Oscillations in systems of difference equations (Q2882527)

The authors study the the following two systems of difference equations: NEWLINE\[NEWLINE\Delta x_i(n)+\sum_{j=1}^{N}p_{ij}(n)x_j(\tau(n))=0, \;\;i=1,2,\dots,N,\;n\geq 0,NEWLINE\]NEWLINE in which \(p_{ij}(n)\) are sequences of real numbers and \(\tau(n)\) is an increasing sequence of integers such that \(\tau(n)\leq n-1\) and \(\lim_{n\rightarrow\infty}\tau(n)=\infty\), and NEWLINE\[NEWLINE\Delta \left(x_i(n)+cx_i(\sigma(n))\right)+\sum_{j=1}^{N}p_{ij}(n)x_j(\tau(n))=0, \;\;i=1,2,\dots,N,\;n\geq 0,NEWLINE\]NEWLINE where \(c\in (0,1)\) and \(\sigma(n)\) is an increasing sequence of integers such that \(\sigma(n)\geq n+1\). These two systems are the discrete analogues of the corresponding differential systems.NEWLINENEWLINEIn Section 2 the authors obtain a sufficient condition for the oscillation of all solutions of the first system above. Then they illustrate the significance of the new result by an example. In Section 3, they achieve a sufficient condition for the oscillation of solutions of the second system and also followed by an example showing the application of their theorem. Researchers who are studying the oscillations of system of difference equations will be interested in this paper.