Abstract and Applied Analysis
Volume 2011 (2011), Article ID 182827, 16 pages
doi:10.1155/2011/182827

Research Article

Two-Parametric Conditionally Oscillatory Half-Linear Differential Equations

Ondřej Došlý¹ and Simona Fišnarová²

¹Department of Mathematics and Statistics, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic
²Department of Mathematics, Mendel University in Brno, Zemĕdĕlská 1, 613 00 Brno, Czech Republic

Received 2 November 2010; Accepted 5 January 2011

Academic Editor: Miroslava Růžičková

Copyright © 2011 Ondřej Došlý and Simona Fišnarová. Ondřej Došlý and . 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 study perturbations of the nonoscillatory half-linear differential equation , , . We find explicit formulas for the functions , such that the equation is conditionally oscillatory, that is, there exists a constant such that the previous equation is oscillatory if and nonoscillatory if . The obtained results extend the previous results concerning two-parametric perturbations of the half-linear Euler differential equation.

1. Introduction

Conditionally oscillatory equations play an important role in the oscillation theory of the Sturm-Liouville second-order differential equation with positive continuous functions , . Equation (1.1) with instead of is said to be conditionally oscillatory if there exists , the so-called oscillation constant of (1.1), such that this equation is oscillatory for and nonoscillatory for . A typical example of a conditionally oscillatory equation is the Euler differential equation which has the oscillation constant as can be verified by a direct computation when looking for solutions of (1.2) in the form . This leads to the classical Kneser (non)oscillation criterion which states that (1.1) with is oscillatory provided and nonoscillatory if This shows that the potential is the border line between oscillation and nonoscillation. Note that the concept of conditional oscillation of (1.1) was introduced in [1].

The linear oscillation theory extends almost verbatim to the half-linear differential equation including the definition of conditional oscillation. The half-linear version of Euler equation (1.2) is the equation which has the oscillation constant , and (non)oscillation criteria (1.3), (1.4) extend in a natural way to (1.5) with . A complementary concept to the conditional oscillation is the concept of strong (non)oscillation. Equation (1.5) with instead of is said to be strongly (non)oscillatory if it is (non)oscillatory for every . Sometimes, strongly oscillatory equations are regarded as conditionally oscillatory with the oscillation constant and strongly nonoscillatory as conditionally oscillatory with the oscillation constant . We refer to [2] for results along this line.

In our paper, we are motivated by a statement presented in [3, 4], where the two-parametric perturbation of the Euler differential equation with the critical coefficient is investigated. It is shown there that the equation is oscillatory if and nonoscillatory in the opposite case. Note that an important role in proving the results of [4] is played by the fact that we know explicitly the solution of (1.7).

Here, we treat the problem of conditional oscillation in the following general setting. We suppose that (1.5) is nonoscillatory and that is its eventually positive solution. We find explicit formulas for the functions , such that the equation is conditionally oscillatory, that is, there exists a constant such that (1.9) is oscillatory if and nonoscillatory if .

The setup of the paper is as follows. In the next section, we present some statements of the half-linear oscillation theory. Section 3 is devoted to the so-called modified Riccati equation associated with (1.5) and (1.9). The main result of the paper, the construction of the functions , such that (1.9) is two-parametric conditionally oscillatory, is presented in Section 4.

2. Auxiliary Results

As we have already mentioned in the previous section, the linear oscillation theory extends almost verbatim to half-linear equation (1.5). The word “almost” reflects the fact that not all linear methods can be extended to (1.5), some results for (1.5) are the same as those for (1.1), but to prove them, one has to use different methods than in the linear case. A typical method of this kind is the following transformation formula. If is a sufficiently smooth function and functions , are related by the formula , then we have the identity where In particular, is a solution of (1.1) if and only if is a solution of the equation . The transformation identity (2.1) does not extend to (1.5).

To illustrate the meaning of this fact in the conditional oscillation of (1.1) and (1.5), suppose that (1.1) is nonoscillatory and let be its so-called principal solution (see [5, Chapter XI]), that is, a solution such that . We would like to find a function such that the equation is conditionally oscillatory and to find its oscillation constant. The transformation transforms (1.1) into the one term equation and the transformation of independent variable further to the equation . Now, from (1.2), we know that the “right” perturbation term in the last equation is with the oscillation constant . Substituting back for , we get the conditionally oscillatory equation and the back transformation results in the conditionally oscillatory equation with the oscillation constant . The previous result is one of the main statements of [6], but it was proved there by a different method.

