Abstract
This paper deals with the oscillation properties of higher-order nonlinear differential equations with distributed delay
under the condition
We obtain new oscillation criteria by employing a refinement of the generalized Riccati transformations and new comparison principles. We provide some examples to illustrate the main results.
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1 Introduction
In this work, we investigate the oscillation behavior of solutions to the higher-order nonlinear differential equation with distributed delay of the form
We assume that the following assumptions hold:
- \(( A_{1} ) \) :
-
\(b\in C^{1}[\ell_{0},\infty)\), \(b^{\prime } ( \ell ) \geq0, b ( \ell ) >0, \gamma\) is a quotient of odd positive integers;
- \(( A_{2} ) \) :
-
\(q(\ell,\xi), q(\ell,\xi)\in C ( [ \ell_{0},\infty ) \times{}[ c,d],\mathbb{R} ), q(\ell,\xi)\) is positive, \(g(\ell,\xi)\) is a nondecreasing function in ξ, \(g(\ell,\xi)\leq\ell\), and \(\underset{\ell \rightarrow \infty}{\lim}g(\ell,\xi)=\infty\).
By a solution of equation (1.1) we mean a function \(y \in C^{n-1}[\ell _{y},\infty), L_{y}\geq\ell_{0}\), that has the property \(b ( \ell ) ( y^{n-1} ( \ell ) ) ^{\gamma}\in C^{1}[L_{y},\infty) \) and satisfies equation (1.1) on \([L_{y},\infty)\). We consider only solutions y of equation (1.1) that satisfy \(\sup \{ \vert y ( \ell ) \vert :\ell\geq L \} >0 \) for all \(\ell>L_{y}\). We assume that (1.1) possesses such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on \([L_{y},\infty)\), and otherwise it is called nonoscillatory. Equation (1.1) is said to be oscillatory if all its solutions are oscillatory.
The problem of the oscillation of fourth- and higher-order differential equations have been widely studied by many authors, who have provided many techniques for obtaining oscillatory criteria for fourth- and higher-order differential equations. We refer the reader to the related books [1, 4, 9,10,11, 19] and to the papers [2, 3, 5, 8, 12,13,14,15,16,17,18,19,20,21,22,23,24]. In the follows, we present some related results that served as a motivation for the contents of this paper.
Zhang et al. [22] studied the oscillation behavior of the higher-order nonlinear differential equation
Tunc and Bazighifan [21] studied the oscillatory behavior of the fourth-order nonlinear differential equation with a continuously distributed delay
Bazighifan [4] considered the oscillatory properties of the higher-order differential equation
under the conditions
and
Moaaz et al. [19] considered the fourth-order differential equations of the form
Elabbasy et al. [6, 7] and Zhang et al. [24] examined the oscillation of the fourth-order nonlinear delay differential equation
Our aim in the present paper is to employ the Riccatti technique to establish some new conditions for the oscillation of all solutions of equation (1.1) under condition (1.2). We present some examples to illustrate our main results.
The proof of our main results are essentially based on the following lemmas.
Lemma 1.1
(See [1], Lemma 2.2.3)
Let \(z\in ( C^{n} [ \ell_{0},\infty ],\mathbb{R} ^{+} ) \) and assume that \(z^{ ( n ) }\) is of fixed sign and not identically zero on a subray \([ \ell_{0},\infty ]\). If, moreover, \(z ( \ell ) >0, z^{ ( n-1 ) } ( \ell ) z^{ ( n ) } ( \ell ) \leq0\), and \(\underset {\ell \rightarrow\infty}{\lim}z ( \ell ) \neq0\), then, for every \(\lambda\in ( 0,1 ) \), there exists \(\ell_{\lambda}\geq\ell _{0} \) such that
Lemma 1.2
(See [19], Lemma 1.1)
If the function z satisfies \(z^{(i)}>0\), \(i=0,1,\ldots,n\), and \(z^{ ( n+1 ) }<0\), then
Lemma 1.3
(See [21], Lemma 1)
Let \(\beta\geq1 \) be a ratio of two numbers, and let U and V be constants. Then
2 Main results
In this section, we establish some oscillation criteria for equation (1.1). We are now ready to state and prove the main results. For convenience, we denote
and
Theorem 2.1
Let \(( A_{1} )\), \(( A_{2} )\), and (1.2) hold. Assume that there exists a positive function \(\delta\in C^{1}[\ell_{0},\infty) \) such that
If
for some constant \(\lambda_{2}\in ( 0, 1 ) \), then every solution of (1.1) is oscillatory.
Proof
Assume that (1.1) has a nonoscillatory solution y. Without loss of generality, we can assume that \(y ( \ell ) >0\). It follows from (1.1) that there exist tow possible cases for \(\ell\geq\ell _{1}\), where \(\ell_{1}\geq\ell_{0}\) is sufficiently large:
-
Case 1:
\(y^{\prime} ( \ell ) >0, y^{ ( n-1 ) } ( \ell ) >0, y^{ ( n ) } ( \ell ) <0, \bigl( b \bigl( y^{ ( n-1 ) } \bigr) ^{\gamma} \bigr) ^{\prime } ( \ell ) \leq0\);
-
Case 2:
\(y^{\prime} ( \ell ) >0, y^{ ( n-2 ) } ( \ell ) >0, y^{ ( n-1 ) } ( \ell ) <0, \bigl( b \bigl( y^{ ( n-1 ) } \bigr) ^{\gamma} \bigr) ^{\prime } ( \ell ) \leq0\).
