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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 738472, 19 pages
http://dx.doi.org/10.1155/2012/738472
Research Article

Perturbations of Half-Linear Euler Differential Equation and Transformations of Modified Riccati Equation

Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic

Received 10 July 2012; Accepted 17 September 2012

Academic Editor: Allan Peterson

Copyright © 2012 Ondřej Došlý and Hana Funková. 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 investigate transformations of the modified Riccati differential equation and the obtained results we apply in the investigation of oscillatory properties of perturbed half-linear Euler differential equation. A perturbation is also allowed in the differential term.

1. Introduction

The half-linear Euler differential equation with the so-called oscillation constant plays an important role in the oscillation theory of the half-linear differential equation with the continuous functions , , and . The reason is that (1.1) represents a kind of borderline between oscillation and nonoscillation in the half-linear oscillation theory. More precisely, if in (1.2), then this equation is oscillatory provided and nonoscillatory if see, for example, [1]. Formulas (1.3), (1.4) show what “borderline” means. The potential “separates” potentials in (1.2) with for which this equation is oscillatory or nonoscillatory. Criteria (1.3), (1.4) can be extended to the general case . In this general setting, the Kneser type criterion is formulated in terms of the lower and upper limit of the expression if , being the conjugate exponent of , and of the expression if . The constant in this criterion remains the same. In the linear case , (1.3) and (1.4) are the classical Kneser (non)oscillation criteria, see [2].

Our investigation is mainly motivated by the papers [35]. In [4], perturbations of (1.1) of the form, were investigated. Here, the notation is used. It was shown that the crucial role in (1.7) plays the constant . In particular, if in (1.7), that is, this equation reduces to the so-called Riemann-Weber half-linear differential equation, then this equation is oscillatory if and nonoscillatory in the opposite case. In general, if for , then (1.7) is oscillatory if and only if .

In [5], the perturbations of the linear Euler differential equation were investigated and a perturbation was also allowed in the term-involving derivative. More precisely, the differential equation was considered. It was shown that if there exists such that for , and , then (1.9) is oscillatory if and only if . If for all , then (1.11) is nonoscillatory. This result was partially extended to half-linear equations in [3]. There, was investigated and it was shown that (1.10) is oscillatory if and only . For some related results see also [6].

In this paper we deal with perturbations of the Euler half-liner differential equation in full generality. We consider and we find an explicit formula for the relationship between constants , in (1.11) which implies (non)oscillation of this equation. In the last section of the paper we explain why perturbations are just in the above considered form. Our result is based on a new method which consists in transformations of the modified Riccati equations associated with (1.2). The main result along this line is established in Section 3, while its application to the perturbed Euler equation is presented in Section 4. In the last section we present some remarks and comments concerning the results of our paper. In the next section we recall some essentials of the half-linear oscillation theory.

2. Preliminaries

It is a well-known fact that many of the results of the linear oscillation theory can be directly extended to half-linear equation (1.2), even if, in contrast to the case (then (1.2) is a linear equation), the additivity of the solution space is lost and only homogeneity remains. In particular, the so-called Riccati technique, consisting in the relationship between (1.2) and its associated Riccati type equation (related to (1.2) by the substitution ) extends almost literally to (1.2). More precisely, the following statement holds (see [1, Theorem 2.2.1]).

Proposition 2.1. Equation (1.2) is nonoscillatory if and only if there exists a differentiable function such that for large .

The modified Riccati equation associated with (1.2) is introduced explicitly in [7], but it can be found implicitly already in some earlier papers, for example, [810]. Suppose that (1.2) is nonoscillatory (i.e., every its nontrivial solution is eventually positive or negative) and let be a positive differentiable function. Consider the substitution where is a solution of (2.1). Then is a solution of the modified Riccati equation with being the inverse function of , and Note that the function satisfies for every and if and only if . Observe also that Riccati equation (2.1) is a special case of (2.3) with , that is, .

In the investigation of perturbations of the half-linear Euler equation we will need the following criteria for (non)existence of a proper solution of (2.3). Recall that a solution of (2.3) is called proper if it exists on some interval . Nonexistence of a proper solution of (2.3) is equivalent to oscillation of (1.2) since it eliminates (via the transformation ) proper solutions of (2.1). For more details concerning this method, as well as the proof of the next two propositions, we refer to [11].

For the sake of the later application, we will write (2.3) in the form with continuous functions , , and .

Proposition 2.2. (i) If for large , then (2.6) possesses a (nonnegative) proper solution.
In the remaining part of the proposition suppose that Denote and suppose that
(ii) If then (2.6) has a proper solution.
(iii) If then (2.6) possesses no proper solution.

