# Asymptotic behavior of third-order functional differential equations with a negative middle term

## 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

*γ*is a quotient of odd positive integers. Throughout this paper, we assume that

- (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 \(xy>0\).

*y*of (1.1) which exist on \(\mathcal{I}\) and satisfy the condition

*oscillatory*if it has arbitrarily large zeros on \([T_{y},\infty)\) and otherwise it is called

*nonoscillatory*. Equation (1.1) is said to be

*oscillatory*if all its solutions are

*oscillatory*.

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, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 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

*canonical form*(see Trench [24]).

### 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

*a principal solution*at infinity, and such a solution is determined uniquely up to a constant factor. In order to reveal the structure of possible nonoscillatory solutions of (1.1), the following property of a principal solution of (2.1) plays a crucial role.

### Lemma 2

*If*

*then*(2.1)

*has a positive solution*\(v(t)\)

*satisfying*

### Proof

*t*to ∞, we see that

*t*, we get

In the lemma below we recall the adaptation of the generalized Kiguradze lemma [17] to the canonical operator \(L_{3}y(t)\).

### Lemma 3

*Let*\(y(t)\)

*be a real*-

*valued function on*\(\mathcal{I}\)

*which has the property*\(L_{n}y(t)\in\mathcal{C}^{1}(\mathcal{I})\), \(n = 0,1,2\).

*If*

*then there exists*\(t_{1}\in\mathcal{I}\)

*and*\(\ell= \{1,3\}\)

*such that*

### 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

*Let*\(y(t)\in\mathcal{N}_{1}\)

*be a positive solution of*(1.1)

*on*\([t_{1},\infty)\).

*Then*

*for*\(t\ge t_{1}\).

### Proof

*t*to ∞ yields

In the lemma below we shall point out that estimate (2.10) can be improved further.

### Lemma 6

*Let*\(y(t)\in\mathcal{N}_{1}\)

*be a solution of*(1.1)

*on*\([t_{1},\infty )\), \(t_{1}\in\mathcal{I}\).

*Then*

### Proof

*t*to ∞, we see that

### Lemma 7

*Let*\(y(t)\in\mathcal{N}_{3}\)

*be a positive solution of*(1.1)

*on*\([t_{1},\infty)\).

*If*

*then there exists*\(t_{2}>t_{1}\)

*such that*

### Proof

### Remark 3

*q*,

*g*and

*f*satisfy conditions (i), (iii) and (iv), respectively. The following lemma can be found in [1] or, separately for delayed and advanced cases, in [22] and [3], respectively.

## 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

*t*to ∞, we find

### Theorem 2

*Let*(2.5)

*hold and*\(g(t)> t\)

*for*\(t\ge t_{1}\).

*If the first*-

*order advanced differential equation*

*is oscillatory*,

*then*(1.1)

*has Property B*.

### 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

*Let*(2.5)

*hold and*\(g(t)\le t\)

*for*\(t\ge t_{1}\).

*Assume that*

*and the function*

*f*

*satisfies*

*If*

*then*(1.1)

*has Property B*.

### Proof

*t*, we easily find that

*f*and dividing both sides of the latter inequality by \(f^{1/\gamma}y(t)\), one can see that

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

*Let*(2.5), (3.7)

*and*(3.6)

*hold*,

*and*\(g(t)\ge t\)

*for*\(t\ge t_{1}\).

*If*

*then*(1.1)

*has Property B*.

### 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

*Assume that all assumptions of Theorem*2

*are satisfied and*(2.13)

*holds*.

*If*

*then*(1.1)

*is oscillatory*.

### Proof

*t*to

*v*, we obtain

*v*, one gets

*v*once more, we get

*v*last time, we find

## 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).

## Notes

### 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.

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