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Asymptotic behavior of thirdorder functional differential equations with a negative middle term
Advances in Difference Equations volume 2017, Article number: 71 (2017)
Abstract
This paper is concerned with asymptotic and oscillatory properties of the nonlinear thirdorder 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.
Introduction
The purpose of this work is to investigate the asymptotic behavior of the thirdorder functional differential equation with the middleterm
where \(t_{0}\) is fixed and 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 \(xy>0\).
By a solution of equation (1.1) we mean a function \(y\in C^{1}([T_{y},\infty))\), \(T_{y}\in\mathcal{I}\), which has the property \(r_{1}y', r_{2} (r_{1} (y' )^{\gamma})'\in C^{1}([T_{y},\infty),\mathbb {R})\) and satisfies (1.1) on \([T_{y},\infty)\). Our attention is restricted to those solutions y of (1.1) which exist on \(\mathcal{I}\) and satisfy the condition
We make the standing hypothesis that (1.1) admits such a solution. A solution of (1.1) is called 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 longterm 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 thirdorder differential equations in the study of an entryflow 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].
In the recent works [8, 9], the authors used a generalized Riccati transformation and an integral averaging technique in order to establish some sufficient conditions for oscillation of all solutions of a trinomial thirdorder differential equation
where \(p(t)\) and \(q(t)\) are positive functions and the auxiliary equation
is nonoscillatory. They have shown (see [8], Lemma 2.2) that any nonoscillatory solution of (1.2) satisfies
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 firstorder functional or secondorder ordinary differential equations. Their results, however, strongly rely on the knowledge of the auxiliary solution \(z(t)\).
In this work, we would like to study equation (1.1) under assumptions (i)–(v). The organization of the paper is as follows. Using different arguments as those in [8] and by imposing one restrictive condition on coefficients of the corresponding auxiliary equation, we show that any nonoscillatory solution \(y(t)\) of (1.1) satisfies
In the next, we consider separately delay and advanced cases of the argument deviation to establish new sufficient conditions for all solutions of (1.1) to have property B (see Definition 1). Moreover, in the advanced case, we will attain oscillation of all solutions of (1.1).
Some basic definitions and auxiliary lemmas
Following [8], we define
for \(t\in\mathcal{I}\). With this notation, (1.1) can be rewritten as
For the sake of clarity, we list the functions used in this work:
for \(s\ge t\ge t_{1}\), \(t_{1}\in\mathcal{I}\).
Throughout and without further mentioning, it will be assumed that
which means that the operator \(L_{3}y(t)\) is in the socalled canonical form (see Trench [24]).
To give a sense of the definitions of \(P(t)\) and \(\tilde{P}(t)\), we also suppose that
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).
Properties of solutions to equation (1.1) are closely related to those of solutions to an auxiliary secondorder linear ordinary differential equation
as the following lemma says.
Lemma 1
Let \(v(t)\) be a positive solution of (2.1) on \(\mathcal{I}\). Then (1.1) can be written in the form
Proof
It follows from a straightforward calculation that
The proof of the above equivalence is similar to that stated in [5], and so it is omitted. □
We recall that (2.1) always has a couple of nonoscillatory solutions such that, for all \(t\in\mathcal{I}\), either
or
According to a classical work of Hartman [15], a nonoscillatory solution \(v(t)\) of (2.1) satisfying (2.4) is termed 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
Let \(v(t)\) be a principal solution of (2.1) which is positive on \([t_{1},\infty)\). It is clear from the fact that \(v'(t)<0\) and the assumption (v) that the first integral in (2.6) is divergent. On the other hand, since
then \(r_{2}(t)v'(t)\) is increasing and there exists a constant \(\ell\le 0\) such that
We claim that \(\ell= 0\). If not, then
a contradiction. Thus \(\ell= 0\). By integrating (2.1) from t to ∞, we see that
Integrating (2.7) from \(t_{1}\) to t, we get
that yields
It is easy to see that integration of (2.8) from \(t_{1}\) to ∞ together with (2.5) implies that the second integral in (2.6) is divergent. The proof is complete. □
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 realvalued 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
Now we are prepared to state the sign structure of possible nonoscillatory solutions to equation (1.1). We introduce the following classes of nonoscillatory (let us say positive) solutions:
for \(t\ge t_{1}\).
