Bounded and unbounded non-oscillatory solutions of a four-dimensional nonlinear neutral difference system
- Josef Diblík^{1},
- Barbara Łupińska^{2},
- Miroslava Růžičková^{3}Email author and
- Joanna Zonenberg^{2}
https://doi.org/10.1186/s13662-015-0662-9
© Diblík et al. 2015
Received: 2 June 2015
Accepted: 7 October 2015
Published: 15 October 2015
Abstract
The paper is concerned with a four-dimensional nonlinear difference system with delayed arguments where the first equation of the system is of a neutral type. A classification of non-oscillatory solutions is given and results on their boundedness or unboundedness are derived. The results obtained are illustrated by examples.
Keywords
MSC
1 Introduction
The properties of solutions to second-order difference equations (such as oscillatory or non-oscillatory behavior) have been the subject of intensive studies in the last 20 years.
Considerably less attention received the study of similar properties to special classes of fourth-order nonlinear difference equations. Some interesting results concerning asymptotic and oscillatory properties of the fourth-order nonlinear difference equations can be find in [1, 2]. A systematic investigation of the behavior properties of solutions of some classes of fourth-order nonlinear difference equations can be found in [3]. The authors presented sufficient conditions for the oscillation of solutions via comparison with the first and second-order difference equations whose oscillatory behavior is well known. The study of various kinds of fourth-order nonlinear difference equations are brought to the attention of many authors (see, e.g., [4–19] and the references therein).
A four-dimensional system of difference equations, which can be understood as a generalization of fourth-order difference equations, was investigated, e.g., in [20–22]. The boundedness and oscillation of nonlinear three-dimensional difference systems with delays were considered in [23, 24].
By a solution of (1) we mean a vector \(X=X(n)\), \(n\in\mathbb{Z}_{n_{0}-\eta}^{\infty}\) such that, for every \(n\in\mathbb{Z}_{n_{0}}^{\infty}\), (1) is fulfilled. A solution \(X=X(n)\), \(n\in\mathbb{Z}_{n_{0}-\eta}^{\infty}\) of (1) is said to be non-oscillatory if each of its components is non-oscillatory (i.e. it is always positive or is always negative) on \(\mathbb{Z}_{n_{0}-\eta}^{\infty}\). A solution \(X=X(n)\), \(n\in\mathbb{Z}_{n_{0}-\eta}^{\infty}\) of (1) is said to be eventually non-oscillatory if all its components are non-oscillatory for all sufficiently large n. Otherwise (if a solution is neither non-oscillatory nor eventually non-oscillatory), it is called oscillatory. A solution \(X=X(n)\), \(n\in\mathbb{Z}_{n_{0}-\eta}^{\infty}\) of (1) is said to be bounded if all its components are bounded, otherwise, it is called unbounded. In the paper, we often assume or state that a property holds ‘eventually’ meaning that such property is valid for all sufficiently large values of the independent variable n. The rest of the paper is organized as follows. In Section 2, some auxiliary results are introduced. The main results are proved in Section 3 and, in the last Section 4, we illustrate selected results by examples.
2 Auxiliary results
We will employ some auxiliary results to prove the main results of the paper. First, recall two lemmas, which can be found in [27] and in [28], respectively.
Lemma 1
Remark 1
If conditions of Lemma 1 hold when \(l \neq0\), then the sequences \(\{x_{1}(n)\}\) and \(\{z(n)\}\) are both eventually non-oscillatory.
Lemma 2
Remark 2
If conditions of Lemma 2 hold, then the sequence \(\{z(n)\}\) is bounded if and only if the sequence \(\{ x_{1}(n)\}\) is bounded. Thus, the sequence \(\{z(n)\}\) is unbounded if and only if the sequence \(\{x_{1}(n)\}\) is unbounded.
Next, we will prove some properties of the solutions concerning sign, monotonicity and convergence to zero, formulating them as lemmas.
Lemma 3
If conditions (13) and (14) are satisfied and \(l \neq0\), then the sequences \(\{x_{1}(n)\}\) and \(\{z(n)\}\) have the same sign for all sufficiently large n.
Proof
The following lemma can be immediately proved if, in the system (1) written by (3)-(6), single equations are analyzed separately (starting with (6)).
Lemma 4
Let \(X=X(n)\), \(n\in\mathbb{Z}_{n_{0}-\eta}^{\infty}\) be a solution of the system (1) such that the sequence \(\{ x_{1}(n)\}\) is non-oscillatory. If conditions (7)-(9) are satisfied, X is eventually non-oscillatory and the sequences \(\{x_{i}(n)\}\), \(i = 2, 3, 4\), are monotonic for all sufficiently large n. Moreover, the sequence \(\{z(n)\}\) defined by (12) is also monotonic for all sufficiently large n.
