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- Open Access
On some classes of solvable systems of difference equations
- Stevo Stević^{1, 2, 3, 4}Email author,
- Bratislav Iričanin^{5},
- Witold Kosmala^{6} and
- Zdeněk Šmarda^{4}
https://doi.org/10.1186/s13662-019-1959-x
© The Author(s) 2019
- Received: 22 November 2018
- Accepted: 15 January 2019
- Published: 29 January 2019
Abstract
In a recent paper several periodic systems of difference equations have been presented. We show in an elegant way that all the results therein follow from some known ones. We also show how some extensions of the systems of difference equations can be solved.
Keywords
- System of difference equations
- Periodicity
- General solution
- Difference equation with interlacing indices
MSC
- 39A05
- 39A06
1 Introduction
Here we use the following standard notations: \({\mathbb{N}}\) stands for the set of all natural numbers, \({\mathbb{Z}}\) stands for the set of all integers, for an \(l\in {\mathbb{Z}}\), \({\mathbb{N}}_{l}\) is defined as \(\{n\in {\mathbb{Z}}:n\ge l\}\), \({\mathbb{R}}\) denotes the set of all real numbers, and \({\mathbb{C}}\) denotes the set of all complex numbers. Let \(k,l\in {\mathbb{Z}}\), \(k\le l\), then the notation \(j= \overline{k,l}\) denotes the set of all \(j\in {\mathbb{Z}}\) such that \(k\le j\le l\).
Solvability of difference equations has been studied for a long time. Basic classes of solvable difference equations and systems were found during the eighteenth century, and since that time the books which contain results on the topic have more or less presented the old original methods or their modifications and refinements which were obtained during the nineteenth century (see, e.g., [1–11]). For some applications of solvability methods of difference equations, see, e.g., [1, 3–5, 8, 10–19].
We are witnesses that the area of solvability of difference equations and related ones re-attracts some attention. One of the reasons for the recent interest is the use of computer algebra systems, which can help in finding or guessing closed-form formulas for solutions to some difference equations and systems. The computer algebra systems are certainly useful, but in the majority cases the authors obtain results which are known or easily follow from the known ones (see, e.g., some of our comments in [23, 24, 27–29] related to the issue).
Another area of recent interest is concrete systems of difference equations, with a special interest in symmetric and close-to-symmetric ones. The study of such systems was essentially initiated and popularized by Papaschinopoulos and Schinas (see, e.g., [37–44]). Many of our papers are also devoted to the area (see, e.g., [17, 23, 24, 27, 29, 32, 33, 36] and the references therein). For other related results, including the ones on invariants of difference equations, see, e.g., [39, 40, 43, 45, 46] and the references therein. Let us also mention that some more complex solvable difference equations and systems can be found, e.g., in [17, 33, 47–51], but the idea is essentially the same as in the above-mentioned papers, that is, some connections of studied difference equations and systems to some solvable ones are found.
We first study system (6) and then turn to the special cases of the system in (2)–(5). We show that all the results in [46] follow from some known ones. We also show how some extensions of the systems of difference equations can be solved.
2 On system (6) and the results in [46]
In this section we first study the structural form of system (6), and then by using the analysis we discuss systems (2)–(5).
2.1 Analysis of the structural form of system (6)
Here we investigate the structural form of system (6).
Hence, by induction we have proved that in calculation of all terms of the sequences \((x_{m(k+1)+1})_{m\ge -2}\) and \((y_{m(k+1)+1})_{m\ge -2}\) also only terms \(x_{-(2k+1)}\), \(y_{-(2k+1)}\), \(x_{-k}\), and \(y_{-k}\) are used.
The same argument shows that for each fixed \(j\in \{1,2,\ldots ,k+1\}\), in calculation of all terms of the sequences \((x_{m(k+1)+j})_{m\ge -2}\) and \((y_{m(k+1)+j})_{m\ge -2}\), only terms \(x_{j-(2k+2)}\), \(y_{j-(2k+2)}\), \(x_{j-(k+1)}\), and \(y_{j-(k+1)}\) are used.
It has been recently shown in [29] that system (15) is solvable, where also the long-term behavior of its solutions has been described in detail in many cases. Hence, the long-term behavior of solutions to system (6) practically directly follows from the long-term behavior of solutions to system (15).
2.2 On the results on systems (2)–(5) quoted in [46]
Here we discuss in detail the results quoted in [46].
The first result quoted in [46] is the following theorem, which was proved by a long calculatory-inductive argument.
Theorem 1
- (i)
Sequences \((x_{n})_{n\ge -(2k+1)}\) and \((y_{n})_{n\ge -(2k+1)}\) are periodic with period \(6(k+1)\).
