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On the oscillation of odd order advanced differential equations
Boundary Value Problems volume 2014, Article number: 214 (2014)
Abstract
The aim of this paper is to study the asymptotic properties and oscillation of the n th order advanced differential equations
The results obtained are based on the Riccati transformation.
MSC: 34K11, 34C10.
1 Introduction
In this paper, we shall study the asymptotic and oscillation behavior of the solutions of the higher order advanced differential equations
Throughout the paper, we assume , and
(H1): n is odd, γ is the ratio of two positive odd integers,
(H2): , , , .
Whenever, it is assumed
By a solution of Eq. (1.1), we mean a function , , which has the property and satisfies Eq. (1.1) on . We consider only those solutions of (1.1) which satisfy for all . We assume that (1.1) possesses such a solution. A solution of (1.1) is called oscillatory if it has arbitrarily large zeros on , and otherwise it is called to be nonoscillatory.
The problem of the oscillation of differential equations has been widely studied by many authors who have provided many techniques especially for lower order delay differential equations. Dong in [1] improved and extended the Riccati transformation to obtain new oscillatory criteria for the second order delay differential equations
Grace et al. in [2] and the present authors in [3]–[6] used the comparison technique for the third order delay differential equation
that was compared with the oscillation of certain first order differential equation.
On the other hand, there are comparatively less methods established for the advanced differential equations. The aim of the paper is to fill this gap in the oscillation theory.
Remark 1
All functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for all t large enough.
2 Main results
Our results essentially use the following estimate which is due to Philos and Staikos (see [7] and [8]).
Lemma A
Let. Assume thatis of fixed sign and not identically zero on a subray of. If, moreover, , , and, then for everythere existssuch that
holds on.
The following useful result will be used later in the proofs of our main results.
Lemma 1
Assume, , , eventually. Then, for arbitrary,
eventually.
Proof
It follows from the monotonicity of that
That is,
On the other hand, since as , then for any there exists large enough such that
or equivalently,
Using (2.4) in (2.3), we obtain
The proof is complete. □
The positive solutions of (1.1) have the following structure.
Lemma 2
Ifis a positive solution of (1.1), thenis decreasing, all derivatives, , are of constant signs, andsatisfies either
or
Proof
Since is a positive solution of (1.1), then it follows from (1.1) that
Thus, is decreasing, which implies that either or . But the case implies . An integration from to t yields
but in view of (1.2) for . Repeating this procedure, we obtain that and this is a contradiction, and we conclude that . Moreover, implies that either or , but the first case leads to for . Repeating these considerations, we verify that satisfies either (2.5) or (2.6).
On the other hand, since , then using in
we conclude that . The proof is complete. □
Now, we offer some criteria for certain asymptotic behavior of all nonoscillatory solutions. For our further references, we set
and
Theorem 1
Assume that
and
then every nonoscillatory solutionof (1.1) satisfies.
Proof
Assume that is an eventually positive solution of (1.1). First assume that satisfies (2.5). By (2.7), it is easy to see that there exists some such that
We put , then setting (2.2) into (1.1), we get
We define
Differentiating , one gets
On the other hand, Lemma A implies
Setting the last inequality into (2.11), we obtain
Integrating the last inequality from t to ∞, we have
or
eventually, let us say . Since , then
Thus,
From (2.9), we see that there exists some positive η such that
Combining (2.13) together with (2.14), we have
Therefore,
or equivalently,
This contradicts the fact that the function
is nonnegative for all , and we conclude that cannot satisfy (2.5).
Now we assume that satisfies (2.6). Then there exists a finite . We claim that . Assume that . Integrating (1.1) from t to ∞, we obtain
which implies
Integrating the last inequality twice from t to ∞, we get
Repeating this procedure, we arrive at
Now, integrating from to ∞, we see that
which contradicts (2.8), and so we have verified that . □
Example 1
Consider the odd order () nonlinear differential equation
Here and , so that
Consequently,
i.e., (2.8) holds; moreover, (2.7) reduces to
which, by Theorem 1, guarantees that all nonoscillatory solutions of (2.15) tend to zero at infinity.
Let be a sequence of continuous functions defined as follows,
and
Then we have the following result.
Theorem 2
Assume that (2.8) holds and there exists somesuch that
for some. Then every nonoscillatory solutionof (1.1) satisfies.
Proof
Assume that is an eventually positive solution of (1.1). By Lemma 2, satisfies either (2.5) or (2.6). It follows from the proof of Theorem 1 that if satisfies (2.6), then (2.8) insures that it tends to zero at infinity.
Assume that satisfies (2.5). It follows from the proof of Theorem 1 that (2.12) holds for every .
By induction, using (2.12), it is easy to see that the sequence is nondecreasing and . Thus the sequence converges to . By the Lebesgue monotone convergence theorem and letting in (2.16), we get
which in view of implies
eventually, let us say . Therefore,
An integration from to t yields
Letting , we obtain a contradiction. The proof is complete. □
Theorem 3
Assume that (2.8) holds and there exist someandsuch that
Then every nonoscillatory solutionof (1.1) satisfies.
Proof
Assume that is an eventually positive solution of (1.1) satisfying (2.5). It follows from Lemma A that
eventually, where is the same as in . Then
or equivalently,
which contradicts (2.18). □
Letting in Theorem 3, we have the following result.
Corollary 1
Assume that (2.8) holds and
Then every nonoscillatory solutionof (1.1) satisfies.
Proof
It follows from (2.19) that there exists some such that
which is equivalent to
The assertion now follows from Theorem 3. □
Example 2
Consider the third order nonlinear differential equation
A simple calculation leads to
Then (2.8) holds and (2.19) reduces to
and thus, by Corollary 1, every nonoscillatory solution of (2.20) tends to zero as .
Our results are based on Lemma 1, i.e., we essentially utilize the estimate (2.2). It is easy to see that for and , estimate (2.2) does not hold, that is, for
relationship (2.2) fails, and so our result here cannot be applied for n even. Hence, it remains an open problem how to obtain the corresponding results also for n even.
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Baculíková, B., Džurina, J. On the oscillation of odd order advanced differential equations. Bound Value Probl 2014, 214 (2014). https://doi.org/10.1186/s13661-014-0214-3
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DOI: https://doi.org/10.1186/s13661-014-0214-3