On a product-type system of difference equations of second order solvable in closed form
- Stevo Stević^{1, 2}Email author,
- Bratislav Iričanin^{3} and
- Zdeněk Šmarda^{4, 5}
https://doi.org/10.1186/s13660-015-0835-9
© Stević et al. 2015
Received: 14 August 2015
Accepted: 23 September 2015
Published: 12 October 2015
Abstract
Keywords
MSC
1 Introduction
Recently there has been a great interest in studying nonlinear difference equations and systems not stemming from differential ones (see, e.g., [1–30]). The old area of solving difference equations and systems has re-attracted recent attention (see, e.g., [1–6, 12, 15, 19–26, 28–30]). Recent Stević’s idea of transforming complicated equations and systems into simpler solvable ones, used for the first time in explaining the solvability of the equation appearing in [6] (an extension of the original result can be found in [20], see also [22]), was employed in several other papers (see, e.g., [1, 2, 4, 12, 15, 19, 21, 24–26, 29, 30] and the related references therein). Another area of some recent interest, essentially initiated by Papaschinopoulos and Schinas, is studying symmetric and close to symmetric systems of difference equations (see, e.g., [5, 8–10, 13, 14, 19, 21, 23, 25, 27–29]).
Let us mention here that although systems (3) and (4) are similar in appearance, the methods used in dealing with them are quite different.
2 Main result
The main result in this paper is proved in this section.
Theorem 1
Assume that \(a,b,c,d\in \mathbb {Z}\) and \(z_{-1}, z_{0}, w_{-1}, w_{0}\in \mathbb {C}\setminus\{0\}\). Then system (4) is solvable in closed form.
Proof
From (7), (9) and the method of induction we see that (8) holds for every k such that \(1\le k\le n-1\).
Now we have two subcases to consider.
From (18), (20) and the method of induction we see that (19) holds for every k such that \(1\le k\le n-1\).
Now we have two subcases to consider.
Since \(b_{k-1}=a_{k}-a_{1}a_{k-1}\) and equation (44) is linear, we have that the sequence \((b_{k})_{k\in \mathbb {N}}\) is also a solution to equation (44). From this, the linearity of equation (44) and since \(c_{k-1}=b_{k}-b_{1}a_{k-1}\), we have that the sequence \((c_{k})_{k\in \mathbb {N}}\) is also a solution to equation (44). Finally, since \(d_{k}=d_{1}a_{k-1}\), the linearity of equation (44) shows that \((d_{k})_{k\in \mathbb {N}}\) is also a solution to the equation.
Now, we show that these four sequences can be prolonged for some negative indices of use. This enables easier getting formulas for solutions to system (4).
Since difference equation (44) is solvable, it follows that closed form formulas for \((a_{k})_{k\ge-3}\), \((b_{k})_{k\ge-3}\), \((c_{k})_{k\ge-3}\) and \((d_{k})_{k\ge-3}\) can be found. From this fact and (43) we see that equation (29) is solvable too.
Also the sequences \((a_{k})_{k\in \mathbb {N}}\), \((b_{k})_{k\in \mathbb {N}}\), \((c_{k})_{k\in \mathbb {N}}\) and \((d_{k})_{k\in \mathbb {N}}\) satisfy the difference equation (44) with initial conditions in (49), respectively.
As above the solvability of equation (44) shows that closed form formulas for \((a_{k})_{k\ge-3}\), \((b_{k})_{k\ge-3}\), \((c_{k})_{k\ge-3}\) and \((d_{k})_{k\ge-3}\) can be found. This fact along with (55) implies that equation (52) is solvable too. A direct calculation shows that the sequences \((z_{n})_{n\ge-1}\) in (43) and \((w_{n})_{n\ge-1}\) in (55) are solutions to system (4) with initial values \(z_{-1}\), \(z_{0}\), that is, \(w_{-1}\), \(w_{0}\) respectively. Hence, system (4) is also solvable in this case, finishing the proof of the theorem. □
Remark 1
Remark 2
Since we are interested in those initial values \(z_{-1}, z_{0}, w_{-1}, w_{0}\in \mathbb {C}\) which uniquely define solutions to system (4), to avoid multi-valued solutions to the system, we posed the condition \(a,b,c,d\in \mathbb {Z}\).
