Large time behavior of a linear difference equation with rationally non-related delays (Q1936222)

The author studies the equation with discrete delays \[ x\left( t\right) =\sum_{j=1}^{N}a_{j}x\left( t-r_{j}\right),~ t\geq 0, \] where \(a_{j}>0,\) \(1\leq j\leq N\) and \(0<r_{1}<\dots<r_{N}.\) The case of rationally non-related delays is considered. Under some specific hypotheses, he finds the limit as \(t\rightarrow\infty\) of \(x\left( t\right) e^{-\lambda t}\), where \(\lambda\) is the unique real root of the characteristic function \(h\) defined through \[ h\left( s\right) =0, ~h\left( s\right) =1- \sum_{j=1}^{N}a_{j}e^{-sr_{j}}. \] If the sum of the coefficients in the above equation is one, then he deduces a result on asymptotic constancy.