\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 308, pp. 1--14.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/308\hfil Product-type systems of difference equations]
{First-order product-type systems of difference equations solvable
in closed form}

\author[S. Stevi\'c \hfil EJDE-2015/308\hfilneg]
{Stevo Stevi\'c}

\address{Stevo Stevi\'c \newline
 Mathematical Institute of the Serbian Academy of Sciences,
 Knez Mihailova 36/III, 11000 Beograd, Serbia. \newline
Operator theory and  Applications Research Group,
 Department of Mathematics, King Abdulaziz University, P.O. Box 80203,
Jeddah 21589, Saudi Arabia}
\email{sstevic@ptt.rs}

\thanks{Submitted August 29, 2015. Published December 21, 2015.}
\subjclass[2010]{39A20, 39A45}
\keywords{Difference equation; first order system;product-type system; 
\hfill\break\indent solvable in closed form}

\begin{abstract}
 We show that the first-order system of difference equations
 $$
 z_{n+1}=\alpha z_n^aw_n^b,\quad
 w_{n+1}=\beta z_n^cw_n^d,\quad n\in\mathbb{N}_0,
 $$
 where $a,b,c,d\in\mathbb{Z}$, $\alpha,\beta \in\mathbb{C}\setminus\{0\}$,
 $z_0, w_0\in\mathbb{C}\setminus\{0\}$, is solvable in closed form,
 by finding closed form formulas of its solutions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{corollary}[theorem]{Corollary}
\allowdisplaybreaks

\section{Introduction}

The study of nonlinear difference equations and systems is of a
great recent interest (see, for example, \cite{al0}-\cite{c},
\cite{mj}, \cite{ps2}-\cite{508523}). The classical area of
solving difference equations and systems has re-attracted a quite
recent attention (see, for example, \cite{al0}-\cite{amc218-sde},
\cite{c}, \cite{pst2}, \cite{ejqtde-fib}-\cite{ejqtde1}, \cite{508523} 
and the related references therein). Our recent idea
of transforming some complicated difference equations and systems
into simpler solvable ones, used for the first time in explaining
the solvability of the equation appearing in \cite{c}, has been
employed recently in several papers (see, for example, 
\cite{al0,amc218-sde,pst2,ejqtde-fib,ejqtde1,508523} 
and the 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, for example,
\cite{amc218-sde,cm,ps2,ps3,sps,ps-cana}, \cite{ejqtde2015}-\cite{508523}). 
Our important observation in some of above quoted papers on solvability of
difference equations was that suitable changes of variables
transform relatively complicated equations considered therein into
special cases of the linear first-order difference equation 
\begin{equation}
x_n=a_nx_{n-1}+b_n,\quad n\in\mathbb{N},\label{lfo}
\end{equation}
which is a basic solvable difference equation (for a nice presentation of some
methods for solving equation \eqref{lfo} and some related ones
see, for example, monograph \cite{mk}). This was also essentially
the case with some of the equations in
\cite{al0, pst2,ejqtde-fib}. Actually, in this or that way, many
equations and systems are related to equation \eqref{lfo} or to
the corresponding difference inequalities or to the corresponding
system of linear difference equations (for example, some of the
equations, inequalities and systems in
\cite{amc218-sde}-\cite{cm}, \cite{mj,mk,ejqtde1,508523}
are of this type). For some results on general
theory of difference equations and systems or on some other types
of results on various classes of difference equations and systems
see, for example, \cite{cp,cm}, \cite{ll}-\cite{mk}, \cite{ra,rr1},
and the references therein.

On the other hand, the present author also essentially triggered a
systematic study of non-rational concrete difference equations and
systems, from one side those obtained by the scalar translation
operator (see, for example, \cite{na1} and the references therein)
and from the other side those obtained by using some max-type
operators (see, for example, \cite{ejqtde-maxsde}). It can be
noticed that behavior of only positive solutions of the difference
equations and systems in \cite{na1,ejqtde-maxsde}, are
investigated. As we have mentioned in \cite{ejde2015}, in
\cite{ejqtde-maxsde} was studied the boundedness character of
positive solutions to the system 
$$
x_{n+1}=\max\bigg\{a,\frac{y_n^p}{x_{n-1}^q}\bigg\},\quad
y_{n+1}=\max\bigg\{a,\frac{x_n^p}{y_{n-1}^q}\bigg\},\quad
n\in\mathbb{N}_0,
$$ 
with $\min\{a, p, q\}>0$, which is obtained from the
product-type one 
\begin{equation}
x_{n+1}=\frac{y_n^p}{x_{n-1}^q},\quad
y_{n+1}=\frac{x_n^p}{y_{n-1}^q},\quad n\in\mathbb{N}_0,\label{a2}
\end{equation}
 by acting with a max-type operator onto the right-hand sides of both
equations in \eqref{a2} (\cite{na1} deals with a related scalar
equation). If initial values of system \eqref{a2} are positive,
then it can be solved by taking the logarithm to the both sides of
both equations (this transforms the system to a solvable linear
second-order system of difference equations, whose general theory
can be found in \cite{ll}). However, the method is not possible if
initial values are not positive, due to the fact that the
logarithm of a complex number is not uniquely defined. Another
reason for a detailed study of product-type difference equations
and systems is found in the fact that behavior of their solutions
are not so rarely related to the ones of the equations and systems
obtained from them by acting with the translation, max-type or
some other natural operators.

These observations lead us to the investigation of some
product-type difference equations and systems with real and/or
complex initial values. Namely, in \cite{ejqtde2015} and
\cite{ejde2015}, the present author and his collaborators started
studying such systems by modifying methods and ideas from above
mentioned papers, not only those on solving difference equations
and systems, but also using some ideas on non-rational difference
equations and systems appearing, for example, in \cite{na1} and
\cite{ejqtde-maxsde}.

In this article we continue our investigation of solvability of
nonlinear difference equations and systems by studying the
solvability of the following product-type system of difference
equations 
\begin{equation}
z_{n+1}=\alpha z_n^aw_n^b,\quad w_{n+1}=\beta
z_n^cw_n^d,\quad n\in\mathbb{N}_0,\label{ms1}
\end{equation}
where $a,b,c,d\in\mathbb{Z}$,
$\alpha,\beta\in\mathbb{C}$ and initial values $z_0, w_0\in\mathbb{C}$.

The reason why the parameters $a,b,c$ and $d$ are chosen to be
integers is that solutions to system \eqref{ms1} are uniquely
defined.

Note that if any of numbers $a,b,c,d$ is a negative integer then
the domain of undefinable solutions to system \eqref{ms1} is a
subset of the set
$$
\mathcal{U}=\{(z_0, w_0)\in\mathbb{C}^2: z_0=0\mbox{ or } w_0=0\}.
$$ 
Otherwise, if additionally $\alpha,\beta\in\mathbb{C}\setminus\{0\}$,
system \eqref{ms1} is defined on the whole complex space $\mathbb{C}^2$.
Hence, from now on we will assume that our initial values belong
to the set $\mathbb{C}^2\setminus\mathcal{U}$.

Throughout the paper we use the following standard convention
$\sum_{j=k}^{k-1}a_j=0$, if $k\in\mathbb{Z}$.

