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On a close to symmetric system of difference equations of second order

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Abstract

Closed form formulas of the solutions to the following system of difference equations:

$$x_{n}=\frac{y_{n-1}y_{n-2}}{x_{n-1}(a_{n}+b_{n}y_{n-1}y_{n-2})},\qquad y_{n}=\frac{x_{n-1}x_{n-2}}{y_{n-1}(\alpha _{n}+\beta _{n}x_{n-1}x_{n-2})},\quad n\in \mathbb {N}_{0}, $$

where \(a_{n}\), \(b_{n}\), \(\alpha _{n}\), \(\beta _{n}\), \(n\in \mathbb {N}_{0}\), and initial values \(x_{-i}\), \(y_{-i}\), \(i\in\{1,2\}\) are real numbers, are found. The domain of undefinable solutions to the system is described. The long-term behavior of its solutions is studied in detail for the case of constant \(a_{n}\), \(b_{n}\), \(\alpha _{n}\) and \(\beta _{n}\), \(n\in \mathbb {N}_{0}\).

Introduction

Studying concrete nonlinear difference equations and systems is a topic of a great recent interest (see, e.g., [146] and the references therein). Studying systems of difference equations, especially symmetric and close to symmetric ones, is a topic of considerable interest (see, e.g., [2, 6, 7, 10, 1216, 18, 19, 23, 24, 2629, 3138, 40, 41, 44, 46]). Another topic of interest is solvable difference equations and systems and their applications (see, e.g., [15, 7, 17, 20, 21, 2327, 2937, 3946]). Renewed interest in the area started after the publication of [20] where a formula for a solution of a difference equation was theoretically explained. The most interesting thing in [20] was a change of variables which reduced the equation to a linear one with constant coefficients. Related ideas were later used, e.g., in [1, 4, 7, 17, 21, 2327, 2937, 3945].

Quite recently in [2] the following systems of difference equations were presented:

$$ \begin{aligned} &x_{n}=\frac{y_{n-1}y_{n-2}}{x_{n-1}(\pm1\pm y_{n-1}y_{n-2})},\\ &y_{n}= \frac{x_{n-1}x_{n-2}}{y_{n-1}(\pm1\pm x_{n-1}x_{n-2})},\quad n\in \mathbb {N}_{0}, \end{aligned} $$
(1)

where \(x_{-i}\), \(y_{-i}\), \(i\in\{1,2\}\) are real numbers, and some formulas for their solutions are given, some of which are proved by induction.

The next system of difference equations

$$ \begin{aligned} &x_{n}=\frac{y_{n-1}y_{n-2}}{x_{n-1}(a_{n}+b_{n}y_{n-1}y_{n-2})},\\ &y_{n}=\frac{x_{n-1}x_{n-2}}{y_{n-1}(\alpha _{n}+\beta _{n}x_{n-1}x_{n-2})},\quad n\in \mathbb {N}_{0}, \end{aligned} $$
(2)

where \(a_{n}\), \(b_{n}\), \(\alpha _{n}\), \(\beta _{n}\), \(n\in \mathbb {N}_{0}\), and initial values \(x_{-i}\), \(y_{-i}\), \(i\in\{1,2\}\), are real numbers, is a generalization of the system in (1). Our aim is to show that more general system (2) is solvable by giving a natural method for getting its solutions. The domain of undefinable solutions to the system is also described. For the case when \(a_{n}\), \(b_{n}\), \(\alpha _{n}\), \(\beta _{n}\), \(n\in \mathbb {N}_{0}\), are constant, the long-term behavior of its solutions is investigated in detail.

A solution \((x_{n}, y_{n})_{n\ge-2}\) of system (2) is called periodic, or eventually periodic, with period p if there is \(n_{0}\ge-2\) such that

$$x_{n+p}=x_{n}\quad \mbox{and}\quad y_{n+p}=y_{n} \quad \mbox{for } n\ge n_{0}. $$

For some results in the area, see, e.g., [6, 911, 19, 21, 22, 28].

Solutions to system (2) in closed form

Assume first that \(x_{-i}\ne0\), \(y_{-i}\ne0\), \(i\in\{1,2\}\). Then, by the method of induction and the equations in (2), it follows that for every well-defined solution to system (2), \(x_{n}\ne0\) and \(y_{n}\ne0\), for every \(n\in \mathbb {N}_{0}\). On the other hand, if \(x_{n_{0}}=0\) for some \(n_{0}\in \mathbb {N}\), then the first equation in (2) implies that \(y_{n_{0}-1}=0\) or \(y_{n_{0}-2}=0\). If \(y_{n_{0}-1}=0\), then \(x_{n_{0}-2}=0\) or \(x_{n_{0}-3}=0\), while if \(y_{n_{0}-2}=0\), then \(x_{n_{0}-3}=0\) or \(x_{n_{0}-4}=0\). Repeating this procedure, we get that \(x_{-i}=0\) or \(y_{-i}=0\) for some \(i\in\{1,2\}\). Similarly, if \(y_{n_{1}}=0\) for some \(n_{1}\in \mathbb {N}\), we get \(x_{-i}=0\) or \(y_{-i}=0\) for some \(i\in\{1,2\}\). Hence, for a well-defined solution \((x_{n},y_{n})_{n\ge-2}\) of system (2), we have that

$$\begin{aligned} x_{n}y_{n}\ne0,\quad n\ge-2 \end{aligned}$$
(3)

if and only if \(x_{-i}y_{-i}\ne0\), \(i\in\{1,2\}\).

Assume now that \((x_{n},y_{n})_{n\ge-2}\) is a solution to system (2) such that (3) holds. Then, by multiplying the first equation in (2) by \(x_{n-1}\) and the second one by \(y_{n-1}\), and using the following changes of variables

$$\begin{aligned} u_{n}=\frac{1}{x_{n}x_{n-1}},\qquad v_{n}= \frac{1}{y_{n}y_{n-1}}, \end{aligned}$$
(4)

\(n\ge-1\), system (2) is transformed in the following one:

$$\begin{aligned} u_{n}=a_{n}v_{n-1}+b_{n}, \qquad v_{n}=\alpha _{n}u_{n-1}+\beta _{n},\quad n \in \mathbb {N}_{0}. \end{aligned}$$
(5)

From (5) it follows that

$$\begin{aligned} &u_{n}=a_{n}\alpha _{n-1}u_{n-2}+a_{n} \beta _{n-1}+b_{n}, \end{aligned}$$
(6)
$$\begin{aligned} &v_{n}=\alpha _{n}a_{n-1}v_{n-2}+ \alpha _{n}b_{n-1}+\beta _{n},\quad n\in \mathbb {N}. \end{aligned}$$
(7)

This means that \((u_{2n})_{n\in \mathbb {N}_{0}}\), \((u_{2n-1})_{n\in \mathbb {N}_{0}}\), \((v_{2n})_{n\in \mathbb {N}_{0}}\), and \((v_{2n-1})_{n\in \mathbb {N}_{0}}\) are solutions to two linear first-order difference equations, which are solvable.

Solving these equations, we get

$$\begin{aligned}& u_{2n} =u_{0}\prod_{j=1}^{n}a_{2j} \alpha _{2j-1} +\sum_{i=1}^{n} (a_{2i}\beta _{2i-1}+b_{2i} ) \prod _{s=i+1}^{n}a_{2s}\alpha _{2s-1}, \end{aligned}$$
(8)
$$\begin{aligned}& u_{2n-1} =u_{-1}\prod_{j=1}^{n}a_{2j-1} \alpha _{2j-2} +\sum_{i=1}^{n} (a_{2i-1}\beta _{2i-2}+b_{2i-1} ) \prod _{s=i+1}^{n}a_{2s-1}\alpha _{2s-2}, \end{aligned}$$
(9)
$$\begin{aligned}& v_{2n} =v_{0}\prod_{j=1}^{n} \alpha _{2j}a_{2j-1} +\sum_{i=1}^{n} (\alpha _{2i}b_{2i-1}+\beta _{2i} ) \prod _{s=i+1}^{n}\alpha _{2s}a_{2s-1}, \end{aligned}$$
(10)
$$\begin{aligned}& v_{2n-1} =v_{-1}\prod_{j=1}^{n} \alpha _{2j-1}a_{2j-2} +\sum_{i=1}^{n} (\alpha _{2i-1}b_{2i-2}+\beta _{2i-1} ) \prod _{s=i+1}^{n}\alpha _{2s-1}a_{2s-2}. \end{aligned}$$
(11)

Using (4) we obtain

$$\begin{aligned} x_{2n+i}=\frac{1}{u_{2n+i}x_{2n+i-1}}=\frac {u_{2n+i-1}}{u_{2n+i}}x_{2(n-1)+i},\quad i\in \{0,1\}, \end{aligned}$$

and

$$\begin{aligned} y_{2n+i}=\frac{1}{v_{2n+i}y_{2n+i-1}}=\frac {v_{2n+i-1}}{v_{2n+i}}y_{2(n-1)+i},\quad i\in \{0,1\}, \end{aligned}$$

for \(2n+i\ge0\), from which it follows that

$$\begin{aligned}& x_{2m+i}=x_{i-2}\prod_{j=0}^{m} \frac{u_{2j+i-1}}{u_{2j+i}}, \end{aligned}$$
(12)
$$\begin{aligned}& y_{2m+i}=y_{i-2}\prod_{j=0}^{m} \frac{v_{2j+i-1}}{v_{2j+i}} \end{aligned}$$
(13)

for every \(m\in \mathbb {N}_{0}\), \(i\in\{0,1\}\).

