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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 29, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2009/29\hfil Oscillation criteria]
{Oscillation criteria for first-order systems of linear difference
equations}

\author[A. K. Tripathy\hfil EJDE-2009/29\hfilneg]
{Arun Kumar Tripathy}

\address{Arun Kumar Tripathy \newline
Department of Mathematics \\
Kakatiya Institute of Technology and Science \\
Warangal-506015, India}
\email{arun\_tripathy70@rediffmail.com}

\thanks{Submitted November 29, 2008. Published February 9, 2009.}
\subjclass[2000]{39A10, 39A12}
\keywords{Oscillation; linear system; difference equation}

\begin{abstract}
 In this article, we obtain conditions for the oscillation of
 vector solutions to the first-order systems of linear difference
 equations
 \begin{gather*}
 x(n+1)=a(n)x+b(n)y  \\
 y(n+1)=c(n)x+d(n)y
 \end{gather*}
 and
 \begin{gather*}
 x(n+1)=a(n)x+b(n)y+f_1(n) \\
 y(n+1)=c(n)x+d(n)y+f_2(n)
 \end{gather*}
 where $a(n), b(n), c(n), d(n) $ and $f_i(n),  i=1,  2$ are real valued
 functions defined for $n \geq 0$.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

\section{Introduction}

 Consider the system of $k$-equations of the form
\begin{equation} \label{e1}
X(n+1)=AX(n),
\end{equation}
where $A=(a_{ij})_{k\times k}$
is a constant matrix. The characteristic equation of \eqref{e1} is
given by
\[
\det (\lambda I-A)=0;
\]
that is,
\begin{equation} \label{e2}
\lambda^k+(-1)^k  b_1  \lambda^{k-1} + \dots +(-1)^k  b_k = 0,
\end{equation}
where $b_k = \det A$. If $ k $ is odd, then from the theory of algebraic
equations (see e.g. \cite{b1}), it follows that \eqref{e2} admits at least
one real root $\lambda_1$ such that the sign of $\lambda_1$ is opposite
to that of the last term, namely $(-1)^k  b_k$. Hence we have the
following result.

\begin{theorem} \label{thm1.1}
Let $ k $ be odd. If $\det A < 0$, then \eqref{e1} admits at least one
oscillatory solution; if $\det A > 0$, then \eqref{e1} admits at least
one nonoscillatory solution.
\end{theorem}

\begin{proof}
When $\det A < 0$, we find a real root $\lambda_1$ of \eqref{e2} such
that $\lambda_1 < 0$ and $X(n) =(\lambda_1)^nC$, where
$C=(C_1  C_2 \dots C_k)^T$ is a column vector of constants.
Thus $X(n) $ is oscillatory. Similarly for $\det A > 0$.
\end{proof}

\noindent\textbf{Remark.}
If $\det A = 0$, then \eqref{e1} admits a nonoscillatory solution.
Indeed, $\det A = 0$, implies that $\lambda = 0$ is a solution of
\eqref{e2} and hence $X(n) = C$ is a solution of \eqref{e1},
where $ C $ is a non-zero constant vector. We note that $AC=0 $
always admits a nontrivial solution.

\begin{theorem} \label{thm1.2}
Let $ k $ be even. If $\det A < 0$, then \eqref{e1} admits an oscillatory
solution and a nonoscillatory solution.
\end{theorem}

The proof is simple and can be obtained from the following
Theorem in \cite{b1}.

\begin{theorem}\label{thm0}
\begin{itemize}
\item[(I)] Every equation of an even degree, whose constant term is
negative has at least two real roots one positive and the other negative.

\item[(II)]
If the equation contains only even powers of $x$ and the coefficients
are all of the same sign, then the equation has no real root; that is,
all roots are complex.
\end{itemize}
\end{theorem}

\noindent\textbf{Remarks.}
If the last term of an even degree equation is positive, no definite
conclusion can be drawn regarding the roots of the equation.
If $\det A > 0$, then no definite conclusion can be drawn regarding
the oscillation of solutions of \eqref{e1} when $ k $ is even.


\begin{theorem} \label{thm1.3}
 Let $ k $ be even and $ A $ be such that
$b_1 = b_3 = \dots = b_{k-1} = 0$,  $b_2 > 0$,  $b_4 > 0$ \dots $b_k > 0$.
Then every component of the vector solution of \eqref{e1} is oscillatory.
\end{theorem}

The proof of the above theorem follows from the above Theorem \ref{thm0}(II).