\section{Main result}

In this section we formulate and prove the main results of this paper.
Before this we quote two lemmas which will be frequently used in
the rest of the paper.
The following lemma was essentially proved in \cite{ejde2015}, so
we will omit its proof.


\begin{lemma} \label{lem1} 
 Let $\eta\in\mathbb{C}\setminus\{0\}$,
$f\in \mathbb{Z}\setminus\{0\}$ and $u_0\in\mathbb{C}\setminus\{0\}$. Then the
difference equation 
\begin{equation}
u_n=u_{n-1}^f\eta,\quad n\in\mathbb{N},\label{c1}
\end{equation}
is solvable and
\begin{equation}
u_n=u_0^{f^n}\eta^{\sum_{j=0}^{n-1}f^j},\quad
n\in\mathbb{N}_0.\label{c2}
\end{equation}
\end{lemma}

\begin{remark} \label{rmk1} 
Note that if $\eta=0$, then from
\eqref{c1} we have that $u_n=0$, $n\in\mathbb{N}$, while if $f=0$ then
$u_n=\eta$, $n\in\mathbb{N}$. If $u_0=0$, then $u_n=0$, $n\in\mathbb{N}$, if
$f\in\mathbb{N}$, while if $f<0$ then such a solution is not defined.
\end{remark}

The following elementary lemma is well-know (see, e.g., \cite{mk}).

\begin{lemma} \label{lem2} 
 Let
$S_k(z)=1+2z+3z^2+\cdots+kz^{k-1}$. Then
$$
S_k(z)=\frac{1-(k+1)z^k+kz^{k+1}}{(1-z)^2},
$$
for $z\in\mathbb{C}\setminus\{1\}$.
\end{lemma}

Now, first note that if $\alpha=0$, then from the first equation in
\eqref{ms1} we have $z_n=0$, $n\in\mathbb{N}$, from which along with the
second equation in \eqref{ms1} it follows that $w_n=0$ for 
$n\ge 2$, if $c>0$, while if $c=0$ we have 
$$
w_n=w_{n-1}^d\beta,\quad n\in\mathbb{N},
$$ 
so by Lemma \ref{lem1}, if $\beta\neq 0$ and $w_0\neq 0$, we have 
$$
w_n=w_0^{d^n}\beta^{\frac{1-d^n}{1-d}},
$$ 
if $d\ne 1$, and
$$
w_n=w_0\beta^n,\quad n\in\mathbb{N}_0,
$$ 
if $d=1$.

Similarly, if $\beta=0$, then from the second equation in \eqref{ms1}
we have $w_n=0$, $n\in\mathbb{N}$, from which along with the first
equation in \eqref{ms1} it follows that $z_n=0$ for $n\ge 2$, if
$b>0$, while if $b=0$ we have 
$$
z_n=z_{n-1}^a\alpha,\quad n\in\mathbb{N},
$$
so by Lemma \ref{lem1}, if $\alpha\ne 0\ne z_0$ we have that
$$
z_n=z_0^{a^n}\alpha^{\frac{1-a^n}{1-a}},
$$ 
if $a\ne 1$, and
$$
z_n=z_0\alpha^n,\quad n\in\mathbb{N}_0,
$$ 
if $a=1$.
Hence, from now on we may also assume that $\alpha\ne0\ne\beta$.

Our first result deals with the case when all the parameters
$a,b,c$ and $d$ are integers different from zero.

\begin{theorem} \label{thm1}
 Assume that $a,b,c,d\in\mathbb{Z}\setminus\{0\}$, 
$\alpha,\beta\in\mathbb{C}\setminus\{0\}$, and
$z_0, w_0\in\mathbb{C}\setminus\{0\}$. Then system \eqref{ms1} is
solvable in closed form.
\end{theorem}

\begin{proof}
 First note that the assumption $\alpha,\beta,
z_0,w_0\in\mathbb{C}\setminus\{0\}$ along with a simple inductive
argument shows that all such solutions are well-defined.

From the first equation in \eqref{ms1}, we have that for every
well-defined solution of the system 
\begin{equation}
w_n^b=\alpha^{-1}z_{n+1}z_n^{-a},\quad n\in\mathbb{N}_0.\label{d1}
\end{equation}
Taking the second equation in \eqref{ms1} to the power $b$ (the
condition $b\ne 0$ is essential here), we obtain
\begin{equation}
w_{n+1}^b=\beta^b z_n^{bc}w_n^{bd},\quad n\in\mathbb{N}_0.\label{d2}
\end{equation}
Employing equality \eqref{d1} into \eqref{d2}, we obtain
$$
\alpha^{-1}z_{n+2}z_{n+1}^{-a}=\beta^b z_n^{bc}\alpha^{-d}z_{n+1}^dz_n^{-ad},
\quad n\in\mathbb{N}_0,
$$
from which it follows that
\begin{equation}
z_{n+2}=z_{n+1}^{a+d}z_n^{bc-ad}\alpha^{1-d}\beta^b,\quad
n\in\mathbb{N}_0.\label{d3}
\end{equation}
Note also that
\begin{equation}
z_0\in\mathbb{C}\setminus\{0\},\quad z_1=z_0^aw_0^b\alpha.\label{d5}
\end{equation}

Let $\gamma=\alpha^{1-d}\beta^b$ and \begin{equation} a_1=a+d,\quad b_1=bc-ad,\quad
c_1=1.\label{ic}\end{equation} Then equation \eqref{d3} can be written in
the following form
\begin{equation}
z_n=z_{n-1}^{a_1}z_{n-2}^{b_1}\gamma^{c_1},\quad n\ge 2.\label{d6}
\end{equation}
By using the equality
$$
z_{n-1}=z_{n-2}^{a_1}z_{n-3}^{b_1}\gamma^{c_1},\quad n\ge 3,
$$
in \eqref{d6}, it follows that
\begin{equation} \label{d8}
\begin{aligned}
z_n
&=(z_{n-2}^{a_1}z_{n-3}^{b_1}\gamma^{c_1})^{a_1}z_{n-2}^{b_1}\gamma^{c_1}\\
&=z_{n-2}^{a_1a_1+b_1}z_{n-3}^{a_1b_1}\gamma^{a_1c_1+c_1} \\
&=z_{n-2}^{a_2}z_{n-3}^{b_2}\gamma^{c_2},
\end{aligned}
\end{equation}
 for $n\ge 3$, where
\begin{equation}
a_2:=a_1a_1+b_1,\quad b_2:=a_1b_1,\quad
c_2:=a_1c_1+c_1.\label{d85}
\end{equation}


If we use the equality
$$
z_{n-2}=z_{n-3}^{a_1}z_{n-4}^{b_1}\gamma^{c_1},\quad n\ge 4,
$$ 
in \eqref{d8}, we obtain
\begin{equation}
z_n=(z_{n-3}^{a_1}z_{n-4}^{b_1}\gamma^{c_1})^{a_2}z_{n-3}^{b_2}\gamma^{c_2}
=z_{n-3}^{a_1a_2+b_2}z_{n-4}^{b_1a_2}\gamma^{c_1a_2+c_2}
=z_{n-3}^{a_3}z_{n-4}^{b_3}\gamma^{c_3},\label{d15}
\end{equation}
for $n\ge 4$, where
\begin{equation} a_3:=a_1a_2+b_2, \quad b_3:=b_1a_2, \quad
c_3:=c_1a_2+c_2.\label{d16}
\end{equation}
Now, assume that  the following equality was proven,
\begin{equation}
z_n=z_{n-k}^{a_k}z_{n-k-1}^{b_k}\gamma^{c_k},\label{d9}
\end{equation}
for some
$k\in\mathbb{N}$ such that $n\ge k+1$, and that
\begin{equation}
a_k=a_1a_{k-1}+b_{k-1},\quad b_k=b_1a_{k-1},\quad
c_k=c_1a_{k-1}+c_{k-1}.\label{d10}
\end{equation}
Then, employing the  equality
$$
z_{n-k}=z_{n-k-1}^{a_1}z_{n-k-2}^{b_1}\gamma^{c_1},
$$
for $n\ge k+2$, into \eqref{d9} we obtain
\begin{equation} \label{d11}
\begin{aligned}
z_n
&=(z_{n-k-1}^{a_1}z_{n-k-2}^{b_1}\gamma^{c_1})^{a_k}z_{n-k-1}^{b_k}\gamma^{c_k} \\
&=z_{n-k-1}^{a_1a_k+b_k}z_{n-k-2}^{b_1a_k}\gamma^{c_1a_k+c_k} \\
&=z_{n-k-1}^{a_{k+1}}z_{n-k-2}^{b_{k+1}}\gamma^{c_{k+1}},
\end{aligned}
\end{equation}
for $n\ge k+2$, where
\begin{equation}
a_{k+1}:=a_1a_k+b_k,\quad b_{k+1}:=b_1a_k,\quad
c_{k+1}:=c_1a_k+c_k.\label{d12}
\end{equation}
From
\eqref{d8}, \eqref{d85}, \eqref{d11}, \eqref{d12} and by using the
method of induction it follows that \eqref{d9} and \eqref{d10}
hold for all natural numbers $k$ and $n$ such that
$2\le k\le n-1$.


