscillation criteria of fourth-order quasi-linear differential equations with neutral term. Adv Differ Equ 2021, 401 (2021). https://doi.org/ten.1186/s13662-021-03555-10

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• Received: 16 June 2021

• Accepted: 17 August 2021

• Published: 30 August 2021

• DOI : https://doi.org/10.1186/s13662-021-03555-10

Keywords

• 4th-order differential equations
• Neutral delay
• Oscillation
• Philos-type oscillation

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