One method for the investigation of linear functional-differential equations (Q2440072)

Consider linear functional-differential equations with delay of the form \[ \dot{x}(t)=ax(t-1)+bx(t/q)+\bar{f}(t),\quad q>1 \] and the initial condition \(x(0)=x_0\). Here, \(a\) and \(b\) are constants and \(\bar{f}=\sum_{1}^F \bar{f}_nt^n\). It is shown that if some matrix \(M_N\) (constructed from the constants \(a\), \(b\), \(q\) and binomial coefficients) is non-singular then the delay equation with initial condition has a unique polynomial quasisolution of the form \[ x(t)=\sum_0^N x_nt^n \text{ for } N=F+1. \] If \(a+bq^{-k}=0\) for some \(k<N\) and some matrices \(M_N\) and \(M_{k_N}\) are non-singular then the initial-value problem for the delay equation for all \(x_0\neq0\) has a unique polynomial solution of the form \(x(t)=\sum_0^k x_nt^n\).