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ISRN Mathematical Analysis
Volume 2013 (2013), Article ID 946453, 11 pages
http://dx.doi.org/10.1155/2013/946453
Research Article

Asymptotic Solutions of nth Order Dynamic Equation and Oscillations

Kent State University at Stark, 6000 Frank Avenue NW, Canton, OH 44720-7599, USA

Received 6 June 2013; Accepted 25 July 2013

Academic Editors: M. McKibben, A. Peris, and C. Zhu

Copyright © 2013 Gro Hovhannisyan. 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.

Abstract

We establish a new asymptotic theorem for the nth order nonautonomous dynamic equation by its transformation to the almost diagonal system and applying Levinson's asymptotic theorem. Our transformation is given in the terms of unknown phase functions and is chosen in such a way that the entries of the perturbation matrix are the weighted characteristic functions. The characteristic function is defined in the terms of the phase functions and their choice is exible. Further applying this asymptotic theorem we prove the new oscillation and nonoscillation theorems for the solutions of the nth order linear nonautonomous differential equation with complex-valued coefficients. We show that the existence of the oscillatory solutions is connected with the existence of the special pairs of phase functions.

1. Introduction

Consider an ordinary nonautonomous differential equation of the th order with complex-valued continuous variable coefficients ,  .

A solution of (1) is said to be oscillatory if it has an infinite sequence of zeros in and nonoscillatory otherwise. Equation (1) is said to be nonoscillatory if all nontrivial solutions are nonoscillatory.

Oscillation theorems for ordinary differential equation of the th order in the case of real variable coefficients have been studied in many papers (see [1, 2] and references therein). To the best of the author’s knowledge the oscillations of the solutions of nonautonomous th order equations with complex coefficients have not been studied yet (except [3]).

Let be the set of times differentiable functions on and the set of Lebesgue absolutely integrable functions on .

To consider the case of complex coefficients we are using asymptotic solutions of (1) in Euler form . Define the characteristic function of (1) depending on a phase function

By direct calculations for we get where For example, Define the auxiliary square matrix with the entries (see (4)) and denote by ,   the minors of the matrix .

2. Main Theorems

The basic method of this paper is a new version of Levinson’s asymptotic theorem (see [4, 5]).

Theorem 1. Assume there exist complex-valued functions ,   such that for all expressions , do not change a sign; that is, where characteristic functions ,   are defined in (2), (3), and
Then solutions of (1) may be represented in the form where

Note that condition (9) is the well-known Abel’s identity that is satisfied if (12) are exact solutions of (1).

Theorem 1 means that the error functions as if the weighted characteristic functions are absolutely integrable in .

In Theorem 1 the weighted characteristic functions are the entries of the perturbation matrix from Levinson’s Theorem (see Remark 18 in Section 3).

Note that the error functions in representation (11) may be estimated via the characteristic function (see, e.g., Theorem 2.2 in [6]).

Theorem 1 may be used also for the applications mentioned in [5], the stability theory, and the Dirac equation with complex coefficients (see, e.g., [68]).

We will say that (1) has the asymptotic solutions corresponding to the phase functions , if (7)–(9) are satisfied.

Theorem 2. The asymptotic solution of (1) (corresponding to the phase function ) generates the oscillatory solution of (1) if there exists another asymptotic solution with the phase such that and one of the following conditions is satisfied:

The interesting question is to obtain the converse of Theorem 2.

Conditions (13) and (14) mean that the phase functions are complex congugate. For the equations with real coefficients the complex phase functions appear in complex conjugate pairs. But for the equations with complex coefficients this is not true.

Since every solution of (1) with the constant complex coefficients (without repeated characteristic roots) is the linear combination of the exponents, in view of Theorem 2, the following questions arise.

Question 1. Describe the complex numbers such that

Question 2. For which complex constant coefficients ,   a polynomial ,   has at least one pair of complex conjugate zeros with nonzero imaginary parts?

Question 1 is complicated in the multidimensional case (see [9]), but by the substitution one can reduce dimensional equation to the two equations: ()-dimensional and 3-dimensional .

For the two dimensional case there is a simple answer to Question 1: if and only if

One can answer Question 2 for quadratic, cubic, and quartic equations with the complex coefficients since the simple formulas for solutions are available for these cases (see Remark 14, Theorem 15 below).

Theorem 2 shows that the oscillations of the solutions could be produced not only by the complex conjugate phase functions, which one can see from the following example.