The literature on study of system of difference equations does not conisder
the case when $k $ is even. Therefore the present work is devoted to
study the system of equations
\begin{equation} \label{e3}
\begin{gathered}
x(n+1)=a(n)x+b(n)y \\
y(n+1)=c(n)x+d(n)y
\end{gathered}
\end{equation}
and the corresponding nonhomogeneous system of equations
\begin{equation} \label{e4}
\begin{gathered}
x(n+1)=a(n)x+b(n)y+f_1(n) \\
y(n+1)=c(n)x+d(n)y+f_2(n),
\end{gathered}
\end{equation}
where $a(n),   b(n),   c(n),   d(n), f_1(n),  f_2(n)$ are real-valued
functions defined for $n \geq n_0 \geq 0$. One may think
of systems \eqref{e3} and \eqref{e4} as being a discrete analoge of
the differential systems
\begin{equation} \label{e5}
\begin{gathered}
x'(t)=a(t)x+b(t)y\\
y'(t)=c(t)x+d(t)y
\end{gathered}
\end{equation}
and
\begin{equation} \label{e6}
\begin{gathered}
x'(t)=a(t)x+b(t)y+f_1(t) \\
y'(t)=c(t)x+d(t)y+f_2(t)
\end{gathered}
\end{equation}
respectively, where $a, b,  c,  d,  f_1, f_2 $ are in
$C(\mathbb{R},\mathbb{R})$.
If  $a(n) \equiv a$, $b(n) \equiv b$, $c(n) \equiv c$ and $d(n) \equiv d$,
then the characteristic equation of \eqref{e3} is
\begin{equation} \label{e7}
\lambda^2-(a+c)\lambda+(ad-bc)=0.
\end{equation}
We note that this equation is the same for both the systems \eqref{e3}
and \eqref{e5}. Hence the oscillatory behaviour of solutions of these
systems are comparable. Clearly, the components of the vector solution
of \eqref{e5} are oscillatory only if \eqref{e7} has complex roots.
Otherwise, it is nonoscillatory. On the other hand, the behaviour of the
components of the vector solution of \eqref{e3} is given below.

\begin{proposition} \label{prop1.4}
Let $\lambda_1$ and $\lambda_2$ be the roots of \eqref{e7}. If any one
of the following three conditions
\begin{enumerate}
\item $(a-d)^2 + 4bc < 0$,
\item $(a-d)^2 + 4bc > 0 $ but $ (a+d)\pm [(a-d)^2+4bc]^{\frac{1}{2}}<0$,
\item $(a-d)^2 + 4bc = 0 $ and $ (a+d) < 0$
\end{enumerate}
hold, then every component of the vector solution of \eqref{e3} is
oscillatory. Otherwise, there exists a nonoscillatory solution
to \eqref{e3}.
\end{proposition}

The proof is simple and hence it is omitted.

The object of this work is to establish the sufficient conditions for
the oscillation of all solutions of the systems \eqref{e3} and \eqref{e4}.
Proposition 1.4 which demonstrate the difference in the behaviour of
the solutions of the systems \eqref{e3}-\eqref{e4} and \eqref{e5}-\eqref{e6}
motivate us to study further for the oscillatory behaviour of solutions
of \eqref{e3}-\eqref{e4}. Furthermore, an attempt is made here to
apply some of the results of \cite{p1} for the oscillatory behaviour of solutions
of the systems \eqref{e3} and \eqref{e4}.

A close observation reveals that, all most all works in difference
equations / system of equations are the discrete analogue of the
differential equations / system of equations see for e.g.
\cite{a1,e1,g1} and the references cited therein.
Agarwal and Grace \cite{a1} have discussed the oscillatory behavour of solutions
of the system of equations of the form
\[
(-1)^{m+1} \Delta^m y_i (n)+\sum_{j=1}^{N} q_{ij}  y_j (n-\tau_{jj})
= 0, \quad m \geq 1,\;  i=1, 2, \dots, N
\]
which is the discrete analogue of the functional differential equations
\[
\frac{d^m}{dt^m} y_i (t)+\sum_{j=1}^{N} q_{ij}  y_j (t-\tau_{jj}) = 0, \quad
 m \geq 1,  i=1, 2 ,\dots, N,
\]
where $q_{ij}$ and $\tau_{jj}$ are real numbers and $\tau_{jj} > 0$.
It seems that the results in \cite{a1} are the discrete analog results of
the continuous case. We note that, in this work an investigation
is made to study the system of equations \eqref{e3}/\eqref{e4} without
following any results of the continuous case.

By a solution of \eqref{e3}/\eqref{e4} we mean a real valued
vector function $X(n)$ for $n=0,  1,  2  \dots $ which satisfies
\eqref{e3}/\eqref{e4}. We say that the solution
$X(n) = [x(n), y(n)]^T$ oscillates componentwise or simply oscillates
if each component oscillates. Otherwise, the solution $X(n)$ is called
non-oscillatory. Therefore a solution of \eqref{e3}/\eqref{e4} is
non-oscillatory if it has a component which is eventually positive
or eventually negative.

