Abstract
In this paper, we study the oscillation of solutions for an even-order differential equation with middle term, driven by a p-Laplace differential operator of the form
The oscillation criteria for these equations have been obtained. Furthermore, an example is given to illustrate the criteria.
Similar content being viewed by others
1 Introduction
It is worth mentioning in this context that delay differential equations have many real-life applications in all branches of science and engineering; see [1, 2]. On the other hand, the p-Laplace equations have crucial applications in different areas such as in elasticity theory, see, for example, Aronsson–Janfalk [3], and in general nonlinear phenomena, see Vetro [4]. Therefore, the literature reveals results of various studies concerning the oscillatory behavior of equations driven by a p-Laplace differential operator; see, by way of example not exhaustive enumeration, Li–Baculikova–Dzurina–Zhang [5], Liu–Zhang–Yu [6], Zhang–Agarwal–Li [7]. Additionally, the oscillatory properties of differential equations are studied intensively by many scientists; see, for example, [8–22].
The aim of this work is to investigate the oscillatory behavior of the even-order delay differential equation (DDE) with damping of the form
under the following conditions:
-
(G1)
\(\Phi _{p}[s]=|s|^{p-2}s\);
-
(G2)
\(r, a, q_{i}\in C ( [x_{0},\infty ),[0,\infty ) ) \), \(q_{i} ( x ) >0\), \(i=1,2,\dots ,j\) are such that
$$ \int _{x_{0}}^{\infty } \biggl[ \frac{1}{r ( s ) }\exp \biggl( - \int _{x_{0}}^{s} \frac{a ( u ) }{r ( u ) }\,du \biggr) \biggr] ^{1/ ( p-1 ) }\,ds< \infty ; $$(2) -
(G3)
\(\delta _{i}\in C ( [ x_{0},\infty ) , ( 0, \infty ) ) \), \(\delta _{i} ( x ) \leq x\) and \(\lim_{x\rightarrow \infty }\delta _{i} ( x ) =\infty \), \(i=1,2, \dots ,j\);
-
(G4)
\(f, h\in C ( \mathbb{R} ,\mathbb{R} ) \), \(f ( x ) \geq m \vert x \vert ^{p-2}x>0\), \(h ( x ) \geq \ell \vert x \vert ^{p-2}x>0\) for \(x\neq 0\), \(m\geq 1\) and \(\ell >0\),
where the first term of equation (1) means the p-Laplace-type operator with \(1< p<\infty \).
To achieve our target, we implemented several relevant facts and auxiliary results from the existing literature [7, 23–26]. Notice that Liu–Zhang–Yu [6] provided some theoretical information on the oscillation of half-linear functional differential equations with damping, i.e.,
where n is even. The authors used the comparison method with second order equations. In Bazighifan–Poom [23] and Bazighifan–Abdeljawad [24], the comparison method with the first and second order equations is used to establish oscillation criteria for
where n is even and p is a real number greater than 1, in the case where \(\delta _{i} ( x ) \geq \upsilon \), \(\alpha \leq \beta \) (with \(r\in C^{1} ( (0,\infty ),\mathbb{R} ) \), \(q_{i}\in C ( [0,\infty ), \mathbb{R} ) \), \(i=1,2,\dots ,j\)).
For the special case when \(p=1\), Elabbasy et al. [16] provided some information on the asymptotic behavior of (1). The authors used the comparison method with second order equations to achieve their targets. We must point out that Li et al. [5] had used the Riccati transformation, together with the integral averaging technique, to discuss the oscillation of the following equation:
In Park et al. [26], the Riccati technique is used to obtain oscillation criteria of
where n is even. Zhang et al. in [7] studied the equation
where
As a matter of fact, the investigation of the half-linear/p-Laplace equation (1) has become an important area of research due to the fact that such equations arise in a variety of real-world problems such as in the study of non-Newtonian fluid theory, the turbulent flow of a polytrophic gas in a porous medium, etc.; see the following papers for more details [27–33]. In this work, we will partially use the tools and findings of [7, 23–26] to obtain new oscillation conditions for (1). Theoretical results will be illustrated via an example.
