Solvability of a class of product-type systems of difference equations
- Stevo Stević^{1, 2}Email author
https://doi.org/10.1186/s13662-016-1031-z
© The Author(s) 2016
Received: 5 October 2016
Accepted: 17 November 2016
Published: 25 November 2016
Abstract
A solvable class of product-type systems of difference equations with two dependent variables on the complex domain is presented. The main results complement some recent ones in the literature, while their proofs contain some refined methodological details. We provide closed form formulas for general solutions to the system or give procedures for how to get them.
Keywords
MSC
1 Introduction
Nonlinear difference equations and systems have been studied a lot in the last few decades (see, e.g., [1–29]). Two of the topics of recent interest are symmetric and closely related systems (see, for example, [3, 6–12, 15, 16, 19, 20, 22, 23, 26–29]), whose investigation was considerably influenced by some papers by Papaschinopoulos and Schinas (see, for example, [8–10]), and the solvable difference equations and systems (see, for example, [3, 14, 18–22, 24–29] and the references therein). For some classical methods for solving the equations and systems see, for example, [1, 30–33]. It has been shown recently that many nonlinear equations and systems can be solved by transforming them to linear ones (see, for example, [3, 14, 18, 21, 24, 25] and the related references therein).
Some of the equations and systems that we have studied recently, such as the ones in [17] and [23] (see also [13]), are obtained by adding constants to the right-hand sides of some product-type equations/systems or by taking the maximum of some constants and the right-hand sides of the equations/systems. This means that they are related to the product-type ones, which are usually some kind of boundary cases. The case of positive initial values and multipliers is simple, since in that case the equations/systems can easily be treated by one of the simplest transformation methods. The case of complex initial values is more complex due to the fact that complex functions need not be single valued. Hence, our methods in [3, 18, 21] and other related papers cannot be applied. This has motivated us to start investigating product-type systems on the complex domain. In a series of papers, see [19, 20, 22, 26–29], we have obtained some results in the area (during the study of the equation in [21] we came across a product-type equation). In our first papers on the topic (see [20, 22, 26]) the systems have not had coefficients different from one. However, not long after that we have introduced two coefficients and also got solvable systems (this was done for the first time in [19], and somewhat later in [27–29]). We have also observed the fact that there are only a few solvable product-type systems of difference equations with two dependent variables. Hence, our aim is to describe all such product-type systems and present formulas for the general solutions for each of them.
2 Main results
- (i)
\(c=0\), \(ac=bd\);
- (ii)
\(c\ne0\), \(ac=bd\);
- (iii)
\(ac\ne bd\).
Clearly, in case (i) from \(c=0\) and \(ac=bd\) it immediately follows that \(bd=0\), but we have chosen to write \(ac=bd\) at this point, to point out that the whole analysis essentially depends on the values of the quantities c and \(ac-bd\), that is, if they are equal to zero or not.
First, we will consider case (i), then case (iii) and at the end case (ii), for the presentational reasons.
Theorem 1
- (a)If \(a\ne1\), then the general solution to system (1) is given by the following formulas:and$$\begin{aligned} z_{n}=\alpha ^{\frac{1-a^{n+1}}{1-a}}\beta ^{b\frac {1-a^{n-1}}{1-a}}z_{-1}^{a^{n+1}}w_{-2}^{ba^{n}}w_{-1}^{ba^{n-1}},\quad n\ge2, \end{aligned}$$(2)$$\begin{aligned} w_{n}=\alpha ^{d\frac{1-a^{n}}{1-a}}\beta z_{-1}^{da^{n}},\quad n \ge2. \end{aligned}$$(3)
- (b)If \(a=1\), then the general solution to system (1) is given by the following formulas:and$$\begin{aligned} z_{n}=\alpha ^{n+1}\beta ^{b(n-1)}z_{-1}w_{-2}^{b}w_{-1}^{b},\quad n\ge 2, \end{aligned}$$(4)$$\begin{aligned} w_{n}=\alpha ^{dn}\beta z_{-1}^{d}, \quad n \ge2. \end{aligned}$$(5)
Proof
By using equation (9) along with the formula for the sum of the geometric progression we see that equation (2) holds when \(a\ne1\), while equation (4) is directly obtained for \(a=1\).