Plugging $k=n-1$ into \eqref{d9}, we obtain
\begin{align}
z_n
&=z_1^{a_{n-1}}z_0^{b_{n-1}}\gamma^{c_{n-1}}\label{a16}\\
&=\big(z_0^aw_0^b\alpha\big)^{a_{n-1}}z_0^{b_{n-1}}\gamma^{c_{n-1}} \nonumber \\
&=z_0^{aa_{n-1}+b_{n-1}}w_0^{ba_{n-1}}\alpha^{a_{n-1}}\gamma^{c_{n-1}} \nonumber \\
&=z_0^{aa_{n-1}+b_{n-1}}w_0^{ba_{n-1}}\alpha^{a_{n-1}+(1-d)c_{n-1}}\beta^{bc_{n-1}},\quad
n\in\mathbb{N}_0.\label{d19}
\end{align}
From the first two equations in \eqref{d10} we see that
$$
a_k=a_1a_{k-1}+b_1a_{k-2},\quad k\ge 3,
$$
which by \eqref{ic} can be rewritten as 
 \begin{equation}
a_{k+2}-(a+d)a_{k+1}+(ad-bc)a_k=0,\quad k\in\mathbb{N}.\label{d17}
\end{equation}
\smallskip

\noindent\textbf{Case $ad\ne bc$}.
To calculate $a_k$ more easily in this case, note that from the first 
two relations in \eqref{d12} with $k=0$, we have
$$
a_1=a_1a_0+b_0,\quad b_1=b_1a_0,
$$
from which along with the assumption $b_1=bc-ad\ne 0$, it follows
that 
\begin{equation}
a_0=1,\quad b_0=0.\label{a3}
\end{equation}
The characteristic polynomial associated to equation \eqref{d17}
is
$$
P(\lambda)=\lambda^2-(a+d)\lambda+ad-bc,
$$
from which it follows that the characteristic values are
$$
\lambda_{1,2}=\frac{a+d\pm\sqrt{(a+d)^2-4(ad-bc)}}2,
$$
which implies that
$$
a_k=\hat c_1\lambda^k_1+\hat c_2\lambda^k_2,\quad k\in\mathbb{N}_0,
$$
for some constants $\hat c_1$ and $\hat c_2$, if
$\Delta:=(a+d)^2-4(ad-bc)\ne 0$, while
$$
a_k=(\hat c_3+k\hat c_4)\lambda^k_1,\quad k\in\mathbb{N}_0,
$$
for some constants $\hat c_3$ and $\hat c_4$, if $\Delta=0$ (in
this case $\lambda_1=\lambda_2=(a+d)/2$).

Using the initial conditions $a_0=1$ and $a_1=a+d$, after some
calculations, it is not difficult to see that 
\begin{equation}
a_k=\frac{\lambda_1^{k+1}-\lambda_2^{k+1}}{\lambda_1-\lambda_2},\quad
k\in\mathbb{N}_0,\label{a5}
\end{equation}
when $\Delta\ne 0$, while
\begin{equation}
a_k=(k+1)\lambda_1^k,\quad k\in\mathbb{N}_0,\label{a6}
\end{equation}
when $\Delta=0$.

From \eqref{a5}, \eqref{a6} and the second equation in
\eqref{d10}, we have
 \begin{equation}
b_k=(bc-ad)\frac{\lambda_1^k-\lambda_2^k}{\lambda_1-\lambda_2},\quad
k\in\mathbb{N}_0,\label{a7}
\end{equation}
 when $\Delta\ne 0$, while
\begin{equation}
b_k=(bc-ad)k\lambda_1^{k-1},\quad k\in\mathbb{N}_0,\label{a8}
\end{equation}
when $\Delta=0$.

From the third equation in \eqref{d10} and since $c_1=1$, we
obtain
$$
c_k=c_{k-1}+a_{k-1},\quad k\ge 2,
$$
from which it follows that 
\begin{equation}
c_k=1+\sum_{j=1}^{k-1}a_j,\quad k\ge 2.\label{a10}
\end{equation}

Using \eqref{a5} in \eqref{a10} we have 
 \begin{equation}
c_k=\frac{(\lambda_2-1)(\lambda_1^{k+1}-1)
-(\lambda_1-1)(\lambda_2^{k+1}-1)}{(\lambda_1-\lambda_2)(\lambda_1-1)(\lambda_2-1)},\quad
k\in\mathbb{N},\label{a11}
\end{equation}
if $\Delta\ne 0$, while by Lemma \ref{lem1}
\begin{equation}
c_k=\frac{1-(k+1)\lambda_1^k+k\lambda_1^{k+1}}{(1-\lambda_1)^2},\quad
k\in\mathbb{N},\label{a12}
\end{equation}
if $\Delta=0$.


Hence, if $\Delta\ne0$ using \eqref{a5}, \eqref{a7} and
\eqref{a11} in \eqref{d19}, we obtain 
\begin{equation}
\begin{aligned}
z_n&=z_0^{a\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}+(bc-ad)
 \frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}}
w_0^{b\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}} \\
&\quad \times\alpha^{\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}+(1-d)
 \big(\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)}\big)}
\\
&\quad \times\beta^{b\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)}},
\end{aligned} \label{a14}
\end{equation}
for $n\in\mathbb{N}_0$, while if $\Delta=0$ using \eqref{a6}, \eqref{a8}
and \eqref{a12} in \eqref{d19}, we obtain
\begin{equation} \label{a15}
\begin{aligned}
z_n
&=z_0^{an\lambda_1^{n-1}+(bc-ad)(n-1)\lambda_1^{n-2}}w_0^{bn\lambda_1^{n-1}} \\
&\quad \times\alpha^{n\lambda_1^{n-1}+(1-d)\frac{1-n\lambda_1^{n-1}+(n-1)\lambda_1^n}
 {(1-\lambda_1)^2}}\beta^{b\frac{1-n\lambda_1^{n-1}+(n-1)\lambda_1^n}{(1-\lambda_1)^2}},
\end{aligned}
\end{equation}
for $n\in\mathbb{N}_0$.

On the other hand, from the second equation in \eqref{ms1}, we
have that for every well-defined solution of the system 
\begin{equation}
z_n^c=\beta^{-1} w_n^{-d}w_{n+1},\quad n\in\mathbb{N}_0.\label{f1}
\end{equation}
Taking the first equation in \eqref{ms1} to the $c$\,th power
(the condition $c\ne 0$ is essential here), we obtain
\begin{equation}
z_{n+1}^c=z_n^{ac}w_n^{bc}\alpha^c,\quad n\in\mathbb{N}_0.\label{f2}
\end{equation}
Employing \eqref{f1} in \eqref{f2}, we obtain
$$
\beta^{-1}w_{n+1}^{-d}w_{n+2}=\beta^{-a}w_n^{-ad}w_{n+1}^aw_n^{bc}\alpha^c,
\quad n\in\mathbb{N}_0,
$$
which can be written as
\begin{equation}
w_{n+2}=w_{n+1}^{a+d}w_n^{bc-ad}\beta^{1-a}\alpha^c,\quad
n\in\mathbb{N}_0.\label{f3}
\end{equation}
Let $\delta=\beta^{1-a}\alpha^c$ and $a_1$, $b_1$ and $c_1$ are given by
\eqref{ic}. Then  \eqref{f3} can be written as
\begin{equation}
w_{n+2}=w_{n+1}^{a_1}w_n^{b_1}\delta^{c_1},\quad
n\in\mathbb{N}_0.\label{a21}
\end{equation}
Note also that
\begin{equation}
w_0\in\mathbb{C}\setminus\{0\},\quad  w_1=z_0^cw_0^d\beta.\label{f5}
\end{equation}

