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Solvability of a class of producttype systems of difference equations
Advances in Difference Equations volume 2016, Article number: 302 (2016)
Abstract
A solvable class of producttype 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.
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 righthand sides of some producttype equations/systems or by taking the maximum of some constants and the righthand sides of the equations/systems. This means that they are related to the producttype 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 producttype 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 producttype 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 producttype systems of difference equations with two dependent variables. Hence, our aim is to describe all such producttype systems and present formulas for the general solutions for each of them.
This paper complements our previous results on the solvability of producttype systems of difference equations with two dependent variables, by studying the following one:
where \(a,b,c,d\in {\mathbb {Z}}\), \(\alpha ,\beta \in {\mathbb {C}}\) and \(z_{1}, w_{2}, w_{1}\in {\mathbb {C}}\). To do this we will modify ideas and methods from our previous papers, for example, the ones in [19, 27–29].
If the initial values belong to the following set:
and if some of the exponents a, b, c, d are negative, then such solutions are not defined. Hence, this set of initial values will not be taken into consideration. Besides, if \(\alpha =0\) or \(\beta =0\), we get \(z_{n}=0\) and \(w_{n}=0\) for every \(n\in {\mathbb {N}}_{0}\), respectively, which are quite simple cases, or also obtain solutions which are not well defined, so, we will also assume that \(\alpha \ne0\ne \beta \). We use the convention \(\sum_{i=k}^{l}a_{i}=0\), when \(l< k\), throughout the paper.
Main results
Our main results are presented here. The following three cases will be considered separately:

(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 \(acbd\), 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
Assume that \(a,b,d\in {\mathbb {Z}}\), \(c=0\), \(bd=0\), \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\) and \(z_{1}, w_{2}, w_{1}\in {\mathbb {C}}\setminus\{0\}\). Then the following statements hold.

(a)
If \(a\ne1\), then the general solution to system (1) is given by the following formulas:
$$\begin{aligned} z_{n}=\alpha ^{\frac{1a^{n+1}}{1a}}\beta ^{b\frac {1a^{n1}}{1a}}z_{1}^{a^{n+1}}w_{2}^{ba^{n}}w_{1}^{ba^{n1}},\quad n\ge2, \end{aligned}$$(2)and
$$\begin{aligned} w_{n}=\alpha ^{d\frac{1a^{n}}{1a}}\beta z_{1}^{da^{n}},\quad n \ge2. \end{aligned}$$(3) 
(b)
If \(a=1\), then the general solution to system (1) is given by the following formulas:
$$\begin{aligned} z_{n}=\alpha ^{n+1}\beta ^{b(n1)}z_{1}w_{2}^{b}w_{1}^{b},\quad n\ge 2, \end{aligned}$$(4)and
$$\begin{aligned} w_{n}=\alpha ^{dn}\beta z_{1}^{d}, \quad n \ge2. \end{aligned}$$(5)
Proof
Since \(c=0\), we have
From (6) and \(bd=0\) is obtained
From (7) we easily get
which, along with
yields
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 \(a=1\).
From (9), the equality \(w_{n}=\beta z_{n1}^{d}\), and the condition \(bd =0\), we obtain
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
Since \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\) and \(z_{1}, w_{2}, w_{1}\in {\mathbb {C}}\setminus\{0\}\), using (1) it easily follows that \(z_{n}\ne0\) for \(n\ge1\) and \(w_{n}\ne0\) for \(n\ge2\). Thus, from (1) we have
and
Let \(\mu=\alpha ^{1c}\beta ^{b}\),
Then
From (15) is obtained
for \(n\in {\mathbb {N}}\), where
Suppose that
for a \(k\in {\mathbb {N}}\) and every \(n\ge k1\), and
From (15) with \(n\to nk\) and (18), it follows that
for \(n\ge k\), where
Equalities (16), (17), (21), (22), (23), and the induction show that (18), (19), and (20), hold for every \(k, n\in {\mathbb {N}}\) such that \(2\le k\le n+1\). Note that (18) holds also for \(k=1\).
