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Abstract and Applied Analysis
Volume 2013 (2013), Article ID 432936, 12 pages
http://dx.doi.org/10.1155/2013/432936
Research Article

Oscillation and Nonoscillation of Asymptotically Almost Periodic Half-Linear Difference Equations

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

Received 18 March 2013; Accepted 20 May 2013

Academic Editor: Ondrej Dosly

Copyright © 2013 Michal Veselý and Petr Hasil. 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 analyse half-linear difference equations with asymptotically almost periodic coefficients. Using the adapted Riccati transformation, we prove that these equations are conditionally oscillatory. We explicitly find a constant, determined by the coefficients of a given equation, which is the borderline between the oscillation and the nonoscillation of the equation. We also mention corollaries of our result with several examples.

1. Introduction

We intend to study the oscillation behaviour of the second-order half-linear difference equation where , , and for all considered . Throughout this paper, we consider integers for a sufficiently large . Concerning the oscillation and nonoscillation of these equations, many results have been obtained in the last years. The basic facts of the oscillation theory for (1) are established in [1]. Several criteria for (non)oscillation of (1) are proved, for example, in [26] (for (1) with , see [79]; for the continuous case, see [1012]). Oscillatory properties of slightly more general difference equations are investigated, for example, in [13, 14]. A detailed literature overview can be found in [15].

The oscillation theory of half-linear difference equations is motivated by the continuous case and by the linear case.

Equation (1) is the discretization of the half-linear differential equation The first occurrence of an equation of this form is dated back to 1961 (see [16]). But [17, 18] are regarded as the basic papers in this field. For a comprehensive overview of half-linear differential equations, we refer to [19] with references cited therein. Note that techniques needed in the discrete theory are often more complicated than their continuous counterparts. In this paper, we apply the Riccati method which is used in the continuous case as well. Comparing this paper with its continuous counterpart [20], one can see that the discrete method requires several nontrivial extra steps and is technically more exacting. Particularly, we cannot use the fact that the ranges of considered solutions are intervals.

Equation (1) is a generalization of the linear equation whose oscillation theory is well described in the literature (see [21, 22] and references therein). There are limitations of the tools known from the theory of linear equations (e.g., the transformation theory and the Casoratian identity). These limitations are more or less consequences of the fact that the solution space of half-linear equations is not additive. Nevertheless, it remains homogeneous (likewise in the linear case). This observation motivates the term “half-linear.”

Actually, we consider (1) in the form where , are positive sequences, , and stands for the so-called generalized power function. Our objective is to prove that (4) is conditionally oscillatory and to find the corresponding oscillation constant. We recall that (4) is conditionally oscillatory if there exists with the property that (4) is oscillatory for and nonoscillatory for . In this case, is called the oscillation constant of (4). Note that we consider the form with the generalized power function because we want to be consistent with previous results and our method is much more transparent in this case.

According to the best of our knowledge, the first result about the conditional oscillation of a second-order equation was proved by Kneser back in 1893 (see [23]). In [23], it is shown that the differential equation is conditionally oscillatory with the oscillation constant . The corresponding difference equation is conditionally oscillatory with the oscillation constant as well (see [24]). In [25], the previous Kneser result is generalized for the equation where are positive periodic continuous functions. We can also mention [26] and more generally [27, 28]. For the discrete linear equation we refer to [29], where the corresponding result is obtained. Nevertheless, the first discrete version of the result of [25] was proved in [30]. The oscillation constant from [30] coincides with the one from [29] if the considered coefficients are asymptotically constant. In fact, the conditional oscillation of (8) with almost periodic coefficients is proved in [29]. The oscillation constant for the periodic half-linear equation is derived in [31] (see also [32]) and the oscillation constant for (9) with asymptotically almost periodic coefficients is obtained in [20]. Our aim is to prove the discrete counterpart of the main result of [20]; that is, we find the oscillation constant for (4) with asymptotically almost periodic coefficients. We add that this result is new in the periodic case and in the linear case as well.

The paper is organized as follows. In Section 2, we recall elements of the theory of almost periodic and asymptotically almost periodic sequences, which we will use later. In Section 3, we mention basic definitions from the oscillation theory and we derive the adapted Riccati equation. Using the prepared Riccati technique, we prove the announced result in Section 4. Then, we state its corollaries and we illustrate them by examples. We also show a way how the main result can be applied for equations with coefficients which change their signs.

Henceforth, let be arbitrarily given and let be such that Thus, for , we have . For the reader’s convenience, we recall the generalized power function (sometimes called the falling factorial power) where and the identities which we will often apply later. We also refer to [22, Chapter 2].

2. Asymptotic Almost Periodicity

We begin with the notion of almost periodicity. The fundamental results of the theory of almost periodic sequences can be found, for example, in [33, Section ].