2 Oscillation criteria
For further convenience, we denote:
Next, we recall some technical tools useful throughout the paper:
Lemma 2.1
([34])
Let \(z\in C^{\kappa } ( [ x_{0},\infty ) , ( 0, \infty ) ) \). If \(\lim_{x\rightarrow \infty }z ( x ) \neq 0\) and
then
Lemma 2.2
([35])
Let \(C>0\) and D be constants. Then
Lemma 2.3
([34])
Let \(z\in C^{\kappa } ( [ x_{0},\infty ) , ( 0, \infty ) ) \). If \(z^{ ( \kappa -1 ) } ( x ) z^{ ( \kappa ) } ( x ) \leq 0\), then for every \(\theta \in ( 0,1 ) \) and \(\kappa >0\) one has
Lemma 2.4
([36])
Let \(z\in C^{n-1}([x_{z},\infty ),\mathbb{R})\) be an (eventually) positive solution of (1). Then, we distinguish the following situations:
Lemma 2.5
Let \(( I_{1} ) \) hold and \(z ( x ) >0\). If
where \(\delta _{i}\in C^{1} ( [ x_{0},\infty ) ) \), then there exists a constant \(\kappa >0\) such that
Proof
Let \(( I_{1} ) \) hold and \(z ( x ) >0\). Using Lemma 2.3, we obtain
From (3), we get
From (1), we get
Hence, we find
The proof is complete. □
Lemma 2.6
Let \(( I_{2} ) \) hold and \(z ( x ) >0\). If
then there exists a constant \(\mu \in ( 0,1 ) \) such that
Proof
Assume that \(( I_{2} ) \) holds and \(z ( x ) >0\). Since
we deduce that \(-r ( x ) ( -z^{ ( \kappa -1 ) } ( x ) ) ^{p-1}\sigma ( x_{0},x )\) is decreasing. Thus, for \(s\geq x\geq x_{1}\), one has
Dividing both sides of (10) by \(( r ( s ) \sigma ( x_{0},s ) ) ^{1/ ( p-1 ) }\) and integrating the resulting inequality from x to u, we get
Letting \(u\rightarrow \infty \), we arrive at
which yields
Hence,
From (8), we have
and
Using Lemma 2.1, we find
Thus, from (11) and (13), we get
The proof is complete. □
Theorem 2.1
Let functions \(\delta _{i},\zeta \in C^{1} ( [ x_{0},\infty ) , ( 0,\infty ) ) \) and \(\kappa >0\), \(\mu \in ( 0,1 )\) be such that
and
Then all solutions of (1) are oscillatory.
Proof
Let z be a nonoscillatory solution of equation (1) and \(z ( x ) >0\). Applying Lemma 2.2 to (4) and setting
we have
Integrating from \(x_{1}\) to x, we find
which contradicts (14).
Now, multiplying (9) by \(\zeta ^{p-1} ( x ) \sigma ( x_{0},x )\) and integrating the resulting inequality from \(x_{1}\) to x, we get
In view of Lemma 2.2, we put
Thus, we get
Hence, by (11), we obtain
which contradicts (15). The proof is complete. □
Remark 2.1
For interested researchers, there is a good problem of finding new results in the following cases:
- \(( \mathbf{S}_{1} ) \):
-
\(z ( x ) >0\), \(z^{\prime } ( x ) >0\), \(z^{ ( \kappa -2 ) } ( x ) >0\), \(z^{ ( \kappa -1 ) } ( x ) \leq 0\), \(( r ( x ) ( z^{ ( m-1 ) } ( x ) ) ^{p-1} ) ^{\prime }\leq 0 \),
- \(( \mathbf{S}_{2} ) \):
-
\(z ( x ) >0\), \(z^{(r)}(x)<0\), \(z^{(r+1)}(x)>0\) for all odd integer \(r\in \{1,3,\dots ,\kappa -3\}\), \(z^{ ( \kappa -1 ) } ( x ) <0\), \(( r ( x ) ( w^{ ( \kappa -1 ) } ( x ) ) ^{p-1} ) ^{\prime } \leq 0\).