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
Assume that \(y(t)\) is a positive solution of (1.1) on \(\mathcal {I}\). As a consequence of Lemma 1, we may rewrite (1.1) in an equivalent binomial form (2.2). In view of Lemma 2, there exists a positive solution \(v(t)\) of (2.1) which satisfies (2.6); and therefore, we see that the operator
is in a canonical form. Then, by Lemma 3, \(y(t)\) satisfies either
or
for \(t\ge t_{1}\). Note that in both cases we have \(y'(t)>0\) and, by virtue of (1.1), we can see that \(L_{3}y(t)>0\). The rest sign properties of quasiderivatives \(L_{i}y(t)\), \(i=1,2\), immediately follow from Lemma 3. □
Consequently, if we assume (2.5), the set \(\mathcal{N}\) of all positive solutions of (1.1) has the following decomposition:
According to the wellknown 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
Assume that \(y(t)\in\mathcal{N}_{1}\) is a positive solution of (1.1) for \(t\ge t_{1}\). It follows from the monotonicity of \(L_{1}y(t)\) that
Therefore,
and so \(y(t)/R_{1}(t,t_{1})\) is nonincreasing.
On the other hand, integration of (1.1) from t to ∞ yields
By repeated integration, we obtain
The proof is complete. □
In the lemma below we shall point out that estimate (2.10) can be improved further.
Define
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
Proceeding as in the proof of Lemma 5, we obtain (2.10). Setting (2.10) into (1.1) and integrating twice from t to ∞, we see that
By induction, we can show that (2.12) holds for any \(n\in \mathbb{N}\). □
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
Assume that \(y(t)\in\mathcal{N}_{3}\) is a positive solution of (1.1) for \(t\ge t_{1}\). Since \(L_{2}y(t)\) is increasing, \(L_{2}y(t)\ge L_{2}y(t_{1}) = :\ell\). Obviously,
We claim that (2.13) implies \(\lim_{t\to\infty}L_{2}y(t)=\infty\). Setting the above estimates into (1.1), we obtain
By integrating (2.15) from \(t_{1}\) to ∞, we see that the claim holds. Therefore, for any \(t\ge t_{2}>t_{1}\),
which leads to
and consequently, \(L_{1}y(t)/R_{2}(t,t_{1})\) is nondecreasing on \([t_{2},\infty)\).
In the same way, for any \(t\ge t_{3}>t_{2}\),
It follows from l’Hospital’s rule that \(\lim_{t\to\infty }L_{1}y(t)/R_{2}(t,t_{1}) = \lim_{t\to\infty}L_{2}y(t) = \infty\), and so we have
Then
Thus \(y(t)/R_{12}(t,t_{1})\) is nondecreasing on \([t_{3},\infty)\). The proof is complete now. □
Remark 3
It is easy to see that if (1.1) has Property B, then any positive solution of (1.1) satisfies
which gives us information about the rate of convergence of possible positive solutions.
We conclude the introductory part by recalling a useful relationship between the existence of positive solutions of the firstorder functional differential inequalities
and the corresponding firstorder functional differential equations
where 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.
Lemma 8
Let \(g(t)< t\) (\(g(t)>t\)). If inequality (2.16) (inequality (2.17)) has an eventually positive solution, then so does equation (2.18) (equation (2.19)).
Main results
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
Let (2.5) hold and \(g(t)< t\) for \(t\ge t_{1}\). If the firstorder delay differential equation
is oscillatory, then (1.1) has Property B.
Proof
Let \(y(t)\) be a positive solution of (1.1) on \(\mathcal{I}\). It follows from Lemma 4 that there exists \(t_{1}\in\mathcal{I}\) such that either \(y(t)\in\mathcal{N}_{1}\) or \(y(t)\in\mathcal{N}_{3}\) on \([t_{1},\infty)\). If \(y(t)\in\mathcal{N}_{1}\), then by virtue of (1.1) and (2.10), we have
Integrating (3.2) from t to ∞, we find
Using (2.11) in the latter inequality yields
Letting \(z(t) = L_{1}y(t)\), we see that the differential inequality
has a positive solution. By Lemma 8, we see that (3.1) also has a positive solution, which contradicts the hypothesis. Therefore \(y(t)\in\mathcal{N}_{3}\), which means that (1.1) has Property B. The proof is complete. □
Theorem 2
Let (2.5) hold and \(g(t)> t\) for \(t\ge t_{1}\). If the firstorder advanced differential equation
is oscillatory, then (1.1) has Property B.