Case of constant coefficients

In this section we consider the case when all the coefficients in system (2) are constant, that is, when

$$a_{n}=a, \qquad b_{n}=b,\qquad \alpha _{n}=\alpha ,\qquad \beta _{n}=\beta , \quad n\in \mathbb {N}_{0}. $$

Then (2) is

$$ \begin{aligned} &x_{n}=\frac{y_{n-1}y_{n-2}}{x_{n-1}(a+by_{n-1}y_{n-2})},\\ & y_{n}=\frac{x_{n-1}x_{n-2}}{y_{n-1}(\alpha +\beta x_{n-1}x_{n-2})}, \quad n\in \mathbb {N}_{0}. \end{aligned} $$
(14)

Assume that \((x_{n},y_{n})_{n\ge-2}\) is a solution to system (2) such that (3) holds. Then we have

$$\begin{aligned} u_{n}=av_{n-1}+b, \qquad v_{n}=\alpha u_{n-1}+\beta ,\quad n\in \mathbb {N}_{0}, \end{aligned}$$
(15)

and

$$\begin{aligned}& u_{n}=a\alpha u_{n-2}+a\beta +b, \end{aligned}$$
(16)
$$\begin{aligned}& v_{n}=a\alpha v_{n-2}+\alpha b+\beta ,\quad n\in \mathbb {N}. \end{aligned}$$
(17)

From (8)-(11), we obtain

$$\begin{aligned} u_{2n-l}&=u_{-l}(a\alpha )^{n}+(a\beta +b) \frac{1-(a\alpha )^{n}}{1-a\alpha } \\ &=\frac{a\beta +b+(a\alpha )^{n}(u_{-l}(1-a\alpha )-a\beta -b)}{1-a\alpha } \end{aligned}$$
(18)

for \(n\in \mathbb {N}_{0}\), \(l\in\{0,1\}\) when \(a\alpha \ne 1\), while if \(a\alpha =1\), we have

$$\begin{aligned} u_{2n-l}=u_{-l}+(a\beta +b)n,\quad n\in \mathbb {N}_{0}, l\in\{0,1\}, \end{aligned}$$
(19)

and we also have

$$\begin{aligned} v_{2n-l}&=v_{-l}(a\alpha )^{n}+(\alpha b+\beta ) \frac{1-(a\alpha )^{n}}{1-a\alpha } \\ &=\frac{\alpha b+\beta +(a\alpha )^{n}(v_{-l}(1-a\alpha )-\alpha b-\beta )}{1-a\alpha }, \end{aligned}$$
(20)

\(n\in \mathbb {N}_{0}\), \(l\in\{0,1\}\) if \(a\alpha \ne1\), while if \(a\alpha =1\), we have

$$\begin{aligned} v_{2n-l}=v_{-l}+(\alpha b+\beta )n,\quad n\in \mathbb {N}_{0}, l\in \{0,1\}. \end{aligned}$$
(21)

Now we present formulae for solutions to system (14).

Case \(a\alpha \ne1\). We have

$$\begin{aligned}& x_{2m}= x_{-2}\prod_{j=0}^{m} \frac{u_{2j-1}}{u_{2j}}=x_{-2}\prod_{j=0}^{m} \frac{a\beta +b+(a\alpha )^{j}(u_{-1}(1-a\alpha )-a\beta -b)}{a\beta +b+(a\alpha )^{j}(u_{0}(1-a\alpha )-a\beta -b)}, \end{aligned}$$
(22)
$$\begin{aligned}& x_{2m+1} =x_{-1}\prod_{j=0}^{m} \frac{u_{2j}}{u_{2j+1}}=x_{-1}\prod_{j=0}^{m} \frac{a\beta +b+(a\alpha )^{j}(u_{0}(1-a\alpha )-a\beta -b)}{a\beta +b+(a\alpha )^{j+1}(u_{-1}(1-a\alpha )-a\beta -b)}, \end{aligned}$$
(23)
$$\begin{aligned}& y_{2m} =y_{-2}\prod_{j=0}^{m} \frac{v_{2j-1}}{v_{2j}}=y_{-2}\prod_{j=0}^{m} \frac{\alpha b+\beta +(a\alpha )^{j}(v_{-1}(1-a\alpha )-\alpha b-\beta )}{\alpha b+\beta +(a\alpha )^{j}(v_{0}(1-a\alpha )-\alpha b-\beta )}, \end{aligned}$$
(24)
$$\begin{aligned}& y_{2m+1} =y_{-1}\prod_{j=0}^{m} \frac{v_{2j}}{v_{2j+1}}=y_{-1}\prod_{j=0}^{m} \frac{\alpha b+\beta +(a\alpha )^{j}(v_{0}(1-a\alpha )-\alpha b-\beta )}{\alpha b+\beta +(a\alpha )^{j+1}(v_{-1}(1-a\alpha )-\alpha b-\beta )} \end{aligned}$$
(25)

for every \(m\in \mathbb {N}_{0}\).

Case \(a\alpha =1\). We have

$$\begin{aligned}& x_{2m} =x_{-2}\prod_{j=0}^{m} \frac{u_{2j-1}}{u_{2j}}=x_{-2}\prod_{j=0}^{m} \frac{u_{-1}+(a\beta +b)j}{u_{0}+(a\beta +b)j}, \end{aligned}$$
(26)
$$\begin{aligned}& x_{2m+1} =x_{-1}\prod_{j=0}^{m} \frac{u_{2j}}{u_{2j+1}}=x_{-1}\prod_{j=0}^{m} \frac{u_{0}+(a\beta +b)j}{u_{-1}+(a\beta +b)(j+1)}, \end{aligned}$$
(27)
$$\begin{aligned}& y_{2m} =y_{-2}\prod_{j=0}^{m} \frac{v_{2j-1}}{v_{2j}}=y_{-2}\prod_{j=0}^{m} \frac{v_{-1}+(\alpha b+\beta )j}{v_{0}+(\alpha b+\beta )j}, \end{aligned}$$
(28)
$$\begin{aligned}& y_{2m+1}=y_{-1}\prod_{j=0}^{m} \frac{v_{2j}}{v_{2j+1}}=y_{-1}\prod_{j=0}^{m} \frac{v_{0}+(\alpha b+\beta )j}{v_{-1}+(\alpha b+\beta )(j+1)} \end{aligned}$$
(29)

for every \(m\in \mathbb {N}_{0}\).

Long-term behavior of solutions to system (14)

Before we formulate and prove the main results regarding the long-term behavior of well-defined solutions to system (14), we quote the following well-known asymptotic formula which will be used in the proofs of the main results:

$$\begin{aligned} (1+x)^{-1}=1-x+O\bigl(x^{2}\bigr),\quad \mbox{as } x\to 0. \end{aligned}$$
(30)

We also define the following quantities:

$$\begin{aligned}& L_{1}:=\frac{u_{-1}(1-a\alpha )-a\beta -b}{u_{0}(1-a\alpha )-a\beta -b},\qquad L_{2}:=\frac {u_{0}(1-a\alpha )-a\beta -b}{a\alpha (u_{-1}(1-a\alpha )-a\beta -b)}, \\& L_{3}:=\frac{v_{-1}(1-a\alpha )-\alpha b-\beta }{v_{0}(1-a\alpha )-\alpha b-\beta }, \qquad L_{4}:= \frac {v_{0}(1-a\alpha )-\alpha b-\beta }{a\alpha (v_{-1}(1-a\alpha )-\alpha b-\beta )}. \end{aligned}$$

Finally, we give another auxiliary result.

Lemma 1

If \(a\alpha \ne1\), \(a\beta +b\ne0\ne \alpha b+\beta \). Then system (14) has two-periodic solutions.

Proof

The equilibrium solution to system (15) is

$$\begin{aligned} u_{n}=\bar{u}=\frac{a\beta +b}{1-a\alpha }\ne0,\qquad v_{n}=\bar{v}= \frac{\alpha b+\beta }{1-a\alpha }\ne0,\quad n\in \mathbb {N}_{0}. \end{aligned}$$
(31)

From (4) and (31) it follows that

$$\begin{aligned} x_{n}=\frac{1-a\alpha }{(a\beta +b)x_{n-1}}=x_{n-2},\quad n\in \mathbb {N}_{0}, \end{aligned}$$
(32)

and

$$\begin{aligned} y_{n}=\frac{1-a\alpha }{(\alpha b+\beta )y_{n-1}}=y_{n-2},\quad n\in \mathbb {N}_{0}, \end{aligned}$$
(33)

as desired. □

The next three results are devoted to the long-term behavior of well-defined solutions to system (14).