By using equation (10) along with the formula for the sum of the geometric progression we see that equation (3) holds when \(a\ne1\), while equation (5) is directly obtained for \(a=1\), completing the proof of the theorem. □
Theorem 2
Assume that \(a,b,c,d\in {\mathbb {Z}}\), \(ac\ne bd\), \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\) and \(z_{-1}, w_{-2}, w_{-1}\in {\mathbb {C}}\setminus\{0\}\). Then system (1) is solvable in closed form.
Proof
The solvability of (25) along with (27) shows that for \((a_{k})_{k\ge-2}\) we can find a closed form formula, from which along with (29) the formulas for \(y_{k}\) can also be obtained, as described above. These facts along with (36) imply the solvability of equation (33). It is not difficult to show that formulas (24) and (36) really represent solutions to system (1). Thus, system (1) is also solvable in this case, as claimed. □
Corollary 1
Consider system (1) with \(a,b,c,d\in {\mathbb {Z}}\), \(ac\ne bd\), \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\), and \(z_{-1}, w_{-2}, w_{-1}\in {\mathbb {C}}\setminus\{0\}\). Then the general solution to system (1) is given by (24) and (36), where \((a_{k})_{k\in {\mathbb {N}}}\) satisfies equation (25) with initial conditions (27), while \((y_{k})_{k\in {\mathbb {N}}}\) is given by (29) and can be found by using formulas for the sums in (30).
Theorem 3
- (a)If \(a^{2}\ne-4c\) and \(a+c\ne1\), then the general solution to system (1) is given by the following formulas:and$$\begin{aligned} z_{n}={}&\alpha ^{\frac{a(\lambda _{2}-1)\lambda _{1}^{n+1}-a(\lambda _{1}-1)\lambda _{2}^{n+1}+(\lambda _{1}-\lambda _{2})(1-c)}{(\lambda _{1}-1)(\lambda _{2}-1)(\lambda _{1}-\lambda _{2})}} \beta ^{b\frac{(\lambda _{2}-1)\lambda _{1}^{n}-(\lambda _{1}-1)\lambda _{2}^{n}+\lambda _{1}-\lambda _{2}}{(\lambda _{1}-1)(\lambda _{2}-1)(\lambda _{1}-\lambda _{2})}} \\ &\times z_{-1}^{a\frac{\lambda _{1}^{n+1}-\lambda _{2}^{n+1}}{\lambda _{1}-\lambda _{2}}}w_{-2}^{b\frac{\lambda _{1}^{n+1}-\lambda _{2}^{n+1}}{\lambda _{1}-\lambda _{2}}}w_{-1}^{b\frac{\lambda _{1}^{n}-\lambda _{2}^{n}}{\lambda _{1}-\lambda _{2}}} \end{aligned}$$(37)where$$\begin{aligned} w_{n}={}&\alpha ^{d\frac{(\lambda _{2}-1)\lambda _{1}^{n+1}-(\lambda _{1}-1)\lambda _{2}^{n+1}+\lambda _{1}-\lambda _{2}}{(\lambda _{1}-1)(\lambda _{2}-1)(\lambda _{1}-\lambda _{2})}} \beta ^{\frac{c(\lambda _{2}-1)\lambda _{1}^{n}-c(\lambda _{1}-1)\lambda _{2}^{n}+(\lambda _{1}-\lambda _{2})(1-a)}{(\lambda _{1}-1)(\lambda _{2}-1)(\lambda _{1}-\lambda _{2})}} \\ &\times z_{-1}^{d\frac{\lambda _{1}^{n+1}-\lambda _{2}^{n+1}}{\lambda _{1}-\lambda _{2}}}w_{-2}^{c\frac{\lambda _{1}^{n+1}-\lambda _{2}^{n+1}}{\lambda _{1}-\lambda _{2}}}w_{-1}^{c\frac{\lambda _{1}^{n}-\lambda _{2}^{n}}{\lambda _{1}-\lambda _{2}}}, \end{aligned}$$(38)$$\begin{aligned} \lambda _{1,2}=\frac{a+\sqrt{a^{2}+4c}}{2}. \end{aligned}$$(39)