For \(k=n+1\), (18) becomes
where we have also used the fact \(z_{0}=\alpha z_{1}^{a}w_{2}^{b}\), (8), (19), and (20).
Further note that (19) implies that \((a_{k})_{k\ge4}\) is a solution to the equation
Since \(c_{1}\ne0\), from (25), we get
from which it follows that we can calculate \(a_{l}\) for every nonpositive l, that is, for \(k\le3\). A direct calculation shows that
Moreover, it is shown that \((a_{k})_{k\ge2}\), \((b_{k})_{k\ge2}\), \((c_{k})_{k\ge2}\) are solutions to (25) such that
respectively, whereas \((y_{k})_{k\ge2}\) satisfies (20) and
From (20) and since \(a_{0}=1\), we get
The solvability of (25) shows that for \((a_{k})_{k\ge2}\) can be found a closed form formula. Therefore, using equation (29) and known formulas for the following sums:
where \(j=\overline {0,2}\) (see, e.g., [31, 33]), the closed form formula for \((y_{k})_{k\in {\mathbb {N}}}\) is easily obtained. This along with (24) shows the solvability of (13).
From (1), we also have
and
Equalities (31) and (32) yield
We have
Similarly as above we get
where \(\eta=\alpha ^{d}\beta ^{1a}\), \((a_{k})_{k\in {\mathbb {N}}}\), \((b_{k})_{k\in {\mathbb {N}}}\) and \((c_{k})_{k\in {\mathbb {N}}}\) are defined by (14) and (19), whereas \((y_{k})_{k\in {\mathbb {N}}}\) satisfies (20) and (28), so it is given by (29).
From (35) with \(k=n+1\) and (34) is obtained
for \(n\in {\mathbb {N}}_{0}\).
The solvability of (25) along with (27) shows that for \((a_{k})_{k\ge2}\) 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
Assume that \(a,b,c,d\in {\mathbb {Z}}\), \(c\ne 0\), \(ac=bd\), \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\) and \(z_{1}, w_{2}, w_{1}\in {\mathbb {C}}\setminus\{0\}\). Then the following statements hold.

(a)
If \(a^{2}\ne4c\) and \(a+c\ne1\), then the general solution to system (1) is given by the following formulas:
$$\begin{aligned} z_{n}={}&\alpha ^{\frac{a(\lambda _{2}1)\lambda _{1}^{n+1}a(\lambda _{1}1)\lambda _{2}^{n+1}+(\lambda _{1}\lambda _{2})(1c)}{(\lambda _{1}1)(\lambda _{2}1)(\lambda _{1}\lambda _{2})}} \beta ^{b\frac{(\lambda _{2}1)\lambda _{1}^{n}(\lambda _{1}1)\lambda _{2}^{n}+\lambda _{1}\lambda _{2}}{(\lambda _{1}1)(\lambda _{2}1)(\lambda _{1}\lambda _{2})}} \\ &\times z_{1}^{a\frac{\lambda _{1}^{n+1}\lambda _{2}^{n+1}}{\lambda _{1}\lambda _{2}}}w_{2}^{b\frac{\lambda _{1}^{n+1}\lambda _{2}^{n+1}}{\lambda _{1}\lambda _{2}}}w_{1}^{b\frac{\lambda _{1}^{n}\lambda _{2}^{n}}{\lambda _{1}\lambda _{2}}} \end{aligned}$$(37)and
