Oscillation constants for half-linear difference equations with coefficients having mean values
- Petr Hasil^{1} and
- Michal Veselý^{1}Email author
https://doi.org/10.1186/s13662-015-0544-1
© Hasil and Veselý 2015
Received: 25 February 2015
Accepted: 17 June 2015
Published: 9 July 2015
Abstract
We investigate second order half-linear Euler type difference equations whose coefficients have mean values. We show that these equations are conditionally oscillatory and we explicitly identify the corresponding oscillation constants given by the coefficients. Our results generalize the known ones concerning equations with positive constant, periodic, or (asymptotically) almost periodic coefficients. We also demonstrate the obtained results on examples and we give corollaries. In particular, we get new results even for linear difference equations.
Keywords
MSC
1 Introduction
Since the main result of [5] is one of the basic motivations for the research presented here, we reformulate it as follows. We remark that the symbol \(M(\cdot)\) stands for the mean values of the indicated sequences clarified in Definition 2 below and that the definition of asymptotic almost periodicity is mentioned in Definition 4 below.
Theorem 1
Theorem 2
In this paper, we intend to generalize Theorem 1 to the case that the coefficients have mean values and the second coefficient can change sign. It means that our aim is to prove the discrete counterpart of Theorem 2. For this purpose, we improve the method from [5]. Since we study equations with coefficients from more general classes, we have to prove some new auxiliary results and inequalities (especially, we need Lemmas 1 and 2 below). Note that we partially apply the processes used in [5] (see the proof of Theorem 5 below, where it is explicitly mentioned).
The paper is organized as follows. In the next section, we state the necessary background and we recall the so-called Riccati technique, which is essential for our investigation. In Section 3, the reader can find preparatory lemmas, results, and corollaries. In Section 4, we collect illustrative examples.
2 Preliminaries
In this section, we mention the needed background concerning the oscillation theory of half-linear difference equations. For more details, we refer to Chapter 3 in [21] and Chapter 8 in [22] with references cited therein. In addition, we recall the concept of mean values which is necessary to find general oscillation constants. We also state the concept of the (adapted) half-linear Riccati equation which is the main tool in our investigation.
Definition 1
Equation (2.1) is called non-oscillatory if there exists \(l \in\mathbb{N}\) with the property that (2.1) is disconjugate on \(\{l, l+ 1, \ldots, l + n\}\) for all \(n \in\mathbb{N}\). In the opposite case, (2.1) is called oscillatory.
The Sturm type separation theorem (see, e.g., Theorem 3.3.6 in [21]) enables us to give Definition 1, because the oscillation of an arbitrary non-zero solution of (2.1) implies the oscillation of all solutions of (2.1). We will also use a consequence of the Sturm type comparison theorem. We mention only the form that is suitable for our purpose.
Theorem 3
Proof
The theorem follows, e.g., from Theorem 3.3.5 in [21]. □
To obtain explicit oscillation constants, we need the definition of the mean value of a sequence.
Definition 2
An important class of sequences having mean values is formed by asymptotically almost periodic sequences (see also [5]). Hence, we formulate the next definitions.
Definition 3
A sequence \(\{f_{k}\}_{k \in\mathbb{Z}} \subset \mathbb{R}\) is called almost periodic if, for any \(\varepsilon > 0\), there exists \(P(\varepsilon) \in\mathbb{N}\) such that any set of the form \(\{i, i+ 1, \ldots, i + P(\varepsilon) - 1 \} \subset \mathbb{Z}\) contains an integer l for which \(\vert f_{k} - f_{k + l} \vert < \varepsilon\), \(k \in\mathbb {Z}\).
Definition 4
We say that a sequence \(\{f_{k}\}_{k \in\mathbb{N}_{a}} \subset\mathbb{R}\) is asymptotically almost periodic if there exists a pair of sequences \(\{f_{k}^{1}\}_{k \in\mathbb{Z}}, \{f_{k}^{2}\}_{k \in\mathbb {N}_{a}} \subset\mathbb{R}\) such that \(\{f_{k}^{1}\}\) is almost periodic, \(\{f_{k}^{2}\}\) satisfies \(\lim_{k \to\infty} f_{k}^{2} = 0\), and \(\{f_{k} \}_{k \in \mathbb{N}_{a}} \equiv\{ f_{k}^{1} + f_{k}^{2} \}_{k \in\mathbb{N}_{a}}\).
The following theorem is typically known as the Riccati method. It shows the way in which the non-oscillation of (2.1) is connected to the solvability of (2.6).
Theorem 4
Equation (2.1) is non-oscillatory if and only if there exist an integer b and a sequence of \(w_{k}\) which solves (2.6) and satisfies \(w_{k}+r_{k}>0\) for \(k \in\mathbb{N}_{b}\).
Proof
The theorem is a consequence of the well-known roundabout theorem (see, e.g., Theorem 3.3.4 in [21] or directly Theorem 8.2.5 in [22]). □
3 Results
To prove the main results, we need the following lemmas.
Lemma 1
Let a sequence \(\{f_{k}\}_{k \in\mathbb{N}_{a}} \subset\mathbb{R}\) have mean value \(M(\{f_{k}\})\). There exists a number \(K(\{f_{k}\}) > 0\) for which \(|f_{k}| < K(\{f_{k}\})\), \(k \in\mathbb{N}_{a}\).