We need the following two results from \cite{p1} for our use in the sequel.

\begin{theorem} \label{thm1.5}
If $a_n > 0$,  $b_n > 0$ and
\[
a_n \leq \frac{b_{n+1}}{a_{n+1}} + \frac{b_n}{a_{n-1}}
\]
for large $ n $, then
$ y_{n+2} - a_n y_{n+1} +b_n y_n = 0$ is oscillatory.
\end{theorem}

\begin{theorem} \label{thm1.6}
Let $0 \leq a_n \leq 1$ and $c_n \geq 0$. Let $f_n = g_{n+2} - g_{n+1}$,
where for each $n\geq 1$, there exists $m > n$ such that $g_ng_m < 0$. If
\[
\sum_{n=1}^\infty [(1-a_n) g^+_{n+1} + C_n g_n^+ ] = \infty,
\quad
\sum_{n=1}^\infty [(1-a_n) g^-_{n+1} + C_n g_n^- ] = \infty,
\]
then all solutions of
\[
y_{n+2}-a_n y_{n+1} + c_n y_n = f_n
\]
oscillate, where $g^+_n = \max\{g_n,  0\}$ and $g^-_n = \max\{-g_n,  0\}$.
\end{theorem}

\section{Oscillation for System \eqref{e3}}

Consider the system of equations \eqref{e3} as
\[
X(n+1) = A(n) X,
\]
where $X(n) = [x(n),  y(n)]^T$ and
\[
A(n) = \begin{bmatrix}
a(n) & b(n) \\
c(n) & d(n)
\end{bmatrix}.
\]
We assume that $a(n),  b(n),  c(n), d(n) $ are real valued functions
defined for $n \geq n_0 > 0$. Let $b(n) \ne 0$ for all $n \geq n_0$.
Then it follows from \eqref{e3} that
\[
y(n) = \frac{x(n+1)}{b(n)} - \frac{a(n)}{b(n)} x(n);
\]
that is,
\[
y(n+1) = \frac{x(n+2)}{b(n+1)} - \frac{a(n+1)}{b(n+1)} x(n+1).
\]
Hence
\[
c(n)  x(n)+d(n) y(n) = \frac{x(n+2)}{b(n+1)} - \frac{a(n+1)} {b(n+1)} x(n+1);
\]
that is,
\begin{equation} \label{e8}
x(n+2) - P_1(n)  x(n+1)+Q_1(n)  x(n) = 0,
\end{equation}
where we define
\begin{gather*}
P_1(n)=a(n+1)+\frac{d(n) b(n+1)}{b(n)},\\
Q_1(n)=\frac{b(n+1)}{b(n)} [a(n) d(n) - b(n)  c(n)]
\end{gather*}
for all $n \geq n_0$. Similarly, if $c(n) \ne 0$ for all $n \geq n_0$, then
\begin{equation} \label{e9}
y(n+2) - P_2(n)  y(n+1)+Q_2(n)  y(n) = 0,
\end{equation}
where we define
\begin{gather*}
P_2(n)=d(n+1)+\frac{a(n) d(n)}{c(n)},\\
Q_2(n)=\frac{c(n+1)}{c(n)} [a(n) d(n) - b(n)  c(n)]
\end{gather*}

\begin{theorem} \label{thm2.1}
Let $P_i(n) > 0$,   $Q_i(n) > 0$,   $i=1, 2$ be such that
\begin{equation} \label{e10}
P_i(n) \leq \frac{Q_i(n+1)}{P_i(n+1)} + \frac{Q_i(n)}{P_i(n-1)}
\end{equation}
for all large $n$, then every solution $X(n)$ of \eqref{e3} oscillates.
\end{theorem}

\begin{proof}
Suppose, on the contrary, that $X(n)$ is a nonoscillatory solution of
\eqref{e3}. Then there exists $n_0 > 0$ such that at least one component
of $X(n)$ is nonoscillatory for $n \geq n_0$. Let $x(n)$ be the
nonoscillatory component of $X(n)$ such that $x(n)$ is eventually
positive for $n\geq n_0$. Then applying Theorem 1.5, we have a
 contradiction to \eqref{e10}. Similarly, one can proceed for $y(n)$,
if we assume that $y(n)$ is a nonoscillatory component of $X(n)$ for
$n\geq n_0$. Hence the proof is complete.
\end{proof}

\noindent\textbf{Remark.}
If \eqref{e10} holds for either $i=1$ or $i=2$,
then there could be a possibility for the existence of nonoscillatory
solution. However, we are not sure about the fact. We note that \eqref{e8}
and \eqref{e9} are non self-adjoint difference equations.
Hence the above possibility may be true.

