Trichotomy of nonoscillatory solutions to second-order neutral difference equation with quasi-difference
- Agata Bezubik^{1},
- Małgorzata Migda^{2},
- Magdalena Nockowska-Rosiak^{3} and
- Ewa Schmeidel^{1}Email author
https://doi.org/10.1186/s13662-015-0531-6
© Bezubik et al. 2015
Received: 13 February 2015
Accepted: 8 June 2015
Published: 23 June 2015
Abstract
In this paper the nonlinear second-order neutral difference equation of the following form: \(\Delta ( a_{n}\Delta(x_{n}-p_{n}x_{n-1}) ) + q_{n}f(x_{n-\tau})=0\) is considered. By suitable substitution the above equation is transformed into a new one, which is a third-order non-neutral difference equation. Using results obtained for the new equation, the asymptotic properties of the neutral difference equation are studied. Some classification of nonoscillatory solutions is presented, as well as an estimation of the solutions. Finally, we present necessary and sufficient conditions for the existence of solutions to both considered equations being asymptotically equivalent to the given sequences.
Keywords
MSC
1 Introduction
By a solution to (1) we mean a sequence \((x_{n})\) which satisfies (1) for n sufficiently large. We consider only solutions which are nontrivial for all large n. A solution to (1) is called nonoscillatory if it is eventually positive or eventually negative. Otherwise it is called oscillatory.
Neutral type difference equations have been widely studied in the literature. Some recent results on the asymptotic behavior of second-order neutral difference equations can be found, for example, in [1–7]. The higher-order neutral difference equations were studied in [8–13].
For results concerning the oscillatory and asymptotic behavior of the third-order difference equation we refer to [14, 15], for equations with quasi-differences to [16–19], and to the references cited therein. Many results on the oscillation of second- and third-order functional differential and difference equations can also be found in [20].
The purpose of this paper is to study the asymptotic properties of the neutral difference equation (1). Transforming the considered equation into a new one, which is a third-order difference equation of type (6), we get various results concerning the asymptotic behavior of solutions to this equation. These results are then used to establish some properties of the solutions to (1). In particular, we obtain necessary and sufficient conditions for the existence of solutions asymptotically equivalent to the given sequences.
Fourth-order non-neutral difference equations with one quasi-difference, by the techniques here used, were studied in [21–23]. Some generalizations of the results presented in these papers were published in [24, 25]. Even so, there is not a full analogy to the results since the Kneser type classification of the nonoscillatory solution is different for odd- or even-order equations, and of neutral or non-neutral type as well.
The following definitions and theorems will be used in the sequel.
We say that the sequence \((u_{n})\) is asymptotically constant if this sequence has a nonzero limit, and we say that it is an asymptotically zero sequence if the limit of this sequence equals zero. We say that the sequence \((u_{n})\) is asymptotically equivalent to \((v_{n})\) if \((\frac {u_{n}}{v_{n}})\) has a nonzero limit. In the present paper, we study the three types of solutions: asymptotically zero solutions, asymptotically constant solutions, and unbounded solutions. It is called a trichotomy of nonoscillatory solutions.
Definition 1
(Uniformly Cauchy subset [26])
A subset S of the Banach space B is said to be uniformly Cauchy if for every \(\varepsilon>0\) there exists a positive integer N such that \(\vert x_{i}-x_{j} \vert<\varepsilon\) whenever \(i,j >N\) for any \((x_{n})\in B\).
Lemma 1
(Arzela-Ascoli’s theorem [26])
Each bounded and uniformly Cauchy subset of B is relatively compact.
Theorem 1
(Schauder theorem [27])
Let S be a nonempty, closed, and convex subset of a Banach space B and \(T \colon S \to S\) be a continuous mapping such that \(T(S)\) is a relatively compact subset of B. Then T has at least one fixed point in S.
The following theorem of Stolz-Cesáro is a discrete analog of l’Hospital’s rule.
Theorem 2
(Stolz-Cesáro theorem [28])
2 Existence of nonoscillatory solutions
In this section, we obtain necessary and sufficient conditions for the existence of nonoscillatory solutions to (1) with certain asymptotic properties. We start with the following lemmas.
