Systems of multidimensional discrete inequalities
The paper is devoted to the study of the Gronwall's type discrete inequality \[ u(x)\leq a(x)+\sum^{x-1}_{s=0}K(x,s)u(s), \] where u and a are m-vector functions, K is a \(m\times m\) nonnegative matrix, \(x\in N^ n.\) In the main part of the paper the kernel K is separable, that is \(K(x,s)=G(x)H(s),\) where G,H are \(m\times m\) matrices. The best possible estimation is presented by a discrete analogue of \(m\times m\) matrix Riemann's function. Using the method of splitting or the method of maxima the author gives explicit bounds for the function u(x) or \(\max_{1\leq j\leq m}u_ j(x),\) respectively.
scientific article; zbMATH DE number 4107137
scientific article