The paper concerns the oscillatory behavior of the linear system \[ (-1)^{m+1}\Delta^{m}y_{i}(n)+ \sum_{j=1}^{N}q_{ij}y_{j}(n-\tau_{jj})=0, \quad m\geq 1, \quad i=1,\ldots,N, \tag{1} \] where \(q_{ij}\) are real numbers and \(\tau_{jj}\) are positive integers, \(\Delta\) is the first-order forward difference operator and \(\Delta^{m}y(n)=\Delta^{m-1}(\Delta y(n))\), \(m\geq 2\). A solution \(y(n)=(y_{1}(n),\ldots,y_{N}(n))^{T}\) of (1) is said to be oscillatory if for some \(i\in\{1,\ldots, N\}\) and every integer \(n_{0}>0\) there exists \(n>n_{0}\) such that \(y_{i}(n)y_{i}(n+1)<0\).
The authors present several theorems on sufficient conditions for all bounded solutions to be oscillatory. These results are discrete analogs of some of the known results for the corresponding system of delay differential equations \[ (-1)^{m+1}\frac{d^{m}y_{i}(t)}{dt^{m}}+ \sum_{j=1}^{N}q_{ij}y_{j}(t-\tau_{jj})=0, \quad m\geq 1,\;i=1,2,\dots,N. \]
The same question is investigated for the system of neutral difference equations of the form \[ (-1)^{m+1}\Delta^{m}y_{i}(n)+cy_{i}(n-a\sigma)+ \sum_{j=1}^{N}q_{ij}y_{j}(n-\tau)=0, \quad m\geq 1, \quad i=1,\ldots,N. \]
Remark: In the introduction, a wrong definition of the nonoscillatory solution is given by analogy with the same property of the differential equation solutions.