Abstract and Applied Analysis

Volume 2014 (2014), Article ID 258159, 14 pages

http://dx.doi.org/10.1155/2014/258159

Research Article

Conditional Oscillation of Half-Linear Differential Equations with Coefficients Having Mean Values

Petr Hasil,¹ Robert Mařík,² and Michal Veselý¹

¹Department of Mathematics and Statistics, Faculty of Science, Masaryk University, Kotlářská 2, 611 37 Brno, Czech Republic

²Department of Mathematics, Faculty of Forestry and Wood Technology, Mendel University in Brno, Zemědělská 1, 613 00 Brno, Czech Republic

Received 3 February 2014; Accepted 12 June 2014; Published 8 July 2014

Academic Editor: Yuriy Rogovchenko

Copyright © 2014 Petr Hasil et al. 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 prove that the existence of the mean values of coefficients is sufficient for second-order half-linear Euler-type differential equations to be conditionally oscillatory. We explicitly find an oscillation constant even for the considered equations whose coefficients can change sign. Our results cover known results concerning periodic and almost periodic positive coefficients and extend them to larger classes of equations. We give examples and corollaries which illustrate cases that our results solve. We also mention an application of the presented results in the theory of partial differential equations.

1. Introduction

In this paper, we analyse oscillatory properties of the half-linear differential equation

where and are continuous functions, is positive, and . To describe our main interest, let us consider (1) with for a continuous function and . We say that such an equation is conditionally oscillatory if there exists the so-called oscillation constant such that the equation under consideration is oscillatory for and nonoscillatory for . In fact, the oscillation constant depends on coefficients and .

Looking back to the history (according to our best knowledge), the first attempt to this problem was made by Kneser in [1], where the oscillation constant for the linear equation

has been identified as . Later, in [2, 3], it has been shown that the conditional oscillation remains preserved also for periodic coefficients. More precisely, the equation

where , are positive -periodic continuous functions, is conditionally oscillatory for

We also refer to more general results in [4–8].

Since a lot of results from the linear oscillation theory are extendable to the half-linear case (see, e.g., [9, 10]), it is reasonable to suppose that the oscillation constant can be found for the corresponding Euler-type half-linear equations as well. This hypothesis has been shown to be true for

in [11] (see also [12]). Later, this result has been extended in a number of papers (e.g., [13–17]), where equations of the below given form (6) have been treated with coefficients , replaced by perturbations consisting of constant or periodic functions and iterated logarithms. Nevertheless, the most general result (concerning the topic of this paper) can be found in [18], where the equation

is treated for positive asymptotically almost periodic functions and satisfying

It is shown that (6) is conditionally oscillatory with the oscillation constant

where stands for the mean value of function . This result is the main motivation of our current research. Our goal is to remove the condition of positivity of function and, at the same time, to extend the class of functions , as much as possible applying the used methods. We present an oscillation criterion which is new in the half-linear case as well as in the linear one.

We should mention some relevant references from the discrete and time scale theory. In this paper, we give only the most relevant references concerning the topic. The reader can find more comprehensive literature overview together with historical references in our previous article [18]. Here, we refer at least to [19, 20] for the corresponding results about difference equations (see also, e.g., [21, 22]) and to [23–25] for results about dynamical equations on time scales.

The paper is organized as follows. In the next section, we mention the necessary background and we recall the basics of the Riccati technique. In Section 3, we prove preparatory lemmas and our results. We also state several corollaries, concluding remarks, and examples. In the last section, we give an application in the theory of partial differential equations.

2. Preliminaries

Let be arbitrarily given and let be the real number conjugated with satisfying

As usual, for given , the symbol stands for .

To prove the main results, we will apply the Riccati technique for (1), where the transformation

leads to the half-linear Riccati differential equation

whenever . For details, we refer to [10]. The fundamental connection between the nonoscillation of (1) and the solvability of (11) is described by the following theorem.

Theorem 1. Equation (1) is nonoscillatory if and only if there exists a function which solves (11) on some interval .

Proof. The theorem is a consequence of the well-known roundabout theorem (see, e.g., [10, Theorem ]).

We will also use the Sturmian comparison theorem in the form given below.

Theorem 2. Let , be continuous functions satisfying for all sufficiently large . Let one consider (1) and the equation

(i)If (1) is oscillatory, then (12) is oscillatory as well.(ii)If (12) is nonoscillatory, then (1) is nonoscillatory as well.

Proof. The theorem follows, for example, from [10, Theorem ].