Proof
Lemma 2
Proof
Considering Theorem 4, the non-oscillation of (2.9) implies that there exist \(L \in\mathbb{N}\) and a solution \(\{w_{k}\}_{k \in\mathbb{N}_{L}}\) of (2.11) such that \(w_{k}+r_{k}>0\) for \(k \geq L\). Considering (2.7), it gives the solution \(\{\zeta_{k}\}_{k \in\mathbb{N}_{L}} \equiv\{- w_{k} k^{(p-1)}\}_{k \in\mathbb{N}_{L}}\) of (2.12). We show that this solution \(\{\zeta_{k}\}\) is negative and satisfies (3.6).
We remark that, in the case that the sequence of \(s_{k}\) is positive, the statement of Lemma 2 follows from Lemma 1, (v) and Theorem 1 in [23] combined with Lemma 3.5.9 in [21] or with Lemma 8.2.2 in [22].
Lemma 3
If there exists a negative solution \(\{\zeta_{k}\}_{k \in\mathbb{N}_{L}}\) of (2.12), then (2.9) is non-oscillatory.
Proof
A negative solution \(\{\zeta_{k}\}_{k \in\mathbb{N}_{L}}\) of (2.12) gives \(\{w_{k}\}_{k \in\mathbb{N}_{L}} \equiv\{- \zeta_{k}/k^{(p-1)}\}_{k \in \mathbb{N}_{L}}\), which is a positive solution of (2.11). Thus, the lemma follows from Theorem 4. □
Applying the above lemmas, we can obtain the announced result. For the reader’s convenience, we recall the assumptions on the coefficients.
Theorem 5
Let sequence \(\{r_{k}\}_{k \in\mathbb {N}_{a}}\) have mean value \(M(\{r_{k}^{1-q}\}) = 1\) and satisfy (2.10) and let sequence \(\{s_{k}\}_{k \in\mathbb{N}_{a}}\) have mean value \(M(\{ s_{k}\})> 0\). Then (2.9) is oscillatory for \(M(\{s_{k}\}) > q^{-p}\) and non-oscillatory for \(M(\{s_{k}\}) < q^{-p}\).
Proof
We slightly improve Theorem 5 into the following form (more common in the literature). In particular, we remove the requirement on sequence \(\{s_{k}\}\) that it has a positive mean value.
Theorem 6
Proof
Since the presented results are new also for linear difference equations (the case that \(p=q=2\)), we mention the following direct corollary of Theorem 6.
Corollary 1
Based on results of [24] (see also [25, 26]), the conjecture is given in our previous paper [5] that the border case \(M ( \{{r_{k}^{-1}} \} ) M(\{s_{k}\}) = 1/4 \) from Corollary 1 is not solvable for general coefficients; i.e., in the border case, there exist oscillatory equations in the form of (3.41) and, at the same time, there exist non-oscillatory equations in this form.
In addition, using the Sturm type comparison theorem, we get the next new result concerning non-oscillatory half-linear difference equations when the coefficient in the difference term does not need to be bounded.
Theorem 7
Let us consider (3.36) and Γ introduced in (3.37). Let the coefficients \(\{\tilde{r}_{k}\}_{k \in\mathbb{N}_{a}}\), \(\{\tilde{s}_{k}\}_{k \in\mathbb{N}_{a}}\) be such that the mean values of sequences \(\{\tilde{r}_{k}^{1-q} \}\), \(\{ \tilde{s}_{k}\}\) exist and \(\{\tilde{r}_{k}\}\) is positive. Then (3.36) is non-oscillatory if \(M(\{\tilde{s}_{k}\}) < \Gamma\).
Proof
Again, from the theorem above, we obtain a new result in the linear case. The linear version of Theorem 7 reads as follows.
4 Examples
In this section, we give some simple examples of oscillatory and non-oscillatory equations whose oscillatory properties do not follow from any previously known oscillation or non-oscillation criteria. To illustrate Theorems 5, 6, 7 and Corollaries 1, 2, we mention Examples 1, 2, 4 and Examples 3, 5, respectively.
Example 1
Example 2
Example 3
Example 4
Example 5
Now we briefly explain why the oscillatory problems in the above examples are not covered by any previously known results (see also Theorem 1). In both of Examples 1 and 2, the second coefficient changes its sign. In Example 3, the coefficients are not asymptotically almost periodic. In Example 4, the coefficient in the difference term is not bounded. In the last example, the first coefficient is not asymptotically almost periodic and, at the same time, it is not bounded.
As a final remark, we focus our attention on the denominators of the potentials considered in Examples 2 and 4, where \((k+1)^{(p)}\) and \((k+1)^{(3/2)}\) is replaced by \(k^{(p)}\) and \(\sqrt {k^{3}}\), respectively. In fact, all presented results remain true if we replace the coefficients \(\{s_{k}\}\) or \(\{\tilde{s}_{k}\}\) by \(\{f_{k} \cdot s_{k}\}\) or \(\{f_{k} \cdot\tilde{s}_{k}\}\) for any sequence of real numbers \(f_{k}\) satisfying \(\lim_{k \to\infty } f_{k} = 1\). Indeed, the existence of \(M(\{h_{k}\})\) implies that \(M(\{h_{k} \cdot g_{k}\}) = M(\{h_{k}\})\) whenever \(\lim_{k \to\infty} g_{k} = 1\) (consider Definition 2 and Lemma 1). Note that we consider the denominator \((k+1)^{(p)}\) due to the form of previously known results (see Section 1).
Declarations
Acknowledgements
The first author is supported by Grant P201/10/1032 of the Czech Science Foundation. The second author is supported by the project ‘Employment of Best Young Scientists for International Cooperation Empowerment’ (CZ.1.07/2.3.00/30.0037) co-financed from European Social Fund and the state budget of the Czech Republic.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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