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On certain inequalities and their applications in the oscillation theory
Advances in Difference Equations volume 2013, Article number: 165 (2013)
In the paper, we offer a set of inequalities involving delayed argument and offer their application for higher-order differential equations of the form
to be oscillatory. The conditions obtained essentially improve many other known results.
The paper is organized as follows. In the first part we consider only properties of functions and their derivatives, and later we connect the estimate obtained with properties of solutions of differential equations. We shall investigate the properties of a couple of functions and , .
Lemma 1 Assume that ℓ is a positive integer such that
eventually. Then for any constant and for every ,
Proof Assume that holds for . Using the monotonicity of , it is easy to see that for any ,
eventually, let us say, for . We define a sequence of functions as follows:
It follows from (1.2) that . An integration of this from to t yields
On the other hand, since as , we see that
Combining the last two inequalities, we conclude that
Proceeding as above, we verify that , eventually, for all . Therefore,
or in other words
Setting , the last inequalities imply (1.1) and the proof is complete. □
Lemma 2 Assume that and that ℓ is a positive integer such that holds. Then, for any constant ,
Proof Taylor’s theorem implies that
Employing (1.1), we have
The proof is complete. □
The obtained estimates can be used, e.g., in the theory of functional equations. In the paper, we present their application in discussing oscillatory and asymptotic properties of higher-order delay differential equations.
2 Main results
We consider higher-order delay differential equation
(H1) , .
Denote by the set of all nonoscillatory solutions of (E). It follows from the classical lemma of Kiguradze  that the set has the following decomposition:
where the nonoscillatory solution , let us say positive, satisfies
A nonoscillatory solution of (E) is said to be of degree ℓ if . Following Kondratiev and Kiguradze, we say that (E) has property (A) provided that
The investigation of oscillatory properties of the second- and higher-order linear differential equations started with the Sturm comparison theorem . Later Mahfoud  essentially contributed to the subject and presented a very useful comparison technique for studying the properties of a delay differential equation from those of a differential equation without delay. A new impetus to investigation in this direction was given by papers of Chanturia and Kiguradze , Kusano and Naito  and Koplatadze et al. [6, 7]. See also [1–20]. In the paper, we employ Lemma 2 to establish new criteria for oscillation of (E).
It is interesting to note that the condition
Theorem 1 Assume that (E) has a solution of degree , then for any so does the ordinary equation
Proof Assume that (E) possesses a nonoscillatory solution . We may assume that is positive. Then condition (1.3) of Lemma 1 implies that is a positive solution of the differential inequality
On the other hand, it follows from Theorem 2 of  that the corresponding equation () has also a solution of degree ℓ. The proof is complete. □
So, if we eliminate solutions of degree ℓ of equations (), we get property (A) of studied equation (E). To do it, we recall the following comparison result which is due to Chanturia .
Theorem 2 Assume that
If the differential equation
has no solution of degree ℓ, neither does the equation
In view of Theorem 1, we apply this comparison theorem to equations () and the Euler equation
to obtain new criteria for property (A) of (E). Properties of (2.3) are connected with properties of the polynomial . Let us denote
where for n odd, while for n even. In other words, represents all local maxima of the polynomial (see Figure 1). Then it is easy to verify (see also ) that the following criterion for the to be empty holds true.
Lemma 3 Let . If
where for n odd and for n even, then (2.3) has no solution of degree ℓ.
Employing Theorem 2 to (2.3) and (), in view of Theorem 1, one gets the following theorem.
Theorem 3 Assume that
Then (E) has property (A).
Proof Assume that n is odd. Observing that and for every , it follows from (P) that for every ,
On the other hand, () implies that there exists a couple of constants and such that
Since , Euler equation (2.3) has no solution of degree ℓ. On the other hand, taking (2.5) into account, Theorem 2 ensures that () has no solution of degree ℓ. Finally, Theorem 1 guarantees that (E) has property (A). The proof is complete. □
For , , the previous result simplifies to the following.
Corollary 1 Assume that
Then the delay differential equation
has property (A).
Theorem 4 Let n be odd. Assume that (E) has property (A). Then every nonoscillatory solution of (E) satisfies
Proof First note that property (A) of (E) implies (2.1). Moreover, it follows from the definition of property (A) that every nonoscillatory solution , which implies that there exists . We claim that . If not, then . An integration of (E) from t to ∞ yields
Having repeated this procedure, we are led to
which contradicts (2.1) and we conclude that . □
We support our results with the following illustrative example.
Example 1 Consider the fifth-order delay differential equation
The graph of the polynomial that corresponds to the fifth-order equation is presented in Figure 1. Employing Matlab, we easily evaluate that
Consequently, criterion () for property (A) of (E) reduces for () to
Theorem 2 essentially improves Chanturia’s test  that guarantees property (A) of
Kiguradze’s test  that for property (A) of (E) requires
and Koplatadze’s test  for property (A) of (E) that claims the condition
where is nondecreasing.
We provide details while comparing those criteria with our one.
Example 2 Consider once more the fifth-order delay differential equation (). It is easy to see that Chanturia’s test can be applied only when and requires
for property (A) of (). Kiguradze’s test fails. On the other hand, Koplatadze’s test simplifies for and to
respectively, while our criterion needs only
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This work was supported by the Slovak Research and Development Agency under the contract No. APVV-0008-10.
The authors declare that they have no competing interests.
The authors have made the same contribution. All authors read and approved the final manuscript.
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