\noindent\textbf{Remark.}
If $P_i(n)=p_i$ and $Q_i(n)=q_i,  i=1,   2$ then \eqref{e10} becomes
$p_i^2 \leq 2q_i$,   $i=1,  2$. Hence the inequalities
$p_1^2 \leq 2q_1$ and $p_2^2 \leq 2q_2$ reduce to $(a+d)^2 \leq 2(ad-bc)$.
Thus we have the following corollary.


\begin{corollary} \label{coro2.2}
If $A(n)\equiv A$ and $(\mathop{\rm tr}  A)^2 \leq 2   \det A$,
then \eqref{e3} is oscillatory.
\end{corollary}

\noindent\textbf{Example.} Consider
\begin{equation} \label{e11}
\begin{bmatrix} x(n+1)\\ y(n+1)
\end{bmatrix}
= \begin{bmatrix} 1 &-1 \\ 2 & 1
\end{bmatrix}
\begin{bmatrix} x(n) \\ y(n)
\end{bmatrix}
\end{equation}
Indeed, $\mathop{\rm tr}  A=2$ and $\det A=3$.
$\lambda_1 = 1+i \sqrt{2}$ and $\lambda_2 = 1-i \sqrt{2} $ are two
 characteristic roots of the coefficient matrix $ A $. Clearly,
\begin{align*}
x(n) &=  \lambda_1^n \begin{bmatrix} 1 \\ i\sqrt{2} \end{bmatrix} \\
&=  (1+i\sqrt{2})^n \begin{bmatrix} 1\\ i\sqrt{2} \end{bmatrix} \\
&=  3^{n/2} (\cos n \theta + i \sin n\theta)
\begin{bmatrix} 1\\ i\sqrt{2} \end{bmatrix}\\
&= \begin{bmatrix}  3^{n/2}(\cos n\theta + i \sin n\theta \\
 -3^{n/2} (\sin n\theta - i \cos n\theta) \end{bmatrix}
\end{align*}
and
\begin{align*}
y(n) &=  \lambda_2^n \begin{bmatrix} 1\\ -i\sqrt{2} \end{bmatrix} \\
&=  (1-i\sqrt{2})^n \begin{bmatrix} 1\\-i\sqrt{2} \end{bmatrix} \\
&=  3^{n/2} (\cos n \theta - i \sin n\theta) \begin{bmatrix} 1\\
-i\sqrt{2} \end{bmatrix} \\
&= \begin{bmatrix}  3^{n/2}(\cos n\theta + i \sin n\theta \\
3^{n/2} (\sin n\theta - i \cos n\theta) \end{bmatrix},
\end{align*}
where $\theta = \tan^{-1} (\sqrt{2})$. By  Corollary 2.2, the
system \eqref{e11} is oscillatory.

If we define $a(n)=\frac{r(n)}{r(n+1)}$ and
$d(n)= \frac{t(n)} {t(n+1)}$, then $r(n+1)=\frac{r(n)}{a(n)}$ and
$t(n+1)= \frac{t(n)} {d(n)}$ and hence solving the two relations we get
\[
r(n)=\frac{r(0)}{\prod _{i=0}^{n-1} a(i)},\quad
t(n)=\frac{d(0)}{\prod _{j=0}^{n-1} d(j)} ,
\]
where $r(0)$ and $d(0)$ are non-zero constants if $a(n) \ne 0 \ne d(n)$
for $n\geq n_0 > 0$. From \eqref{e3} it follows that
\[
r(n+1)  x(n+1) - r(n)  x(n) = b(n)   r(n+1)   y(n);
\]
that is,
\[
\Delta (r(n)  x(n)) = b(n)   r(n+1)   y(n).
\]
Consequently,
\[
\sum_{s=0}^{n-1}\Delta [r(s) x(s)]=\sum_{s=0}^{n-1} b(s)  r(s+1)  y(s);
\]
that is,
\begin{align*}
x(n) &= \frac{r(0) x(0)}{r(n)}+\frac{1}{r(n)} \sum_{s=0}^{n-1} b(s)  r(s+1)
 y(s) \\
&=  \prod_{i=0}^{n-1} a(i) \Big[ x(0)+ \sum_{s=0}^{n-1} \frac{b(s)
y(s)}{\prod_{i=0}^{s} a(i)} \Big].
\end{align*}
Similarly,
\[
y(n)=\prod_{j=0}^{n-1}d(j)\Big[y(0)+ \sum_{s=0}^{n-1} \frac{c(s)
 x(s)}{\prod_{j=0}^{s} d(j)} \Big].
\]
Hence or otherwise the following theorem holds


\begin{theorem} \label{thm2.3}
 Let $A(n)$ be a real valued coefficient matrix such that
$a(n) \ne 0 \ne d(n)$ for $n \geq n_0 > 0$. Then \eqref{e3} is either
oscillatory or nonoscillatory.
\end{theorem}

\begin{theorem} \label{thm2.4}
Suppose that $a(n) = 0 = d(n)$ and $c(n)\ne 0 \ne b(n)$ for all
$n\geq n_0 > 0$. If $\liminf_{n\to\infty} b(n)=\alpha \ne 0$ and
$\liminf_{n\to\infty} c(n)=\beta \ne 0$ such that $\alpha   \beta < 0$,
then \eqref{e3} is oscillatory.
\end{theorem}