Since \eqref{a21} has the same form as equation \eqref{d6}, where
only $\gamma$ is replaced by $\delta$, we have that the recurrent
relations in \eqref{d10} hold, and consequently formulas
\eqref{a5}-\eqref{a8}, \eqref{a11} and \eqref{a12}. Also we have
that \eqref{a16} holds with $\gamma$ replaced by $\delta$, and $z$
replaced by $w$. From this, by using \eqref{f5} and the definition
of $\delta$, we have 
\begin{equation} \label{a17}
\begin{aligned}
w_n&=w_1^{a_{n-1}}w_0^{b_{n-1}}\delta^{c_{n-1}} \\
&=(z_0^cw_0^d\beta)^{a_{n-1}}w_0^{b_{n-1}}\delta^{c_{n-1}} \\
&=z_0^{ca_{n-1}}w_0^{da_{n-1}+b_{n-1}}\beta^{a_{n-1}+(1-a)c_{n-1}}\alpha^{cc_{n-1}},\quad
n\in\mathbb{N}_0.
\end{aligned}
\end{equation}
Thus, if $\Delta\ne0$ using \eqref{a5}, \eqref{a7} and \eqref{a11}
in \eqref{a17}, we obtain
\begin{equation} \label{a18}
\begin{aligned}
w_n
&=z_0^{c\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}}
w_0^{d\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}+(bc-ad)
 \frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}} \\
&\quad \times\beta^{\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}+(1-a)
 \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)}} \\
&\quad \times \alpha^{c\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)}},
\end{aligned}
\end{equation}
for $n\in\mathbb{N}_0$, while if $\Delta=0$ using \eqref{a6}, \eqref{a8}
and \eqref{a12} in \eqref{a17}, we obtain
\begin{equation} \label{a19}
\begin{aligned}
w_n
&= z_0^{cn\lambda_1^{n-1}}w_0^{dn\lambda_1^{n-1}+(bc-ad)(n-1)\lambda_1^{n-2}} \\
&\quad \times\beta^{n\lambda_1^{n-1}+(1-a)\frac{1-n\lambda_1^{n-1}
 +(n-1)\lambda_1^n}{(1-\lambda_1)^2}}\alpha^{c\frac{1-n\lambda_1^{n-1}
 +(n-1)\lambda_1^n}{(1-\lambda_1)^2}},
\end{aligned}
\end{equation}
for $n\in\mathbb{N}_0$.
\smallskip

\noindent\textbf{Case $ad=bc$}.
 Since $b\ne 0\ne c$, we see that \eqref{d3}
and \eqref{f3} hold, from which along with the assumption we have
that
\begin{gather}
z_{n+1}=z_n^{a+d}\alpha^{1-d}\beta^b=z_n^{a_1}\gamma,\label{a22}\\
w_{n+1}=w_n^{a+d}\beta^{1-a}\alpha^c=w_n^{a_1}\delta,\label{a23}
\end{gather}
for $n\in\mathbb{N}$.
By Lemma \ref{lem1} we have 
\begin{equation}
z_n=z_1^{a_1^{n-1}}\gamma^{\sum_{j=0}^{n-2}a_1^j}=(\alpha
z_0^aw_0^b)^{a_1^{n-1}}\gamma^{\sum_{j=0}^{n-2}a_1^j},\label{a25}
\end{equation}
for $n\in\mathbb{N}$.
Hence, if $a_1\ne 1$, then from \eqref{a25} we obtain
\begin{align} 
z_n&=(\alpha z_0^aw_0^b)^{a_1^{n-1}}\gamma^{\frac{1-a_1^{n-1}}{1-a_1}} \nonumber\\
&=z_0^{a(a+d)^{n-1}}w_0^{b(a+d)^{n-1}}
 \alpha^{\frac{1-d-a(a+d)^{n-1}}{1-a-d}}
 \beta^{b\frac{1-(a+d)^{n-1}}{1-a-d}}, \label{a26}
\end{align}
for $n\in\mathbb{N}$, while if $a_1=1$, from \eqref{a25} we obtain
\begin{equation}
z_n=\alpha z_0^aw_0^b\gamma^{n-1}=z_0^aw_0^b\alpha^{(1-d)n+d}\beta^{b(n-1)},\label{a27}
\end{equation}
for $n\in\mathbb{N}$.

On the other hand, from \eqref{a23} and by using Lemma \ref{lem1}, we obtain
\begin{equation}
 w_n=w_1^{a_1^{n-1}}\delta^{\sum_{j=0}^{n-2}a_1^j}=(\beta
z_0^cw_0^d)^{a_1^{n-1}}\delta^{\sum_{j=0}^{n-2}a_1^j},\label{a28}
\end{equation}
for $n\in\mathbb{N}$.
Hence, if $a_1\ne 1$, then from \eqref{a28} we obtain 
\begin{align}
w_n&=(\beta z_0^cw_0^d)^{a_1^{n-1}}\delta^{\frac{1-a_1^{n-1}}{1-a_1}} \nonumber\\
&=z_0^{c(a+d)^{n-1}}w_0^{d(a+d)^{n-1}}
 \alpha^{c\frac{1-(a+d)^{n-1}}{1-a-d}}
 \beta^{\frac{1-a-d(a+d)^{n-1}}{1-a-d}}, \label{a29}
\end{align}
for $n\in\mathbb{N}$, while if $a_1=1$, from \eqref{a28} we obtain
\begin{equation}
w_n=\beta z_0^cw_0^d\delta^{n-1}=z_0^cw_0^d\alpha^{c(n-1)}\beta^{(1-a)n+a},\label{a30}
\end{equation}
for $n\in\mathbb{N}$.

It is easy to see that formulas \eqref{a14} and \eqref{a18} in the
case $ad\ne bc$ and $\Delta\ne 0$, \eqref{a15} and \eqref{a19} in
the case $ad\ne bc$ and $\Delta=0$, \eqref{a26} and \eqref{a29} in
the case $ad=bc$ and $a+d\ne 1$, and \eqref{a27} and \eqref{a30}
in the case $ad=bc$ and $a+d=1$, annihilate system \eqref{ms1}.
So, they are solutions to the system, and it is solvable indeed,
as claimed.
\end{proof}

\begin{remark} \label{rmk2} \rm
In the case $ad\ne bc$, this condition
enables to prolong solutions to system \eqref{d10} for every
non-positive integer $k$. For example, using \eqref{a3} in the
third equation in \eqref{d10} it is obtained $c_0=0$. From this
along with \eqref{a3} and \eqref{d10} with $k=0$ it is further
easily obtained $a_{-1}=0=c_{-1}$ and $b_{-1}=1$. This fact, among
others, shows that equalities \eqref{d19} and \eqref{a17} really
hold for every $n\in\mathbb{N}_0$.
\end{remark}


The following corollary is a consequence of Theorem \ref{thm1}.

\begin{corollary} \label{coro1}
 Consider system \eqref{ms1} with
$a,b,c,d\in\mathbb{Z}\setminus\{0\}$ and $\alpha,\beta\in\mathbb{C}\setminus\{0\}$.
Assume that $z_0, w_0\in\mathbb{C}\setminus\{0\}$. Then the following
statements are true.
\begin{itemize}

\item[(a)] If $ad\ne bc$ and $\Delta\ne 0$, then the general solution
to system \eqref{ms1} is given by \eqref{a14} and \eqref{a18}.

\item[(b)] If $ad\ne bc$ and $\Delta=0$, then the general solution to
system \eqref{ms1} is given by \eqref{a15} and \eqref{a19}.

\item[(c)] If $ad=bc$ and $a+d\ne 1$, then the general solution to
system \eqref{ms1} is given by \eqref{a26} and \eqref{a29}.