Now we recall the concept of mean values which is necessary to find an explicit oscillation constant for general half-linear equations.

Definition 3. Let continuous function be such that the limit

is finite and exists uniformly with respect to . The number is called the mean value of .

In fact, we will study (1) in the form

where is a continuous function having mean value and satisfying

and is a continuous function having mean value . We repeat that the basic motivation comes from [18], where asymptotically almost periodic half-linear equations are analysed. Since positive nonvanishing asymptotically almost periodic functions have positive mean values and they are bounded, we will consider more general equations (cf. (15) with (7) as well).

The Riccati equation associated to (14) has the form (see (11))

Finally, using the substitution , we obtain the adapted Riccati equation

which will play a crucial role in the proof of the announced result (see the below given Theorem 8).

3. Results

To prove the announced result, we need the following lemmas.

Lemma 4. If there exists a solution of (17) on some interval , then (14) is nonoscillatory.

Proof. A solution of (17) on an interval gives the solution of (16) on the same interval. Thus, the lemma follows from Theorem 1.

Lemma 5. Let (14) be nonoscillatory and let there exist such that

For any solution of (16) on , it holds

Proof. The lemma follows, for example, from [10, Theorem ], where it suffices to use (15).

Lemma 6. If (14) is nonoscillatory, then there exists a solution of (17) on some interval with the property that for all and for some .

Proof. Considering Theorem 1, the nonoscillation of (14) implies that there exists a solution of (16) on some interval which gives the solution of (17) on the interval. We show that this solution is bounded above.

At first, we prove the convergence of the integral

and the inequality

Evidently, it suffices to prove (20) and

Let be such that

where we use directly Definition 3 (the existence of ). The symbols and will denote the positive and negative parts of function , respectively. We choose . We can express

For an arbitrarily given positive integer , we have

if , and

if . Using

and using (23), (25), and (26), we obtain the existence of such that it holds

Since is arbitrary, it also holds

for all sufficiently large .

Hence, the integral is convergent because

where is sufficiently large. Particularly,

Moreover, we have (see (29))

Thus, (22) is valid; that is, there exists for which (21) is valid.

Integrating (16), we obtain

We know that

Indeed, considering (18) together with (31), one can get (19) from Lemma 5. From (20) and (33) it follows that there exists the limit . In addition, the convergence of the integral in (34) gives

Again, we consider arbitrarily given . We can rewrite (33) into (or see directly (16))

Putting , from (20), (34), (35), and (36), we obtain

Finally, let us denote , where

We know that (see (21) and (37))

We denote . If is positive, then the statement of the lemma is true for all . Therefore, we can assume that . Since is nonincreasing and , function is nonnegative. From (39) it follows

Hence, we have

and, consequently, we obtain that , . It means that the statement of the lemma is valid for .

Remark 7. Let (14) be nonoscillatory. If the considered function is positive for all , then the statement of Lemma 6 is true for a negative solution of (17). See, for example, [10, Lemma ].

Theorem 8. Equation (14) is oscillatory if and nonoscillatory if .

Proof. The proof is organized as follows. In the first part, we derive upper bounds for two integrals involving function . Then we prove the oscillatory part and, finally, the nonoscillatory part.

At first, we use the existence of and the continuity of function . Considering Definition 3, there exists with the property that

and, consequently, there exists with the property that

We can rewrite (43) into the form

Using (42), we obtain

that is,

Combining (44) and (47), we have

where . Since the function is decreasing and positive on , it holds

for all and for some . Analogously, for any , there exists such that

Hence, from (48) it follows

Now we prove the oscillatory part. Let . By contradiction, in this part of the proof, we will suppose that (14) is nonoscillatory. Lemma 6 says that there exists a solution of (17) on some interval and that for all and for a certain number . Evidently, we can assume that .

We show that there exists satisfying

On the contrary, let us assume that . Let for all from some interval , where for some , and let be such that (see (15))

Indeed, . We can assume that is increasing for . Using (17), (51), and (53), it holds

Thus, for all which proves (52). Indeed, it suffices to consider . In addition, we can assume that ; that is,

Thus (see directly (17) and (51)), we have

for all , where . The previous inequality implies

Considering Definition 3 and , there exist and such that

and, at the same time, such that

For such an integer , we define

Since

we have

Hence, to prove the first implication in the statement of the theorem, it suffices to show that (62) is not true.