\begin{proof} Let $X(n)$ be a nonoscillatory solution of \eqref{e3}
for $n\geq n_0$. Let $x(n)$ be a component of $X(n)$ such that $x(n)$
is eventually positive for $n\geq n_0$. Clearly, from \eqref{e3} we
obtain that, $x(n)$ is a solution of
\begin{equation} \label{e12}
z(n+2) -b(n+1)  c(n)  z(n)=0 .
\end{equation}
Without any loss of generality, we may assume that $z(n) >0 $ for
$n\geq n_0$. Equation \eqref{e12} can be written as
\[
\frac{z(n+2)}{z(n+1)} \frac{z(n+1)}{z(n)} = b(n+1) c(n)
\]
for $n\geq n_0$. If we denote $u(n)=\frac{z(n+1)}{z(n)} > 0$ for
$n\geq n_1$, then the above equation yields
\begin{equation} \label{e13}
\begin{aligned}
\liminf_{n\to\infty} [u(n+1)  u(n)]
&=  \liminf_{n\to\infty} [b(n+1)  c(n)]   \\
&=  [\liminf_{n\to\infty} b(n+1)]   [ \liminf_{n\to\infty} c(n)]
=  \alpha   \beta .
\end{aligned}
\end{equation}
Since $\alpha   \beta \ne 0$, then $\liminf_{n\to\infty} [u(n)  u(n+1)]$
exists. Let $\lambda = \liminf_{n\to\infty} u(n)$. From \eqref{e13},
it follows that $f(\lambda) = \lambda^2 - \alpha   \beta = 0$.
It is easy to see that $f(\lambda)$ attains minimum at $\lambda = 0$.
Consequently, $\min f(\lambda) \leq f(\lambda)$ implies that
$\alpha   \beta \geq 0$, a contradiction. Hence \eqref{e12} is oscillatory.
Similarly, we can show that $y(n)$ is a solution of
\begin{equation} \label{e14}
w(n+2) - b(n) c(n+1) w(n) = 0,
\end{equation}
and \eqref{e14} is oscillatory. This completes the proof.
\end{proof}

\noindent\textbf{Example}
Consider the system of equations
\begin{equation} \label{e15}
\begin{bmatrix} x(n+1)\\ y(n+1)
\end{bmatrix} = \begin{bmatrix} 0 & -2+(-1)^n \\ 2+(-1)^n & 0
\end{bmatrix}
\begin{bmatrix} x(n) \\ y(n)
\end{bmatrix},\quad  n \geq 0.
\end{equation}
Indeed,
\begin{equation} \label{e16}
y(n+2)+(5-4(-1)^n)  y(n) = 0, \quad n \geq 0
\end{equation}
and $\alpha =-3$,  $\beta =1$,  $\alpha \beta=-3 < 0$. From Theorem 2.4,
it follows that \eqref{e15} is oscillatory. We note that
\[
y(n)=y(0)(-1)^{n/2} \prod_{i=0}^{n-2} [5-4(-1)^i]
\]
is one of the solution of \eqref{e16}, where $(n/2)$ is an odd positive
integer.

We conclude this section with the following result.

\begin{theorem} \label{thm2.5}
Let $X(n_0) \in R\times R$ for $n_0 \in Z^+$. If $\det A(n) \ne 0$,
then \eqref{e3} is oscillatory if and only if every component of the
matrix $\prod_{i=n_0}^{n-1} A(i)$ is oscillatory, where
\[
\prod_{i=n_0}^{n-1} A(i)=
\begin{cases}
A(n-1)   A(n-2) \dots A(n_0) & n > n_0 \\
I & n = n_0.
\end{cases}
\]
\end{theorem}

The proof of the above theorem follows from the proof of the
\cite[Theorem 3.3]{e1} and hence it is omitted.

\noindent\textbf{Remark,} If \eqref{e3} is an autonomous system,
then $\prod_{i=n_0}^{n-1} A(i)=A^{n-n_0}$ and Theorem 2.5 holds
for $A^{n-n_0}$ for all $n> n_0$.

\section{Oscillation for System \eqref{e4}}

 This section presents  sufficient conditions for the oscillation
of all solutions of the system of equations \eqref{e4}.
If we assume that $b(n)\ne0$ for all $n\geq n_0$, then
\[
y(n) = \frac{x(n+1)}{b(n)} -\frac{a(n)}{b(n)} x(n)
 - \frac{f_1(n)} {b(n)};
\]
that is,
\[
y(n+1) = \frac{x(n+2)}{b(n+1)} -\frac{a(n+1)}{b(n+1)} x(n+1)
- \frac{f_1(n+1)} {b(n+1)}.
\]
Consequently,
\[
c(n)x(n)+d(n)y(n)+f_2(n)=y(n+1)
\]
implies that
\begin{equation} \label{e17}
x(n+2)-P_1(n) x(n+1)+Q_1(n)  x(n)=G_1(n),
\end{equation}
where $G_1(n)=f_2(n)+\frac{f_1(n+1)} {b(n+1)}$, for $n\geq n_0$
and $P_1(n), Q_1(n)$ are same as in \eqref{e8}. Similarly, if we assume
that $c(n) \ne 0$ for all $n \geq n_0$, then we find
\begin{equation} \label{e18}
y(n+2) - P_1(n) y(n+1) +Q_2(n)  y(n) = G_2(n),
\end{equation}
where $P_2(n)$ and $Q_2(n)$ are same as in \eqref{e9} and
$G_2(n) = f_1(n) +\frac{f_2(n+1)} {c(n+1)}$. We note that $G_i(n)$ could
be oscillatory or could be nonoscillatory for $i=1, 2$.