Definition 1. A sequence is called almost periodic if, for every , there exists an integer with the property that any set consisting of consecutive integers contains at least one number for which

Definition 2. A continuous function is called almost periodic if, for every , there exists a number such that any interval of length of the real line contains at least one point satisfying

Theorem 3. A necessary and sufficient condition for a sequence to be almost periodic is the existence of an almost periodic function with the property that for all .

Proof. See [33, Theorem 1.27].

Theorem 4. Any almost periodic sequence is bounded.

Proof. The statement of the theorem follows from Theorem 3 and from the boundedness of any almost periodic function on the real line (see, e.g., [33, Theorem 1.2]).

Theorem 5. A sequence is almost periodic if and only if any sequence of the form , where , has a uniformly convergent subsequence with respect to .

Proof. See [33, Theorem 1.26].

The following result is a consequence of Theorems 4 and 5.

Corollary 6. The sum and the product of two almost periodic sequences are almost periodic as well.

Theorem 7. If is almost periodic, then the limit is finite and exists uniformly with respect to .

Proof. See [33, Theorem 1.28].

Theorem 8. If is almost periodic and is a uniformly continuous function on the set , then the composition is almost periodic.

Proof. The theorem follows, for example, from [34, Theorem 2.4].

As a direct generalization of almost periodicity, we consider the asymptotic almost periodicity. We refer to [35], where asymptotically almost periodic sequences were introduced with their basic characteristics.

Definition 9. We say that a sequence is asymptotically almost periodic if it can be represented in the form , , where is almost periodic and has the property that . The sequence is called the almost periodic part of .

Considering Theorem 7, we immediately obtain the following.

Corollary 10. Let an asymptotically almost periodic sequence be given, and let be its almost periodic part. The limit is finite and exists uniformly with respect to .

Definition 11. The number introduced in (17) is called the mean value of an asymptotically almost periodic sequence .

Remark 12. According to Definition 1 and (17), we have for any positive asymptotically almost periodic sequence with the property that .

Corollary 13. If is an asymptotically almost periodic sequence satisfying then the sequence , defined by the formula , is asymptotically almost periodic for arbitrarily given .

Proof. Let , , where is the almost periodic part of . Considering Definition 1, we have and therefore (see (18)) If we put , , then Theorems 4 and 8 imply the almost periodicity of . Since we know that can be expressed as the sum of the almost periodic sequence and the sequence vanishing at infinity. It means that is asymptotically almost periodic.

3. Half-Linear Riccati Equation

In this section, we recall the oscillation behaviour of half-linear difference equations and we introduce the (adapted) half-linear Riccati equation. For more details, we refer to [15, Chapter 3] or [19, Chapter 8] (with references cited therein). We remark that the (adapted) Riccati technique is an essential tool to prove the following given Theorem 17.

Let us consider the equation where for all considered and the sequence is bounded. We say that an interval , , contains the generalized zero of a solution of (22) if and . Equation (22) is said to be disconjugate on a set provided any solution of (22) has at most one generalized zero on and the solution given by the initial value has no generalized zero on . Otherwise, (22) is said to be conjugate on .

The Sturm type separation theorem is valid for (22) (see [1, Theorem 3]). Thus, the oscillation of an arbitrary solution of (22) implies the oscillation of all solutions of (22). It means that we can categorize (22) as oscillatory or nonoscillatory.

Definition 14. Equation (22) is called nonoscillatory if there exists with the property that (22) is disconjugate on for all . In the opposite case, (22) is called oscillatory.

Now, we turn our attention to the half-linear Riccati equation. Using the substitution we transform (22) into According to [15, Lemma , (I8)], if , then where is between and ; that is, for , we can rewrite (24) in the following form:

Next, putting we obtain and then, applying (26), we obtain the adapted Riccati equation where is between and .

4. Conditional Oscillation

To prove the announced theorem, we need the following two lemmas.

Lemma 15. If (22) is nonoscillatory, then there exists a negative solution of (29) such that

Proof. Applying [1, Theorem 1] (see also [15, Theorem ]), we know that the nonoscillation of (22) implies the existence of a solution of (26) satisfying , . Considering [36, Lemma 1, (v) and Theorem 1], we obtain . Since the sequence is decreasing (see directly (26)), we have for all . Thus, there exists a negative solution of (29). Finally, (27) gives

Lemma 16. If there exists a negative solution of (29), then (22) is nonoscillatory.

Proof. The statement of the lemma follows from [1, Theorem 1].

Now we can formulate and prove the main result.

Theorem 17. Let be given, and let and be arbitrary positive asymptotically almost periodic sequences such that Let Consider the equation If , then (34) is oscillatory. If , then (34) is nonoscillatory.

