Abstract and Applied Analysis

Volume 2012 (2012), Article ID 471435, 19 pages

doi:10.1155/2012/471435

Research Article

Critical Oscillation Constant for Difference Equations with Almost Periodic Coefficients

Petr Hasil ¹ and Michal Veselý ²

¹Department of Mathematics, Mendel University in Brno, Zemědělská 1, 613 00 Brno, Czech Republic

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

Received 28 May 2012; Revised 10 September 2012; Accepted 13 September 2012

Academic Editor: Elena Braverman

Copyright © 2012 Petr Hasil and Michal Veselý. 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 a type of the Sturm-Liouville difference equations with almost periodic coefficients. We prove that there exists a constant, which is the borderline between the oscillation and the nonoscillation of these equations. We compute this oscillation constant explicitly. If the almost periodic coefficients are replaced by constants, our result reduces to the well-known result about the discrete Euler equation.

1. Introduction

In this paper, we analyse the second-order Sturm-Liouville equation

whose oscillation properties are widely studied over the last few decades. We begin with a short literature overview concerning the (non)oscillation of and of some direct generalizations (including half-linear equations and dynamic equations on time scales).

Basic necessary and sufficient conditions for in order to be oscillatory are derived in [1–3]. In [4] (see also [5]), the concept of a phase is established to obtain other oscillation criteria. For the matrix difference equations of the form of , we refer to [6]. Several oscillation criteria for slightly more general equations are presented in [7, 8]. The oscillation theory for the corresponding higher-order two-term Sturm-Liouville difference equations can be found in [9–11] (for differential case, see [12]).

Fundamental aspects of Sturmian theory (and some oscillation criteria) for second-order Sturm-Liouville equations on arbitrary time scales are formulated in [13]. Oscillation criteria for second-order difference equations can be obtained from oscillation criteria for more general dynamic equations. The oscillation properties of second-order linear dynamic equations, which have the Sturm-Liouville difference equations as special cases, are considered, for example, in [14].

As an illustration, we mention a particular Sturm-Liouville equation of which the complete oscillation classification is done as a consequence of general results on time scales. Using the comparison theorem for second-order linear dynamic equations, it is shown in [15] that the difference equation

is oscillatory for any and . Further, it is obtained in [16] (based on results of [17, 18]) that the equation

is oscillatory if and only if . Finally in [16], applying the Willett-Wong-type theorems for second-order linear dynamic equations, there is given the full oscillation analysis of for

with regard to arbitrary .

The importance of the oscillation results about second-order equations lies among others in the fact that such results can be used to study the oscillation and nonoscillation properties of solutions of different equations. For example (see [19] and also [20]), all solutions of the delay equation oscillate if and only if all solutions of a certain type of with oscillate.

The main aim of this paper is to present a sharp oscillation constant for the Euler-type difference equation

where , , and are positive almost periodic sequences. More precisely, we show that is the so-called conditionally oscillatory; that is, we prove that there exists a positive constant (the oscillation constant) such that is oscillatory for and non-oscillatory for .

Our research is motivated by the continuous case. It is a famous result due to Kneser [21] that the differential Euler equation

is conditionally oscillatory with the oscillation constant . It is known (see [22]) that the equation

where , are positive periodic continuous functions, is conditionally oscillatory as well. We also refer to [23] and [24–29] which generalize [23] (for the discrete case, see [30]). Since the Euler difference equation

is conditionally oscillatory with the oscillation constant (see [31]), it is natural to analyse the conditional oscillation of . Note that the announced result is more general than the results known in the continuous case, because has almost periodic coefficients. The conditional oscillation of discrete equations with constant coefficients can be generalized in other ways. Point out [32], where an oscillation constant is characterized. The constant from [32] coincides with our oscillation constant if the considered coefficients are asymptotically constant.

Solutions of the second-order Sturm-Liouville difference equations with periodic coefficients are studied in [33] (see also [34, 35]). In [36], the half-linear differential equations of the second order with the Besicovitch almost periodic coefficients are considered and an oscillation theorem for these equations is obtained.

In the last years, many results dealing with the conditional oscillation of second-order equations and two-term equations of even order appeared. The two-term difference equation of even order

where denotes the gamma function, is studied in [9, 10]. Results concerning the half-linear difference equation

where

can be found in [37] for , and also in [38, 39] (for dynamic half-linear equations on time scales, see [40–42]).

The paper is organized as follows. In Section 2, we mention only necessary preliminaries and an auxiliary result. Our main result is proved in Section 3, where the particular case concerning the equation with periodic coefficients is formulated as well. The paper is finished by concluding remarks and simple examples.

2. Preliminaries

We begin this section recalling some elements of the oscillation theory of the Sturm-Liouville difference equation

For more details, we can refer to books [43, 44] and references cited therein.