Theorem 1

Assume that \(\vert a\alpha \vert \ne1\) and \((x_{n}, y_{n})_{n\ge-2}\) is a well-defined solution to system (14). Then the following statements are true.

  1. (a)

    If \(a\beta +b\ne0\ne \alpha b+\beta \) and \(\vert a\alpha \vert <1\), then \((x_{n},y_{n})\) converges to a, not necessarily prime, two-periodic solution.

  2. (b)

    If \(u_{-1}=u_{0}=(a\beta +b)/(1-a\alpha )\), then the sequences \((x_{2m})_{m\ge-1}\) and \((x_{2m+1})_{m\ge-1}\) are constant.

  3. (c)

    If \(v_{-1}=v_{0}=(\alpha b+\beta )/(1-a\alpha )\), then the sequences \((y_{2m})_{m\ge-1}\) and \((y_{2m+1})_{m\ge-1}\) are constant.

  4. (d)

    If \(\vert a\alpha \vert >1\) and \(u_{-1}=(a\beta +b)/(1-a\alpha )\ne u_{0}\), then \(x_{2m}\to0\) and \(\vert x_{2m+1}\vert \to\infty\), as \(m\to\infty\).

  5. (e)

    If \(\vert a\alpha \vert >1\) and \(u_{-1}\ne(a\beta +b)/(1-a\alpha )=u_{0}\), then \(x_{2m+1}\to0\) and \(\vert x_{2m}\vert \to\infty\), as \(m\to\infty\).

  6. (f)

    If \(\vert a\alpha \vert >1\) and \(v_{-1}=(a\beta +b)/(1-a\alpha )\ne v_{0}\), then \(y_{2m}\to0\) and \(\vert y_{2m+1}\vert \to\infty\), as \(m\to\infty\).

  7. (g)

    If \(\vert a\alpha \vert >1\) and \(v_{-1}\ne(a\beta +b)/(1-a\alpha )=v_{0}\), then \(y_{2m+1}\to0\) and \(\vert y_{2m}\vert \to\infty\), as \(m\to\infty\).

  8. (h)

    If \(\vert a\alpha \vert >1\), \(u_{-1}\ne(a\beta +b)/(1-a\alpha )\ne u_{0}\) and \(\vert L_{1}\vert <1\), then \(x_{2m}\to0\), as \(m\to\infty\).

  9. (i)

    If \(\vert a\alpha \vert >1\), \(u_{-1}\ne(a\beta +b)/(1-a\alpha )\ne u_{0}\) and \(\vert L_{1}\vert >1\), then \(\vert x_{2m}\vert \to\infty\), as \(m\to\infty\).

  10. (j)

    If \(\vert a\alpha \vert >1\), \(u_{-1}\ne(a\beta +b)/(1-a\alpha )\ne u_{0}\) and \(L_{1}=1\), then \((x_{2m})_{m\ge-1}\) is constant.

  11. (k)

    If \(\vert a\alpha \vert >1\), \(u_{-1}\ne(a\beta +b)/(1-a\alpha )\ne u_{0}\) and \(L_{1}=-1\), then \((x_{4m})_{m\ge-1}\) and \((x_{4m+2})_{m\ge-1}\) are convergent.

  12. (l)

    If \(\vert a\alpha \vert >1\), \(u_{-1}\ne(a\beta +b)/(1-a\alpha )\ne u_{0}\) and \(\vert L_{2}\vert <1\), then \(x_{2m+1}\to0\), as \(m\to\infty\).

  13. (m)

    If \(\vert a\alpha \vert >1\), \(u_{-1}\ne(a\beta +b)/(1-a\alpha )\ne u_{0}\) and \(\vert L_{2}\vert >1\), then \(\vert x_{2m+1}\vert \to\infty\), as \(m\to\infty\).

  14. (n)

    If \(\vert a\alpha \vert >1\), \(u_{-1}\ne(a\beta +b)/(1-a\alpha )\ne u_{0}\) and \(L_{2}=1\), then \((x_{2m+1})_{m\ge-1}\) is constant.

  15. (o)

    If \(\vert a\alpha \vert >1\), \(u_{-1}\ne(a\beta +b)/(1-a\alpha )\ne u_{0}\) and \(L_{2}=-1\), then \((x_{4m+1})_{m\ge-1}\) and \((x_{4m+3})_{m\ge-1}\) are convergent.

  16. (p)

    If \(\vert a\alpha \vert >1\), \(v_{-1}\ne(\alpha b+\beta )/(1-a\alpha )\ne v_{0}\) and \(\vert L_{3}\vert <1\), then \(y_{2m}\to0\), as \(m\to\infty\).

  17. (q)

    If \(\vert a\alpha \vert >1\), \(v_{-1}\ne(\alpha b+\beta )/(1-a\alpha )\ne v_{0}\) and \(\vert L_{3}\vert >1\), then \(\vert y_{2m}\vert \to\infty\), as \(m\to\infty\).

  18. (r)

    If \(\vert a\alpha \vert >1\), \(v_{-1}\ne(\alpha b+\beta )/(1-a\alpha )\ne v_{0}\) and \(L_{3}=1\), then \((y_{2m})_{m\ge-1}\) is constant.

  19. (s)

    If \(\vert a\alpha \vert >1\), \(v_{-1}\ne(\alpha b+\beta )/(1-a\alpha )\ne v_{0}\) and \(L_{3}=-1\), then \((y_{4m})_{m\ge-1}\) and \((y_{4m+2})_{m\ge-1}\) are convergent.

  20. (t)

    If \(\vert a\alpha \vert >1\), \(v_{-1}\ne(\alpha b+\beta )/(1-a\alpha )\ne v_{0}\) and \(\vert L_{4}\vert <1\), then \(y_{2m+1}\to0\), as \(m\to\infty\).

  21. (u)

    If \(\vert a\alpha \vert >1\), \(v_{-1}\ne(\alpha b+\beta )/(1-a\alpha )\ne v_{0}\) and \(\vert L_{4}\vert >1\), then \(\vert y_{2m+1}\vert \to\infty\), as \(m\to\infty\).

  22. (v)

    If \(\vert a\alpha \vert >1\), \(v_{-1}\ne(\alpha b+\beta )/(1-a\alpha )\ne v_{0}\) and \(L_{4}=1\), then \((y_{2m+1})_{m\ge-1}\) is constant.

  23. (w)

    If \(\vert a\alpha \vert >1\), \(v_{-1}\ne(\alpha b+\beta )/(1-a\alpha )\ne v_{0}\) and \(L_{4}=-1\), then \((y_{4m+1})_{m\ge-1}\) and \((y_{4m+3})_{m\ge-1}\) are convergent.

Proof

Let

$$\begin{aligned} &p_{m}=\frac{a\beta +b+(a\alpha )^{m}(u_{-1}(1-a\alpha )-a\beta -b)}{a\beta +b+(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)}, \\ &\hat{p}_{m}=\frac{a\beta +b+(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)}{a\beta +b+(a\alpha )^{m+1}(u_{-1}(1-a\alpha )-a\beta -b)}, \\ &q_{m}=\frac{\alpha b+\beta +(a\alpha )^{m}(v_{-1}(1-a\alpha )-\alpha b-\beta )}{\alpha b+\beta +(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )}, \\ &\hat{q}_{m}=\frac{\alpha b+\beta +(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )}{\alpha b+\beta +(a\alpha )^{m+1}(v_{-1}(1-a\alpha )-\alpha b-\beta )} \end{aligned}$$

for \(m\in \mathbb {N}_{0}\).

(a) By using (30) we have

$$\begin{aligned}& \begin{aligned}[b] p_{m}&=\frac{1+(a\alpha )^{m}(u_{-1}(1-a\alpha )-a\beta -b)(a\beta +b)^{-1}}{1+(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)(a\beta +b)^{-1}} \\ &=1+(u_{-1}-u_{0}) (1-a\alpha ) (a\beta +b)^{-1}(a \alpha )^{m}+o\bigl((a\alpha )^{m}\bigr), \end{aligned} \end{aligned}$$
(34)
$$\begin{aligned}& \begin{aligned}[b] \hat{p}_{m}&=\frac{1+(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)(a\beta +b)^{-1}}{1+(a\alpha )^{m+1}(u_{-1}(1-a\alpha )-a\beta -b)(a\beta +b)^{-1}} \\ &=1+\frac{(1-a\alpha )(u_{0}-a\alpha u_{-1}-a\beta -b)}{a\beta +b}(a\alpha )^{m}+o\bigl((a\alpha )^{m} \bigr), \end{aligned} \end{aligned}$$
(35)
$$\begin{aligned}& \begin{aligned}[b] q_{m}&=\frac{1+(a\alpha )^{m}(v_{-1}(1-a\alpha )-\alpha b-\beta )(\alpha b+\beta )^{-1}}{1+(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )(\alpha b+\beta )^{-1}} \\ &=1+(v_{-1}-v_{0}) (1-a\alpha ) (\alpha b+\beta )^{-1}(a \alpha )^{m}+o\bigl((a\alpha )^{m}\bigr), \end{aligned} \end{aligned}$$
(36)
$$\begin{aligned}& \begin{aligned}[b] \hat{q}_{m}&=\frac{1+(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )(\alpha b+\beta )^{-1}}{1+(a\alpha )^{m+1}(v_{-1}(1-a\alpha )-\alpha b-\beta )(\alpha b+\beta )^{-1}} \\ &=1+\frac{(1-a\alpha )(v_{0}-a\alpha v_{-1}-\alpha b-\beta )}{\alpha b+\beta }(a\alpha )^{m}+o\bigl((a\alpha )^{m} \bigr) \end{aligned} \end{aligned}$$
(37)

for sufficiently large m.