Proof. We put (see Theorem 4) For an arbitrarily given number , we will consider an integer such that That is, From Remark 12 and Corollaries 10 and 13, we know that such a number exists.
At first, we consider the oscillatory part of the theorem. By contradiction, we suppose that is given and that (34) is nonoscillatory. Lemma 15 says that there exists a negative solution of the equation where Note that (39) is the adapted Riccati equation associated with (34) and that (30) holds. Particularly, from (30), we know and (see (40)) Using (consider, e.g., the Stirling formula) we also obtain Combining (39), (41), (42), and (44), we can find an integer with the property that and that whenever . Let for some . In this case, we estimate and hence, by (45), we get . Thus, there exists satisfying Therefore (see (45) and (46)), we have where Particularly, (49) implies
We define From (48), we can see
For , we compute (see (39)) if , or if .
Since it holds which follows from the Young inequality.
Using (36), (37), and (38), we have That is,
It remains to estimate the following: Applying (41), (42), and (44), we can express where the sequence of has the property that . From (48), it follows and, from (51), we obtain Since the function is continuously differentiable on , there exists for which Now it suffices to consider (32), (48), (53), (61), (62), (63), and (64) which imply the existence of an integer such that
Altogether (see (54), (57), (59), and (65)), we have Summing (66) from to an integer , we obtain Thus (see also (53)), it is valid that Particularly (consider (52)), has to be positive for infinitely many . This contradiction proves that (34) is oscillatory if .
Let , and let be given. In this part of the proof, our objective is to show that there exists a negative solution of (39) for all , where Evidently, it is true that for all for which is negative. Thus (consider (41), (42), and (44)), we can assume that is so large that In addition, as in the first part of the proof (see (49)), we can estimate for some . From (72), it follows
If we denote then Again (see (54)), we estimate if , or if , where
If then it is valid that That is, For any , we can find with the property that Thus, for a given number , considering (73) and (79), we can assume that is so large that
It holds (see (36), (37), and (38)) That is,
As in the first part of the proof (compare (48) with (71) and (63) with (75)), we can show that Let be so large that
Finally (consider (77), (83), (85), and (87)), we have Hence, That is, Particularly, if then (71) and (90) give
Since (73) remains true if is replaced by an arbitrary integer , to complete the proof, it suffices to find a number such that implies Indeed, in this case, we have (92) and, consequently, we have that for all . Then, Lemma 16 says that (34) is nonoscillatory. Of course, such a number exists because the concrete initial value is not applied to obtain (77), (85), and (87) and because (83) can be proved for a sufficiently small perturbation of the initial value (the left side of (83) continuously depends on the initial value (69)).

Remark 18. The statement of Theorem 17 remains true for . It suffices to consider the nonoscillatory equation where is defined in (35). We refer to [1, Theorem 2].

Remark 19. We conjecture that, for in the general case, it is not possible to decide whether (34) is oscillatory or nonoscillatory. Our conjecture is based on the existence of positive almost periodic sequences , satisfying Such sequences can be constructed applying, for example, [34, Theorem 3.5].

Example 20. Let , , and . We can use Theorem 17 for the following equation: with asymptotically almost periodic coefficients (consider Definition 2, Theorem 3, and Corollaries 6 and 13). It holds (, i.e., ) Thus, the equation is oscillatory for and nonoscillatory for

Theorem 17 gives a new result also for (i.e., for the linear equations with asymptotically almost periodic coefficients). This situation is illustrated by the next example. Note that, in the below given corollaries, we explicitly mention two other new results which follow from Theorem 17.

Example 21. For any positive continuous functions and , the equation has the form of (34) for . Considering [29, Example 3.4], we have Hence, the oscillation constant is

Corollary 22. Consider (34), where and are positive periodic sequences with period . Denote Then, (34) is oscillatory for and nonoscillatory for .

Example 23. We apply Corollary 22 for the equation where is an odd integer. By the following computations: we get the oscillation constant

For the general nonzero sequence in (34), we have to use the full definition of the generalized zero. An interval , , contains the generalized zero of a solution of (34) if and . This definition is necessary for the correct formulation of Corollary 24.

Corollary 24. Consider (34), where and are nonzero asymptotically almost periodic sequences satisfying Let (i)If for all and , then (34) is nonoscillatory. (ii)If for all and , then (34) is oscillatory.

Proof. Since is asymptotically almost periodic if is asymptotically almost periodic, it suffices to use the half-linear Sturm type comparison theorem (see [1, Theorem 2]) and Theorem 17.

Example 25. For an arbitrarily given , Corollary 24 says that the equation is nonoscillatory if and that the equation is oscillatory if

Acknowledgments

The authors thank the referees for their remarks and suggestions which improved the final version of the paper. The first author is supported by the Programme of “Employment of Best Young Scientists for International Cooperation Empowerment” (grant number CZ.1.07/2.3.00/30.0037) co-financed from European Social Fund and the state budget of the Czech Republic. The second author is supported by the Czech Science Foundation under Grant P201/10/1032.

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