In the next section, we will show how to modify this method to be applicable to half-linear equations. At this moment, we present the result of [7] with the classical (i.e., one parametric) conditional oscillation of (1.5). Let be a positive solution of (1.5) such that for large . We denote

Theorem 2.1. Suppose that (1.5) possesses a nonoscillatory solution such that for large , and , are given by (2.6). If then the equation is conditionally oscillatory, and its oscillation constant is , where is the conjugate exponent to , that is, .

Note that in the linear case , the function is a solution of (2.9) with . In the general half-linear case, an explicit solution of (2.9) is no longer known, but we are able to “estimate” this solution. The next statement, which is also taken from [7], presents a result along this line.

Theorem 2.2. Suppose that (2.8) holds and let , then a solution of (2.9) with is of the form and (suppressing the argument ) as .

The last statement presented in this section is the so-called reciprocity principle. Let be a solution of (1.5) and let be its quasiderivative, then is a solution of the reciprocal equation where is the inverse function of .

3. Modified Riccati Equation

Suppose that and in (1.9) are such that . Let in an interval be a solution of (1.9), and let . Then, solves in the “standard” Riccati equation More precisely, the following statement holds.

Lemma 3.1 ([8, Theorem  2.2.1]). The following statements are equivalent: (i)equation (1.9) is nonoscillatory; (ii)equation (3.1) has a solution on an interval ; (iii)there exists a continuously differentiable function such that on an interval .

In the linear case, if is a solution of (1.1), , and is the Riccati variable corresponding to the equation on the right-hand side in (2.1), then where , . This suggests to investigate the function in the half-linear case, and this leads to the modified Riccati equation introduced in the next statement which is taken from [4] with a modification from [3].

Lemma 3.2. Suppose that is a positive differentiable function, , and is a continuously differentiable function, and put , then the following identity holds: where In particular, if is a solution of (3.1), then is a solution of the modified Riccati equation Conversely, if is a solution of (3.6), then is a solution of (3.1).

Observe that in case , the modified Riccati equation (3.6) reduces to the standard Riccati equation (3.1).

Next, we will investigate the function in (3.5). First, we present a result from [4, Lemmas  5 and 6].

Lemma 3.3. The function defined in (3.5) has the following properties. (i) with the equality if and only if . (ii)If , one has the inequality

Now, we concentrate on an estimate of the function in case .

Lemma 3.4. Suppose that and , then there is a constant such that for , , and large

Proof. Consider the function First of all, Now, we compute local extrema of with respect to . We have (suppressing the argument ) Denote the function in braces on the last line of the previous computation. We have , if and only if and , and This means that is the local minimum and is the local maximum of the function . Using this result, an examination of the graph of the function shows that this function has the local minimum at and a local maximum in the interval if , and this maximum is in if . Next, denote the value for which . Consequently, for any , it follows from (3.10) that where Next, we want to investigate the dependence of this infimum on when . To this end, we investigate the function for , being a parameter. We have (using the expansion formula for ) as . Consequently, if , there exists a constant such that (3.8) holds.

Now, we are ready to formulate a complement of [9, Theorem  2] which is presented in that paper under the assumption that the function is bounded.

Theorem 3.5. Let be a positive continuously differentiable function such that for large . Suppose that , where , for large , and , then all possible proper solutions (i.e., solutions which exist on some interval of the form ) of the equation are nonnegative.

Proof. First consider the case . Let be arbitrary. By Lemma 3.4, there exists and such that for and , Suppose that is the solution of (3.17) such that for some , then for for which the solution exists. Now, we use the same argument as in the proof of Theorem  2 in [9]. Consider the equation This is the standard Riccati equation corresponding to the half-linear equation Assumptions of theorem imply, by [8, Corollary  4.2.1], that all proper solutions of (3.20) are nonnegative. It means that any solution of (3.20) which starts with a negative initial condition blows down to in a finite time. Inequality (3.19) implies that if is the solution of (3.20) satisfying , that is, starts with the same initial value as the solution of (3.17), then decreases faster than . In particular, if blows down to at a finite time, then does as well. This means that all proper solutions of (3.17), if any, are nonnegative.
In case , we proceed in a similar way. We use (3.7) and we compare (3.17) with the equation which is the standard Riccati equation corresponding to the linear equation Then, reasoning in the same way as in case , we obtain the conclusion that all proper solutions of (3.17) are nonnegative also in this case.