From the proof of Theorem 1 we obtain the following corollary.
Corollary 1
- (a)
- (b)
- (c)
- (d)
- (e)
In order to find, in this case, a general solution to system (4) in closed form, we will need the following known lemma. We give a proof of it for the completeness and benefit of the reader.
Lemma 1
Proof
From this and since \(\operatorname{Res}_{z=\infty}f_{l}(z)\) is equal to the negative value of the coefficient at \(1/z\) in the Laurent expansion, it follows that \(\operatorname{Res}_{z=\infty}f_{l}(z)=0\) when \(l=\overline {0,k-2}\) and \(\operatorname{Res}_{z=\infty}f_{k-1}(z)=-1/\alpha _{k}\). Using these facts in (58) the lemma follows. □
By using (59), (63), (64) and (65) into (43) and (55), we get formulas for general solutions to system (4) in closed form.
Formulas obtained in this section can be used in describing the long-term behavior of solutions to system (4) in many cases. We will formulate and prove here only one result, just as an example. The formulations and proofs of other results, which are similar and whose proofs use standard techniques, we leave to the reader as some exercises.
Theorem 2
- (a)
If \(a=1\), then every solution to system (4) is eventually constant.
- (b)
If \(a=0\), then \(z_{n}=w_{n}=1\), \(n\ge3\).
- (c)
If \(a=-1\), then every solution to system (4) is two-periodic.
- (d)
If \(a>1\) and \(|z_{0}|<1\), then \(z_{n}\to0\) as \(n\to\infty\).
- (e)
If \(a>1\) and \(|z_{0}|>1\), then \(|z_{n}|\to\infty\) as \(n\to\infty\).
- (f)
If \(a>1\) and \(|z_{0}^{d}|<1\), then \(|w_{n}|\to\infty\) as \(n\to\infty\).
- (g)
If \(a>1\) and \(|z_{0}^{d}|>1\), then \(w_{n}\to0\) as \(n\to\infty\).
- (h)
If \(a<-1\) and \(|z_{0}|<1\), then \(z_{2n}\to0\) as \(n\to\infty\) and \(|z_{2n+1}|\to\infty\) as \(n\to\infty\).
- (i)
If \(a<-1\) and \(|z_{0}|>1\), then \(z_{2n+1}\to0\) as \(n\to\infty\) and \(|z_{2n}|\to\infty\) as \(n\to\infty\).
- (j)
If \(a<-1\) and \(|z_{0}^{d}|>1\), then \(w_{2n}\to0\) as \(n\to\infty\) and \(|w_{2n+1}|\to\infty\) as \(n\to\infty\).
- (k)
If \(a<-1\) and \(|z_{0}^{d}|<1\), then \(w_{2n+1}\to0\) as \(n\to\infty\) and \(|w_{2n}|\to\infty\) as \(n\to\infty\).
Proof
(a) If we replace \(a=1\) and \(c=0\) in (6) and (7), we obtain \(z_{n}=z_{0}\) and \(w_{n+2}=1/z_{0}^{d}\), \(n\in \mathbb {N}_{0}\), from which the statement follows.
(b) By replacing \(a=0\) and \(c=0\) in (6) and (7), we get \(z_{n}=1\), \(n\in \mathbb {N}\) and \(w_{n}=1\), \(n\ge3\), from which the statement follows.
(c) By replacing \(a=-1\) and \(c=0\) in (6) and (7), we get \(z_{2n}=z_{0}\), \(z_{2n+1}=\frac{1}{z_{0}}\), \(w_{2n}=1/{z_{0}^{d}}\) and \(w_{2n+1}=z_{0}^{d}\), \(n\in \mathbb {N}\), from which the statement follows.
Declarations
Acknowledgements
The work of Stevo Stević is supported by the Serbian Ministry of Education and Science projects III 41025 and III 44006. The work of Bratislav Iričanin is supported by the Serbian Ministry of Education and Science projects III 41025 and OI 171007. The work of Zdeňek Šmarda was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund. He was also supported by the project FEKT-S-14-2200 of Brno University of Technology.
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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