- (ii)We have$$\begin{aligned} &x_{6(k+1)m+l}= \textstyle\begin{cases} \frac{x_{l-k-1}y_{l-2k-2}}{y_{l-2k-2}-y_{l-k-1}} , & 1\le l\le k+1, \\ \frac{y_{l-3k-3}(-x_{l-3k-3}+x_{l-2k-2})}{y_{l-3k-3}-y_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{y_{l-3k-3}(-x_{l-4k-4}+x_{l-3k-3})}{y_{l-4k-4}-y_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{y_{l-4k-4}x_{l-5k-5}}{-y_{l-5k-5}+y_{l-4k-4}}, & 3k+4\le l \le 4k+4, \\ x_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \\ &y_{6(k+1)m+l}= \textstyle\begin{cases} \frac{y_{l-k-1}x_{l-2k-2}}{x_{l-2k-2}-x_{l-k-1}} , & 1\le l\le k+1, \\ \frac{x_{l-3k-3}(-y_{l-3k-3}+y_{l-2k-2})}{x_{l-3k-3}-x_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{x_{l-3k-3}(-y_{l-4k-4}+y_{l-3k-3})}{x_{l-4k-4}-x_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{x_{l-4k-4}y_{l-5k-5}}{-x_{l-5k-5}+x_{l-4k-4}}, & 3k+4\le l \le 4k+4, \\ y_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \end{aligned}$$
for \(m\in {\mathbb{N}}_{0}\).
In [29, Theorem 4] (see also Example 4 therein for details) it was proved that every well-defined solution to system (15) with \(a=-b=c=-d=1\) is six-periodic. Bearing in mind this fact, as well as the above consideration which shows that the sequences defined in (12) and (13) are \(k+1\) independent solutions to the system, it immediately follows that for every well-defined solution to system (2) the sequences \((x_{n})_{n\ge -(2k+1)}\) and \((y_{n})_{n\ge -(2k+1)}\) are periodic with period \(6(k+1)\). So, Theorem 1(i) is a very simple consequence of known results.
The second result in [46] is the following theorem, for which it was only said that it is proved similarly to Theorem 1 (by an inductive argument).
Theorem 2
- (i)
Sequences \((x_{n})_{n\ge -(2k+1)}\) and \((y_{n})_{n\ge -(2k+1)}\) are periodic with period \(6(k+1)\).
- (ii)We have$$\begin{aligned} &x_{6(k+1)m+l}= \textstyle\begin{cases} -\frac{x_{l-k-1}y_{l-2k-2}}{y_{l-2k-2}+y_{l-k-1}} , & 1\le l\le k+1, \\ \frac{y_{l-3k-3}(x_{l-3k-3}+x_{l-2k-2})}{y_{l-3k-3}+y_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{y_{l-3k-3}(x_{l-4k-4}+x_{l-3k-3})}{y_{l-4k-4}+y_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ -\frac{y_{l-4k-4}x_{l-5k-5}}{y_{l-5k-5}+y_{l-4k-4}}, & 3k+4\le l \le 4k+4, \\ x_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \\ &y_{6(k+1)m+l}= \textstyle\begin{cases} -\frac{y_{l-k-1}x_{l-2k-2}}{x_{l-2k-2}+x_{l-k-1}} , & 1\le l\le k+1, \\ \frac{x_{l-3k-3}(y_{l-3k-3}+y_{l-2k-2})}{x_{l-3k-3}+x_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{x_{l-3k-3}(y_{l-4k-4}+y_{l-3k-3})}{x_{l-4k-4}+x_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ -\frac{x_{l-4k-4}y_{l-5k-5}}{x_{l-5k-5}+x_{l-4k-4}}, & 3k+4\le l \le 4k+4, \\ y_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \end{aligned}$$
Using again Theorem 4 in [29], we see that every well-defined solution to system (30) is six-periodic, from which along with (29) it follows that every well-defined solution to system (28) is six-periodic. Hence, we have that every well-defined solution to system (3) is periodic with period \(6(k+1)\). So, Theorem 2(i) is also a very simple consequence of known results.
The next two results quoted in [46, Corollary 4.1] are the following theorems, for which it was also only said that they can be proved similarly to Theorem 1.
Theorem 3
- (i)
Sequences \((x_{n})_{n\ge -(2k+1)}\) and \((y_{n})_{n\ge -(2k+1)}\) are periodic with period \(6(k+1)\).
