Bounded solutions of a class of linear delay and advanced partial difference equations (Q2753276)

The following partial difference equation NEWLINE\[NEWLINEu_{i,j+1}= p_{i,j}+ \sum^N_{n=1} \alpha^{(n)}_{i,j} u_{i-\sigma,j- \tau_n}+ \sum^M_{m=1} \beta_{i, j}^{(m)}u_{i+ \xi_m,j+\eta_m},\;i,j=1,2,\dots, \tag{1}NEWLINE\]NEWLINE is considered, where \(\sigma_n,\tau_n, \xi_m,\eta_m\) are nonnegative integers. By introducing the (linear) Hilbert space \(l^2_{\mathbb{N} \times\mathbb{N}}\) and the (linear) Banach space \(l_{\mathbb{N}\times \mathbb{N}}\) of double sequences of the form \(\{u_{i,j}\}^\infty_{i,j=1}\) defined in the usual manners, and by introducing the shift operators and their adjoints, the authors claim that the above equation can be written as operator equations. Then the application of standard results in functional analysis leads to stability of the solutions. There, however, seem to be some basic problems. First of all, a solution of (1) is in general not a double sequence of the form \(\{u_{i,j}\}^\infty_{i,j=1}\). Second a left shift operator is in general not well defined. There are other errors in Remark 3.1, Application 1, etc. Therefore, the main stability result Theorem 3.1, if not completely wrong, will need some serious modifications before it can be used. NEWLINENEWLINENEWLINE[Editor's remark: The authors reply that in their opinion the operator is well-defined and that an erratum is in preparation.]