$$\begin{aligned} w_{n}={}&\alpha ^{d\frac{(\lambda _{2}1)\lambda _{1}^{n+1}(\lambda _{1}1)\lambda _{2}^{n+1}+\lambda _{1}\lambda _{2}}{(\lambda _{1}1)(\lambda _{2}1)(\lambda _{1}\lambda _{2})}} \beta ^{\frac{c(\lambda _{2}1)\lambda _{1}^{n}c(\lambda _{1}1)\lambda _{2}^{n}+(\lambda _{1}\lambda _{2})(1a)}{(\lambda _{1}1)(\lambda _{2}1)(\lambda _{1}\lambda _{2})}} \\ &\times z_{1}^{d\frac{\lambda _{1}^{n+1}\lambda _{2}^{n+1}}{\lambda _{1}\lambda _{2}}}w_{2}^{c\frac{\lambda _{1}^{n+1}\lambda _{2}^{n+1}}{\lambda _{1}\lambda _{2}}}w_{1}^{c\frac{\lambda _{1}^{n}\lambda _{2}^{n}}{\lambda _{1}\lambda _{2}}}, \end{aligned}$$(38)where
$$\begin{aligned} \lambda _{1,2}=\frac{a+\sqrt{a^{2}+4c}}{2}. \end{aligned}$$(39) 
(b)
If \(a^{2}\ne4c\) and \(a+c=1\), then the general solution to system (1) is given by the following formulas:
$$ z_{n}= \alpha ^{\frac{a\lambda _{1}^{n+1}+((c1)n2)\lambda _{1}+(1c)n+1+c}{(1\lambda _{1})^{2}}}\beta ^{b\frac{n1n\lambda _{1}+\lambda _{1}^{n}}{(1\lambda _{1})^{2}}} z_{1}^{a\frac{\lambda _{1}^{n+1}1}{\lambda _{1}1}}w_{2}^{b\frac{\lambda _{1}^{n+1}1}{\lambda _{1}1}}w_{1}^{b\frac{\lambda _{1}^{n}1}{\lambda _{1}1}} $$(40)and
$$\begin{aligned} w_{n}={}&\alpha ^{d\frac{n(n+1)\lambda _{1}+\lambda _{1}^{n+1}}{(1\lambda _{1})^{2}}}\beta ^{\frac{c\lambda _{1}^{n}+((a1)n+a2)\lambda _{1}+(1a)n+1}{(1\lambda _{1})^{2}}} \\ &\times z_{1}^{d\frac{\lambda _{1}^{n+1}1}{\lambda _{1}1}}w_{2}^{c\frac{\lambda _{1}^{n+1}1}{\lambda _{1}1}}w_{1}^{c\frac{\lambda _{1}^{n}1}{\lambda _{1}1}}, \end{aligned}$$(41)where
$$\begin{aligned} \lambda _{1}=c. \end{aligned}$$(42) 
(c)
If \(a^{2}=4c\) and \(a+c\ne1\), then the general solution to system (1) is given by the following formulas:
$$\begin{aligned} z_{n}={}&\alpha ^{\frac{an\lambda _{1}^{n+1}a(n+1)\lambda _{1}^{n}+1c}{(1\lambda _{1})^{2}}}\beta ^{b\frac{1n\lambda _{1}^{n1}+(n1)\lambda _{1}^{n}}{(1\lambda _{1})^{2}}} \\ &\times z_{1}^{a(n+1)\lambda _{1}^{n}}w_{2}^{b(n+1)\lambda _{1}^{n}}w_{1}^{bn\lambda _{1}^{n1}} \end{aligned}$$(43)and
$$\begin{aligned} w_{n}={}&\alpha ^{d\frac{1(n+1)\lambda _{1}^{n}+n\lambda _{1}^{n+1}}{(1\lambda _{1})^{2}}}\beta ^{\frac{c(n1)\lambda _{1}^{n}cn\lambda _{1}^{n1}+1a}{(1\lambda _{1})^{2}}} \\ &\times z_{1}^{d(n+1)\lambda _{1}^{n}}w_{2}^{c(n+1)\lambda _{1}^{n}}w_{1}^{cn\lambda _{1}^{n1}}, \end{aligned}$$(44)where
$$\begin{aligned} \lambda _{1}=\frac{a}{2}. \end{aligned}$$(45) 
(d)
If \(a^{2}=4c\) and \(a+c=1\), then the general solution to system (1) is given by the following formulas:
$$ z_{n}=\alpha ^{\frac{(1c)n^{2}+(c+3)n+2}{2}}\beta ^{b\frac {(n1)n}{2}}z_{1}^{a(n+1)}w_{2}^{b(n+1)}w_{1}^{bn} $$(46)and
$$ w_{n}=\alpha ^{d\frac{n(n+1)}{2}}\beta ^{\frac{(n+1)((1a)n+2)}{2}} z_{1}^{d(n+1)}w_{2}^{c(n+1)}w_{1}^{cn}. $$(47)
Proof
Similar to the proof of Theorem 2 we have \(z_{n}\ne0\) for \(n\ge1\) and \(w_{n}\ne0\) for \(n\ge2\), and for every such solution (13) holds, from which, along with the condition \(ac=bd\), it follows that
for \(n\in {\mathbb {N}}_{0}\).