From (57) it follows

where

Particularly (see (60)), (63) gives

Next, we consider the function

If for some , then . Henceforth (in this paragraph), we consider the case when , . Let us define

It can be directly verified that function has the global minimum

It means that , . Particularly, it gives the inequality

Considering (59) and (69), we have

Applying (63), the inequalities , , and the fact that the function has the Lipschitz property on any bounded set, there exists such that

Hence (see also (59)), we get

From (70) and (72), we know that

Of course, (73) remains true for as well.

Let us consider for which

Note that the existence of such a number follows from (65). It is seen that (73) and (74) imply

Evidently, we can consider the solution in an arbitrarily given neighbourhood of . Hence, we can assume that

From (48) and (76), we see that

from (55) and (77), we have

and, analogously, from (15), (55), and (78) it follows

For all , using (58), (75), (79), (80), and (81), we obtain

Thus, it holds

Since

we obtain that . The contradiction with (62) proves the first implication.

In the nonoscillatory part of the proof, we consider . Let and satisfy

Let us consider solution of (17) given by for some sufficiently large . Since the right-hand side of (17) is continuous, the considered solution can be defined on an interval , where . In addition, if , we can assume that

If , then the considered solution of (17) satisfies the condition of Lemma 4. It means that it suffices to find for which

As in the oscillatory part of the proof (see (52)), we can prove that for some and for all . Indeed, we can analogously show that the inequality cannot be valid for any , where is taken from (48) and from (53). We want to prove that . On the contrary, let (86) be valid for some . Particularly, solution has to be positive on some interval in this case.

We denote

and we compute

We know that is negative on an interval . Let have the property that . For all , , we have (see (51))

Thus, for general satisfying , we have

We can assume that is so large that

if (see (88)). Particularly (), we can define the function

for all and for all when . Particularly, let be so large that .

We repeat that we assume the positivity of which implies the inequality for from some interval. The continuity of gives the existence of such that

From (91) it follows that, for any , one can choose so large that

Thus (see (93)), we can assume that

Consequently, let

At the same time, we can assume that was chosen in such a way that it is valid

Using (97) and (98), we have

Since (see (89), (94))

we have

Let be so large that (see (48) and also (79))

and (see (15) together with (96) and also (81))

Considering (85), (101), (102), and (103), we obtain

This contradiction (see (94)) means that (87) is true for and . Since (86) cannot be valid for any , the considered solution exists on interval . We repeat that the nonoscillation of (14) actually follows from Lemma 4.

The following theorem is a version of Theorem 8 which is ready for applications to the half-linear equations written in the form common in the literature.

Theorem 9. Let be a continuous function, for which mean value exists and for which it holds

and let be a continuous function having mean value . Let

Consider the equation

Equation (107) is oscillatory if and nonoscillatory if .

Proof. Let . Equation(107) can be rewritten into the form

that is,

Equation (109) has the form of (14) for

Note that and and that (15) follows from (105). Thus (see Theorem 8), (107) is oscillatory for

and nonoscillatory if the opposite inequality holds.

It remains to consider the case when . Of course, there exists such that . We know that the equation

is nonoscillatory. Now it suffices to use Theorem 2 (ii).

Remark 10. For reader’s convenience, we consider (107) (instead of (14)) in Theorem 9. The form of (107) shows how the presented result improves the known ones (see Section 1). Particularly, we get new results in two important cases, when function changes sign and when it is unbounded. For details, we refer to our previous paper [18].

Remark 11. For , it is not possible to decide whether (107) is oscillatory or nonoscillatory for general functions , satisfying the conditions from the statement of Theorem 9. It follows, for example, from the main results of [13, 16]. One of the most studied classes of functions which have mean values is formed by almost periodic functions. Based on the constructions from [26], it is conjectured in [18] that the case is not generally solvable (in the sense whether it is oscillatory or nonoscillatory) even for almost periodic coefficients of (107). It means that there exist almost periodic functions , such that and (107) is oscillatory. At the same time, there exist different almost periodic functions , satisfying with the property that (107) is nonoscillatory. We add that the case of periodic functions , was proved to be nonoscillatory (see again [13, 16]).

To illustrate Theorem 9, we mention at least two examples.

Example 12. For , , and , let us consider the equation

Equation (113) has the form of (107) for

It can be directly verified that

and that

Hence, (113) is oscillatory if and nonoscillatory if ; that is, (113) is oscillatory for and nonoscillatory for , where . Since is oscillatory, the other related results in the literature give no conclusion for (107).

Example 13. Let and be arbitrarily given. We define the function by

It is seen that and that . Analogously, for and , we define as

We have (105) and . Thus, (107) for the above given functions , is oscillatory if and nonoscillatory if . Since is not bounded, the other related results in the literature give no conclusion for this equation.