\begin{theorem} \label{thm3.1}
Let $P_i(n) < 0, Q_i(n) > 0$ for $n\geq n_0$ and $i=1,  2$.
Assume that $G_i(n)$ changes sign. In addition, there exists $g_i(n)$
which changes sign such that $G_i(n) = g_i(n+2) -g_i(n+1),  i=1,  2$. If
\begin{gather} \label{e19}
\sum_{n=0}^{\infty} [Q_i(n)  g_i^+(n)-P_i(n)g_i^+(n+1)] = \infty ,\\
\label{e20}
\sum_{n=0}^{\infty} [Q_i(n)  g_i^-(n)-P_i(n)g_i^-(n+1) ] = \infty
\end{gather}
hold, then \eqref{e4} is oscillatory, where
\[
g_i^+(n)=\max\{g_i(n), 0\}   and   g_i^-(n)=\max\{0,  -g_i(n)\}
\]
\end{theorem}

\begin{proof}
Suppose on the contrary that $X(n) = [x(n), y(n)]^T$ is a nonoscillatory
solution of \eqref{e4}. Then there exists $n_0 > 0$ such that at least
one component of $X(n)$ is nonoscillatory for $n\geq n_0$. Let $x(n) $
be the nonoscillatory component of $X(n)$ such that $x(n) > 0 $ for
$n\geq n_0$. Consequently, $x_1(n)$ and $x_2(n)$ are two solutions
of \eqref{e17}. Applying Theorem 1.6, we obtain a contradiction
to our hypothesis \eqref{e19}. A contradiction can be obtained
to \eqref{e20} if we assume that $x(n) < 0$ eventually for $n\geq n_0$.
Similar observations can be dealt with the solution $y(n)$ if we assume
that $y(n)$ is a nonoscillatory component of $X(n)$ for $n\geq n_0$.
 Hence or otherwise the proof of the theorem is complete.
\end{proof}

\begin{theorem} \label{thm3.2}
Let $0 \leq P_i(n) <1$,  $Q_i(n) > 0$ and $G_i(n)$ changes sign for
$i=1,2$. Assume that there exists $g_i(n)$ which changes sign such
that $G_i(n)=g_i(n+2) -g_i(n+1)$,   $i=1,2$. If
\begin{gather*}
\sum_{n=0}^{\infty} [Q_i(n)  g_i^+(n)+(1-P_i(n)) g_i^+(n+1)] = \infty,\\
\sum_{n=0}^{\infty} [Q_i(n)  g_i^-(n)+(1-P_i(n)) g_i^-(n+1)] = \infty
\end{gather*}
hold, then \eqref{e4} is oscillatory, where $g_i^+(n)$ and $g_i^-(n)$
are same as in Theorem 3.1.
\end{theorem}

The proof of the above theorem follows from the Theorem 3.1 and Theorem 1.6
and hence it is omitted.

\begin{theorem} \label{thm3.3}
Let $P_i(n)<0$ and $Q_i(n)>0$ for all $n\geq n_0$ and $i=1, 2$.
 Assume that $G_i(n)$ is nonoscillatory for all large $n$. Furthermore,
assume that there exists $g_i(n)$ such that $G_i(n) = g_i(n+2) - g_i(n+1)$
and $0< \lim_{n\to\infty} |g_i(n)| < \infty$. If
\begin{gather} \label{e21}
\sum_{n=0}^{\infty}[Q_i(n) g_i(n)-P_i(n) g_i(n+1)]=+ \infty,\\
 \label{e22}
\sum_{n=0}^{\infty} [Q_i(n)-P_i(n)] =+ \infty
\end{gather}
hold, then \eqref{e4} is oscillatory.
\end{theorem}