We recall that an interval , , contains the generalized zero of a solution of (2.1) if and . Equation (2.1) is said to be conjugate on , , if there exists a solution which has at least two generalized zeros on or if the solution satisfying has at least one generalized zero on . Otherwise, (2.1) is said to be disconjugate on . Since Sturmian theory is valid for difference equations, all solutions of (2.1) have either a finite or an infinite number of generalized zeros on . Hence, we can categorize these equations as oscillatory and non-oscillatory.

Definition 2.1. Equation (2.1) is called non-oscillatory provided a solution of (2.1) is disconjugate at infinity, that is, there exists such that (2.1) is disconjugate on any set , . Otherwise, we say that (2.1) is oscillatory.

Since we study a special case of (2.1), when the coefficients are almost periodic, we also mention the basics of the theory of almost periodic sequences. Here, we refer to each one of books [45, 46].

Definition 2.2. A real sequence is called almost periodic if, for any , there exists a positive integer such that any set consisting of consecutive integers contains at least one integer with the property that

We say that a sequence is almost periodic if there exists an almost periodic sequence for which , .

The above definition of almost periodicity is based on the Bohr concept. Now we formulate a necessary and sufficient condition for a sequence to be almost periodic. The following theorem is often used as an equivalent definition (the Bochner one) of almost periodicity for .

Theorem 2.3. 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 [45, Theorem 1.26].

Corollary 2.4. Let be almost periodic. The sequence is almost periodic if and only if

Proof. The corollary follows from [45, Theorem 1.27] and [47, Theorem 1.9] (or directly from Theorem 2.3). It suffices to use that (2.3) implies for any almost periodic sequence if .

Note that there exist nonzero almost periodic sequences for which (2.3) is not satisfied (see, e.g., [48, Theorem 3]).

Theorem 2.5. If is an almost periodic sequence, then the limit

exists uniformly with respect to .

Proof. See [45, Theorem 1.28].

Definition 2.6. Let be almost periodic. The number introduced in (2.4) is called the mean value of .

Remark 2.7. For any positive almost periodic sequence , we have . Indeed, if we put and find a corresponding in Definition 2.2, then we obtain

In the proof of our main result, we use an adapted Riccati technique. The classical Riccati technique deals with the so-called Riccati difference equation, which we obtain from (2.1) using the substitution , that is, we obtain the equation

Putting , we adapt (2.6) to our purposes. A direct calculation leads to the equation

We also mention two lemmas which we use to prove the main result.

Lemma 2.8. Let the equation

where and , , , be non-oscillatory. For any solution of the associated equation (2.7), there exists such that, if for some , then , .

Proof. Let be a solution of the non-oscillatory equation (2.8) for which , . From [43, Lemma  6.6.1] it follows that the sequence , where , is decreasing. Further, [43, Theorem  6.6.2] implies that . Thus, the sequence is positive, that is, , .

Lemma 2.9. If there exists a solution of the associated equation (2.7) satisfying for all , then (2.8) is non-oscillatory.

Proof. The statement of the lemma follows from [44, Theorem 6.16].

3. Oscillation Constant

This section is devoted to the main result of our paper. After its proof, within the concluding remarks, we formulate as a corollary the result which deals with periodic equations. This corollary is the discrete counterpart of the main result of [49].

Theorem 3.1. Let the equation

where and and are positive almost periodic sequences satisfying

be arbitrarily given. Let

Then, (3.1) is oscillatory for and non-oscillatory for .

Proof. At first, let us prepare several estimates which we will use to prove the theorem. Henceforth, for given , we will consider and such that

The fact that such numbers , exist follows from Theorem 2.5 and Remark 2.7 (consider also Corollary 2.4 with (3.2)). We put

The adapted Riccati equation associated to (3.1) has the form (see (2.7))

Since one can express

it is valid that

and that ( is arbitrarily given)

Particularly, if for all sufficiently large , then there exists such that

Indeed, it follows directly from (3.10) and (3.11).

Similarly, applying (3.10) and (3.11), it is seen that there exists for any and with the property that the solution of the Cauchy problem

where , satisfies

and hence there exists (consider again (3.11)) for which

Now we can proceed to the oscillatory part of the theorem. By contradiction, we suppose that and that (3.1) is non-oscillatory. According to Lemma 2.8, any solution of the associated adapted Riccati equation (3.8) for which satisfies , , if is enough large. Further, (3.12) gives the existence of with the property that for all . Using (3.11), we obtain

Our goal is to achieve a contradiction with by estimating the arithmetic mean of subsequent values of . We denote

and compute (for )

or (for )

where

Note that we can choose . For reader’s convenience, we will estimate stepwise.

Step 1. We show that there exist and such that

Applying for , we have (see (3.4) and (3.6))

Thus, there exist and with the property that (3.21) is satisfied for all .

Step 2. It holds (see (3.17) and (3.20))

Step 3. We prove that there exists satisfying

where is taken from Step 1. Considering (3.16), we obtain

for each , . Thus, it is true

where

Now we can calculate (see (3.17))

Let us discuss the inequality before the final one in more detail. If we denote

then we easily get (applying (3.26))

Of course, (3.28) implies the existence of such that (3.24) is satisfied.

