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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 471435, 19 pages
Critical Oscillation Constant for Difference Equations with Almost Periodic Coefficients
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.
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.
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  (see also ), the concept of a phase is established to obtain other oscillation criteria. For the matrix difference equations of the form of , we refer to . 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 ).
Fundamental aspects of Sturmian theory (and some oscillation criteria) for second-order Sturm-Liouville equations on arbitrary time scales are formulated in . 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 .
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  that the difference equation is oscillatory for any and . Further, it is obtained in  (based on results of [17, 18]) that the equation is oscillatory if and only if . Finally in , 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  and also ), 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  that the differential Euler equation is conditionally oscillatory with the oscillation constant . It is known (see ) that the equation where , are positive periodic continuous functions, is conditionally oscillatory as well. We also refer to  and [24–29] which generalize  (for the discrete case, see ). Since the Euler difference equation is conditionally oscillatory with the oscillation constant (see ), 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 , where an oscillation constant is characterized. The constant from  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  (see also [34, 35]). In , 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  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.
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.
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
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, , .
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 .
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.
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 ) and other cases which generalize the discrete equation with constant coefficients (see, e.g.,  with references cited therein), we conjecture that (3.66) (with periodic coefficients) is non-oscillatory even for .
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 .
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 . 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  (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).
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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