Proposition 2.3. Together with (2.6) consider the equation of the same form with the function satisfying for large . If the (majorant) equation (2.12) has a proper solution, then (2.6) has a proper solution as well.

Next, we recall basic properties of solutions of the “critical” half-linear Euler and Riemann-Weber differential equations as presented, for example, in [4]. Consider the half-linear Euler differential equation This equation is nonoscillatory if and only if . In the critical case , (2.13) has the solution , and every linearly independent solution is asymptotically equivalent (up to a multiplicative factor) to the function . The Riemann-Weber half-linear differential equation is nonoscillatory if and only if . In the critical case , (2.14) has the (so-called principal) solution which is asymptotically equivalent (up to a multiplicative factor) to the function , and every linearly independent solution is asymptotically equivalent to the function , see [4].

Finally, we recall the transformation method of the investigation of (1.9) which we extend in a modified form to half-linear equations. The Sturm-Liouville differential equation is the special case in (1.2). The transformation gives the identity (suppressing the argument ) In particular, if , then is a solution of (2.15) if and only if is a solution of the equation Let us emphasize at this moment that we have in disposal no half-linear version of transformation identity (2.16).

Let us denote First we apply the transformation to (1.9). Using (2.16) and the fact that is a solution of the critical Euler linear equation , we find that is a solution of the equation Now, we change the independent variable , the resulting equation is Here we take . Equation (2.20) is oscillatory by Kneser oscillation criterion if and nonoscillatory if . Indeed, since as , we have in (1.5) with as , and hence If , we can repeat the previous transformations and we obtain here . Now it should be clear how one can obtain the result of [5] concerning oscillation of (1.9). We repeat the transformation of dependent variable followed by the change of independent variable as long as the condition is satisfied.

As we have emphasized above, we have no half-linear version of the linear transformation identity (2.16). Consequently, the above procedure cannot be applied directly to (1.2). However, as observed, for example, in [6, 11], the modified Riccati equation in the linear case is which is just the Riccati equation associated with differential equation (2.17). Hence, modified Riccati equation can be regarded, in a certain sense, as a half-linear substitution for the linear transformation identity (2.16). This is just the idea which we develop in the next section and apply it in the investigation of the perturbed Euler equation.

3. Transformation of Modified Riccati Equation

As a starting point of this section we consider the modified Riccati equation in the form where the function is given by (2.4), the functions , are supposed to be continuous and . In this equation, we call the function the absolute term (since this term does not contain the unknown function ).

We consider the transformation with a positive differentiable function and with a function which we determine as follows. We have (again suppressing the argument , this argument we will suppress also now and then in the next parts of the paper) the following: Next we determine the function in such a way that the differential equation for is again an equation of the form (3.1) (in which ). Denote . The terms on the fourth line of the previous computation we will take as the first two terms in the function of the same form as in (3.1). Differentiating (3.4) with respect to , substituting , and setting the obtained expression equal to zero, we obtain hence Consequently, we obtain the transformed modified Riccati equation where

4. Perturbations of Euler Differential Equation

Now we apply the results of the previous section to the perturbed Euler half-linear differential equation where , and with .

To simplify the next computations, we denote The Riccati equation associated with (4.1) is

In order to better understand the next transformation procedure, we recommend the reader to compare it with the linear transformation idea described at the end of the previous section. The transformation with specified later, transforms (4.4) into with given by (3.8), that is, with , , and . This means that , where Hence, by a direct computation we obtain where

In (4.6), with the above given , we change the independent variable and the resulting equation is with

As the next step, we consider the modified Riccati equation where now

We apply the transformation , the quantity is again determined in such a way that we obtain a modified Riccati equation containing type function for . Hence, using the results from formula (3.8), with , , and , we have and using the binomial expansion Futher, Hence, the absolute term in the resulting modified Riccati equation is Observe that the term in and later in other can be differentiated because of its special form. Hence, if , we obtain In this equation we apply again the change of independent variable and the resulting equation is with , , and We use the notation in the next computations. With this notation, we have We apply the transformation to (4.22). We obtain where, with and , with Substituting into (4.25) the above computed quantities, we have Consequently, if , we obtain In this equation, the change of independent variable results Here, and also in the sequel, we use the notation where is the integer in (4.1).

Now we are already in a position to make the induction step in transformations of modified Riccati equations. We suppose that for for some , so we have with where is given by (4.21). We will also use the notation Then and with given by (4.3).