Using the previous steps, it is possible to prove the following result. If tends to infinity, then so do . Combining (3.21), (3.23), and (3.24), we obtain

We use the estimate (3.18) because . Summing inequality (3.31) from to an integer , we have

This estimate implies that

Particularly, for sufficiently large which means that for infinitely many . This contradiction gives that (3.1) is oscillatory for .

To prove the non-oscillatory part of Theorem 3.1, we will consider the initial value problem

for some integer

where satisfies (3.14) and (3.15). Let . Analogously as in the first part of the proof, we put

and we express (for )

or (for )

Again, we estimate stepwise. Using (3.14) and

we have

Similarly to the first part of the proof, we can show that

Thus (consider (3.4)), there exist and

such that

for . Henceforth, let .

Now we want to estimate

Firstly, consider that, for , , it is valid

and hence (see (3.5) and (3.42))

because

We repeat that (see (3.15))

which gives

Considering (3.40), (3.46), (3.48), and (3.49) together for general , we have

Since it suffices to consider very large , we can assume that .

Altogether, we obtain

The resulting inequality (3.51) implies

Partially, if

then from (3.14) it follows

In fact (see the below given), this result remains true also if

for a number which depends only on and . Considering (3.15) for large , it is seen that the solution of the Cauchy problem (3.34) satisfies

and hence (see (3.54))

Therefore, (3.57) and Lemma 2.9 say that (3.1) is non-oscillatory for .

It means that, to complete the proof, it suffices to find which guaranties the above-mentioned generalization, that is, we need to prove (3.51) for (3.55) with . The concrete initial value was not used in the proof of (3.43). Thus, depends only on and . Let

In the estimate of , since (3.14) and (3.15) remain true, we have (consider also (3.39), (3.48), and (3.49))

where , and (see again (3.39))

which confirms and then the validity of (3.51).

Remark 3.2. Let us point out that the constant arises from the calculations in Step 1.

Remark 3.3. If , , then (see (3.3)); that is, Theorem 3.1 reduces to the result about the discrete Euler equation.

Example 3.4. For arbitrarily given continuous function and , , let us consider

The almost periodicity of and follows from Corollary 2.4 and from, for example, [45, Theorem 1.27] and [47, Theorem 1.9]. It is seen that

Thus, (3.61) is oscillatory if

and non-oscillatory if .

Analogously, under the additional condition , the oscillation constant for the equation

is

Evidently, any periodic sequence is almost periodic. Thus, we also obtain this new result.

Corollary 3.5. The equation

where and and are positive sequences with period , is oscillatory if

and non-oscillatory if .

Remark 3.6. The border case given by remains open. Nevertheless, based on the corresponding continuous case (see [23]) and other cases which generalize the discrete equation with constant coefficients (see, e.g., [10] with references cited therein), we conjecture that (3.66) (with periodic coefficients) is non-oscillatory even for .

Example 3.7. Let an odd integer be given. We can use Corollary 3.5 for the equation

Since we can choose , we obtain

Hence, the oscillation constant for (3.68) is

We add that we can use Theorem 3.1 also in the case when one of the sequences and in (3.1) changes its sign. If the sequence in (3.71) changes its sign, then we have to generalize the definition of the generalized zeros as follows. An interval , , contains the generalized zero of a solution of (3.71) if and .

Corollary 3.8. Let the equation

where and and are nonzero almost periodic sequences, be given.

(i) If and , then (3.71) is non-oscillatory. (ii) If , is positive and , then (3.71) is oscillatory.

Proof. Since the almost periodicity of implies the almost periodicity of , it suffices to apply the discrete Sturm comparison theorem and Theorem 3.1.

At the end we remark that it is possible to find several definitions of almost periodicity for in the literature. For example, concerning almost periodic sequences with indices , we refer to [50]. There is proved that, for any precompact sequence , there exists a permutation of the set of positive integers such that the sequence is almost periodic. In fact, the so-called asymptotically almost periodic sequences are considered in [50] (based on the Bochner concept), where a bounded sequence is called asymptotically almost periodic if the set of sequences , , is precompact in the space of all bounded sequences. We add that a sequence is asymptotically almost periodic if and only if it is the sum of an almost periodic sequence and a sequence which approaches zero as . One finds that this representation is unique. See, for example, [51, 52].

We consider difference equations with almost periodic coefficients given by the limitation of almost periodic sequences on because this approach is the standard one. But we conjecture that the main result can be similarly proved for almost periodic coefficients defined in other ways (e.g., for the above-mentioned asymptotically almost periodic sequences).

Acknowledgments

The authors thank the anonymous referees for their suggestions and references which improved the final version of the paper. This research is supported by the Czech Science Foundation under Grant P201/10/1032.

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