From (34)-(37), by using the condition \(\vert a\alpha \vert <1\) and a well-known criterion for the convergence of products, the statement easily follows.

(b) By using the condition \(u_{-1}=u_{0}=(a\beta +b)/(1-a\alpha )\) in (22) and (23), the statement immediately follows.

(c) By using the condition \(v_{-1}=v_{0}=(\alpha b+\beta )/(1-a\alpha )\) in (24) and (25), the statement immediately follows.

(d) By using the condition \(u_{-1}=(a\beta +b)/(1-a\alpha )\ne u_{0}\), we get

$$\begin{aligned} &p_{m}=\frac{a\beta +b}{a\beta +b+(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)}, \end{aligned}$$
(38)
$$\begin{aligned} &\hat{p}_{m}=\frac{a\beta +b+(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)}{a\beta +b}. \end{aligned}$$
(39)

Letting \(m\to\infty\) in (38) and (39) and using the condition \(\vert a\alpha \vert >1\), we have \(p_{m}\to0\) and \(\vert \hat{p}_{m}\vert \to\infty\), from which along with (22) and (23) the statement easily follows.

(e) By using the condition \(u_{-1}\ne(a\beta +b)/(1-a\alpha )=u_{0}\), we get

$$\begin{aligned} &p_{m}=\frac{a\beta +b+(a\alpha )^{m}(u_{-1}(1-a\alpha )-a\beta -b)}{a\beta +b}, \end{aligned}$$
(40)
$$\begin{aligned} &\hat{p}_{m}=\frac{a\beta +b}{a\beta +b+(a\alpha )^{m+1}(u_{-1}(1-a\alpha )-a\beta -b)}. \end{aligned}$$
(41)

Letting \(m\to\infty\) in (40) and (41) and using the condition \(\vert a\alpha \vert >1\), we have \(\vert p_{m}\vert \to\infty\) and \(\hat{p}_{m}\to0\), from which along with (22) and (23) the statement easily follows.

(f) By using the condition \(v_{-1}=(a\beta +b)/(1-a\alpha )\ne v_{0}\), we get

$$\begin{aligned} &q_{m}=\frac{\alpha b+\beta }{\alpha b+\beta +(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )}, \end{aligned}$$
(42)
$$\begin{aligned} &\hat{q}_{m}=\frac{\alpha b+\beta +(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )}{\alpha b+\beta }. \end{aligned}$$
(43)

Letting \(m\to\infty\) in (42) and (43) and using the condition \(\vert a\alpha \vert >1\), we have \(q_{m}\to0\) and \(\vert \hat{q}_{m}\vert \to\infty\), from which along with (24) and (25) the statement easily follows.

(g) By using the condition \(v_{-1}\ne(a\beta +b)/(1-a\alpha )= v_{0}\), we get

$$\begin{aligned} &q_{m}=\frac{\alpha b+\beta +(a\alpha )^{m}(v_{-1}(1-a\alpha )-\alpha b-\beta )}{\alpha b+\beta }, \end{aligned}$$
(44)
$$\begin{aligned} &\hat{q}_{m}=\frac{\alpha b+\beta }{\alpha b+\beta +(a\alpha )^{m+1}(v_{-1}(1-a\alpha )-\alpha b-\beta )}. \end{aligned}$$
(45)

Letting \(m\to\infty\) in (44) and (45) and using the condition \(\vert a\alpha \vert >1\), we have \(\vert q_{m}\vert \to\infty\) and \(\hat{q}_{m}\to0\), from which along with (24) and (25) the statement easily follows.

(h), (i) Note that \(\lim_{m\to\infty}p_{m}=L_{1}\). Hence, from the assumptions \(\vert L_{1}\vert <1\), that is, \(\vert L_{1}\vert >1\) along with (22), the statements easily follow.

(j) The statement immediately follows by using the condition \(L_{1}=1\) in (22).

(k) Since \(L_{1}=-1\) and by using (30), we have that

$$\begin{aligned} p_{m}&=\frac{a\beta +b+(a\alpha )^{m}(u_{-1}(1-a\alpha )-a\beta -b)}{a\beta +b-(a\alpha )^{m}(u_{-1}(1-a\alpha )-a\beta -b)} \\ &=-\frac{1+\frac{a\beta +b}{(a\alpha )^{m}(u_{-1}(1-a\alpha )-a\beta -b)}}{1-\frac {a\beta +b}{(a\alpha )^{m}(u_{-1}(1-a\alpha )-a\beta -b)}} \\ &=- \biggl(1+\frac{2(a\beta +b)}{(a\alpha )^{m}(u_{-1}(1-a\alpha )-a\beta -b)}+o \biggl(\frac{1}{(a\alpha )^{m}} \biggr) \biggr). \end{aligned}$$
(46)

From (46), by using the condition \(\vert a\alpha \vert >1\) and a well-known criterion for the convergence of products, the statement easily follows.

(l), (m) Note that \(\lim_{m\to\infty}\hat{p}_{m}=L_{2}\). Hence, from the assumptions \(\vert L_{2}\vert <1\), that is, \(\vert L_{2}\vert >1\) along with (23), the statements easily follow.

(n) The statement immediately follows by using the condition \(L_{2}=1\) in (23).

(o) Since \(L_{2}=-1\) and by using (30), we have that

$$\begin{aligned} \hat{p}_{m}&=\frac{a\beta +b+(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)}{a\beta +b-(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)} \\ &=-\frac{1+\frac{a\beta +b}{(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)}}{1-\frac{a\beta +b}{(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)}} \\ &=- \biggl(1+\frac{2(a\beta +b)}{(a\alpha )^{m}(u_{0}(1-a\alpha )-a\beta -b)}+o \biggl(\frac{1}{(a\alpha )^{m}} \biggr) \biggr). \end{aligned}$$
(47)

From (47), by using the condition \(\vert a\alpha \vert >1\) and a well-known criterion for the convergence of products, the statement easily follows.

(p), (q) Note that \(\lim_{m\to\infty}q_{m}=L_{3}\). Hence, from the assumptions \(\vert L_{3}\vert <1\), that is, \(\vert L_{3}\vert >1\) along with (24), the statements easily follow.

(r) The statement immediately follows by using the condition \(L_{3}=1\) in (24).

(s) Since \(L_{3}=-1\) and by using (30), we have that

$$\begin{aligned} q_{m}&=\frac{\alpha b+\beta +(a\alpha )^{m}(v_{-1}(1-a\alpha )-\alpha b-\beta )}{\alpha b+\beta -(a\alpha )^{m}(v_{-1}(1-a\alpha )-\alpha b-\beta )} \\ &=-\frac{1+\frac{\alpha b+\beta }{(a\alpha )^{m}(v_{-1}(1-a\alpha )-\alpha b-\beta )}}{1-\frac{\alpha b+\beta }{(a\alpha )^{m}(v_{-1}(1-a\alpha )-\alpha b-\beta )}} \\ &=- \biggl(1+\frac{2(\alpha b+\beta )}{(a\alpha )^{m}(v_{-1}(1-a\alpha )-\alpha b-\beta )}+o \biggl(\frac{1}{(a\alpha )^{m}} \biggr) \biggr). \end{aligned}$$
(48)

From (48), by using the condition \(\vert a\alpha \vert >1\) and a well-known criterion for the convergence of products, the statement easily follows.

(t), (u) Note that \(\lim_{m\to\infty}\hat{q}_{m}=L_{4}\). Hence, from the assumptions \(\vert L_{4}\vert <1\), that is, \(\vert L_{4}\vert >1\) along with (25), the statements easily follow.

(v) The statement immediately follows by using the condition \(L_{4}=1\) in (25).

(w) Since \(L_{4}=-1\) and by using (30), we have that

$$\begin{aligned} \hat{q}_{m}&=\frac{\alpha b+\beta +(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )}{\alpha b+\beta -(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )} \\ &=-\frac{1+\frac{\alpha b+\beta }{(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )}}{1-\frac {\alpha b+\beta }{(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )}} \\ &=- \biggl(1+\frac{2(\alpha b+\beta )}{(a\alpha )^{m}(v_{0}(1-a\alpha )-\alpha b-\beta )}+o \biggl(\frac{1}{(a\alpha )^{m}} \biggr) \biggr). \end{aligned}$$
(49)

From (49), by using the condition \(\vert a\alpha \vert >1\) and a well-known criterion for the convergence of products, the statement easily follows. □

Let

$$M_{1}:=\frac{u_{-1}(u_{-1}-b-a\beta )}{u_{0}(u_{0}-b-a\beta )},\qquad M_{2}:=\frac {v_{-1}(v_{-1}-\beta -\alpha b)}{v_{0}(v_{0}-\beta -\alpha b)}. $$

Theorem 2

Assume that \(a\alpha =-1\) and \((x_{n}, y_{n})_{n\ge-2}\) is a well-defined solution to system (14). Then the following statements are true.