Example 3. The equation is oscillatory, but there is no complex conjugate pair among its phase functions with nonzero imaginary parts: Note that , are exact solutions of (19), is an oscillatory solution of (19), and (13), (15) are satisfied.

Example 4. If ,   are a real-valued functions such that then the equation is oscillatory if and only if

The following theorem is inspired by the asymptotic theorems for two-term differential equation from [10, 11]. We deduce it from Theorem 1 by choosing the phase function as an approximate solution of the characteristic equation (see [10]): Note that Theorem 1 may be used to obtain other oscillation theorems by taking different approximate solutions of the characteristic equation (see [10, 12]).

Theorem 5. Assume there exists a complex-valued function ,  , and complex numbers ,   such that the set contains a set of pairs of complex conjugate functions, and
Then (1) with has oscillatory and nonoscillatory linearly independent solutions.

By taking from Theorem 5 we get the following corollary.

Corollary 6. Assume that for some complex numbers , the set contains a set of pairs of complex conjugate numbers, and conditions (25), (26), are true for all . Then (1) with has oscillatory and nonoscillatory linearly independent solutions.

From Corollary 6 follows the well-known result.

Corollary 7 (see [1]). Assume that conditions are satisfied. Then (1) with is nonoscillatory.

For the fourth order equation we deduce the following result.

Corollary 8. Assume that for some real number and for the fixed number j from the set Then equation is nonoscillatory.

Remark 9. If conditions (33) turn to

Remark 10. To compare condition (35) with the well-known real coefficient case, note that (34) with and real valued is nonoscillatory if (see [1])

By taking , one can get another result.

Corollary 11. Assume that for some real number conditions are satisfied. Then (34) has 2 oscillatory and 2 nonoscillatory (linearly independent) solutions.

For two terms equation: with real valued coefficient by choosing from Theorem 5 one can deduce the following theorem.

Theorem 12. Assume , and for all Then (38) has oscillatory and nonoscillatory linearly independent solutions. Here is the integral part of a real number.

Remark 13. If , then conditions (40), (41) are simplified to

Remark 14. For the quadratic equations with complex coefficients , it is easy to see that it has one pair of complex conjugate zeros with nonzero imaginary parts if and only if the coefficients are real, and .

Theorem 15. The cubic equation with complex constant coefficients has a pair of complex conjugate solutions with nonzero imaginary parts if and only if one of the following conditions is true: or where are the discriminants of (43) and correspondingly,

Example 16. For the equation we have and condition (45) is satisfied.

3. Proofs

We are going to use Levinson’s asymptotic theorem as it appears in [5].

Theorem 17 (see [5]). Let be an diagonal matrix-function which satisfies dichotomy condition.
For each pair of integers and in () exist constants such that for all and , Let the matrix satisfy or by which one means that each entry in has an absolutely convergent infinite integral. Then the system has a vector solution with the asymptotic form where is the identity matrix, is the error matrix-function, and is a constant column-vector.

Proof of Theorem 1. Rewrite (1) as a system By transformation where matrix-function is defined via phase functions : and the entries of the matrix are defined in (4), (6), we get or, in view of identity, we have where Here and further we often suppress the time variable for the simplicity.

Remark 18. Remarkable formulas (61)–(63) show that any th order (1) may be transformed to the first order system , where the matrix is a sum of diagonal matrix and perturbation matrix ; that is, (Levinson form). Moreover, if the diagonal matrix is chosen in terms of phase functions as in (62), then the perturbation matrix is the weighted characteristic function (i.e., the entries of matrix B are proportional to the characteristic functions ). Our conjecture is that it is true not only for (1), but also for any first order system. For planar systems it was proved in [7].

To prove basic formula (60) note that by using Laplace expansion by the minors of the determinants: we get identities: or By direct calculations we get and the same way we get

To apply Theorem 17 to system (61) note that from (7) follows dichotomy condition (51) of Theorem 17. Condition (52) of Theorem 17 turns to : or (8) in view of  .

From Theorem 17 applied to system (61) we get From (57): we get representation (11). If functions are exact solutions of , , then is the fundamental solution of (55). Abel’s identity may be written in the form (9). We always choose (approximate) phase functions , such that (9) is satisfied.