Lemma 5
Proof
3 Main results
We focus on the study of boundedness and unboundedness of non-oscillatory solutions of system (1). Before we state sufficient conditions for the solutions to be bounded or unbounded, we give a classification of non-oscillatory solutions of system (1). Investigation of properties of solutions to difference equations and systems relates to the signs of their solutions. Many authors use, e.g., auxiliary results based on the well-known Kiguradze theorem, useful in the theory of functional differential equations. Although our approach is different, it is also based on the signs of solutions.
3.1 Classification of non-oscillatory solutions
Theorem 1
Let conditions (7)-(9) and (14) be satisfied. Then the coordinates of every eventually non-oscillatory solution \(X=X(n)\), \(n\in\mathbb{Z}_{n_{0}-\eta}^{\infty}\) of the system (1) satisfy exactly one of the cases (I)-(III) for all sufficiently large n.
Proof
Cases (5) and (8). Suppose that \(x_{2}(n)<0\) for all sufficiently large n. Hence, using (3), we can prove that the sequence \(\{z(n)\}\) is eventually nonincreasing. Since \(\{z(n)\}\) is eventually positive, there exists a finite limit of this sequence. By Lemma 2, the sequence \(\{x_{1}(n)\}\) is bounded and, by Lemma 1, \(\{x_{1}(n)\}\) has a finite limit. Thus, by Lemma 5, \(\lim_{n\to\infty} x_{2}(n)= 0\). We conclude that the sequence \(\{x_{2}(n)\}\) is nondecreasing. By (4), the sequence \(\{x_{3}(n)\}\) is eventually positive. This statement excludes the case (8).
Finally, we show that case (5) is not possible either. Suppose that it holds. Then \(x_{4}(n)\) is eventually positive and, by (5), the sequence \(\{x_{3}(n)\}\) is eventually nondecreasing, i.e., \(\Delta x_{3}(n)\ge0\) eventually.
By Lemma 5, \(\lim_{n\to\infty} x_{3}(n)= 0\). Since the sequence \(\{x_{3}(n)\}\) is assumed to be eventually positive, it implies \(\Delta x_{3}(n_{k}^{*})<0\), \(k=1,2,\ldots\) , for a sequence of indices \(n_{0}< n_{1}^{*}< n_{2}^{*}<\cdots\) such that \(\lim_{k\to\infty}n_{k}^{*}=\infty\). This contradicts not only the above derived inequality \(\Delta x_{3}(n)\ge0\) for all sufficiently large n but, in the end, case (5), too. □
3.2 Bounded and unbounded solutions
In this part, we give sufficient conditions for the boundedness or unboundedness of non-oscillatory solutions of system (1).
Theorem 2
Proof
If we take \(i=3\) or \(i=4\), as above, we get \(\lim_{n\to\infty} x_{2}(n)=0\) or \(\lim_{n\to\infty} x_{3}(n)=0\), respectively. □
Theorem 3
If conditions (7)-(9) and (14) are satisfied, then every eventually non-oscillatory solution \(X=X(n)\), \(n\in\mathbb{Z}_{n_{0}-\eta}^{\infty}\) of (1) fulfilling one of conditions (I), (II) is unbounded.
Proof
Let \(X=X(n)\), \(n\in\mathbb{Z}_{n_{0}-\eta}^{\infty}\) be an eventually non-oscillatory solution of (1). Without loss of generality, assume that the sequence \(\{x_{1}(n)\}\) is eventually positive.
If (I) or (II) hold, then \(x_{2}(n)>0\) and \(x_{3}(n)>0\) eventually. From (4), since the sequence \(\{x_{3}(n)\}\) is eventually positive, the sequence \(\{x_{2}(n)\}\) is eventually nondecreasing. Thus, \(\lim_{n\to\infty} x_{2}(n)>0\). By Theorem 2 again, we get the assertion. □
Theorem 4
If conditions (7)-(9) and (14) are satisfied, every eventually non-oscillatory solution \(X=X(n)\), \(n\in\mathbb{Z}_{n_{0}-\eta}^{\infty}\) of (1) fulfilling condition (III) is bounded.
Proof
4 Examples
The following examples illustrate Theorem 3 and Theorem 4. Among others, the examples demonstrate that the sets of solutions (I)-(III) defined in Theorem 1 are nonempty.
Example 1
Example 2
Example 3
Declarations
Acknowledgements
The first author has been supported by the project No. LO1408, AdMaS UP-Advanced Materials, Structures and Technologies, (supported by Ministry of Education, Youth and Sports of the Czech Republic under the National Sustainability Programme I). The third author was supported by the project KEGA 004ŽU-4/2014 of the Cultural and Educational Grant Agency of the Ministry of Education, Science, Research and Sport of the Slovak 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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