  1. (a)

    If \(\vert M_{1}\vert <1\), then \(x_{2m}\to0\) and \(\vert x_{2m+1}\vert \to\infty\), as \(m\to\infty\).

  2. (b)

    If \(\vert M_{1}\vert >1\), then \(x_{2m+1}\to0\) and \(\vert x_{2m}\vert \to\infty\), as \(m\to\infty\).

  3. (c)

    If \(M_{1}=1\), then \((x_{n})_{n\ge-2}\) is four-periodic.

  4. (d)

    If \(M_{1}=-1\), then \((x_{n})_{n\ge-2}\) is eight-periodic.

  5. (e)

    If \(\vert M_{2}\vert <1\), then \(y_{2m}\to0\) and \(\vert y_{2m+1}\vert \to\infty\), as \(m\to\infty\).

  6. (f)

    If \(\vert M_{2}\vert >1\), then \(y_{2m+1}\to0\) and \(\vert y_{2m}\vert \to\infty\), as \(m\to\infty\).

  7. (g)

    If \(M_{2}=1\), then \((y_{n})_{n\ge-2}\) is four-periodic.

  8. (h)

    If \(M_{2}=-1\), then \((y_{n})_{n\ge-2}\) is eight-periodic.

Proof

First, note that since \(a\alpha =-1\), from (22)-(25) we have

$$\begin{aligned}& x_{4m}=x_{0}M_{1}^{m},\qquad x_{4m+2}=x_{-2}M_{1}^{m+1},\qquad x_{4m+1}=\frac{x_{1}}{M_{1}^{m}},\qquad x_{4m+3}=\frac{x_{-1}}{M_{1}^{m+1}}, \end{aligned}$$
(50)
$$\begin{aligned}& y_{4m}=y_{0}M_{2}^{m},\qquad y_{4m+2}=y_{-2}M_{2}^{m+1},\qquad y_{4m+1}=\frac{y_{1}}{M_{2}^{m}},\qquad y_{4m+3}=\frac{y_{-1}}{M_{2}^{m+1}}, \end{aligned}$$
(51)

for \(m\in \mathbb {N}_{0}\). From (50) and (51) all the statements easily follow. □

Let

$$N_{1}:=\frac{u_{-1}}{u_{0}},\qquad N_{2}:=\frac{v_{-1}}{v_{0}}. $$

Theorem 3

Assume that \(a\alpha =1\) and \((x_{n}, y_{n})_{n\ge-2}\) is a well-defined solution to system (14). Then the following statements hold true.

  1. (a)

    If \(a\beta +b=0\) and \(\vert N_{1}\vert <1\), then \(x_{2m}\to0\) and \(\vert x_{2m+1}\vert \to\infty\), as \(m\to\infty\);

  2. (b)

    If \(a\beta +b=0\) and \(\vert N_{1}\vert >1\), then \(\vert x_{2m}\vert \to\infty\) and \(x_{2m+1}\to0\), as \(m\to\infty\);

  3. (c)

    If \(a\beta +b=0\) and \(N_{1}=1\), then \((x_{2m})_{m\ge-1}\) and \((x_{2m+1})_{m\ge-1}\) are constant;

  4. (d)

    If \(a\beta +b=0\) and \(N_{1}=-1\), then \((x_{4m+i})_{m\ge-1}\), \(i=\overline {0,3}\), are constant.

  5. (e)

    If \(a\beta +b\ne0\) and \((u_{-1}-u_{0})/(a\beta +b)>0\), then \(\vert x_{2m}\vert \to\infty\), as \(m\to\infty\);

  6. (f)

    If \(a\beta +b\ne0\) and \((u_{-1}-u_{0})/(a\beta +b)<0\), then \(x_{2m}\to0\), as \(m\to\infty\);

  7. (g)

    If \(a\beta +b\ne0\) and \(u_{-1}=u_{0}\), then \((x_{2m})_{m\ge-1}\) is constant;

  8. (h)

    If \(a\beta +b\ne0\) and \((u_{0}-u_{-1})/(a\beta +b)>1\), then \(\vert x_{2m+1}\vert \to\infty\), as \(m\to\infty\);

  9. (i)

    If \(a\beta +b\ne0\) and \((u_{0}-u_{-1})/(a\beta +b)<1\), then \(x_{2m+1}\to0\), as \(m\to\infty\);

  10. (j)

    If \(a\beta +b\ne0\) and \(u_{-1}-u_{0}=a\beta +b\), then \((x_{2m+1})_{m\ge-1}\) is constant;

  11. (k)

    If \(\alpha b+\beta =0\) and \(\vert N_{2}\vert <1\), then \(y_{2m}\to0\) and \(\vert y_{2m+1}\vert \to\infty\), as \(m\to\infty\);

  12. (l)

    If \(\alpha b+\beta =0\) and \(\vert N_{2}\vert >1\), then \(\vert y_{2m}\vert \to\infty\) and \(y_{2m+1}\to0\), as \(m\to\infty\);

  13. (m)

    If \(\alpha b+\beta =0\) and \(N_{2}=1\), then \((y_{2m})_{m\ge-1}\) and \((y_{2m+1})_{m\ge-1}\) are constant;

  14. (n)

    If \(\alpha b+\beta =0\) and \(N_{2}=-1\), then \((y_{4m+i})_{m\ge-1}\), \(i=\overline {0,3}\), are constant.

  15. (o)

    If \(\alpha b+\beta \ne0\) and \((v_{-1}-v_{0})/(\alpha b+\beta )>0\), then \(\vert y_{2m}\vert \to\infty\), as \(m\to\infty\);

  16. (p)

    If \(\alpha b+\beta \ne0\) and \((v_{-1}-v_{0})/(\alpha b+\beta )<0\), then \(y_{2m}\to0\), as \(m\to\infty\);

  17. (q)

    If \(\alpha b+\beta \ne0\) and \(v_{-1}=v_{0}\), then \((y_{2m})_{m\ge -1}\) is constant.

  18. (r)

    If \(\alpha b+\beta \ne0\) and \((v_{0}-v_{-1})/(\alpha b+\beta )<1\), then \(y_{2m+1}\to0\), as \(m\to\infty\);

  19. (s)

    If \(\alpha b+\beta \ne0\) and \((v_{0}-v_{-1})/(\alpha b+\beta )>1\), then \(\vert y_{2m+1}\vert \to\infty\), as \(m\to\infty\);

  20. (t)

    If \(\alpha b+\beta \ne0\) and \(v_{-1}-v_{0}=\alpha b+\beta \), then \((y_{2m+1})_{m\ge-1}\) is constant.

Proof

Let

$$\begin{aligned}& r_{m} =\frac{u_{-1}+(a\beta +b)m}{u_{0}+(a\beta +b)m},\qquad \hat{r}_{m}= \frac{u_{0}+(a\beta +b)m}{u_{-1}+(a\beta +b)(m+1)}, \\& s_{m} =\frac{v_{-1}+(\alpha b+\beta )m}{v_{0}+(\alpha b+\beta )m}, \qquad \hat{s}_{m}= \frac{v_{0}+(\alpha b+\beta )m}{v_{-1}+(\alpha b+\beta )(m+1)},\quad m\in \mathbb {N}_{0}. \end{aligned}$$

(a)-(d) Since in this case we have

$$x_{2m}=x_{-2} \biggl(\frac{u_{-1}}{u_{0}} \biggr)^{m+1},\qquad x_{2m+1}=x_{-1} \biggl( \frac{u_{0}}{u_{-1}} \biggr)^{m+1},\quad m\in \mathbb {N}_{0}, $$

these statements easily follow.

(e), (f) By using (30) we have

$$\begin{aligned} r_{m}&=\frac{u_{-1}+(a\beta +b)m}{u_{0}+(a\beta +b)m}= \biggl(1+\frac {u_{-1}}{(a\beta +b)m} \biggr) \biggl(1+\frac{u_{0}}{(a\beta +b)m} \biggr)^{-1} \\ &= \biggl(1+\frac{u_{-1}}{(a\beta +b)m}+O \biggl(\frac{1}{m^{2}} \biggr) \biggr) \biggl(1-\frac {u_{0}}{(a\beta +b)m}+O \biggl(\frac{1}{m^{2}} \biggr) \biggr) \\ &=1+\frac{u_{-1}-u_{0}}{(a\beta +b)m}+O \biggl(\frac{1}{m^{2}} \biggr) \end{aligned}$$
(52)

for sufficiently large m.

