Stability of linear mixed functional-differential equations with commensurable deviations of the space argument (Q1873539)

The paper presents a linear spatially homogeneous mixed functional-differential equation (MFDE) of the form \[ \dot u(x,t)=\sum_{j}a_j(t)u(x+h_j,t), \quad -\infty<x<+\infty, \;0\leq t<+\infty,\tag \(*\) \] where \(h_j\) is an integer and the index \(j\) runs over a finite set of values. The author discusses the continuous and the discrete case depending on whether \(x\) is a continuous or a discrete variable, respectively. In both cases, sufficient and necessary conditions are provided for the quadratic stability of the solutions. These are done in the first five sections, while in the last section similar stability results are given for a multi-dimensional MFDE of the form \((*)\).