\item[(d)] If $ad=bc$ and $a+d=1$, then the general solution to system
\eqref{ms1} is given by \eqref{a27} and \eqref{a30}.
\end{itemize}
\end{corollary}

Now we  consider the cases when some of the coefficients
$a,b,c,d$ are equal to zero.
\smallskip

\noindent\textbf{Case $a=0$.} 
Since $a=0$, then the first equation in \eqref{ms1} becomes 
\begin{equation}
z_{n+1}=\alpha w_n^b,\quad n\in\mathbb{N}_0.\label{b1}
\end{equation}
By substituting \eqref{b1} into the
second equation in \eqref{ms1}, we obtain
\begin{equation}
w_{n+1}=\alpha^c\beta w_n^dw_{n-1}^{bc},\quad n\in\mathbb{N}.\label{b2}
\end{equation}
Equation \eqref{b2} is nothing but equation \eqref{a21} with $a=0$. Hence,
we can apply formulas for $w_n$ obtained in the proof of Theorem \ref{thm1}
along with \eqref{b1} and get the following result.


\begin{theorem} \label{thm2} 
Consider system \eqref{ms1} with
$b,c,d\in\mathbb{Z}$, $a=0$ and $\alpha,\beta\in\mathbb{C}\setminus\{0\}$. Assume that
$z_0, w_0\in\mathbb{C}\setminus\{0\}$. Then the following statements are
true.
\begin{itemize}

\item[(a)] If $bc\ne 0$ and $\Delta\ne 0$, then the general solution to
system \eqref{ms1} is 
\begin{align*}
z_n&=z_0^{bc\frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}}
w_0^{bd\frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}+b^2
 c\frac{\lambda_1^{n-2}-\lambda_2^{n-2}}{\lambda_1-\lambda_2}} \\
&\quad \times\beta^{b\frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}
+b\frac{(\lambda_2-1)(\lambda_1^{n-1}-1)-(\lambda_1-1)(\lambda_2^{n-1}-1)}
{(\lambda_1-\lambda_2)(\lambda_1-1)(\lambda_2-1)}} \\
&\quad \times \alpha^{1+bc\frac{(\lambda_2-1)(\lambda_1^{n-1}-1)-(\lambda_1-1)(\lambda_2^{n-1}-1)}
 {(\lambda_1-\lambda_2)(\lambda_1-1)(\lambda_2-1)}}, 
\end{align*}
\begin{align*}
w_n&=z_0^{c\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}}
w_0^{d\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}
 +bc\frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}}\beta^{\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}
+\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)}} \\
&\quad \times \alpha^{c\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)}}, 
\end{align*}
for $n\in\mathbb{N}$, where
$$
\lambda_{1,2}=\frac{d\pm\sqrt{d^2+4bc}}2.
$$


\item[(b)] If $bc\ne 0$ and $\Delta=0$, then the general solution to
system \eqref{ms1} is 
\begin{align*}
z_n&=z_0^{bc(n-1)\lambda_1^{n-2}}w_0^{bd(n-1)\lambda_1^{n-2}+b^2c(n-2)\lambda_1^{n-3}} \\
&\quad \times\beta^{b(n-1)\lambda_1^{n-2}+b\frac{1-(n-1)\lambda_1^{n-2}
 +(n-2)\lambda_1^{n-1}}{(1-\lambda_1)^2}}\alpha^{1+bc
 \frac{1-(n-1)\lambda_1^{n-2}+(n-2)\lambda_1^{n-1}}{(1-\lambda_1)^2}}, 
\end{align*}
\begin{align*}
w_n&=z_0^{cn\lambda_1^{n-1}}w_0^{dn\lambda_1^{n-1}+bc(n-1)\lambda_1^{n-2}}\\
&\quad\times \beta^{n\lambda_1^{n-1}+\frac{1-n\lambda_1^{n-1}+(n-1)\lambda_1^n}{(1-\lambda_1)^2}}
\alpha^{c\frac{1-n\lambda_1^{n-1}+(n-1)\lambda_1^n}{(1-\lambda_1)^2}},
\end{align*}
for $n\in\mathbb{N}$, where $\lambda_1=d/2$.

\item[(c)] If $bc=0$ and $d\ne 1$, then the general solution to system
\eqref{ms1} is 
\begin{gather*}
z_n= w_0^{bd^{n-1}}\alpha\beta^{b\frac{1-d^{n-1}}{1-d}} \\
w_n = z_0^{cd^{n-1}}w_0^{d^n}\alpha^{c\frac{1-d^{n-1}}{1-d}}\beta^{\frac{1-d^n}{1-d}},
\end{gather*}
for $n\in\mathbb{N}$.


\item[(d)] If $bc=0$ and $d=1$, then the general solution to system
\eqref{ms1} is 
\begin{gather*}
z_n=w_0^b\alpha\beta^{b(n-1)}\\
w_n=z_0^cw_0\alpha^{c(n-1)}\beta^n,
\end{gather*} 
for $n\in\mathbb{N}$.
\end{itemize}
\end{theorem}



\noindent\textbf{Case $b=0$.} 
Since $b=0$, then the first equation in
\eqref{ms1} becomes 
\begin{equation}
z_{n+1}=\alpha z_n^a,\quad n\in\mathbb{N}_0.\label{b3}
\end{equation}
By using Lemma \ref{lem1} see that
\begin{equation}
z_n=z_0^{a^n}\alpha^{\sum_{j=0}^{n-1}a^j},\quad
n\in\mathbb{N}_0.\label{b4}
\end{equation}
From \eqref{b4} we have that
\begin{equation}
z_n=z_0^{a^n}\alpha^{\frac{1-a^n}{1-a}},\quad
n\in\mathbb{N}_0,\label{b5}
\end{equation}
when $a\ne 1$, while
\begin{equation}
z_n=z_0\alpha^n,\quad n\in\mathbb{N}_0,\label{b6}
\end{equation}
when $a=1$.

Hence, if $a\ne 1$, substituting \eqref{b5} into the second
equation in \eqref{ms1}, we obtain 
\begin{equation}
w_n=w_{n-1}^d\beta
z_0^{ca^{n-1}}\alpha^{c\frac{1-a^{n-1}}{1-a}},\quad
n\in\mathbb{N}.\label{b7}
\end{equation}
By repeating use of \eqref{b7} we obtain that
for $n\ge 3$,
\begin{equation} \label{b8}
\begin{aligned}
w_n&=\Big(w_{n-2}^d\beta
z_0^{ca^{n-2}}\alpha^{c\frac{1-a^{n-2}}{1-a}}\Big)^d\beta
z_0^{ca^{n-1}}\alpha^{c\frac{1-a^{n-1}}{1-a}} \\
&=w_{n-2}^{d^2}\beta^{1+d}
z_0^{ca^{n-1}+cda^{n-2}}\alpha^{c\frac{1-a^{n-1}}{1-a}+cd\frac{1-a^{n-2}}{1-a}} \\
&=\Big(w_{n-3}^d\beta
z_0^{ca^{n-3}}\alpha^{c\frac{1-a^{n-3}}{1-a}}\Big)^{d^2}\beta^{1+d}
z_0^{ca^{n-1}+cda^{n-2}}\alpha^{c\frac{1-a^{n-1}}{1-a}+cd\frac{1-a^{n-2}}{1-a}} \\
&= w_{n-3}^{d^3}\beta^{1+d+d^2}
z_0^{ca^{n-1}+cda^{n-2}+cd^2a^{n-3}}\alpha^{c\frac{1-a^{n-1}}{1-a}
+cd\frac{1-a^{n-2}}{1-a}+cd^2\frac{1-a^{n-3}}{1-a}}.
\end{aligned}
\end{equation}
An inductive argument shows that
\begin{equation}
w_n=w_0^{d^n}\beta^{\sum_{j=0}^{n-1}d^j}
z_0^{c\sum_{j=0}^{n-1}a^{n-1-j}d^j}\alpha^{c\sum_{j=0}^{n-1}\frac{1-a^{n-1-j}}{1-a}d^j},
\quad n\in\mathbb{N}.\label{b9}
\end{equation}
Hence, from \eqref{b9} we obtain
\begin{equation}
w_n=w_0^{d^n}\beta^{\frac{1-d^n}{1-d}}z_0^{c\frac{a^n-d^n}{a-d}}
\alpha^{c\big(\frac{1-d^n}{(1-a)(1-d)}-\frac{a^n-d^n}{(1-a)(a-d)}\big)},\quad
n\in\mathbb{N}_0,\label{b10}
\end{equation}
if $d\ne 1$ and $a\ne d$,
\begin{equation}
w_n=w_0^{d^n}\beta^{\frac{1-d^n}{1-d}}z_0^{cnd^{n-1}}
\alpha^{c\big(\frac{1-d^n}{(1-d)^2}-\frac{nd^{n-1}}{1-d}\big)},\quad
n\in\mathbb{N}_0,\label{b11}
\end{equation}
if $d\ne 1$ and $a=d$, and
\begin{equation}
w_n=w_0\beta^nz_0^{c\frac{1-a^n}{1-a}}
\alpha^{c\big(\frac{n}{1-a}-\frac{1-a^n}{(1-a)^2}\big)},
\quad
n\in\mathbb{N}_0,\label{b10a}
\end{equation}
if $d=1$ and $a\ne 1$.