From (52), by using the fact that for every \(k\in \mathbb {N}\)

$$\begin{aligned} \sum_{j=k}^{m}\frac{1}{j}\to\infty, \quad \mbox{as } m\to\infty, \end{aligned}$$
(53)

and a known criterion for convergence of products, the statements easily follow.

(g) Using the condition \(u_{-1}=u_{0}\) in (26), the statement immediately follows.

(h), (i) By using (30) we have

$$\begin{aligned} \hat{r}_{m}&=\frac{u_{0}+(a\beta +b)m}{u_{-1}+(a\beta +b)(m+1)}= \biggl(1+\frac {u_{0}}{(a\beta +b)m} \biggr) \biggl(1+\frac{u_{-1}+a\beta +b}{(a\beta +b)m} \biggr)^{-1} \\ &= \biggl(1+\frac{u_{0}}{(a\beta +b)m} \biggr) \biggl(1-\frac{u_{-1}+a\beta +b}{(a\beta +b)m}+O \biggl( \frac{1}{m^{2}} \biggr) \biggr) \\ &=1+\frac{u_{0}-u_{-1}-a\beta -b}{(a\beta +b)m}+O \biggl(\frac{1}{m^{2}} \biggr) \end{aligned}$$
(54)

for sufficiently large m.

From (54), (53), (27) and a known criterion for convergence of products, the statements easily follow.

(j) Using the condition \(u_{0}=u_{-1}+a\beta +b\) in (27), the statement immediately follows.

(k)-(n) Since in this case we have

$$y_{2m}=y_{-2} \biggl(\frac{v_{-1}}{v_{0}} \biggr)^{m+1},\qquad y_{2m+1}=y_{-1} \biggl( \frac{v_{0}}{v_{-1}} \biggr)^{m+1},\quad m\in \mathbb {N}_{0}, $$

these statements easily follow.

(o), (p) By using (30) we have

$$\begin{aligned} s_{m}&=\frac{v_{-1}+(\alpha b+\beta )m}{v_{0}+(\alpha b+\beta )m}= \biggl(1+\frac {v_{-1}}{(\alpha b+\beta )m} \biggr) \biggl(1+\frac{v_{0}}{(\alpha b+\beta )m} \biggr)^{-1} \\ &= \biggl(1+\frac{v_{-1}}{(\alpha b+\beta )m} \biggr) \biggl(1-\frac{v_{0}}{(\alpha b+\beta )m}+O \biggl( \frac{1}{m^{2}} \biggr) \biggr) \\ &=1+\frac{v_{-1}-v_{0}}{(\alpha b+\beta )m}+O \biggl(\frac{1}{m^{2}} \biggr) \end{aligned}$$
(55)

for sufficiently large m.

From (55), (53), (28) and a known criterion for convergence of products, the statements easily follow.

(q) Using the condition \(v_{0}=v_{-1}\) in (28), the statement immediately follows.

(r), (s) By using (30) we have

$$\begin{aligned} \hat{s}_{m}&=\frac{v_{0}+(\alpha b+\beta )m}{v_{-1}+(\alpha b+\beta )(m+1)}= \biggl(1+\frac {v_{0}}{(\alpha b+\beta )m} \biggr) \biggl(1+\frac{v_{-1}+\alpha b+\beta }{(\alpha b+\beta )m} \biggr)^{-1} \\ &= \biggl(1+\frac{v_{0}}{(\alpha b+\beta )m} \biggr) \biggl(1-\frac{v_{-1}+\alpha b+\beta }{(\alpha b+\beta )m}+O \biggl( \frac{1}{m^{2}} \biggr) \biggr) \\ &=1+\frac{v_{0}-v_{-1}-\alpha b-\beta }{(\alpha b+\beta )m}+O \biggl(\frac{1}{m^{2}} \biggr) \end{aligned}$$
(56)

for sufficiently large m.

From (56), (53), (29) and a known criterion for convergence of products, the statements easily follow.

(t) Using the condition \(v_{0}=v_{-1}+\alpha b+\beta \) in (29), the statement immediately follows. □

Domain of undefinable solutions to system (2)

In Section 2 we proved that solutions to system (2), for which \(x_{-j}=0\) or \(y_{-j}=0\) for some \(j\in\{1,2\}\), are not defined. The set of all such initial values is characterized here.

Definition 1

Consider the system of difference equations

$$\begin{aligned} \begin{aligned} &x_{n}=f(x_{n-1}, \ldots,x_{n-s},y_{n-1},\ldots,y_{n-s},n), \\ &y_{n}=g(x_{n-1},\ldots,x_{n-s},y_{n-1}, \ldots,y_{n-s},n),\quad n\in \mathbb {N}_{0}, \end{aligned} \end{aligned}$$
(57)

where \(s\in \mathbb {N}\), and \(x_{-i},y_{-i}\in \mathbb {R}\), \(i=\overline {1,s}\). The string of vectors

$$(x_{-s},y_{-s}),\ldots,(x_{-1},y_{-1}),(x_{0},y_{0}), \ldots,(x_{n_{0}},y_{n_{0}}), $$

where \(n_{0}\ge-1\), is called an undefined solution of system (57) if

$$x_{j}=f(x_{j-1},\ldots,x_{j-s},y_{j-1}, \ldots,y_{j-s},j) $$

and

$$y_{j}=g(x_{j-1},\ldots,x_{j-s},y_{j-1}, \ldots,y_{j-s},j) $$

for \(0\le j< n_{0}+1\), and \(x_{n_{0}+1}\) or \(y_{n_{0}+1}\) is not a defined number, that is, the quantity

$$f(x_{n_{0}},\ldots,x_{n_{0}-s+1},y_{n_{0}}, \ldots,y_{n_{0}-s+1},n_{0}+1) $$

or

$$g(x_{n_{0}},\ldots,x_{n_{0}-s+1},y_{n_{0}}, \ldots,y_{n_{0}-s+1},n_{0}+1) $$

is not defined.

The set of all initial values \((x_{-s},y_{-s}),\ldots,(x_{-1},y_{-1})\) which generate undefined solutions to system (57) is called domain of undefinable solutions of the system.

The next result characterizes the domain of undefinable solutions to system (2) when \(a_{n}b_{n}\alpha _{n}\beta _{n}\ne0\), \(n\in \mathbb {N}_{0}\).

Theorem 4

Assume that \(a_{n}b_{n}\alpha _{n}\beta _{n}\ne0\), \(n\in \mathbb {N}_{0}\). Then the domain of undefinable solutions to system (2) is the following set:

$$\begin{aligned} {\mathcal{U}}={}&\bigcup_{m\in \mathbb {N}_{0}} \biggl\{ (x_{-2},x_{-1},y_{-2},y_{-1})\in \mathbb {R}^{4}: \\ &{}\frac{1}{x_{-1}x_{-2}}=g_{0}^{-1}\circ f_{1}^{-1} \circ\cdots\circ g_{2m-2}^{-1}\circ f_{2m-1}^{-1} \circ g_{2m}^{-1}\circ f_{2m+1}^{-1}(0) \\ &{} \textit{or } \frac{1}{x_{-1}x_{-2}}=g_{0}^{-1}\circ f_{1}^{-1}\circ\cdots\circ g_{2m-2}^{-1} \circ f_{2m-1}^{-1} \circ g_{2m}^{-1}(0) \\ &{} \textit{or } \frac{1}{y_{-1}y_{-2}}=f_{0}^{-1}\circ g_{1}^{-1}\circ\cdots\circ f_{2m-2}^{-1} \circ g_{2m-1}^{-1} \circ f_{2m}^{-1}(0) \\ &{} \textit{or } \frac{1}{y_{-1}y_{-2}}=f_{0}^{-1}\circ g_{1}^{-1}\circ\cdots\circ g_{2m-1}^{-1} \circ f_{2m}^{-1} \circ g_{2m+1}^{-1}(0) \biggr\} \\ &{} \cup \bigl\{ (x_{-2},x_{-1},y_{-2},y_{-1}) \in \mathbb {R}^{4}: \\ &{}x_{-2}=0\textit{ or }x_{-1}=0\textit{ or }y_{-2}=0\textit{ or }y_{-1}=0 \bigr\} , \end{aligned}$$
(58)

where

$$f_{n}(t)=a_{n}t+b_{n}, \qquad g_{n}(t)= \alpha _{n}t+\beta _{n},\quad n\in \mathbb {N}_{0}. $$

Proof

We have already proved that the set

$$\bigl\{ (x_{-2},x_{-1},y_{-2},y_{-1})\in \mathbb {R}^{4}: x_{-2}=0\mbox{ or }x_{-1}=0\mbox{ or }y_{-2}=0\mbox{ or }y_{-1}=0 \bigr\} $$

belongs to the domain of undefinable solutions to system (2).

