On certain inequalities and their applications in the oscillation theory
© Baculíková and Džurina; licensee Springer 2013
Received: 1 March 2013
Accepted: 23 May 2013
Published: 11 June 2013
In the paper, we offer a set of inequalities involving delayed argument and offer their application for higher-order differential equations of the form
to be oscillatory. The conditions obtained essentially improve many other known results.
The paper is organized as follows. In the first part we consider only properties of functions and their derivatives, and later we connect the estimate obtained with properties of solutions of differential equations. We shall investigate the properties of a couple of functions and , .
Setting , the last inequalities imply (1.1) and the proof is complete. □
The proof is complete. □
The obtained estimates can be used, e.g., in the theory of functional equations. In the paper, we present their application in discussing oscillatory and asymptotic properties of higher-order delay differential equations.
2 Main results
(H1) , .
The investigation of oscillatory properties of the second- and higher-order linear differential equations started with the Sturm comparison theorem . Later Mahfoud  essentially contributed to the subject and presented a very useful comparison technique for studying the properties of a delay differential equation from those of a differential equation without delay. A new impetus to investigation in this direction was given by papers of Chanturia and Kiguradze , Kusano and Naito  and Koplatadze et al. [6, 7]. See also [1–20]. In the paper, we employ Lemma 2 to establish new criteria for oscillation of (E).
On the other hand, it follows from Theorem 2 of  that the corresponding equation () has also a solution of degree ℓ. The proof is complete. □
So, if we eliminate solutions of degree ℓ of equations (), we get property (A) of studied equation (E). To do it, we recall the following comparison result which is due to Chanturia .
where for n odd and for n even, then (2.3) has no solution of degree ℓ.
Employing Theorem 2 to (2.3) and (), in view of Theorem 1, one gets the following theorem.
Then (E) has property (A).
Since , Euler equation (2.3) has no solution of degree ℓ. On the other hand, taking (2.5) into account, Theorem 2 ensures that () has no solution of degree ℓ. Finally, Theorem 1 guarantees that (E) has property (A). The proof is complete. □
For , , the previous result simplifies to the following.
has property (A).
which contradicts (2.1) and we conclude that . □
We support our results with the following illustrative example.
where is nondecreasing.
We provide details while comparing those criteria with our one.
This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0008-10.
- Kiguradze IT:On the oscillation of solutions of the equation . Mat. Sb. 1964, 65: 172-187. (Russian)MathSciNetGoogle Scholar
- Sturm JCF: Mémoire sur les équations différentielles linéaires du second ordre. J. Math. Pures Appl. 1836, 1: 106-186.Google Scholar
- Mahfoud WE: Oscillation and asymptotic behavior of solutions of n th order nonlinear delay differential equations. J. Differ. Equ. 1977, 24: 75-98. 10.1016/0022-0396(77)90171-1MathSciNetView ArticleGoogle Scholar
- Kiguradze IT, Chaturia TA: Asymptotic Properties of Solutions of Nonautonomous Ordinary Differential Equations. Kluwer Academic, Dordrecht; 1993.View ArticleGoogle Scholar
- Kusano T, Naito M: Comparison theorems for functional differential equations with deviating arguments. J. Math. Soc. Jpn. 1981, 3: 509-533. Zbl 0494.34049MathSciNetView ArticleGoogle Scholar
- Koplatadze RG: On differential equations with a delayed argument having properties A and B. Differ. Uravn. (Minsk) 1989, 25: 1897-1909.MathSciNetGoogle Scholar
- Koplatadze RG, Kvinkadze G, Stavroulakis I: Properties A and B of n th order linear differential equations with deviating argument. Georgian Math. J. 1999, 6: 553-566. 10.1023/A:1022962129926MathSciNetView ArticleGoogle Scholar
- Agarwal RP, Grace SR, O’Regan D: Oscillation Theory for Difference and Functional Differential Equations. Kluwer Academic, Dordrecht; 2000.View ArticleGoogle Scholar
- Baculíková B, Džurina J: Oscillation of third-order neutral differential equations. Math. Comput. Model. 2010, 52: 215-226. 10.1016/j.mcm.2010.02.011View ArticleGoogle Scholar
- Baculíková B, Graef J, Džurina J: On the oscillation of higher order delay differential equations. Nonlinear Oscil. 2012, 15: 13-24.Google Scholar
- Baculíková B, Džurina J: Oscillation of third-order nonlinear differential equations. Appl. Math. Lett. 2011, 24: 466-470. 10.1016/j.aml.2010.10.043MathSciNetView ArticleGoogle Scholar
- Baculíková B: Properties of third order nonlinear functional differential equations with mixed arguments. Abstr. Appl. Anal. 2011., 2011: Article ID 857860Google Scholar
- Baculíková B, Džurina J, Rogovchenko Y: Oscillation of third order trinomial differential equations. Appl. Math. Comput. 2012, 218: 7023-7033. 10.1016/j.amc.2011.12.049MathSciNetView ArticleGoogle Scholar
- Baculíková B, Džurina J: Property (A) and oscillation of third order differential equations with mixed arguments. Funkc. Ekvacioj 2012, 55: 239-253.View ArticleGoogle Scholar
- Džurina J: Comparison theorems for nonlinear ODE’s. Math. Slovaca 1992, 42: 299-315.MathSciNetGoogle Scholar
- Erbe L, Kong Q, Zhang BG: Oscillation Theory for Functional Differential Equations. Dekker, New York; 1995.Google Scholar
- Ladde GS, Lakshmikantham V, Zhang BG: Oscillation Theory of Differential Equations with Deviating Arguments. Dekker, New York; 1987.Google Scholar
- Zafer A: Oscillation criteria for even order neutral differential equations. Appl. Math. Lett. 1998, 11: 21-25.MathSciNetView ArticleGoogle Scholar
- Meng FW, Xu R: Oscillation criteria for certain even order quasi-linear neutral differential equations with deviating arguments. Appl. Math. Comput. 2007, 190: 458-464. 10.1016/j.amc.2007.01.040MathSciNetView ArticleGoogle Scholar
- Li T, Han Z, Zhao P, Sun S: Oscillation of even-order neutral delay differential equations. Adv. Differ. Equ. 2010., 2010: Article ID 184180Google Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.