- (b)If \(a^{2}\ne-4c\) and \(a+c=1\), then the general solution to system (1) is given by the following formulas:and$$ z_{n}= \alpha ^{\frac{a\lambda _{1}^{n+1}+((c-1)n-2)\lambda _{1}+(1-c)n+1+c}{(1-\lambda _{1})^{2}}}\beta ^{b\frac{n-1-n\lambda _{1}+\lambda _{1}^{n}}{(1-\lambda _{1})^{2}}} z_{-1}^{a\frac{\lambda _{1}^{n+1}-1}{\lambda _{1}-1}}w_{-2}^{b\frac{\lambda _{1}^{n+1}-1}{\lambda _{1}-1}}w_{-1}^{b\frac{\lambda _{1}^{n}-1}{\lambda _{1}-1}} $$(40)where$$\begin{aligned} w_{n}={}&\alpha ^{d\frac{n-(n+1)\lambda _{1}+\lambda _{1}^{n+1}}{(1-\lambda _{1})^{2}}}\beta ^{\frac{c\lambda _{1}^{n}+((a-1)n+a-2)\lambda _{1}+(1-a)n+1}{(1-\lambda _{1})^{2}}} \\ &\times z_{-1}^{d\frac{\lambda _{1}^{n+1}-1}{\lambda _{1}-1}}w_{-2}^{c\frac{\lambda _{1}^{n+1}-1}{\lambda _{1}-1}}w_{-1}^{c\frac{\lambda _{1}^{n}-1}{\lambda _{1}-1}}, \end{aligned}$$(41)$$\begin{aligned} \lambda _{1}=-c. \end{aligned}$$(42)
- (c)If \(a^{2}=-4c\) and \(a+c\ne1\), then the general solution to system (1) is given by the following formulas:and$$\begin{aligned} z_{n}={}&\alpha ^{\frac{an\lambda _{1}^{n+1}-a(n+1)\lambda _{1}^{n}+1-c}{(1-\lambda _{1})^{2}}}\beta ^{b\frac{1-n\lambda _{1}^{n-1}+(n-1)\lambda _{1}^{n}}{(1-\lambda _{1})^{2}}} \\ &\times z_{-1}^{a(n+1)\lambda _{1}^{n}}w_{-2}^{b(n+1)\lambda _{1}^{n}}w_{-1}^{bn\lambda _{1}^{n-1}} \end{aligned}$$(43)where$$\begin{aligned} w_{n}={}&\alpha ^{d\frac{1-(n+1)\lambda _{1}^{n}+n\lambda _{1}^{n+1}}{(1-\lambda _{1})^{2}}}\beta ^{\frac{c(n-1)\lambda _{1}^{n}-cn\lambda _{1}^{n-1}+1-a}{(1-\lambda _{1})^{2}}} \\ &\times z_{-1}^{d(n+1)\lambda _{1}^{n}}w_{-2}^{c(n+1)\lambda _{1}^{n}}w_{-1}^{cn\lambda _{1}^{n-1}}, \end{aligned}$$(44)$$\begin{aligned} \lambda _{1}=\frac{a}{2}. \end{aligned}$$(45)
- (d)If \(a^{2}=-4c\) and \(a+c=1\), then the general solution to system (1) is given by the following formulas:and$$ z_{n}=\alpha ^{\frac{(1-c)n^{2}+(c+3)n+2}{2}}\beta ^{b\frac {(n-1)n}{2}}z_{-1}^{a(n+1)}w_{-2}^{b(n+1)}w_{-1}^{bn} $$(46)$$ w_{n}=\alpha ^{d\frac{n(n+1)}{2}}\beta ^{\frac{(n+1)((1-a)n+2)}{2}} z_{-1}^{d(n+1)}w_{-2}^{c(n+1)}w_{-1}^{cn}. $$(47)
Proof
Using (65)-(72) into (57) and (64) and by some calculations equations (37), (38), (40), (41), (43), (44), (46), and (47) are obtained. By some standard, but time-consuming calculations, it is shown that these formulas really represent solutions to system (1) in each if these four cases. □
Remark 1
Declarations
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Authors’ Affiliations
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