\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 $$\label{e1} X(n+1)=AX(n),$$ 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, $$\label{e2} \lambda^k+(-1)^k b_1 \lambda^{k-1} + \dots +(-1)^k b_k = 0,$$ 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 $$\label{e3} \begin{gathered} x(n+1)=a(n)x+b(n)y \\ y(n+1)=c(n)x+d(n)y \end{gathered}$$ and the corresponding nonhomogeneous system of equations $$\label{e4} \begin{gathered} x(n+1)=a(n)x+b(n)y+f_1(n) \\ y(n+1)=c(n)x+d(n)y+f_2(n), \end{gathered}$$ 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 $$\label{e5} \begin{gathered} x'(t)=a(t)x+b(t)y\\ y'(t)=c(t)x+d(t)y \end{gathered}$$ and $$\label{e6} \begin{gathered} x'(t)=a(t)x+b(t)y+f_1(t) \\ y'(t)=c(t)x+d(t)y+f_2(t) \end{gathered}$$ 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 $$\label{e7} \lambda^2-(a+c)\lambda+(ad-bc)=0.$$ 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, $$\label{e8} x(n+2) - P_1(n) x(n+1)+Q_1(n) x(n) = 0,$$ 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 $$\label{e9} y(n+2) - P_2(n) y(n+1)+Q_2(n) y(n) = 0,$$ 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 $$\label{e10} P_i(n) \leq \frac{Q_i(n+1)}{P_i(n+1)} + \frac{Q_i(n)}{P_i(n-1)}$$ 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 $$\label{e11} \begin{bmatrix} x(n+1)\\ y(n+1) \end{bmatrix} = \begin{bmatrix} 1 &-1 \\ 2 & 1 \end{bmatrix} \begin{bmatrix} x(n) \\ y(n) \end{bmatrix}$$ 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 $$\label{e12} z(n+2) -b(n+1) c(n) z(n)=0 .$$ 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 \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} 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 $$\label{e14} w(n+2) - b(n) c(n+1) w(n) = 0,$$ and \eqref{e14} is oscillatory. This completes the proof. \end{proof} \noindent\textbf{Example} Consider the system of equations $$\label{e15} \begin{bmatrix} x(n+1)\\ y(n+1) \end{bmatrix} = \begin{bmatrix} 0 & -2+(-1)^n \\ 2+(-1)^n & 0 \end{bmatrix} \begin{bmatrix} x(n) \\ y(n) \end{bmatrix},\quad n \geq 0.$$ Indeed, $$\label{e16} y(n+2)+(5-4(-1)^n) y(n) = 0, \quad n \geq 0$$ 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 $$\label{e17} x(n+2)-P_1(n) x(n+1)+Q_1(n) x(n)=G_1(n),$$ 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 $$\label{e18} y(n+2) - P_1(n) y(n+1) +Q_2(n) y(n) = G_2(n),$$ 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 $$\label{e23} \Delta[x(n+1)-g_1(n+1)] = [P_1(n)-1]x(n+1)-Q_1(n)x(n) \leq 0$$ 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. \begin{thebibliography}{0} \bibitem{a1} R. P. Agarwal, S. R. Grace; \emph{The oscillation of systems of difference equations}, Appl. Math. Lett. 13 (2000), 1-7. \bibitem{b1} W. S. Burnside, A. W. Panton; \emph{The Theory of Equations}, S. Chand and Company Ltd., New Delhi, 1979. \bibitem{e1} S. N. Elaydi; \emph{An Introduction to Difference Equations}, Springer - Verlag, New York, 1996. \bibitem{g1} I. Gyori, G. Ladas; \emph{Oscillation Theory of Delay Differential Equations with Applications}, Clarendon Press, Oxford, 1991. \bibitem{k1} W. G. Kelley, A. C. Peterson; \emph{Difference Equations: An Introduction with Applications}, Academic Press, INC, New York, 1991. \bibitem{p1} N. Parhi, A. K. Tripathy; \emph{Oscillatory behaviour of second order difference equations}, Commu. Appl. Nonlin. Anal. 6(1999), 79 - 100. \end{thebibliography} \end{document}