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Abstract and Applied Analysis
Volume 2011 (2011), Article ID 730128, 10 pages
Asymptotic Properties of Third-Order Delay Trinomial Differential Equations
Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Košice, Letná 9, 042 00 Košice, Slovakia
Received 2 September 2010; Accepted 3 November 2010
Academic Editor: Yuri Rogovchenko
Copyright © 2011 J. Džurina and R. Komariková. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The aim of this paper is to study properties of the third-order delay trinomial differential equation , by transforming this equation onto the second-/third-order binomial differential equation. Using suitable comparison theorems, we establish new results on asymptotic behavior of solutions of the studied equations. Obtained criteria improve and generalize earlier ones.
In this paper, we will study oscillation and asymptotic behavior of solutions of third-order delay trinomial differential equations of the form Throughout the paper, we assume that and(i), , , ,(ii), , (iii) as .
By a solution of (), we mean a function , , that satisfies () on . We consider only those solutions of () which satisfy for all . We assume that () possesses such a solution. A solution of () is called oscillatory if it has arbitrarily large zeros on , and, otherwise, it is nonoscillatory. Equation () itself is said to be oscillatory if all its solutions are oscillatory.
Recently, increased attention has been devoted to the oscillatory and asymptotic properties of second- and third-order differential equations (see [1–22]). Various techniques appeared for the investigation of such differential equations. Our method is based on establishing new comparison theorems, so that we reduce the examination of the third-order trinomial differential equations to the problem of the observation of binomial equations.
In earlier papers [11, 13, 16, 20], a particular case of (), namely, the ordinary differential equation (without delay) has been investigated, and sufficient conditions for all its nonoscillatory solutions to satisfy or the stronger condition are presented. It is known that () has always a solution satisfying (1.1). Recently, various kinds of sufficient conditions for all nonoscillatory solutions to satisfy (1.1) or (1.2) appeared. We mention here [9, 11, 13, 16, 21]. But there are only few results for differential equations with deviating argument. Some attempts have been made in [8, 10, 18, 19]. In this paper we generalize these, results and we will study conditions under which all nonoscillatory solutions of () satisfy (1.1) and (1.2). For our further references we define as following.
Definition 1.1. We say that () has property () if its every nonoscillatory solution satisfies (1.1).
In this paper, we have two purposes. In the first place, we establish comparison theorems for immediately obtaining results for third-order delay equation from that of third order equation without delay. This part extends and complements earlier papers [7, 8, 10, 18].
Secondly, we present a comparison principle for deducing the desired property of () from the oscillation of a second-order differential equation without delay. Here, we generalize results presented in [8, 9, 14, 15, 21].
Remark 1.2. All functional inequalities considered in this paper are assumed to hold eventually;0 that is, they are satisfied for all large enough.
2. Main Results
Proof. The proof follows from the fact that
Now, in the sequel, instead of studying properties of the trinomial equation (), we will study the behavior of the binomial equation (). For our next considerations, it is desirable for () to be in a canonical form; that is, because properties of the canonical equations are nicely explored.
Now, we will study the properties of the positive solutions of () to recognize when (2.2)-(2.3) are satisfied. The following result (see, e.g., [7, 9] or ) is a consequence of Sturm's comparison theorem.
To be sure that () possesses a positive solution, we will assume throughout the paper that (2.4) holds. The following result is obvious.
Lemma 2.3. If is a positive solution of (), then , , and, what is more, (2.2) holds and there exists such that .
Now, we will show that if () is nonoscillatory, then we always can choose a positive solution of () for which (2.3) holds.
Proof. First note that Thus, is a positive solution of (). On the other hand, to insure that (2.3) holds for , let us denote . Then and
Lemma 2.5. Let (2.4) hold. Then trinomial () can be represented in its binomial canonical form ().
We present a structure of the nonoscillatory solutions of (). Since () is in a canonical form, it follows from the well-known lemma of Kiguradze (see, e.g., [7, 9, 14]) that every nonoscillatory solution of () is either of degree 0, that is, or of degree 2, that is,
Definition 2.6. We say that () has property if its every nonoscillatory solution is of degree 0; that is, it satisfies (2.9).
Proof. We suppose that is a positive solution of (). We need to verify that . Since is also a solution of (), then it satisfies (2.9). Therefore, .
Assume that is a positive solution of (). We will verify that (2.9) holds. Since is also a solution of (), we see that ; that is, . It follows from () that . Thus, is decreasing. If we admit eventually, then is decreasing, and integrating the inequality , we get as . Therefore, and (2.9) holds.
Employing any known or future result for property of (), then in view of Corollary 2.9, we immediately obtain that property holds for ().
Example 2.10. We consider the third-order delay trinomial differential equation where and satisfies (2.11). The corresponding equation () takes the form and it has the pair of the solutions and . Thus, is our desirable solution, which permits to rewrite (2.12) in its canonical form. Then, by Corollary 2.9, (2.12) has property if the equation has property .
Now, we enhance our results to guarantee stronger asymptotic behavior of the nonoscillatory solutions of (). We impose an additional condition on the coefficients of () to achieve that every nonoscillatory solution of () tends to zero as .
Proof. Assume that is a positive solution of (). Then, it follows from Corollary 2.9 that . Therefore, . Assume . On the other hand, is also a solution of (), and, in view of Theorem 2.7, it has to be of degree 0; that is, (2.9) is fulfilled. Then, integrating () from to , we get Multiplying this inequality by and then integrating from to , we have Multiplying this by and then integrating from to , we obtain This is a contradiction, and we deduce that . The proof is complete.
Example 2.12. We consider once more the third-order equation (2.12). It is easy to see that (2.15) takes the form Then, by Corollary 2.11, every nonoscillatory solution of (2.12) tends to zero as provided that (2.19) holds and (2.14) has property .
In the second part of this paper, we derive criteria that enable us to deduce property of () from the oscillation of a suitable second-order differential equation. The following theorem is a modification of Tanaka's result .
Proof. Assume that is a positive solution of (), then is either of degree 0 or of degree 2. Assume that is of degree 2; that is, (2.10) holds. An integration of () yields On the other hand, Combining the last two inequalities, we get Integrating the previous inequality from to , we see that satisfies Denoting the right-hand side of (2.24) by , it is easy to see that and By Theorem 2 in , the corresponding equation () also has a positive solution. This is a contradiction. We conclude that is of degree 0; that is, () has property (A).
If (2.20) does not hold, then we can use the following result.
Applying any criterion for oscillation of (), Corollary 2.15 yields a sufficient condition property of ().
Example 2.18. We consider again (2.12). By Corollary 2.17, every nonoscillatory solution of (2.12) tends to zero as provided that (2.19) holds and
For a special case of (2.12), namely, for
with , , and , we get that every nonoscillatory solution of (2.30) tends to zero as provided that
If we set , where , then one such solution of (2.12) is .
On the other hand, if for some we have , then (2.31) is violated and (2.12) has a nonoscillatory solution which is of degree 2.
In this paper, we have introduced new comparison theorems for the investigation of properties of third-order delay trinomial equations. The comparison principle established in Corollaries 2.9 and 2.11 enables us to deduce properties of the trinomial third-order equations from that of binomial third-order equations. Moreover, the comparison theorems presented in Corollaries 2.15–2.17 permit to derive properties of the trinomial third-order equations from the oscillation of suitable second-order equations. The results obtained are of high generality, are easily applicable, and are illustrated on suitable examples.
This research was supported by S.G.A. KEGA 019-025TUKE-4/2010.
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