Subexponential solutions of two partial difference equations with delays (Q1362816)

The double sequence \(\{u_{i,j}\}\) is called subexponential or exponentially bounded if \(|u_{i,j}|\leq \Gamma\beta^i \gamma^j\), \((i,j)\in \Omega^+:= \{(i,j)\mid i,j=0,1,2,\dots\}\) for some positive \(\Gamma\), \(\beta\) and \(\gamma\). It is well known that every solution of a linear difference equation with constant coefficients is exponentially bounded. This is not true for partial difference equations. The main result of the paper is the following Theorem 1. Suppose that \(|p_{i,j}|\leq p\), for \((i,j)\in \Omega^+\) and that the sequence \(\{\varphi_{ij}\}\) and \(\{\psi_{ij}\}\) are subexponential such that \(|\varphi_{ij}|\leq M_i\alpha^i\), if \(i\geq 0\), \(-\tau\leq j\leq 0\), \(|\psi_{ij}|\leq M_2\beta^j\), if \(-\sigma\leq i\leq-1\), \(j\geq-\tau\). Then the solution of the partial difference equation \[ u_{i,j+1}=- \{au_{i+1,j}+ cu_{ij}+ p_{ij} u_{i-\sigma,j- \tau}\}, \qquad (i,j)\in \Omega^+ \] satisfying some initial conditions is subexponential.