If $a=1$, substituting \eqref{b6} into the second equation in
\eqref{ms1}, we obtain 
\begin{equation}
w_n=w_{n-1}^d\beta z_0^c\alpha^{c(n-1)},\quad
n\in\mathbb{N}.\label{b12}
\end{equation}
By repeated use of \eqref{b12} we obtain
\begin{equation} \label{b14}
\begin{aligned}
w_n&=\big(w_{n-2}^d\beta z_0^c
\alpha^{c(n-2)}\big)^d\beta z_0^c\alpha^{c(n-1)} \\
&=w_{n-2}^{d^2}\beta^{1+d}z_0^{c+cd}
\alpha^{c(n-1)+cd(n-2)} \\
&=\big(w_{n-3}^d\beta z_0^c\alpha^{c(n-3)}\big)^{d^2}\beta^{1+d}z_0^{c+cd}\alpha^{c(n-1)+cd(n-2)} \\
&=w_{n-3}^{d^3}\beta^{1+d+d^2}z_0^{c+cd+cd^2}\alpha^{c(n-1)+cd(n-2)+cd^2(n-3)},\quad
n\ge 3.
\end{aligned}
\end{equation}
An inductive argument shows that
\begin{equation}
w_n=w_0^{d^n}\beta^{\sum_{j=0}^{n-1}d^j}z_0^{c\sum_{j=0}^{n-1}d^j}
\alpha^{c\sum_{j=0}^{n-1}d^j(n-1-j)},\label{b15}
\end{equation}
 for $n\in\mathbb{N}$.
Hence, from \eqref{b15} and by using Lemma \ref{lem2}, we obtain
\begin{equation}
w_n=w_0^{d^n}\beta^{\frac{1-d^n}{1-d}}z_0^{c\frac{1-d^n}{1-d}}
\alpha^{c\big((n-1)\frac{1-d^n}{1-d}-d\frac{1-nd^{n-1}+(n-1)d^n}{(1-d)^2}\big)},\quad
n\in\mathbb{N}_0,\label{b16}
\end{equation}
if $d\ne 1$, and
\begin{equation}
w_n=w_0\beta^n z_0^{cn}\alpha^{c\frac{n(n-1)}2},\quad n\in\mathbb{N}_0,\label{b17}
\end{equation}
if $d=1$.
From the above considerations we see that the following result
holds.

\begin{theorem} \label{thm3} 
 Consider system \eqref{ms1} with
$a,c,d\in\mathbb{Z}$, $b=0$ and $\alpha,\beta\in\mathbb{C}\setminus\{0\}$. Assume that
$z_0, w_0\in\mathbb{C}\setminus\{0\}$. Then the following statements are
true.
\begin{itemize}

\item[(a)] If $a\ne 1$, $d\ne 1$ and $a\ne d$, then the general
solution to system \eqref{ms1} is given by \eqref{b5} and
\eqref{b10}.

\item[(b)] If $a\ne 1$, $d\ne 1$ and $a=d$, then the general solution
to system \eqref{ms1} is given by \eqref{b5} and \eqref{b11}.

\item[(c)] If $a\ne 1$ and $d=1$, then the general solution to system
\eqref{ms1} is given by \eqref{b5} and \eqref{b10a}.

\item[(d)] If $a=1$ and $d\ne 1$, then the general solution to system
\eqref{ms1} is given by \eqref{b6} and \eqref{b16}.

\item[(e)] If $a=d=1$, then the general solution to system \eqref{ms1}
is given by \eqref{b6} and \eqref{b17}.

\end{itemize}
\end{theorem}



\noindent\textbf{Case $c=0$.} 
Since $c=0$, then the second equation in
\eqref{ms1} becomes 
\begin{equation}
w_{n+1}=\beta w_n^d,\quad n\in\mathbb{N}_0. \label{b18}
\end{equation}
By using Lemma \ref{lem1} in \eqref{b18} we obtain
\begin{equation}
w_n=w_0^{d^n}\beta^{\sum_{j=0}^{n-1}d^j},\quad
n\in\mathbb{N}_0.\label{b19}
\end{equation}
Then we have
\begin{equation}
w_n=w_0^{d^n}\beta^{\frac{1-d^n}{1-d}},\quad
n\in\mathbb{N}_0,\label{b20}
\end{equation}
when $d\ne 1$, while
\begin{equation}
w_n=w_0\beta^n,\quad n\in\mathbb{N}_0,\label{b21}
\end{equation}
when $d=1$.
Hence, if $d\ne 1$, substituting \eqref{b20} into the first
equation in \eqref{ms1}, we obtain
\begin{equation}
z_n=z_{n-1}^a\alpha
w_0^{bd^{n-1}}\beta^{b\frac{1-d^{n-1}}{1-d}},\quad
n\in\mathbb{N}.\label{b22}
\end{equation}
By repeated use of \eqref{b22} we have
that for $n\ge 3$,
\begin{equation} \label{b23}
\begin{aligned}
z_n&=\Big(z_{n-2}^a\alpha
w_0^{bd^{n-2}}\beta^{b\frac{1-d^{n-2}}{1-d}}\Big)^a\alpha
w_0^{bd^{n-1}}\beta^{b\frac{1-d^{n-1}}{1-d}} \\
&=z_{n-2}^{a^2}\alpha^{1+a}
w_0^{bd^{n-1}+bad^{n-2}}\beta^{b\frac{1-d^{n-1}}{1-d}+ba\frac{1-d^{n-2}}{1-d}} \\
&=\Big(z_{n-3}^a\alpha
w_0^{bd^{n-3}}\beta^{b\frac{1-d^{n-3}}{1-d}}\Big)^{a^2}\alpha^{1+a}
w_0^{bd^{n-1}+bad^{n-2}}\beta^{b\frac{1-d^{n-1}}{1-d}+ba\frac{1-d^{n-2}}{1-d}} \\
&=z_{n-3}^{a^3}\alpha^{1+a+a^2}
w_0^{bd^{n-1}+bad^{n-2}+ba^2d^{n-3}}\beta^{b\frac{1-d^{n-1}}{1-d}
+ba\frac{1-d^{n-2}}{1-d}+ba^2\frac{1-d^{n-3}}{1-a}}.
\end{aligned}
\end{equation}
An inductive argument shows that
\begin{equation}
z_n=z_0^{a^n}\alpha^{\sum_{j=0}^{n-1}a^j}
w_0^{b\sum_{j=0}^{n-1}d^{n-1-j}a^j}\beta^{b\sum_{j=0}^{n-1}
\frac{1-d^{n-1-j}}{1-d}a^j},\label{b25}
\end{equation}
for $n\in\mathbb{N}$.
Hence, from \eqref{b25} we obtain
\begin{equation}
z_n=z_0^{a^n}\alpha^{\frac{1-a^n}{1-a}}w_0^{b\frac{a^n-d^n}{a-d}}
\beta^{b\big(\frac{1-a^n}{(1-a)(1-d)}-\frac{a^n-d^n}{(1-d)(a-d)}\big)},
\quad
n\in\mathbb{N}_0,\label{b26}
\end{equation}
if $a\ne 1$ and $a\ne d$,
\begin{equation}
z_n=z_0^{d^n}\alpha^{\frac{1-d^n}{1-d}}w_0^{bnd^{n-1}}
\beta^{b\big(\frac{1-d^n}{(1-d)^2}-\frac{nd^{n-1}}{1-d}\big)},\quad
n\in\mathbb{N}_0,\label{b27}
\end{equation}
if $a\ne 1$ and $a=d$, and
\begin{equation}
z_n=z_0\alpha^nw_0^{b\frac{1-d^n}{1-d}}\beta^{b\big(\frac{n}{1-d}
-\frac{1-d^n}{(1-d)^2}\big)},\quad
n\in\mathbb{N}_0,\label{b27a}
\end{equation}
if $a=1$ and $d\ne 1$.