We put . We have and with , and given by (4.32) Then, using that and we have The last formula is the result of a direct computation where one needs to show that all terms with the faster growth than vanish. Further, again by a direct computation

Consequently, in the resulting modified Riccati equation for with we have as .

Now we can summarize the previous computations as follows.

Theorem 4.1. Suppose that there exists such that and . Then (1.11) is oscillatory if and nonoscillatory if . If (4.42) holds for all , (1.11) is nonoscillatory.

Proof. We apply Proposition 2.2 to the modified Riccati equation (4.40) for . In this equation, with the notation from Proposition 2.2, hence and Here (and also earlier), for a pair of functions , means . Consequently, if , what happens if and only if , modified Riccati equation (4.40) for has no proper solution in view of Proposition 2.2 (iii). Now, via the “back” transformations the same holds for the Riccati equation associated with (1.11) and hence this equation is oscillatory by Proposition 2.1.
If , nonoscillation of (1.11) follows from parts (i) (when ) and (ii) (when ) of Proposition 2.2 since the existence of a proper solution of the modified Riccati equation for implies the existence of a proper solution for the Riccati equation (4.4) associated with (1.11), hence this equation is nonoscillatory by Proposition 2.1.
Finally, if (4.42) holds for all , then the absolute term in the modified Riccati equation for is and replacing by its nonnegative part , we get a majorant of the modified Riccati equation for (in the sense of Proposition 2.3). The function satisfies the same asymptotic estimate as . To estimate the integral we proceed as follows. We have, via the substitution , , using the inequality , and followed by integration by parts, Consequently, hence the modified Riccati equation with for with instead of possesses a proper solution by Proposition 2.2 and, by Proposition 2.3, the Riccati equation for has the same property. This implies that (1.11) is nonoscillatory using the same argument as in the previous part of the proof.

5. Remarks and Comments

(i) In the previous section, we applied successively the transformation to the modified Riccati equation, followed by the change of independent variable . This change of the independent variable was motivated by the linear case and also by the fact that upon this transformation the modified Riccati equation simplifies. Without this change of independent variable, the transformation procedure can be “reformulated” as follows. As shown at the beginning of the previous section, the transformation (2.2), that is, , , transforms the Riccati equation (2.1) associated with (1.2) into the modified Riccati equation (2.3). The transformation (3.2), that is, , transforms (2.3) into an equation of the same form, with the function given by a relatively complicated formula (3.8). The composition of these transformations gives and by a direct computation, using (3.9), we have . So, the resulting modified Riccati equation for is just the modified Riccati equation resulting from (2.1) via (2.2) with replaced by . In this equation, the function is given by (2.5) with replaced by , that is, .

Now, consider the function In view of the previous consideration, the application of transformation (2.2) with this can be decomposed into the successive transformations , , . Hence, the successive transformations treated in the previous section can be replaced by just one transformation, with the transformation function (5.2).

This idea has been used in [3] in the case that in (1.11) and in (5.2). However, as shown in the computations of that paper (where also substantially the results of [12] have been used), this method is technically complicated even in this relatively simple case. This is also the reason why we developed the method of successive transformations of modified Riccati equation presented in the previous section.

(ii) The reason why the perturbation terms in (1.11) are just in the differential term and by is motivated by the fact that in this form they “match together”. More precisely if we replace some of them by a term with a faster asymptotic growth, then this term “overrules” the remaining terms and the equation becomes (non)oscillatory for any positive value of the corresponding parameter or . On the other hand, functions with slower asymptotic growth have no influence on the oscillatory behavior. These considerations are closely related to concepts of strong (non)oscillation of half-linear equations as treated for example in [13].

(iii) In [6], and partially also in [11], we have considered as a perturbation of (1.2). We found assumptions of the functions , , , (which are satisfied in case of the perturbed Euler equation) which guarantee that there exists a constant such that (5.3) is oscillatory if and nonoscillatory if . The limiting case remained undecided, mainly because of technical computational problems. In view of perturbations of Euler equation with , , , and (then ) we hope to prove that (5.3) is nonoscillatory also in the limiting case . We also hope that the method of transformations of modified Riccati equation elaborated in this paper can be applied to treat the “multiparametric” general case, not only for perturbations of Euler equation.

Acknowledgment

The authors thank the referee for his/her remarks which contributed to better presentation of the result of the paper. Research supported by the Grant 201/11/0768 of the Grant Agency of the Czech Republic and the Research Project MUNI/A/0964/2009 of Masaryk University.

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