- (ii)We have$$\begin{aligned} &x_{6(k+1)m+l}= \textstyle\begin{cases} \frac{x_{l-k-1}y_{l-2k-2}}{y_{l-2k-2}+y_{l-k-1}} , & 1\le l\le k+1, \\ \frac{y_{l-3k-3}(-x_{l-3k-3}+x_{l-2k-2})}{y_{l-3k-3}+y_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{y_{l-3k-3}(x_{l-4k-4}-x_{l-3k-3})}{y_{l-4k-4}+y_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{y_{l-4k-4}x_{l-5k-5}}{y_{l-5k-5}+y_{l-4k-4}}, & 3k+4\le l\le 4k+4, \\ x_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \\ &y_{6(k+1)m+l}= \textstyle\begin{cases} \frac{y_{l-k-1}x_{l-2k-2}}{-x_{l-2k-2}+x_{l-k-1}} , & 1\le l\le k+1, \\ \frac{x_{l-3k-3}(y_{l-3k-3}+y_{l-2k-2})}{x_{l-3k-3}-x_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{x_{l-3k-3}(y_{l-4k-4}+y_{l-3k-3})}{-x_{l-4k-4}+x_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{x_{l-4k-4}y_{l-5k-5}}{x_{l-5k-5}-x_{l-4k-4}}, & 3k+4\le l\le 4k+4, \\ y_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \end{aligned}$$
Using Theorem 4 in [29] we see that every well-defined solution to system (45) is six-periodic, from which along with (44) it follows that every well-defined solution to system (43) is six-periodic. Hence, every well-defined solution to system (4) is \(6(k+1)\)-periodic. So, Theorem 3(i) is also a simple consequence of known results.
Theorem 4
- (i)
Sequences \((x_{n})_{n\ge -(2k+1)}\) and \((y_{n})_{n\ge -(2k+1)}\) are periodic with period \(6(k+1)\).
- (ii)We have$$\begin{aligned} &x_{6(k+1)m+l}= \textstyle\begin{cases} \frac{x_{l-k-1}y_{l-2k-2}}{-y_{l-2k-2}+y_{l-k-1}} , & 1\le l\le k+1, \\ \frac{y_{l-3k-3}(x_{l-3k-3}+x_{l-2k-2})}{y_{l-3k-3}-y_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{y_{l-3k-3}(x_{l-4k-4}+x_{l-3k-3})}{-y_{l-4k-4}+y_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{y_{l-4k-4}x_{l-5k-5}}{y_{l-5k-5}-y_{l-4k-4}}, & 3k+4\le l\le 4k+4, \\ x_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \\ &y_{6(k+1)m+l}= \textstyle\begin{cases} \frac{y_{l-k-1}x_{l-2k-2}}{x_{l-2k-2}+x_{l-k-1}} , & 1\le l\le k+1, \\ \frac{x_{l-3k-3}(-y_{l-3k-3}+y_{l-2k-2})}{x_{l-3k-3}+x_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{x_{l-3k-3}(y_{l-4k-4}-y_{l-3k-3})}{x_{l-4k-4}+x_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{x_{l-4k-4}y_{l-5k-5}}{x_{l-5k-5}+x_{l-4k-4}}, & 3k+4\le l\le 4k+4, \\ y_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \end{aligned}$$
Remark 1
Remark 2
Note that the changes of variables (29) in Theorem 2, (44) in Theorem 3, (59) in Remark 1, along with the interlacing argument explained above show that systems (3)–(5) are equivalent to system (2), so essentially only one result was proved in [46], which as we have explained follows from the known ones.
3 Solvability of some extensions of system (6)
Here we show how some extensions of system (6) can be solved.
3.1 A three-dimensional extension to system (6)
As in the previous section, it is proved that system (61) is a system of difference equations with interlacing indices of order \(k+1\).
Using the procedure in Sect. 2, it is seen that equations (68)–(70) are three equations with interlacing indices of order three.
Since bilinear difference equations are solvable, it follows that systems (77)–(79) are also solvable. From this and (71)–(76) it follows that equations (68)–(70) are solvable, from which along with (66) it easily follows that system (61) is also solvable.
3.2 A four-dimensional extension to system (6)
As in Sect. 2 it is proved that system (80) is a system of difference equations with interlacing indices of order \(k+1\).
Using the procedure from Sect. 2, it is seen that equations (88)–(91) are four equations with interlacing indices of order four.
Since bilinear difference equations are solvable in closed form, it follows that systems (100)–(103) are also solvable. From this along with (92)–(99) it follows that equations (88)–(91) are solvable, from which and (86) it follows that system (80) is also solvable.
Remark 3
We will not present closed-form formulas for solutions to systems (61) and (80) since it is done in a standard way by using a method for solving bilinear difference equations along with the changes of variables given in this section. By using such obtained closed-form formulas for solutions to the systems, the long-term behavior of their solutions can be described, which can be done as in our papers [28, 29, 34]. We leave the standard problem to the reader as an exercise.
Remark 4
Declarations
Acknowledgements
The work of Bratislav Iričanin was supported by the Serbian Ministry of Education and Science projects III 41025 and OI 171007, work of Stevo Stević by projects III 41025 and III 44006, while the work of Zdeněk Šmarda was supported by the project FEKT-S-17-4225 of Brno University of Technology.
Availability of data and materials
Not applicable.
Funding
FEKT-S-17-4225 of Brno University of Technology.
Authors’ contributions
All authors contributed equally to the writing of this paper. All authors read and approved the manuscript.
Competing interests
The authors declare that they have no competing interests.
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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