Example 2.1
For \(x\geq 1\), consider the equation
where \(q_{0}>0\) is a constant. Let \(p=2\), \(\kappa =2\), \(x_{0}=1\), \(r ( x ) =x^{2}\), \(a ( x ) =x/2\), \(q ( x ) =q_{0}\), \(\delta _{i} ( x ) =x/2\). We now set \(\delta _{i} ( x ) =m=\ell =1\), then
thus, we get
and, for some \(\mu \in ( 0,1 ) \),
Thus, by Theorem 2.1, every solution of (17) is oscillatory if \(q_{0}>\frac{7}{6\mu }\).
3 Conclusion
In this article, we studied the oscillatory properties of even-order differential equations. New oscillation criteria were established. We used Riccati technique to prove that every solution of (1) is oscillatory. Further, we shall study equation (1) under the condition \(\delta _{i} ( t ) \geq t\) in the future work.
Availability of data and materials
Not applicable.
References
Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)
Rihan, F.A., Al-Mdallal, Q.M., AlSakaji, H.J., Hashish, A.: A fractional-order epidemic model with time-delay and nonlinear incidence rate. Chaos Solitons Fractals 126, 97–105 (2019)
Aronsson, G., Janfalk, U.: On Hele–Shaw flow of power-law fluids. Eur. J. Appl. Math. 3, 343–366 (1992)
Vetro, C.: Pairs of nontrivial smooth solutions for nonlinear Neumann problems. Appl. Math. Lett. 103, 106171 (2020)
Li, T., Baculikova, B., Dzurina, J., Zhang, C.: Oscillation of fourth order neutral differential equations with p-Laplacian like operators. Bound. Value Probl. 2014, 56 (2014)
Liu, S., Zhang, Q., Yu, Y.: Oscillation of even-order half-linear functional differential equations with damping. Comput. Math. Appl. 61, 2191–2196 (2011)
Zhang, C., Agarwal, R., Li, T.: Oscillation and asymptotic behavior of higher-order delay differential equations with p-Laplacian like operators. J. Math. Anal. Appl. 409, 1093–1106 (2014)
Agarwal, R., Shieh, S.L., Yeh, C.C.: Oscillation criteria for second order retarde ddifferential equations. Math. Comput. Model. 26, 1–11 (1997)
Baculikova, B., Dzurina, J., Graef, J.R.: On the oscillation of higher-order delay differential equations. Math. Slovaca 187, 387–400 (2012)
Bazighifan, O., Ramos, H.: On the asymptotic and oscillatory behavior of the solutions of a class of higher-order differential equations with middle term. Appl. Math. Lett. 107, 106431 (2020)
Bazighifan, O., Elabbasy, E.M., Moaaz, O.: Oscillation of higher-order differential equations with distributed delay. J. Inequal. Appl. 2019, 55 (2019)
Chatzarakis, G.E., Elabbasy, E.M., Bazighifan, O.: An oscillation criterion in 4th-order neutral differential equations with a continuously distributed delay. Adv. Differ. Equ. 2019, 336 (2019)
Bazighifan, O.: Kamenev and Philos-types oscillation criteria for fourth-order neutral differential equations. Adv. Differ. Equ. 2020, 201 (2020)
Cesarano, C., Bazighifan, O.: Oscillation of fourth-order functional differential equations with distributed delay. Axioms 8, 61 (2019)
Cesarano, C., Bazighifan, O.: Qualitative behavior of solutions of second order differential equations. Symmetry 11, 777 (2019)
Elabbasy, E.M., Thandpani, E., Moaaz, O., Bazighifan, O.: Oscillation of solutions to fourth-order delay differential equations with middle term. Open J. Math. Sci. 3, 191–197 (2019)
Elabbasy, E.M., Cesarano, C., Bazighifan, O., Moaaz, O.: Asymptotic and oscillatory behavior of solutions of a class of higher-order differential equations. Symmetry 11, 1434 (2019)
Grace, S., Agarwal, R., Graef, J.: Oscillation theorems for fourth order functional differential equations. J. Appl. Math. Comput. 30, 75–88 (2009)
Gyori, I., Ladas, G.: Oscillation Theory of Delay Differential Equations with Applications. Clarendon, Oxford (1991)
Moaaz, O., Kumam, P., Bazighifan, O.: On the oscillatory behavior of a class of fourth-order nonlinear differential equation. Symmetry 12, 524 (2020)