Assume that case 1 holds. By Lemma 1.1 we find that \(y ( \ell ) \geq ( \ell/3 ) y^{\prime} ( \ell )\), and hence
Integrating (1.1) from ℓ to u, we obtain
Letting \(u\rightarrow\infty\), we see that
By virtue of \(y^{\prime} ( \ell ) >0\), \(g ( \ell, \xi ) \leq\ell\), and (2.3), we obtain
Integrating again from ℓ to ∞, we obtain
Integrating \(n-3 \) times from ℓ to ∞, we find
Define the function
Then \(\omega ( \ell ) >0 \) for \(\ell\geq\ell_{1}\), and
From (2.5) and (2.6) it follows that
Hence we have
Integrating (2.9) from \(\ell_{1}\) to ℓ, we get
for all large ℓ, which contradicts (2.1).
Assume that case 2 holds. Noting that \(b ( \ell ) ( y^{{ ( n-1 ) }} ( \ell ) ) ^{\gamma}\) is nonincreasing, we have that
for all \(s\geq\ell\geq\ell_{1}\). This yields
Integrating this inequality from ℓ to u, we get
Letting \(u\rightarrow\infty\), we see that
From Lemma 1.2 we get
for all \(\lambda\in ( 0,1 ) \) and every sufficiently large ℓ. Next, we define
We note that \(\varphi ( \ell ) <0\) for \(\ell\geq\ell_{1}\) and
From (1.1) and (2.12) we obtain
Hence (2.13) yields
Multiplying (2.14) by \(R^{\gamma} ( \ell ) \) and integrating from \(\ell_{2}\) to ℓ, we obtain
Set
By Lemma (1.3) we find
for some constant \(\lambda_{2}\in ( 0, 1 )\), which contradicts (2.2).
Theorem (2.1) is proved. □
It is well known (see [3]) that the differential equation
where \(\alpha>0\) is the ratio of odd positive integers and \(a,q\in C[\ell _{0},\infty)\), is nonoscillatory if and only if there exist a number \(\ell \geq\ell_{0}\) and a function υ \(\in C^{1}[\ell,\infty)\) satisfying the inequality
In what follows, we compare the oscillatory behavior of (1.1) with the second-order half-linear equations of type (2.15).
Theorem 2.2
Let \(( A_{1} )\), \(( A_{2} )\), and (1.2) hold. Assume that the differential equations
and
are oscillatory for some constants \(\lambda,\lambda_{1}\in ( 0,1 ) \). Then every solution of (1.1) is oscillatory.
Proof
We proceed as in the proof of Theorem (2.1). Setting \(\delta ( \ell ) =1\) in (2.9), we get
Thus, we see that equation (2.16) is nonoscillatory, a contradiction. From (2.14) we obtain
for every constant \(\lambda_{1}\in ( 0,1 )\). Thus we have that equation (2.17) is nonoscillatory for every constant \(\lambda\in ( 0, 1 )\), a contradiction.
Theorem (2.2) is proved. □
3 Examples
In this section, we give some examples to illustrate our main results.
Example 3.1
Consider the fourth-order differential equation
Note that \(\gamma=3, n=4, b ( \ell ) =\mathbf{e}^{3\ell }\), \(q ( \ell,\xi ) =\ell\mathbf{e}^{\ell(\xi+3)}/ ( \mathbf{e}^{\ell}-1 )\), \(c=0, d=1\), \(g ( \ell,c ) =\ell, R(\ell)=\mathbf{e}^{-\ell}\), and \(Q ( \ell ) =\mathbf{e}^{3\ell}\). If we choose \(\delta ( \ell ) =1\), then it easy to see that conditions (2.1) and (2.2) hold. Hence, by Theorem 2.1, every solution of equation (3.1) is oscillatory.
Example 3.2
Consider the differential equation
where \(\nu>0\) is a constant. Note that \(\gamma=1, n=4, b ( \ell ) =\ell^{3}, q ( \ell, \xi ) =\xi\nu/\ell, c=0, d=1\), \(g ( \ell,c ) =\ell/2,Q ( \ell ) =\nu /2\ell\), and \(R ( \ell ) =1/2s^{2}\). If we set \(\delta ( \ell ) =1\), then we have
for \(\nu>\frac{16}{\lambda_{2}}\) and some constant \(\lambda_{2}\in ( 0, 1 ) \). Hence, by Theorem 2.1, every solution of equation (3.2) is oscillatory if \(\nu>\frac{16}{\lambda_{2}}\).
Remark 3.1
The results of [4] cannot confirm the conclusion in Example 3.2.
Remark 3.2
Our results supplement and improve the results obtained in [25].
4 Conclusion
In this paper, by employing a refinement of the generalized Riccatti transformations and new comparison principles we have established new oscillation criteria for higher-order nonlinear differential equations with distributed delay. Further, we can consider the case of \(g(\ell,\xi )\geq\ell\), and we can try to get some oscillation criteria of equation (1.1) in the future work.
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Bazighifan, O., Elabbasy, E.M. & Moaaz, O. Oscillation of higher-order differential equations with distributed delay. J Inequal Appl 2019, 55 (2019). https://doi.org/10.1186/s13660-019-2003-0
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DOI: https://doi.org/10.1186/s13660-019-2003-0