Necessary and sufficient conditions for oscillations of linear delay partial difference equations (Q1303517)

The authors present oscillation results for the linear partial difference equation \[ A_{m,n}=\sum_{i=1}^up_iA_{m-k_i,n-l_i}+ \sum_{j=1}^vq_jA_{m+\tau_j,n+\sigma_j}, \tag{1} \] where \(p_i, q_j\) are \(r\times r\) matrices, \(A_{m,n}=(a_{m,n}^1,a_{m,n}^2,\dots ,a_{m,n}^r)^T\), \(k_i,l_i,\tau_j,\sigma_j\in\mathbb N_{0}\), \(i=1,2,\dots u\), \(j=1,2,\dots v\) and \(u,v\) are positive integers. The main result yields a sufficient and necessary condition for all proper solutions of (1) to be componentwise oscillatory. This condition is that the characteristic equation \[ \text{ det} \Big (\sum_{i=1}^up_i\lambda^{-k_i}\mu^{-l_i}-I+ \sum_{j=1}^vq_j\lambda^{\tau_j}\mu^{\sigma_j}\Big)=0 \] admits no positive roots. Considering the scalar linear difference equation \[ a_{m,n}=\sum_{i=1}^up_ia_{m-k_i,n-l_i}+ \sum_{j=1}^vq_ja_{m+\tau_j, n+\sigma_j} , \tag{2} \] the corresponding characteristic equation becomes \[ 1=\sum_{i=1}^up_i\lambda^{-k_i}\mu^{-l_i}+ \sum_{j=1}^vq_j\lambda^{\tau_j}\mu^{\sigma_j} . \tag{3} \] Then the main result applied to some special equations (2) enables to formulate oscillation criteria in terms of the coefficients of (3) and thus generalizes some previous results of the authors.