\begin{proof}
Suppose on the contrary that $X(n) = [x(n),  y(n)]^T$ is a nonoscillatory
solution of \eqref{e4}. Proceeding as in the proof of the Theorem 3.1,
we may assume that $x(n)$ and $y(n)$ are nonosillatory solutions
of \eqref{e17} and \eqref{e18} respectively. Assume that there exists
$n_0>0$ such that $x(n) >0$ for $n\geq n_0$. Then from \eqref{e17},
it follows that
\begin{equation} \label{e23}
\Delta[x(n+1)-g_1(n+1)] = [P_1(n)-1]x(n+1)-Q_1(n)x(n) \leq 0
\end{equation}
but not identically zero for $n\geq n_0$. Ultimately,
$(x(n+1)- g_i(n+1))$ is nonincreasing on $[n_0, \infty)$. We consider two
cases viz. $g_1(n) > 0$ and $g_1(n) < 0$ for $n\geq n_0$.
Suppose the former holds. If
$(x(n+1) - g_1(n+1)) > 0$ for $n\geq n_1 > n_0$, then
$\lim_{n\to\infty} (x(n+1) -g_1(n+1))$ exists and hence \eqref{e23} becomes
\[
\sum_{n=n_1}^{\infty} [Q_1(n)  g_1(n)-(P_1(n)) g_1(n+1)] < \infty ,
\]
a contradiction to \eqref{e21}. Thus $(x(n+1) - g_1(n+1)) < 0$ for
$n\geq n_1$. Consequently, $x(n) < 0$ for large $n$, a contradiction.
Let the latter hold. Ultimately, $x(n+1)-g_1(n+1) > 0$ for $n\geq n_1$.
It is easy to verify that $0<\lim_{n\to\infty} x(n+1) < \infty$.
Let $\lim_{n\to\infty} x(n) =\ell $,  $\ell \in (0, \infty)$.
For every $\epsilon > 0$, there exists $n^*>0$ such that
$x(n+1) > \ell -\epsilon > 0$ for $n\geq n^*$.

Hence summing \eqref{e23} from $n_2$ to $\infty$, we get
\[
\sum_{n=n_2}^{\infty} [Q_1(n)  P_1(n)]< \infty , \quad
  n_2 > \max \{n_1,  n^*\},
\]
a contradiction to our assumption \eqref{e22}. Same type of reasoning
can be made if we assume $x(n) < 0$ for $n\geq n_0$.
A similar type of observation can be formulated when $y(n)$ is a
non-oscillatory component of \eqref{e4} for $n\geq n_0$.
This completes the proof.
\end{proof}

\noindent\textbf{Remark.}
Without any loss of generality, we may assume that $g_i(n) > 0$ for $i=1,2$.

\begin{theorem} \label{thm3.4}
Let $0\leq P_i(n)<1$ and $Q_i(n) > 0$ for large $n$. Assume that all
the conditions of Theorem 3.3 hold except \eqref{e21} and \eqref{e22}. If
\begin{gather*}
\sum_{n=0}^{\infty} [Q_i(n)  g_i(n)+(1-P_i(n)) g_i(n+1) ] = \infty,\\
\sum_{n=0}^{\infty}[Q_i(n)  +(1-P_i(n))]=\infty ,
\end{gather*}
hold, then \eqref{e4} is oscillatory.
\end{theorem}

The proof of the above theorem follows from the proof of the Theorem 3.3.

\noindent\textbf{Example} Consider
\[
\begin{bmatrix} x(n+1)\\ y(n+1)
\end{bmatrix} = \begin{bmatrix} 0 & -1 \\ 1/2 & 0
\end{bmatrix}
\begin{bmatrix} x(n) \\ y(n)
\end{bmatrix} +
\begin{bmatrix} (-1)^n \\ (-1)^n
\end{bmatrix}, \quad n \geq 0.
\]
Clearly, $P_1(n)=0=P_2(n)$, $Q_1(n) = \frac{1}{2} = Q_2(n)$,
$ G_1(n) = 2(-1)^n$ and $G_2(n) = (-1)^{n+1}$. Indeed, $x(n)$ and
$y(n)$ are two solutions of
\begin{gather} \label{e24}
z(n+2) + \frac{1}{2}z(n)=2(-1)^n, \\
 \label{e25}
w(n+2) + \frac{1}{2}w(n)=(-1)^{n+1}
\end{gather}
respectively. If we choose $g_1(n)=(-1)^n$ and
$g_2(n)=\frac{1}{2} (-1)^{n+1}$, then $G_1(n) = 2(-1)^n$ and
$G_2(n)=(-1)^{n+1}$ for all $n \geq 0$.

It follows that all the conditions of Theorem 3.2 are satisfied and
hence the given system of equations is oscillatory.
In particular, $x(n)=\frac{4}{3}(-1)^n$ is a solution of \eqref{e24}
and $y(n)=\frac{2} {3}(-1)^{n+1}$ is a solution of \eqref{e25}.


\noindent\textbf{Example.}
Consider
\[
\begin{bmatrix} x(n+1)\\ y(n+1)
\end{bmatrix} = \begin{bmatrix} -1 & 1 \\ 1 & -2
\end{bmatrix}
\begin{bmatrix} x(n) \\ y(n)
\end{bmatrix} +
\begin{bmatrix} 1-2(-1)^n \\ 1-2(-1)^n
\end{bmatrix}, \quad  n \geq 0.
\]
where $P_1(n)=-3=P_2(n)$,   $Q_1(n)=1=Q_2(n)$, $G_1(n)=2=G_2(n)$.
Clearly, $x(n)$ and $y(n)$ are solutions of
\begin{gather} \label{e26}
z(n+2)+3z(n+1)+z(n)=2, \\
 \label{e27}
w(n+2)+3w(n+1)+w(n)=2
\end{gather}
respectively. If we choose $g_1(n)=2(n-1)=g_2(n)$, then
$G_1(n)=2=G_2(n)$ and hence \eqref{e21} and \eqref{e22} hold good.
But we can not apply the Theorem 3.3 due to the fact that
\[
\liminf_{n\to\infty} g_1(n)=\limsup_{n\to\infty} g_1(n)=\infty .
\]
Then $x(n)=\frac{2}{5}+\big(\frac{3+\sqrt{5}}{2}\big)^n(-1)^n $
is a solution of \eqref{e26} and
$y(n)=\frac{2}{5}+\big(\frac{3+\sqrt{5}}{2}\big)^n(-1)^n $
is a solution of \eqref{e27}. We note that the given system of
equations is oscillatory.

