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On the oscillation of odd order advanced differential equations
Boundary Value Problems volume 2014, Article number: 214 (2014)
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.
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  improved and extended the Riccati transformation to obtain new oscillatory criteria for the second order delay differential equations
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.
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
Let. Assume thatis of fixed sign and not identically zero on a subray of. If, moreover, , , and, then for everythere existssuch that
The following useful result will be used later in the proofs of our main results.
Assume, , , eventually. Then, for arbitrary,
It follows from the monotonicity of that
On the other hand, since as , then for any there exists large enough such that
The proof is complete. □
The positive solutions of (1.1) have the following structure.
Ifis a positive solution of (1.1), thenis decreasing, all derivatives, , are of constant signs, andsatisfies either
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
then every nonoscillatory solutionof (1.1) satisfies.
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
eventually, let us say . Since , then
From (2.9), we see that there exists some positive η such that
This contradicts the fact that the function
is nonnegative for all , and we conclude that cannot satisfy (2.5).
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 . □
Consider the odd order () nonlinear differential equation
Here and , so that
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,
Then we have the following result.
Assume that (2.8) holds and there exists somesuch that
for some. Then every nonoscillatory solutionof (1.1) satisfies.
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.
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. □
Assume that (2.8) holds and there exist someandsuch that
Then every nonoscillatory solutionof (1.1) satisfies.
eventually, where is the same as in . Then
which contradicts (2.18). □
Letting in Theorem 3, we have the following result.
Assume that (2.8) holds and
Then every nonoscillatory solutionof (1.1) satisfies.
It follows from (2.19) that there exists some such that
which is equivalent to
The assertion now follows from Theorem 3. □
Consider the third order nonlinear differential equation
A simple calculation leads to
and thus, by Corollary 1, every nonoscillatory solution of (2.20) tends to zero as .
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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The authors declare that they have no competing interests.
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.