Let \(\nu=\alpha ^{1c}\beta ^{b}\),
Then
for \(n\in {\mathbb {N}}_{0}\).
From (50) is obtained
for \(n\in {\mathbb {N}}\), where
Suppose that
holds for some \(k\in {\mathbb {N}}\) and for every \(n\ge k1\), and that
Then from (50), with \(n\to nk\) and (53), we get
for \(n\ge k\), where
Equalities (51), (52), (55), (56), and the induction show that (53) and (54), hold for every \(k,n\in {\mathbb {N}}\) such that \(2\le k\le n+1\). Note that the equality in (53) also holds for \(k=1\) (see (50)).
For \(k=n+1\), (53) becomes
where we have used the fact \(z_{0}=\alpha z_{1}^{a}w_{2}^{b}\), (8), and (54).
Now note that \((\hat{a}_{k})_{k\ge3}\) is a solution to the equation
As in the proof of Theorem 2, it is shown that \((\hat{a}_{k})_{k\ge2}\) and \((\hat{b}_{k})_{k\ge2}\) are solutions to equation (58), satisfying the initial conditions
respectively, whereas \((\hat{y}_{k})_{k\ge1}\) satisfies the third equation in (54) and
This and \(\hat{a}_{0}=1\) imply
Since (33) also holds, using the condition \(ac=bd\), we get
Similar to the above is obtained
for \(k,n\in {\mathbb {N}}\), \(1\le k \le n+1\), where \(\eta=\alpha ^{d}\beta ^{1a}\), sequences \((\hat{a}_{k})_{k\in {\mathbb {N}}}\), and \((\hat{b}_{k})_{k\in {\mathbb {N}}}\) satisfy (49) and (54), whereas \((\hat{y}_{k})_{k\in {\mathbb {N}}}\) satisfies the third equation in (54) and (60), so, formula (61) holds.
From (63) with \(k=n+1\) and (34) it follows that
The characteristic polynomial associated to the linear difference equation (58) is the following:
from which it follows that the corresponding characteristic roots are given by the formulas in (39).
Since \(a_{1}=0\) and \(a_{0}=1\), then, if \(a^{2}\ne4c\) and \(a+c\ne1\), we easily get
and consequently
If \(a^{2}\ne4c\) and \(a+c=1\), then one of the characteristic roots, say \(\lambda _{2}\), is equal to one, and we have
and
If \(a^{2}=4c\) and \(a+c\ne1\), then we have
and
If \(a^{2}=4c\) and \(a+c=1\), then we have
and
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 timeconsuming calculations, it is shown that these formulas really represent solutions to system (1) in each if these four cases. □
Remark 1
Note that if \(a^{2}=4c\) and \(a+c=1\), then \((a2)^{2}=0\), from which it follows that \(a=2\) and consequently \(c=1\). Hence, equations (46) and (47) can also be written in the following, more concrete, forms:
and
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Stević, S. Solvability of a class of producttype systems of difference equations. Adv Differ Equ 2016, 302 (2016). https://doi.org/10.1186/s136620161031z
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DOI: https://doi.org/10.1186/s136620161031z
MSC
 39A20
 39A45
Keywords
 system of difference equations
 producttype system
 solvable system
 solutions in closed form