If \(x_{-j}\ne0\ne y_{-j}\), \(j=\overline {1,2}\) (i.e., \(x_{n}\ne0\ne y_{n}\) for every \(n\ge-2\)), then such a solution \((x_{n},y_{n})_{n\ge-2}\) is not defined if and only if

$$\begin{aligned} a_{n}+b_{n}y_{n-1}y_{n-2}=0\quad \mbox{or} \quad \alpha _{n}+\beta _{n}x_{n-1}x_{n-2}=0 \end{aligned}$$
(59)

for some \(n\in \mathbb {N}_{0}\), which is equivalent to

$$\begin{aligned} v_{n-1}=-b_{n}/a_{n}\quad \mbox{or}\quad u_{n-1}=-\beta _{n}/\alpha _{n} \end{aligned}$$
(60)

for some \(n\in \mathbb {N}_{0}\).

Note that

$$\begin{aligned} f_{n}^{-1}(0)=-b_{n}/a_{n}\quad \mbox{and} \quad g_{n}^{-1}(0)=-\beta _{n}/\alpha _{n},\quad n \in \mathbb {N}_{0}. \end{aligned}$$
(61)

We have

$$\begin{aligned}& v_{2m-1} =(g_{2m-1} \circ f_{2m-2}\circ\cdots\circ f_{2}\circ g_{1}\circ f_{0}) (v_{-1}), \end{aligned}$$
(62)
$$\begin{aligned}& v_{2m} =(g_{2m}\circ f_{2m-1}\circ\cdots\circ g_{2}\circ f_{1}\circ g_{0}) (u_{-1}), \end{aligned}$$
(63)
$$\begin{aligned}& u_{2m-1} =(f_{2m-1} \circ g_{2m-2}\circ\cdots\circ g_{2}\circ f_{1}\circ g_{0}) (u_{-1}), \end{aligned}$$
(64)
$$\begin{aligned}& u_{2m} =(f_{2m}\circ g_{2m-1}\circ\cdots\circ f_{2}\circ g_{1}\circ f_{0}) (v_{-1}) \end{aligned}$$
(65)

for \(m\in \mathbb {N}_{0}\).

From (61) and (62) we have that

$$-\frac{b_{2m}}{a_{2m}}=v_{2m-1}=(g_{2m-1} \circ f_{2m-2} \circ\cdots \circ f_{2}\circ g_{1}\circ f_{0}) (v_{-1}) $$

for some \(m\in \mathbb {N}_{0}\) if and only if

$$\begin{aligned} \frac{1}{y_{-1}y_{-2}}= f_{0}^{-1}\circ g_{1}^{-1} \circ \cdots \circ f_{2m-2}^{-1}\circ g_{2m-1}^{-1} \circ f_{2m}^{-1}(0). \end{aligned}$$
(66)

From (61) and (63) we have that

$$-\frac{b_{2m+1}}{a_{2m+1}}=v_{2m}=(g_{2m}\circ f_{2m-1} \circ\cdots \circ g_{2}\circ f_{1}\circ g_{0}) (u_{-1}) $$

for some \(m\in \mathbb {N}_{0}\) if and only if

$$\begin{aligned} \frac{1}{x_{-1}x_{-2}}= g_{0}^{-1}\circ f_{1}^{-1} \circ \cdots\circ g_{2m-2}^{-1}\circ f_{2m-1}^{-1} \circ g_{2m}^{-1}\circ f_{2m+1}^{-1}(0). \end{aligned}$$
(67)

From (61) and (64) we have that

$$-\frac{\beta _{2m}}{\alpha _{2m}}=u_{2m-1}=(f_{2m-1} \circ g_{2m-2} \circ \cdots \circ g_{2}\circ f_{1}\circ g_{0}) (u_{-1}) $$

for some \(m\in \mathbb {N}_{0}\) if and only if

$$\begin{aligned} \frac{1}{x_{-1}x_{-2}}=g_{0}^{-1}\circ f_{1}^{-1} \circ \cdots\circ g_{2m-2}^{-1}\circ f_{2m-1}^{-1} \circ g_{2m}^{-1}(0). \end{aligned}$$
(68)

From (61) and (65) we have that

$$-\frac{\beta _{2m+1}}{\alpha _{2m+1}}=u_{2m}=(f_{2m}\circ g_{2m-1} \circ \cdots \circ f_{2}\circ g_{1}\circ f_{0}) (v_{-1}) $$

for some \(m\in \mathbb {N}_{0}\) if and only if

$$\begin{aligned} \frac{1}{y_{-1}y_{-2}}= f_{0}^{-1}\circ g_{1}^{-1} \circ \cdots\circ g_{2m-1}^{-1}\circ f_{2m}^{-1} \circ g_{2m+1}^{-1}(0). \end{aligned}$$
(69)

From (66)-(69) we see that the first union in (58) also belongs to the domain of undefinable solutions, finishing the proof of the theorem. □

Remark 1

Quantities

$$\begin{aligned} &g_{0}^{-1}\circ f_{1}^{-1}\circ\cdots \circ g_{2m-2}^{-1}\circ f_{2m-1}^{-1}\circ g_{2m}^{-1}\circ f_{2m+1}^{-1}(0), \end{aligned}$$
(70)
$$\begin{aligned} &g_{0}^{-1}\circ f_{1}^{-1}\circ\cdots \circ g_{2m-2}^{-1}\circ f_{2m-1}^{-1} \circ g_{2m}^{-1}(0), \end{aligned}$$
(71)
$$\begin{aligned} &f_{0}^{-1}\circ g_{1}^{-1}\circ\cdots \circ f_{2m-2}^{-1}\circ g_{2m-1}^{-1} \circ f_{2m}^{-1}(0), \end{aligned}$$
(72)
$$\begin{aligned} &f_{0}^{-1}\circ g_{1}^{-1}\circ\cdots \circ g_{2m-1}^{-1}\circ f_{2m}^{-1} \circ g_{2m+1}^{-1}(0) \end{aligned}$$
(73)

can be calculated for every \(m\in \mathbb {N}_{0}\).

Indeed, note that

$$\begin{aligned} &g_{0}^{-1}\circ f_{1}^{-1}\circ\cdots \circ g_{2m-2}^{-1}\circ f_{2m-1}^{-1}\circ g_{2m}^{-1}\circ f_{2m+1}^{-1}(0)= \Biggl( \prod_{j=0}^{m} \bigl(g_{2j}^{-1} \circ f_{2j+1}^{-1}\bigr) \Biggr) (t) \Big|_{t=0}, \end{aligned}$$
(74)
$$\begin{aligned} &g_{0}^{-1}\circ f_{1}^{-1}\circ\cdots \circ g_{2m-2}^{-1}\circ f_{2m-1}^{-1} \circ g_{2m}^{-1}(0)= \Biggl(\prod_{j=0}^{m-1} \bigl(g_{2j}^{-1}\circ f_{2j+1}^{-1}\bigr) \Biggr) (t) \Big|_{t=g_{2m}^{-1}(0)}, \end{aligned}$$
(75)
$$\begin{aligned} &f_{0}^{-1}\circ g_{1}^{-1}\circ\cdots \circ f_{2m-2}^{-1}\circ g_{2m-1}^{-1} \circ f_{2m}^{-1}(0)= \Biggl(\prod_{j=0}^{m-1} \bigl(f_{2j}^{-1}\circ g_{2j+1}^{-1}\bigr) \Biggr) (t)\Big|_{t=f_{2m}^{-1}(0)}, \end{aligned}$$
(76)
$$\begin{aligned} &f_{0}^{-1}\circ g_{1}^{-1}\circ\cdots \circ g_{2m-1}^{-1}\circ f_{2m}^{-1} \circ g_{2m+1}^{-1}(0)= \Biggl(\prod_{j=0}^{m} \bigl(f_{2j}^{-1}\circ g_{2j+1}^{-1}\bigr) \Biggr) (t) \Big|_{t=0}, \end{aligned}$$
(77)

and also that

$$\begin{aligned} \bigl(g_{2j}^{-1}\circ f_{2j+1}^{-1}\bigr) (t)=\frac{t}{\alpha _{2j}a_{2j+1}}-\frac{b_{2j+1}}{\alpha _{2j}a_{2j+1}}-\frac{\beta _{2j}}{\alpha _{2j}},\quad j\in \mathbb {N}_{0}, \end{aligned}$$
(78)

and

$$\begin{aligned} \bigl(f_{2j}^{-1}\circ g_{2j+1}^{-1}\bigr) (t)=\frac{t}{a_{2j}\alpha _{2j+1}}-\frac{\beta _{2j+1}}{a_{2j}\alpha _{2j+1}}-\frac{b_{2j}}{a_{2j}},\quad j\in \mathbb {N}_{0}. \end{aligned}$$
(79)

On the other hand, if

$$h_{j}(t)=c_{j}t+d_{j},\quad j\in \mathbb {N}_{0}, $$

it is easy to see that

$$\begin{aligned} (h_{0}\circ h_{1}\circ\cdots\circ h_{n}) (t)= \Biggl(\prod_{j=0}^{n} c_{j} \Biggr)t+\sum_{i=0}^{n}d_{j}\prod _{j=0}^{i-1}c_{j},\quad n\in \mathbb {N}_{0}. \end{aligned}$$
(80)

From (74)-(80) explicit formulas for the quantities in (70)-(73) are easily obtained.