\noindent\textbf{Remark.}
In view of the above example, it seems that some additional condition
is necessary to prove the Theorem 3.3 when
$\lim_{n\to\infty} |g_i(n)| =\infty$.

Let $a(n) = 0= d(n)$ for all $n\geq n_0 \geq 0$. Then the system
of equations \eqref{e4} becomes
\begin{gather*}
x(n+1) = b(n)  y(n) + f_1(n),\\
y(n+1) = c(n)  x(n) + f_2(n)
\end{gather*}
Solving the above two equations, it follows that $x(n)$ and $y(n)$
are solutions of
\begin{gather} \label{e28}
z(n+2) - c(n) b(n+1)  z(n) = E_1(n),\\
\label{e29}
w(n+2) -c(n+1)   b(n)  w(n) = E_2(n)
\end{gather}
respectively, where $E_1(n)=f_1(n+1)+f_2(n)  b(n+1)$,
$E_2(n)=f_2(n+1)+f_1(n)  c(n+1)$ and we assume that
$\det A(n) \ne 0$ for all $n\geq n_0 \geq 0$.

\begin{theorem} \label{thm3.5}
Assume that $c(n)  b(n+1) < 0$ for all large $n$. If there exists
$e_i(n)$, $i=1, 2$ which changes sign such that $E_i(n) = \Delta   e_i(n+1)$
and
\begin{gather*}
\sum_{n=0}^{\infty} [c(n) b(n+1)  e^+_1(n) -e^+_1 (n+1)] = - \infty,\\
\sum_{n=0}^{\infty} [b(n) c(n+1)  e^+_2(n) -e^+_2 (n+1)] = - \infty
\end{gather*}
where $e^+_i(n)=\max\{e_i(n),  0\}$,
$e^-_i(n)=\max\{-e_i(n),  0\}$, then \eqref{e4} is oscillatory.
\end{theorem}

It is easy to verify that, \eqref{e28} and \eqref{e29} can be written as
\begin{gather} \label{e30}
\Delta [z(n+1)-e_1(n+1)]=c(n) b(n+1) z(n)-z(n+1),\\
 \label{e31}
\Delta [w(n+1)-e_2(n+1)]=c(n+1) b(n) w(n)-w(n+1)
\end{gather}
respectively. To prove this theorem it is sufficient to prove that
\eqref{e30} and \eqref{e31} are oscillatory. Moreover, the proof
of the theorem can be done as in Theorems 3.1 and  1.6.


\noindent\textbf{Remark.}
$E_i(n)$ could be nonoscillatory also. If $e_i(n)$ is nonoscillatory
such that $E_i(n)=\Delta e_i(n+1)$, then a result corresponding to the
Theorem 3.3 can be formulated under the conditions
$0<\lim_{n\to\infty} |e_i(n)|<\infty$ and $c(n)  b(n+1) < 0$ for all
large $n$.

\subsection*{Concluding Remarks}
In this work, specific results regarding the oscillatory behaviour
of vector solutions of the systems \eqref{e3} and \eqref{e4} have
been  established under the criteria $\det A(n) \ne 0$ subject to
the fundamental matrix $\Phi (n)   (\det \Phi (n) \ne 0)$. Indeed,
the discrete analog of a second order differential equation is
not necessarily a self adjoint difference equation. Since the work
in \cite{p1} based on the oscillatory behaviour of solutions of a
non-self adjoint difference equation and the author has followed
the work of \cite{p1}, then it follows that the present work is not the
analog work of continuous case. Hence the  results developed here
may initiate further study for the system of equations
\eqref{e3}/\eqref{e4}.

Existence of nonoscillatory vector solution of
\eqref{e3}/\eqref{e4} is not discussed in this work. However, the
same can be followed  from \cite{e1} and \cite{g1}.

It is interesting to apply this work to study the system of equations
\[
X(n+1) = A(n)  h(X(n)) \] and
\[
 X(n+1) = A(n)  h(X(n))+F(n),
\]
where $h \in C(R,  R)$.

\subsection*{Acknowledgements}
The author is thankful to the anonymous referee for their
helpful suggestions and remarks.

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\bibitem{e1} S. N. Elaydi;
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\bibitem{g1} I. Gyori, G. Ladas;
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\bibitem{k1} W. G. Kelley, A. C. Peterson;
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\end{thebibliography}
\end{document}