Square-summable stability in parabolic Volterra difference equations (Q674808)

The authors consider some linear and nonlinear parabolic Volterra difference equations of the forms \[ \begin{aligned} &\Delta_2 \Biggl(u_{m,n}- \sum^\infty_{j=1} q_ju_{m,n-r_j} \Biggr)+ \sum^\infty_{i=1} p_iu_{m,n-k_i}= R\Delta^2_1 u_{m-1,n+1}\\ \text{and} &\Delta_2 \Biggl[h(u_{m,n})- \sum^\infty_{j=1} q_jg(u_{m,n-r_j})\Biggr]+ \sum^\infty_{i=1} p_if(u_{m,n-k_i})= R\Delta^2_1 F(u_{m-1,n+1})\end{aligned} \] for \(m=0,1,\dots,M-1\) and \(n=0,1,\dots\) and obtain several sufficient conditions for the square-summable stability and \(\varphi\)-square summable stability of the zero solution. For related results see the paper of \textit{J. S. Yu} and \textit{S.-S. Cheng} [Appl. Math. Lett. 7, No. 6, 75-80 (1994; Zbl 0812.39004)].