Lemma 2
Proof
Condition (H2) implies that \(\prod^{n}_{i=1} p_{i} \leq C_{0} n\), where \(C_{0}\) is a positive constant. It follows that \(\prod^{n}_{i=1} p_{i}^{-1} \geq\frac{1}{C_{0}n}\). Using the notation of (4), the above inequality takes the form \(\frac{1}{b_{n}} \geq\frac {1}{C_{0}n}\). Since the series \(\sum^{\infty}_{n=1} \frac{1}{n}\) diverges, condition (9) is satisfied. □
Remark 1
Lemma 3
Proof
The proof is obvious and hence omitted. □
Lemma 4
- (I)$$\lim_{n \to\infty}\frac{x_{n}}{b_{n}}=0; $$
- (II)
Proof
If \(\lim_{n \to\infty} y_{n}=0\), then condition (I) is satisfied.
As a consequence of Lemma 4 we obtain the following result.
Lemma 5
- (I)$$\lim_{n \to\infty}y_{n}=0; $$
- (II)there exist positive constants \(C_{1}\) and \(C_{2}\) such that$$C_{1} \leq y_{n} \leq C_{2} Q_{n} \quad \textit{for large }n. $$
Before we derive a necessary and sufficient condition for the existence of a solution to (1) that is asymptotically equivalent to \((b_{n})\), the following theorem needs to be proved.
Theorem 3
Proof
Let \((y_{n})\) be an asymptotically constant solution to (6). Then \((y_{n})\) is a nonoscillatory sequence. Without loss of generality, we assume that \((y_{n})\) is an eventually positive solution. By Lemma 3 it is of type (i) or type (ii). Each solution to type (i) tends to infinity. This implies that \((y_{n})\) is of type (ii).
The next example shows that the condition (15) in Theorem 3 is not a necessary condition for (6) to have an asymptotically zero solution.
Example 1
Sufficient conditions, under which, for every real constant, there exists a solution to the higher-order difference equation with quasi-differences convergent to this constant are obtained in Theorem 3.3 in [29]. Hence, for (6), we have the following.
Theorem 4
Assume that (H1), (H2), (H^{∗}3), (H^{∗}4) hold and condition (15) is satisfied. Then for every \(c\in\mathbb{R}\) there exists a solution x to (1) such that \(\lim_{n\to\infty}x(n)=c\).
Corollary 1
Proof
This corollary follows directly from Theorem 3. □
Theorem 5
Proof
Using the notation of (2), (5), and (7) in condition (15) the conclusion of this theorem follows directly from Theorem 3 and Theorem 4. □
Remark 2
Corollary 2
Example 2
Example 3
Finally, we present a necessary and sufficient condition for the existence of an asymptotically \((Q_{n})\) solution to (1). We start with the following theorem.
Theorem 6
Proof
Remark 3
Note that if the sequences \((\frac{1}{a_{n}})\) and \((\frac{1}{b_{n}})\) are both polynomial sequences, then \((Q_{n})\) is a polynomial sequence, too.
Now, let \(\frac{1}{a_{n}}\equiv1\) and \(\frac{1}{b_{n}}\equiv1\). This means that \(a_{n} \equiv1\) and \(b_{n}\equiv1\). Hence \(Q_{n} = \frac{1}{2} n^{2} - \frac{3}{2}n +1\) is a quadratic polynomial. Obviously, by virtue of (9), this case holds only if \(p_{n} \equiv1\).
Theorem 7
Proof
Using the notation of (2), (5), and (7) in condition (28) the conclusion of this theorem follows directly from Theorem 6. □
Note that for particular cases of (1), if \((\frac{1}{a_{n}})\) is a polynomial sequence and \(p_{n} \equiv1\), from Theorem 7 we get the existence of asymptotically polynomial solutions.
Example 4
In Example 3 (25) is considered. In this equation \(a_{n} \equiv1\) and \(p_{n} \equiv1\). All assumptions of Theorem 7 are satisfied. Hence (25) has an asymptotically \((Q_{n})\) solution, where \(Q_{n} = \frac{1}{2} n^{2} - \frac{3}{2}n +1\). It means that (25) has an asymptotically polynomial solution.
Some results concerning asymptotically polynomial solutions to difference equations can be found, for example, in [30–34].
Declarations
Acknowledgements
This work was partially supported by the Ministry of Science and Higher Education of Poland (PB-43-081/14DS).
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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