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Asymptotic behavior of third-order functional differential equations with a negative middle term
- Jozef Džurina^{1} and
- Irena Jadlovská^{1}Email author
https://doi.org/10.1186/s13662-017-1127-0
© The Author(s) 2017
- Received: 21 November 2016
- Accepted: 27 February 2017
- Published: 2 March 2017
Abstract
This paper is concerned with asymptotic and oscillatory properties of the nonlinear third-order differential equation with a negative middle term. Both delay and advanced cases of argument deviation are considered. Sufficient conditions for all solutions of a given differential equation to have property B or to be oscillatory are established. A couple of illustrative examples is also included.
Keywords
- third-order differential equation
- delay
- advance
- oscillation
- comparison
- integral criteria
MSC
- 34K11
1 Introduction
- (i)
\(r_{1}\), \(r_{2}\), \(q \in C(\mathcal{I},(0,\infty))\), where \(\mathcal{I} = [t_{0},\infty)\);
- (ii)
\(p \in C(\mathcal{I},[0,\infty))\);
- (iii)
\(g \in C^{1}(\mathcal{I},\mathbb {R})\), \(g'(t)\ge0\), \(\lim_{t\to \infty}g(t) = \infty\);
- (iv)
\(f\in C^{1}(\mathbb {R},\mathbb {R})\), \(xf(x)>0\), \(f'(x)\ge0\) for \(x\neq0\), \(f(xy)\ge f(x)f(y)\) for \(xy>0\).
Analysis of the asymptotic and oscillatory behavior of solutions to different classes of differential and functional differential equations has experienced long-term interest of many researchers, see, for example, [1–23] and the references cited therein. A huge amount of significant oscillation results has been collected in several excellent monographs, see, e.g., [1, 2, 16, 21]. This interest is caused by the fact that differential equations, especially those with deviating argument, are deemed to be adequate in modeling of countless processes in all areas of science. In particular, it is worthwhile to mention the use of third-order differential equations in the study of an entry-flow phenomenon in a problem of hydrodynamics, or of the propagation of electrical pulses in the nerve of a squid approximated by the famous Nagumo’s equation [21].
Another approach for studying the asymptotic properties of (1.2) has been employed in papers [6, 11] when \(p(t)\) is negative and \(q(t)\) is positive. The authors presented several comparison theorems in which the desired properties of solutions are deduced from those of corresponding first-order functional or second-order ordinary differential equations. Their results, however, strongly rely on the knowledge of the auxiliary solution \(z(t)\).
2 Some basic definitions and auxiliary lemmas
Remark 1
All the functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all t large enough.
Remark 2
In the sequel and without loss of generality, we can restrict our attention only to positive solutions of (1.1).
Lemma 1
Proof
Lemma 2
Proof
In the lemma below we recall the adaptation of the generalized Kiguradze lemma [17] to the canonical operator \(L_{3}y(t)\).
Lemma 3
Lemma 4
Assume that (2.5) holds. If \(y(t)\) is a positive solution of (1.1) on \(\mathcal{I}\), then there exists \(t_{1}\in\mathcal{I}\) such that either \(y(t)\in\mathcal{N}_{1}\) or \(y(t)\in\mathcal{N}_{3}\) on \([t_{1},\infty)\).
Proof
According to the well-known results of Kiguradze and Chanturia [16], the oscillation criteria are often accomplished by introducing the concepts of having property A and/or B. Such properties have been widely studied by many authors, see, e.g., [4, 10, 16, 19] and the references cited therein.
Definition 1
Equation (1.1) is said to have Property B if \(\mathcal{N} = \mathcal{N}_{3}\).
In what follows, we state and prove some useful estimates which will play an important role in the proofs of our main results.
Lemma 5
Proof
In the lemma below we shall point out that estimate (2.10) can be improved further.
Lemma 6
Proof
Lemma 7
Proof
Remark 3
3 Main results
3.1 Criteria for Property B
Now we are prepared to give sufficient conditions under which (1.1) enjoys Property B. We distinguish between delayed and advanced types of the argument deviation.
Theorem 1
Proof
Theorem 2
Proof
Employing some known criteria for oscillation of first-order functional differential equations (3.1) and (3.4), one can easily obtain oscillation criteria for (1.1). The following ones are due to Ladde et al. [20].
Corollary 1
Corollary 2
Now, we present other results for (1.1) to have Property B which are applicable even in the ordinary case \(g(t) = t\).
Theorem 3
Proof
It follows from (3.6) that \(\lim_{t\to\infty}y(t) = \infty\). Taking the lim sup on both sides of (3.10), we are led to the contradiction with (3.8). Therefore \(y(t)\in\mathcal{N}_{3}\), which means that (1.1) has Property B. The proof is complete. □
Theorem 4
Proof
The proof is similar to that of Theorem 3 and so is omitted. □
Remark 4
3.2 Oscillation of (1.1)
If \(g(t)>t\), we are also able to eliminate the remaining class of nonoscillatory solutions and ensure (1.1) to be oscillatory.
Theorem 5
Proof
4 Examples
Example 1
Example 2
5 Summary
Very recently, authors suggested in [8, 12] the investigation of asymptotic and oscillatory properties for (1.1). Thus, in a certain sense, the presented results may be viewed as a complement of earlier obtained ones. We stress that, contrary to [5, 6, 11], these criteria do not depend on solutions of the auxiliary equation (2.1).
Declarations
Acknowledgements
We are grateful to the editors and three anonymous referees for a very careful reading of the manuscript and for pointing out several inaccuracies. The work on this research has been supported by the internal grant project No. FEI-2015-22.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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