If $d=1$, substituting \eqref{b21} into the first equation in
\eqref{ms1}, we obtain 
\begin{equation}
z_n=z_{n-1}^a\alpha w_0^b\beta^{b(n-1)},\quad
n\in\mathbb{N}.\label{b28}
\end{equation}
By repeated use of \eqref{b28} we obtain
\begin{equation} \label{b14b}
\begin{aligned}
z_n&= \big(z_{n-2}^a\alpha w_0^b\beta^{b(n-2)}\big)^a\alpha w_0^b\beta^{b(n-1)} \\
&= z_{n-2}^{a^2}\alpha^{1+a}w_0^{b+ba}\beta^{b(n-1)+ba(n-2)} \\
&= \big(z_{n-3}^a\alpha w_0^b\beta^{b(n-3)}\big)^{a^2}\alpha^{1+a}w_0^{b+ba}\beta^{b(n-1)+ba(n-2)} \\
&= z_{n-3}^{a^3}\alpha^{1+a+a^2}w_0^{b+ba+ba^2}\beta^{b(n-1)+ba(n-2)+ba^2(n-3)},
\end{aligned}
\end{equation}
for $n\ge 3$.
An inductive argument shows that
\begin{equation}
z_n=z_0^{a^n}\alpha^{\sum_{j=0}^{n-1}a^j} w_0^{b\sum_{j=0}^{n-1}a^j}
\beta^{b\sum_{j=0}^{n-1}a^j(n-1-j)},\label{b29}
\end{equation}
for $n\in\mathbb{N}$.
Hence, from \eqref{b29} we obtain \begin{equation}
z_n=z_0^{a^n}\alpha^{\frac{1-a^n}{1-a}}w_0^{b\frac{1-a^n}{1-a}}
\beta^{b\big((n-1)\frac{1-a^n}{1-a}-a\frac{1-na^{n-1}+(n-1)a^n}{(1-a)^2}\big)},\quad
n\in\mathbb{N}_0,\label{b30}
\end{equation}
if $a\ne 1$, and
\begin{equation}
z_n=z_0\alpha^n w_0^{bn}\beta^{b\frac{n(n-1)}2},\quad n\in\mathbb{N}_0,\label{b31}
\end{equation}
if $a=1$.
From the above considerations we see that the following result
holds.

\begin{theorem} \label{thm4} 
 Consider system \eqref{ms1} with
$a,b,d\in\mathbb{Z}$, $c=0$ and $\alpha,\beta\in\mathbb{C}\setminus\{0\}$. Assume that
$z_0, w_0\in\mathbb{C}\setminus\{0\}$. Then the following statements are
true.
\begin{itemize}

\item[(a)] If $d\ne 1$, $a\ne 1$ and $a\ne d$, then the general
solution to system \eqref{ms1} is given by \eqref{b20} and
\eqref{b26}.

\item[(b)] If $d\ne 1$, $a\ne 1$ and $a=d$, then the general solution
to system \eqref{ms1} is given by \eqref{b20} and \eqref{b27}.

\item[(c)] If $d\ne 1$ and $a=1$, then the general solution to system
\eqref{ms1} is given by \eqref{b20} and \eqref{b27a}.

\item[(d)] If $d=1$ and $a\ne 1$, then the general solution to system
\eqref{ms1} is given by \eqref{b21} and \eqref{b30}.

\item[(e)] If $a=d=1$, then the general solution to system \eqref{ms1}
is given by \eqref{b21} and \eqref{b31}.
\end{itemize}
\end{theorem}



\noindent\textbf{Case $d=0$}. 
Since $d=0$, then the second equation in
\eqref{ms1} becomes 
\begin{equation}
 w_{n+1}=\beta z_n^c,\quad n\in\mathbb{N}_0.\label{b32}
\end{equation}
By substituting \eqref{b32} into the
first equation in \eqref{ms1}, we obtain
\begin{equation}
z_{n+1}=\alpha\beta^b z_n^az_{n-1}^{bc},\quad n\in\mathbb{N}.\label{b33}
\end{equation}
This equation is nothing but \eqref{d3} with $d=0$. Hence, we can
use the formulas for $z_n$ obtained in the proof of Theorem \ref{thm1}
along with \eqref{b32} and get the following result.

\begin{theorem} \label{thm5} 
 Consider system \eqref{ms1} with
$a,b,c\in\mathbb{Z}$, $d=0$ and $\alpha,\beta\in\mathbb{C}\setminus\{0\}$. Assume that
$z_0, w_0\in\mathbb{C}\setminus\{0\}$. Then the following statements are
true.
\begin{itemize}

\item[(a)] If $bc\ne 0$ and $\Delta\ne 0$, then the general solution to
system \eqref{ms1} is 
\begin{align*}
z_n&= z_0^{a\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}
 +bc\frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}}
w_0^{b\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}}
\alpha^{\frac{\lambda_1^n-\lambda_2^n}{\lambda_1-\lambda_2}
 +\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)}} \\
&\quad \times\beta^{b\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)}},
\end{align*} 
\begin{align*}
w_n&= z_0^{ac\frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}
 +bc^2\frac{\lambda_1^{n-2}-\lambda_2^{n-2}}{\lambda_1-\lambda_2}}
w_0^{bc\frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}} \\
&\times\alpha^{c\frac{\lambda_1^{n-1}-\lambda_2^{n-1}}{\lambda_1-\lambda_2}
+c\frac{(\lambda_2-1)(\lambda_1^{n-1}-1)-(\lambda_1-1)(\lambda_2^{n-1}-1)}
 {(\lambda_1-\lambda_2)(\lambda_1-1)(\lambda_2-1)}} \\
&\times\beta^{1+cb\frac{(\lambda_2-1)(\lambda_1^{n-1}-1)-(\lambda_1-1)(\lambda_2^{n-1}-1)}
 {(\lambda_1-\lambda_2)(\lambda_1-1)(\lambda_2-1)}}, 
\end{align*}
for $n\in\mathbb{N}$, where
$$
\lambda_{1,2}=\frac{a+\sqrt{a^2+4bc}}2.
$$

\item[(b)] If $bc\ne 0$ and $\Delta=0$, then the general solution to
system \eqref{ms1} is
\[
z_n= z_0^{an\lambda_1^{n-1}+bc(n-1)\lambda_1^{n-2}}w_0^{bn\lambda_1^{n-1}}
\alpha^{n\lambda_1^{n-1}+\frac{1-n\lambda_1^{n-1}+(n-1)\lambda_1^n}{(1-\lambda_1)^2}}
 \beta^{b\frac{1-n\lambda_1^{n-1}+(n-1)\lambda_1^n}{(1-\lambda_1)^2}}
\]
\begin{align*}
w_n&= z_0^{ac(n-1)\lambda_1^{n-2}+bc^2(n-2)\lambda_1^{n-3}}w_0^{bc(n-1)\lambda_1^{n-2}} \\
&\quad \times\alpha^{c(n-1)\lambda_1^{n-2}+c\frac{1-(n-1)\lambda_1^{n-2}
 +(n-2)\lambda_1^{n-1}}{(1-\lambda_1)^2}}\beta^{1+bc\frac{1-(n-1)\lambda_1^{n-2}
 +(n-2)\lambda_1^{n-1}}{(1-\lambda_1)^2}}
\end{align*}
for $n\in\mathbb{N}$, where $\lambda_1=a/2$.