References

  1. 1.

    Aloqeili, M: Dynamics of a kth order rational difference equation. Appl. Math. Comput. 181, 1328-1335 (2006)

  2. 2.

    Alzahrani, EO, El-Dessoky, MM, Elsayed, EM, Kuang, Y: Solutions and properties of some degenerate systems of difference equations. J. Comput. Anal. Appl. 18(2), 321-333 (2015)

  3. 3.

    Andruch-Sobilo, A, Migda, M: On the rational recursive sequence \(x_{n+1}=ax_{n-1}/(b+cx_{n}x_{n-1})\). Tatra Mt. Math. Publ. 43, 1-9 (2009)

  4. 4.

    Bajo, I, Liz, E: Global behaviour of a second-order nonlinear difference equation. J. Differ. Equ. Appl. 17(10), 1471-1486 (2011)

  5. 5.

    Berezansky, L, Braverman, E: On impulsive Beverton-Holt difference equations and their applications. J. Differ. Equ. Appl. 10(9), 851-868 (2004)

  6. 6.

    Berg, L, Stević, S: Periodicity of some classes of holomorphic difference equations. J. Differ. Equ. Appl. 12(8), 827-835 (2006)

  7. 7.

    Berg, L, Stević, S: On some systems of difference equations. Appl. Math. Comput. 218, 1713-1718 (2011)

  8. 8.

    Berg, L, Stević, S: On the asymptotics of the difference equation \(y_{n}(1+y_{n-1}\cdots y_{n-k+1})=y_{n-k}\). J. Differ. Equ. Appl. 17(4), 577-586 (2011)

  9. 9.

    Grove, EA, Ladas, G: Periodicities in Nonlinear Difference Equations. Chapman & Hall/CRC Press, Boca Raton (2005)

  10. 10.

    Iričanin, B, Stević, S: Some systems of nonlinear difference equations of higher order with periodic solutions. Dyn. Contin. Discrete Impuls. Syst. 13a(3-4), 499-508 (2006)

  11. 11.

    Iričanin, B, Stević, S: Eventually constant solutions of a rational difference equation. Appl. Math. Comput. 215, 854-856 (2009)

  12. 12.

    Papaschinopoulos, G, Schinas, CJ: On a system of two nonlinear difference equations. J. Math. Anal. Appl. 219(2), 415-426 (1998)

  13. 13.

    Papaschinopoulos, G, Schinas, CJ: On the behavior of the solutions of a system of two nonlinear difference equations. Commun. Appl. Nonlinear Anal. 5(2), 47-59 (1998)

  14. 14.

    Papaschinopoulos, G, Schinas, CJ: Invariants for systems of two nonlinear difference equations. Differ. Equ. Dyn. Syst. 7(2), 181-196 (1999)

  15. 15.

    Papaschinopoulos, G, Schinas, CJ: Invariants and oscillation for systems of two nonlinear difference equations. Nonlinear Anal. TMA 46(7), 967-978 (2001)

  16. 16.

    Papaschinopoulos, G, Schinas, CJ: On the dynamics of two exponential type systems of difference equations. Comput. Math. Appl. 64(7), 2326-2334 (2012)

  17. 17.

    Papaschinopoulos, G, Stefanidou, G: Asymptotic behavior of the solutions of a class of rational difference equations. Int. J. Difference Equ. 5(2), 233-249 (2010)

  18. 18.

    Stefanidou, G, Papaschinopoulos, G, Schinas, CJ: On a system of two exponential type difference equations. Commun. Appl. Nonlinear Anal. 17(2), 1-13 (2010)

  19. 19.

    Stević, S: A global convergence results with applications to periodic solutions. Indian J. Pure Appl. Math. 33(1), 45-53 (2002)

  20. 20.

    Stević, S: More on a rational recurrence relation. Appl. Math. E-Notes 4, 80-85 (2004)

  21. 21.

    Stević, S: A short proof of the Cushing-Henson conjecture. Discrete Dyn. Nat. Soc. 2006, Article ID 37264 (2006)

  22. 22.

    Stević, S: Periodicity of max difference equations. Util. Math. 83, 69-71 (2010)

  23. 23.

    Stević, S: On a system of difference equations. Appl. Math. Comput. 218, 3372-3378 (2011)

  24. 24.

    Stević, S: On a system of difference equations with period two coefficients. Appl. Math. Comput. 218, 4317-4324 (2011)

  25. 25.

    Stević, S: On the difference equation \(x_{n}=x_{n-2}/(b_{n}+c_{n}x_{n-1}x_{n-2})\). Appl. Math. Comput. 218, 4507-4513 (2011)

  26. 26.

    Stević, S: On a solvable rational system of difference equations. Appl. Math. Comput. 219, 2896-2908 (2012)

  27. 27.

    Stević, S: On a third-order system of difference equations. Appl. Math. Comput. 218, 7649-7654 (2012)

  28. 28.

    Stević, S: On some periodic systems of max-type difference equations. Appl. Math. Comput. 218, 11483-11487 (2012)

  29. 29.

    Stević, S: On some solvable systems of difference equations. Appl. Math. Comput. 218, 5010-5018 (2012)

  30. 30.

    Stević, S: On the difference equation \(x_{n}=x_{n-k}/(b+cx_{n-1}\cdots x_{n-k})\). Appl. Math. Comput. 218, 6291-6296 (2012)

  31. 31.

    Stević, S: Solutions of a max-type system of difference equations. Appl. Math. Comput. 218, 9825-9830 (2012)

  32. 32.

    Stević, S: Domains of undefinable solutions of some equations and systems of difference equations. Appl. Math. Comput. 219, 11206-11213 (2013)

  33. 33.

    Stević, S: On a system of difference equations which can be solved in closed form. Appl. Math. Comput. 219, 9223-9228 (2013)

  34. 34.

    Stević, S: On a system of difference equations of odd order solvable in closed form. Appl. Math. Comput. 219, 8222-8230 (2013)

  35. 35.

    Stević, S: On the system \(x_{n+1}=y_{n}x_{n-k}/(y_{n-k+1}(a_{n}+b_{n}y_{n}x_{n-k}))\), \(y_{n+1}=x_{n}y_{n-k}/(x_{n-k+1}(c_{n}+d_{n}x_{n}y_{n-k}))\). Appl. Math. Comput. 219, 4526-4534 (2013)

  36. 36.

    Stević, S: Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences. Electron. J. Qual. Theory Differ. Equ. 2014, Article ID 67 (2014)

  37. 37.

    Stević, S, Alghamdi, MA, Alotaibi, A, Shahzad, N: On a higher-order system of difference equations. Electron. J. Qual. Theory Differ. Equ. 2013, Article ID 47 (2013)

  38. 38.

    Stević, S, Alghamdi, MA, Alotaibi, A, Shahzad, N: Boundedness character of a max-type system of difference equations of second order. Electron. J. Qual. Theory Differ. Equ. 2014, Article ID 45 (2014)

  39. 39.

    Stević, S, Alghamdi, MA, Maturi, DA, Shahzad, N: On a class of solvable difference equations. Abstr. Appl. Anal. 2013, Article ID 157943 (2013)

  40. 40.

    Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: On a third-order system of difference equations with variable coefficients. Abstr. Appl. Anal. 2012, Article ID 508523 (2012)

  41. 41.

    Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: On some solvable difference equations and systems of difference equations. Abstr. Appl. Anal. 2012, Article ID 541761 (2012)

  42. 42.

    Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: On the difference equation \(x_{n+1}=x_{n}x_{n-k}/(x_{n-k+1}(a+bx_{n}x_{n-k}))\). Abstr. Appl. Anal. 2012, Article ID 108047 (2012)

  43. 43.

    Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: On the difference equation \(x_{n}=a_{n}x_{n-k}/(b_{n}+c_{n}x_{n-1}\cdots x_{n-k})\). Abstr. Appl. Anal. 2012, Article ID 409237 (2012)

  44. 44.

    Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: On a solvable system of rational difference equations. J. Differ. Equ. Appl. 20(5-6), 811-825 (2014)

  45. 45.

    Stević, S, Diblik, J, Iričanin, B, Šmarda, Z: Solvability of nonlinear difference equations of fourth order. Electron. J. Differ. Equ. 2014, Article ID 264 (2014)

  46. 46.

    Tollu, DT, Yazlik, Y, Taskara, N: On fourteen solvable systems of difference equations. Appl. Math. Comput. 233, 310-319 (2014)

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Acknowledgements

The work of the first and the second authors was supported by the Serbian Ministry of Education and Science, project III 41025. The work of the first author was also supported by the Serbian Ministry of Education and Science, project III 44006. The work of the second author was also supported by the Serbian Ministry of Education and Science, project OI 171007. The work of the third author was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by project CZ.1.05/1.1.00/02.0068 financed from the European Regional Development Fund. The third author was also supported by the project FEKT-S-14-2200 of Brno University of Technology.

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Correspondence to Stevo Stević.

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MSC

  • 39A10
  • 39A20

Keywords

  • system of difference equations
  • closed form solution
  • long-term behavior
  • periodic solutions