Proof
Let \(y(t)\) be a positive solution of (1.1) on \(\mathcal{I}\). It follows from Lemma 4 that there exists \(t_{1}\in\mathcal{I}\) such that either \(y(t)\in\mathcal{N}_{1}\) or \(y(t)\in\mathcal{N}_{3}\) on \([t_{1},\infty)\). If \(y(t)\in\mathcal{N}_{1}\), then, as in the proof of Theorem 1, we obtain (3.3) so that by integration from over \([t,\infty)\), we find that
Therefore, it is clear that \(y(t)\) is a positive solution of the advanced differential inequality
By Lemma 8, we see that (3.4) also has a positive solution, a contradiction. Therefore \(y(t)\in\mathcal{N}_{3}\), which means that (1.1) has Property B. The proof is complete. □
Employing some known criteria for oscillation of firstorder 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
Assume that \(f(u) = u^{\gamma}\). Let (2.5) hold and \(g(t)< t\) for \(t\ge t_{1}\). If
then (1.1) has Property B.
Corollary 2
Assume that \(f(u) = u^{\gamma}\). Let (2.5) hold and \(g(t)>t\) for \(t\ge t_{1}\). If
then (1.1) has Property B.
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
Let \(y(t)\) be a positive solution of (1.1) on \(\mathcal{I}\). It follows from Lemma 4 that there exists \(t_{1}\in\mathcal{I}\) such that either \(y(t)\in\mathcal{N}_{1}\) or \(y(t)\in\mathcal{N}_{3}\) on \([t_{1},\infty)\). If \(y(t)\in\mathcal{N}_{1}\), then, the same as in the proof of Theorem 2, we get (3.5). Then, by integrating (3.5) from \(t_{1}\) to t, we easily find that
By virtue of the monotonicity property (2.9) and the fact that \(g(t)\le t\), we have
Using the assumption (iv) posed on the function 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
Note that in view of Lemma 6, the functions \(Q(t)\) and \(\tilde {Q}(t)\) can be replaced by
respectively, for any \(n\in\mathbb{N}\).
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
Let \(y(t)\) be a positive solution of (1.1) on \(\mathcal{I}\). It follows from Lemma 4 that there exists \(t_{1}\in\mathcal{I}\) such that either \(y(t)\in\mathcal{N}_{1}\) or \(y(t)\in\mathcal{N}_{3}\) on \([t_{1},\infty)\). From Theorem 2, we know that (1.1) has Property B, that is, \(y(t)\in\mathcal{N}_{3}\). Integrating (1.1) from t to v, we obtain
By Lemma 7, there exists \(t_{2}> t_{1}\) such that \(y(t)/R_{12}(t,t_{1})\) is nondecreasing for \(t\ge t_{2}\), and hence
Integrating (3.13) in v, one gets
Setting (3.14) into (3.12), we have
Integrating in v once more, we get
Integrating in v last time, we find
Setting \(v = g(t)\), we obtain
Taking the lim sup on both sides of the resulting inequality, we are led to the contradiction with (3.11). Thus \(\mathcal{N}_{3} = \emptyset\) and (1.1) is oscillatory. The proof is complete. □
Examples
Example 1
Consider the thirdorder linear differential equation
A corresponding auxiliary equation
has a principal solution \(v(t) = t^{\alpha}\) (\(\alpha= \frac {1}{2}(1\sqrt{1+4a})\)), which satisfies (2.6) if \(a< 2\). A simple calculation leads to
Then, by Theorems 1 and 2 together with Remark 4, Property B of (4.1) is guaranteed if
By Theorems 3 and 4, the same conclusion holds for (4.1) if
Furthermore, it follows from Theorem 5 that if \(\lambda>1\) and
then (4.1) is oscillatory.
Example 2
Consider the thirdorder nonlinear differential equation
Note that condition (2.5) is satisfied for \(a\le4/3\). Then
and by Theorems 1 and 2 we obtain that (4.2) has Property B provided that
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).
For a particular case of (1.1), namely
Kusano et al. [18] have shown that there always exists a positive solution \(y(t)\in\mathcal{N}_{3}\). If, however, \(g(t)>t\), then we were able to eliminate also this class of solutions. It is clearly the advanced argument that can generate the oscillations.
It remains an open problem for further research to obtain the solution structure and corresponding asymptotic criteria for the equation
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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. FEI201522.
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Džurina, J., Jadlovská, I. Asymptotic behavior of thirdorder functional differential equations with a negative middle term. Adv Differ Equ 2017, 71 (2017). https://doi.org/10.1186/s1366201711270
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MSC
 34K11
Keywords
 thirdorder differential equation
 delay
 advance
 oscillation
 comparison
 integral criteria