Moaaz, O., Furuichi, S., Muhib, A.: New comparison theorems for the Nth order neutral differential equations with delay inequalities. Mathematics 8, 454 (2020)
Philos, C.: On the existence of nonoscillatory solutions tending to zero at ∞ for differential equations with positive delay. Arch. Math. (Basel) 36, 168–178 (1981)
Bazighifan, O., Kumam, O.: Oscillation theorems for advanced differential equations with p-Laplacian like operators. Mathematics 8, 821 (2020)
Bazighifan, O., Abdeljawad, T.: Improved approach for studying oscillatory properties of fourth-order advanced differential equations with p-Laplacian like operator. Mathematics 8, 656 (2020)
Bazighifan, O.: On the oscillation of certain fourth-order differential equations with p-Laplacian like operator. Appl. Math. Comput. 386, 125475 (2020)
Park, C., Moaaz, O., Bazighifan, O.: Oscillation results for higher order differential equations. Axioms 9, 14 (2020)
Bohner, M., Hassan, T.S., Li, T.: Fite–Hille–Wintner-type oscillation criteria for second-order half-linear dynamic equations with deviating arguments. Indag. Math. 29, 548–560 (2018)
Bohner, M., Li, T.: Oscillation of second-order p-Laplace dynamic equations with a nonpositive neutral coefficient. Appl. Math. Lett. 37, 72–76 (2014)
Li, T., Pintus, N., Viglialoro, G.: Properties of solutions to porous medium problems with different sources and boundary conditions. Z. Angew. Math. Phys. 70, 1–18 (2019)
Liu, Q., Bohner, M., Grace, S.R., Li, T.: Asymptotic behavior of even-order damped differential equations with p-Laplacian like operators and deviating arguments. J. Inequal. Appl. 2016, 321 (2016)
Chatzarakis, G.E., Grace, S.R., Jadlovská, I., Li, T., Tunç, E.: Oscillation criteria for third-order Emden–Fowler differential equations with unbounded neutral coefficients. Complexity 2019, Article ID 5691758 (2019)
Džurina, J., Grace, S.R., Jadlovská, I., Li, T.: Oscillation criteria for second-order Emden–Fowler delay differential equations with a sublinear neutral term. Math. Nachr. 293, 910–922 (2020)
Li, T., Rogovchenko, Y.V.: On the asymptotic behavior of solutions to a class of third-order nonlinear neutral differential equations. Appl. Math. Lett. 105, 106293 (2020)
Agarwal, R., Grace, S., O’Regan, D.: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht (2000)
Agarwal, R.P., Zhang, C., Li, T.: Some remarks on oscillation of second order neutral differential equations. Appl. Math. Comput. 274, 178–181 (2016)
Zhang, C., Li, T., Suna, B., Thandapani, E.: On the oscillation of higher-order half-linear delay differential equations. Appl. Math. Lett. 24, 1618–1621 (2011)
Acknowledgements
Thabet Abdeljawad would like to thank the anonymous reviewers for their helpful remarks. The third author would like to thank Prince Sultan University for funding this work through research group Nonlinear Analysis Methods in Applied Mathematics (NAMAM) group number RG-DES-2017-01-17.
Funding
The authors received no direct funding for this work.
Author information
Authors and Affiliations
Contributions
The authors contributed equally to this work. They all read and approved the final version of the manuscript.
Corresponding author
Ethics declarations
Competing interests
The authors declare that they have no competing interests.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bazighifan, O., Abdeljawad, T. & Al-Mdallal, Q.M. Differential equations of even-order with p-Laplacian like operators: qualitative properties of the solutions. Adv Differ Equ 2021, 96 (2021). https://doi.org/10.1186/s13662-021-03254-7
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13662-021-03254-7