4. Two-Parametric Conditional Oscillation

Recall that is a positive solution of (1.5) such that for large , is its quasiderivative, , are given by (2.6), and is given by (2.7). Recall also that the quasiderivative is a solution of the reciprocal equation (2.12), denote by the “reciprocal” analogues of and , and define

Our main result reads as follows.

Theorem 4.1. Suppose that conditions (2.8) hold. Further, suppose that and that there exist limits the second one being finite, where . If , then (1.9) is nonoscillatory; if , then it is oscillatory.

Proof. First consider the case in (1.9), that is, we consider the equation The quantities and defined in (4.1) satisfy hence, integrating by parts, Consequently, conditions (2.8) imply that corresponding conditions for and also hold. This means, in view of Theorem 2.1 (applied to the reciprocal equation (2.12)), that the equation is oscillatory for and nonoscillatory in the opposite case.
The reciprocal equation to (4.5) is the equation Since (4.3) holds, we have as . Hence, we can rewrite (4.9) in the following form: Let what is equivalent to , then, in view of (4.3), there exists such that , hence, for large , This means that the equation is a majorant of (4.9) and this majorant is nonoscillatory by Theorem 2.1 applied to (4.8). So (4.9) is also nonoscillatory, and hence (4.5) is nonoscillatory as well. The same argument implies oscillation of (4.5) if .
Now, we turn our attention to the general case . Let , and consider the term appearing in the modified Riccati equation (3.6), where the operators , are defined by (3.4). In order to use the asymptotic formula from Theorem 2.2, we write , where
Let be the second limit in (4.4), that is, The leading term in the expression is by Theorem 2.2, while, concerning the asymptotics of , as . The existence of the first limit in (4.4) implies that there exists the limit The limit in (4.18) must be 0, which follows from the l'Hospital rule and the fact that the integral of is divergent, while the integral of is convergent by the second assumption in (2.8). This means that the term dominates ; hence, for large if and for large if .
Now, it remains to prove that these inequalities imply (non)oscillation of (1.9) and that .
To prove the nonoscillation, let , that is, for large , and let be defined by (3.5). By Lemma 3.3(i) is a solution of the inequality for large , and by identity (3.3) in Lemma 3.2 we obtain that satisfies the Riccati inequality (3.2), that is, (1.9) is nonoscillatory by Lemma 3.1(iii).
To prove the oscillation, let , that is, for large . Observe that for and hence , which consequently means that . Here, we have used the fact that the integral of the leading term in and also integrals of other terms in the asymptotic formula of Theorem 2.2 are convergent, see [7, page 161]. Suppose, on the contrary, that (1.9) is nonoscillatory. Then by Lemma 3.1, there exists a solution of the associated Riccati equation (3.1) for large and, by Lemma 3.2, the function , where , is a solution of the modified Riccati equation (3.6) for large . Integrating (3.6), we get Now, we use Theorem 3.5. In view of (2.8) and (4.3), we have for , and hence Consequently, by Theorem 3.5. This means that the left-hand side in (4.21) is bounded above as , while the right-hand side tends to which yields the required contradiction proving that (1.9) is oscillatory if .
Finally, consider again the case . In that case, we proved in the first part of the proof that (1.9) is oscillatory or nonoscillatory depending on whether or . This shows that the second limit in (4.4) must be −1.

Remark 4.2. (i) From the proof of Theorem 4.1, it follows that if the first limit in (4.4) exists, then conditions (2.8) imply that this limit is 1, and the assumptions of the theorem imply that if the second limit in (4.4) exists and is finite, then it is −1.
(ii) Theorem 4.1 can be applied to the Euler equation (1.7), and one can obtain the same result for (1.8) as in [4, Corollary  3]. Indeed, in this case, we have , , , where and by a direct computation hence, which mean that conditions (2.8) and (4.3) are satisfied. Concerning the limits in (4.4), we have that is, the first limit in (4.4) is 1. Next, and consequently, Using this formula, and hence, as . This means that the second limit in (4.4) is −1. According to Theorem 4.1, we obtain that the equation is nonoscillatory if and oscillatory if . If we denote and , we see that (1.8) (with , instead of , , resp.) is nonoscillatory if , and it is oscillatory if , that is, we have the statement from [4].
(iii) In [3], it is proved that (1.8) is nonoscillatory also in the limiting case . We conjecture that we have also the same situation in the general case, that is, (1.9) is nonoscillatory also in the case .

Acknowledgments

This research was supported by Grants nos. P201/11/0768 and P201/10/1032 of the Czech Science Foundation and the Research Project no. MSM0021622409 of the Ministry of Education of the Czech Republic.

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