Oscillation criteria for a 2-D discrete system (Q1697723)

This paper is concerned with the oscillatory behavior of the \(2\)-D discrete system \[ u_{m-1,n} + u_{m,n-1} - p u_{m,n} + q u_{m+k,n+l} = 0, \tag{1} \] where \(p, q\) are real numbers and \(m, n, k, l\) are nonnegative integers. The investigation of the problem above is divided into four mutually exclusive cases: (i) \(k \geq 1\) and \(l\geq 1\), (ii) \(k\geq 1\) and \(l = 0\), (iii) \(k = 0\) and \(l\geq 1\), (iv) \(k = 0\) and \(l = 0\). In view of the cases above, the authors prove the following results. Theorem 1. Assume that \(k \geq 1\) and \(l\geq 1\). Then every solution of (1) oscillates if and only if \(p \leq 0\) and \(q\geq 0\) or \(p > 0\) and \[ q> \frac{k^{l}l^{l}}{(k+l+1)^{k+l+1}}p^{k+l+1}. \] Theorem 2. Assume that \(k = 0\) and \(l \geq 1\). Then every solution of (1) oscillates if and only if \(p\leq 0\) and \(q\geq 0\) or \(p > 0\) and \[ q> \frac{l^{l}}{(l+1)^{l+1}}p^{l+1}. \] Theorem 3. Assume that \(k \geq 1\) and \(l = 0\). Then every solution of (1) oscillates if and only if \(p\leq 0\) and \(q\geq 0\) or \(p > 0\) and \[ q> \frac{k^{k}}{(k+1)^{k+1}}p^{k+1}. \] Theorem 4. Assume that \(k = 0\) and \(l = 0\). Then every solution of (1) oscillates if and only if \(q - p \geq 0\).