Proof of Theorem 2. From the assumptions (13), (14) of Theorem 2 one can generate an asymptotic solution as the difference of two asymptotic solutions: which is oscillatory. Indeed if , or , then the infinite sequence of zeros of is given by
From the assumptions (13), (15) for the solution one can generate another infinite sequence of zeros that is given by Indeed we have the following: Theorem 2 is followed from representation (11).

Proof of Example 4. Asymptotic solutions of (22) we choose in the form or From (4) we have the following:
So if (8) is satisfied, that is, then solutions of may be written in the form following: If has a zero for some complex numbers , then or or By taking the derivative by of both sides we get and since we get or By taking the derivative by of the both sides we get or So condition (8) turns to (21).

Proof of Theorem 5. We deduce Theorem 5 from Theorem 1 by choosing as in (24), (25). By calculations we get Note that (9) is satisfied in view of : Further we have the following: and condition (8), in view of Definition (2), turns to (27): Further (7) turns to (26). So under conditions of Theorem 5 conditions (7)–(9) are satisfied and Theorem 1 is applicable. Further from (28) we get (13), (14) is satisfied for , and Theorem 5 is followed from Theorem 1 and Theorem 2.

Proof of Corollary 6. Corollary 6 is followed from Theorem 5 by taking . From (27) we have or, since ,  , we get or condition (27) turns to (31). Further conditions (28),(29) turn to (30).

Proof of Corollary 7. Corollary 7 is followed from Corollary 6 by taking ,  , .

Proof of Corollary 8. We deduce Corollary 8 from Corollary 6.
In the case ,   condition (29) turns to
Considering the case of Corollary 8 assume ,  . Then condition (97) turns to For any real by choosing we get and (97) turns to (33) with .
In the case , assuming , (97) turns to By choosing , and , , and as real solutions of with the negative discriminant (see (46)), we get conditions (33) with : In the case , assume . Condition (97) turns to
By choosing , , and as real solutions of or we get nonoscillation conditions (33) with :

Proof of Corollary 11. Proof of Corollary 11 is similar to proof of Corollary 8 in case with the choice , , , and . Note since ,  ,  and we get .

Proof of Theorem 12. We deduce Theorem 12 from Theorem 5.
From the choice (39) we get
It is easy to see that condition (25) of Theorem 5 is satisfied. Condition (27) turns to (40). To check condition (26) consider Assuming , we have and since or , the sign of does not depend on and (26) is true. Let be the number of indices for which Then (28) is true in view of (41).
If for some then a nonoscillatory solution exists. This condition that is, since or .
Consider the case If is odd, then is integer if and only if which means that the solution with is nonoscillatory, so other solutions are oscillatory.
If and is even, then and all solutions are oscillatory.
Further in the case and is even we have the solutions with are nonoscillatory, and the other solutions are oscillatory.
In the case and is odd we have only one nonoscillatory solution with and .
So or

Proof of Remark 14. If has 1 pair of complex conjugate zeros , , then , are real numbers since and , and quadratic equation has two complex conjugate roots if . Assuming that , are real numbers and from quadratic formula the equation has two complex conjugate roots.

Proof of Theorem 15. Assume there exists a pair of complex conjugate solutions , of ; that is, then denoting we have as well, and Consider 3 cases.
First case: , , , .
In this case is satisfied. The coefficients of are real and (43) has two complex conjugate solutions since discriminant of is positive. We drop the case , , , since in that case all solutions are real.
Second case: .
In this case by solving quadratic equation we get and implies another solution can be found by division: Note that if discriminant of the quadratic equation is negative, then by solving this equation we get two complex conjugate roots with nonzero imaginary parts:
If then is real, and to find another (complex) solution one should solve Since must be a solution as well, we have and is a real number given by Further since , are solutions by division and one can find the third solution : This case does not work since we have two real solutions , and one complex solution of .
Third case: , .
In this case assuming , are complex conjugate solutions of from we get which is a real number given by and .
If is another solution of , by division we get Since is also solution of , we get and we come back to the case 1.
To prove that (45) is sufficient condition in view of , it is enough to show that but it is true since .
It is easy to see that (44) is a sufficient condition as well.

Acknowledgment

The author wish to thank Stefan Born, Oktay Duman, Roberto Gianni, Herbert Homeier, Boris Kazarnovskii, Kus Krisnawan, Christopher Landauer, Octav Olteanu, Stepan Tersian, and Gail Wolkowicz for the fruitful discussion of Question 1 and Question 2 and the anonymous referees for the careful reading of the paper and fruitful comments and suggestions.

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