Some notes on oscillation of two-dimensional system of difference equations. (Q2925969)

The author considers the first-order linear difference system of the formNEWLINENEWLINE\[NEWLINE\Delta u_k=q_kv_k,\quad \Delta v_k=-p_ku_{k+1},\qquad k\in\mathbb{N}.NEWLINE\]NEWLINENEWLINEIt is assumed that \(0<m\leq q_k\leq M<\infty\) for \(k\in\mathbb{N}\) and that \(\sum_{j=1}^\infty p_j\) converges. Conditions are established which guarantee all solutions of the system to be oscillatory. The criteria are in terms of \(\limsup_{k\to\infty}\) and \(\liminf_{k\to\infty}\) of the expressions \(\sum_{j=1}^{k-1}q_j\sum_{j=k}^\infty p_j\) and \(\sum_{j=1}^{k-1}p_j\left(\sum_{i=1}^j q_i\right)^2/\sum_{j=1}^{k-1}q_j\). A key role in the proof is played by a Riccati-type substitution. The results can be seen as a discrete analogue of those of \textit{L. Polák} [Georgian Math. J. 11, No. 1, 137--154 (2004; Zbl 1064.34019)].