\item[(c)] If $bc=0$ and $a\ne 1$, then the general solution to system
\eqref{ms1} is 
\begin{gather*}
z_n= z_0^{a^n}w_0^{ba^{n-1}}\alpha^{\frac{1-a^n}{1-a}}\beta^{b\frac{1-a^{n-1}}{1-a}}\\
w_n= z_0^{ca^{n-1}}\alpha^{c\frac{1-a^{n-1}}{1-a}}\beta,
\end{gather*}
for $n\in\mathbb{N}$.

\item[(d)] If $bc=0$ and $a=1$, then the general solution to system
\eqref{ms1} is 
\begin{gather*}
z_n= z_0w_0^b\alpha^{n}\beta^{b(n-1)}\\
w_n= z_0^{c}\alpha^{c(n-1)}\beta,
\end{gather*} for
$n\in\mathbb{N}$.
\end{itemize}
\end{theorem}


\begin{remark} \label{rmk3} \rm
The formulaes obtained  in this article can be
used in describing the long-term behavior of solutions to system
\eqref{ms1} in many cases. The formulations and proofs of the
results we leave to the reader as some exercises.
\end{remark}

\subsection*{Acknowledgements} 
The work is supported  by
the Serbian Ministry of Education and Science projects III 41025
and III44006.

\begin{thebibliography}{00}

\bibitem{al0} M.~Aloqeili;
 Dynamics of a $k$th order rational difference equation, 
\emph{Appl. Math. Comput.} \textbf{181} (2006), 1328-1335.

\bibitem{am2} A.~Andruch-Sobilo, M.~Migda;
 Further properties of the rational recursive sequence 
$x_{n+1}=ax_{n-1}/(b+cx_nx_{n-1})$, \emph{Opuscula Math.} 
\textbf{26} (3) (2006), 387-394.

\bibitem{amc218-sde} L.~Berg, S.~Stevi\' c;
 On some systems of difference equations, \emph{Appl. Math. Comput.} 
\textbf{218} (2011), 1713-1718.


\bibitem{cp} S.~Castillo, M.~Pinto;
Dichotomy and almost automorphic solution of difference system,
 \emph{Electron. J. Qual. Theory Differ. Equ.} Vol. 2013, Article 
No. 32, (2013), 17 pages.

\bibitem{cm} M.~Chv\'atal;
 Non-almost periodic solutions of limit periodic
and almost periodic homogeneous linear difference systems, 
\emph{Electron. J. Qual. Theory Differ. Equ.} Vol. 2014, Article No. 76,
(2014), 1-20.

\bibitem{c} C.~Cinar;
 On the positive solutions of difference equation, \emph{Appl. Math. Comput.}
 \textbf{150} (1) (2004), 21-24.

\bibitem{ll} H.~Levy, F.~Lessman;
\emph{Finite Difference Equations}, Dover Publications, Inc., New York, 1992.

\bibitem{mj} J.~Migda;
 Approximative solutions of difference equations,
\emph{Electron. J. Qual. Theory Differ. Equ.} Vol. 2014, Article
No. 13, (2014), 26 pages.

\bibitem{mk} D.~S.~Mitrinovi\'c, J.~D.~Ke\v cki\'c;
 \emph{Methods for Calculating Finite Sums}, Nau\v cna Knjiga, Beograd, 1984 (in
Serbian).

\bibitem{ps2} G.~Papaschinopoulos, C.~J.~Schinas;
On the behavior of the solutions of a system
of two nonlinear difference equations, \emph{Comm. Appl. Nonlinear
Anal.} \textbf{5} (2) (1998), 47-59.

\bibitem{ps3} G.~Papaschinopoulos, C.~J.~Schinas;
 Invariants for systems of two nonlinear difference
equations, \emph{Differential Equations Dynam. Systems} \textbf{7} (2)
(1999), 181-196.

\bibitem{pst2} G. Papaschinopoulos, G. Stefanidou;
 Asymptotic behavior of the solutions of a class of rational difference equations,
 \emph{Inter. J. Difference Equations} \textbf{5} (2) (2010), 233-249.

\bibitem{ra} V.~R\u{a}dulescu;
 Nonlinear elliptic equations with variable exponent: old and new, 
\emph{Nonlinear Anal.} \textbf{121} (2015), 336-369.

\bibitem{rr1} V.~R\u{a}dulescu, D.~Repov\v s;
\emph{Partial Differential Equations with Variable Exponents: 
Variational Methods and Qualitative Analysis,}
CRC Press, Taylor and Francis Group, Boca Raton FL, 2015.

\bibitem{sps} G.~Stefanidou, G.~Papaschinopoulos, C.~Schinas;
On a system of max difference equations, \emph{Dynam. Contin.
Discrete Impuls. Systems Ser. A} \textbf{14} (6) (2007), 885-903.

\bibitem{ps-cana} G. Stefanidou, G. Papaschinopoulos, C. J. Schinas;
 On a system of two exponential type difference equations, \emph{Commun.
Appl. Nonlinear Anal.} \textbf{17} (2) (2010), 1-13.

\bibitem{na1} S.~Stevi\' c;
 Boundedness character of a class of difference equations, 
\emph{Nonlinear Anal. TMA} \textbf{70} (2009), 839-848.

\bibitem{ejqtde-fib} S.~Stevi\'c;
 Representation of solutions of bilinear difference equations in terms of
generalized Fibonacci sequences, \emph{Electron. J. Qual. Theory
Differ. Equ.} Vol. 2014, Article No. 67, (2014), 15 pages.

\bibitem{ejqtde2015} S.~Stevi\' c;
 Product-type system of difference equations of second-order solvable in 
closed form, \emph{Electron. J. Qual. Theory Differ. Equ.} Vol. 2015, 
Article No. 56, (2015), 16 pages.

\bibitem{ejde2015} S.~Stevi\'c, M.~A.~Alghamdi, A.~Alotaibi, E.~M.~Elsayed;
 Solvable product-type system of difference equations of second order, 
\emph{Electron. J. Differential Equations} Vol. 2015, Article No. 169, (2015),
 20 pages.

\bibitem{ejqtde1} S.~Stevi\'c, M.~A.~Alghamdi, A.~Alotaibi, N.~Shahzad;
On a higher-order system of difference equations, 
\emph{Electron. J. Qual. Theory Differ. Equ.} Vol. 2013, Article No. 47, (2013), 18 pages.

\bibitem{ejqtde-maxsde} S.~Stevi\' c, M.~A.~Alghamdi, A.~Alotaibi, N.~Shahzad;
 Boundedness character of a max-type system of difference equations of second
order, \emph{Electron. J. Qual. Theory Differ. Equ.} Vol. 2014,
Article No. 45, (2014), 12 pages.


\bibitem{508523} S.~Stevi\' c, J.~Diblik, B.~Iri\v canin, Z.~\v Smarda;
 On a third-order system of difference equations with variable
coefficients, \emph{Abstr. Appl. Anal.} Vol. 2012, Article ID
508523, (2012), 22 pages.

\end{thebibliography}

\end{document}