Theorem 9 gives a new result for linear equations as well. Therefore, we mention the following corollary.

Corollary 14. Let be continuous functions having mean values , and let (15) hold. Then, the equation

is oscillatory if and nonoscillatory if .

Example 15. Using Corollary 14, we can decide about the oscillation and nonoscillation of several equations. Let and be relative prime. Let be arbitrary positive continuous functions with the property that

The equations

are oscillatory for and nonoscillatory for . Note that this simple criterion does not follow from known ones.

Now we consider the case of negative coefficient .

Corollary 16. Let be continuous functions such that function has mean value and that there exists mean value . Let

be fulfilled. The equation

is oscillatory if and nonoscillatory if .

Proof. To prove the corollary, it suffices to use Theorem 9 and the fact that the solution space of half-linear equations is homogeneous.

One can prove a number of consequences using different types of comparison theorems. For example, applying Theorem 9 and [10, Theorem ], we immediately obtain the next new result.

Corollary 17. Let be continuous functions having mean values , . Let (15) hold and let

Let us consider the equation

where is a continuous function satisfying .

(i)If there exists for which

and if , then (125) is oscillatory.(ii)If there exists for which

and if , then (125) is nonoscillatory.

In the following example, we demonstrate that our result is applicable even if the mean value of the second coefficient does not exist.

Example 18. Let and . Putting

we define the continuous function . Evidently, mean value does not exist. Since

we have

Considering

and (129), there exists a positive continuous function with the property that

and that . Using (132), we obtain

In addition, and

It is also seen that

for all sufficiently large . The equations

are oscillatory for and nonoscillatory for (see Theorem 9). Indeed,

Therefore (see Corollary 17 together with (130), (133), (134), and (135)), we know that the equation

is oscillatory if and nonoscillatory if .

4. An Application

In this section, we use our main result to derive a theorem related to elliptic partial differential equations with -Laplacian and the power-type nonlinearity

where , is an elliptic matrix function with differentiable components, and is a Hölder continuous function. As a solution of (139) in , we understand a differentiable function such that is also differentiable and satisfies (139) in .

The following notation is used. We consider the usual Euclidean norm

and the induced matrix norm

and , stand for the smallest and largest eigenvalues of matrix , respectively. From the fact that is positive definite, it follows that .

Denote . We say that a solution of (139) is oscillatory if it has a zero in for every . Equation (139) is said to be oscillatory if every solution of this equation is oscillatory. Otherwise, (139) is said to be nonoscillatory.

In general, we distinguish two types of oscillation in the theory of (139), namely, the (weak) oscillation defined in the previous paragraph and the so-called (strong) nodal oscillation, which is based on nodal domains (i.e., bounded domains such that the equation possesses a nontrivial solution which vanishes on the boundary of this domain). Concerning this concept of oscillation, (139) is said to be nodally oscillatory if every solution has a nodal domain outside of any ball in and to be nodally nonoscillatory in the opposite case. It is known that the nodal oscillation implies oscillation. The opposite implication has been proved only in the linear case (see [27]) and remains an open question in the half-linear multidimensional case. We also refer to [28] which relates the weak oscillation of linear PDEs (and nodal oscillation) with the finiteness of negative spectrum of the Laplace operator, which is of interest in physical applications.

In [29, 30], there is proved a theorem which allows to deduce the oscillation of certain half-linear partial differential equations from the oscillation of ordinary differential equations if is either the identity matrix or a scalar multiple of the identity matrix. This theorem has been later extended in [31] (see also [32]) as follows.

Theorem 19. Let us define the functions

If the equation

is oscillatory, then (139) is oscillatory as well.

We can easily apply the oscillation part of Theorems 9 and 19 to obtain the following result.

Theorem 20. Let and be defined by (142). Suppose that

and that and exist. If

then (139) is oscillatory.

Remark 21. Note that, in contrast to Theorem 9, we lack the nonoscillation part in Theorems 19 and 20 because there is a principal problem with nonoscillation criteria for partial differential equations via the Riccati method. A detailed discussion related to the relationship of the Riccati equation and the nonoscillation of second-order equations (in the multidimensional case) can be found in [33].

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

Petr Hasil and Robert Mařík are supported by Czech Science Foundation under Grant P201/10/1032. Michal Veselý is supported by the project “Employment of Best Young Scientists for International Cooperation Empowerment” (CZ.1.07/2.3.00/30.0037) cofinanced from European Social Fund and the state budget of the Czech Republic.

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