Norm estimates for fundamental solutions of neutral type functional differential equations (Q1930636)

The linear neutral functional differential equation \[ \begin{aligned} y'(t)-\int^{\eta}_0d\tilde{R}(\tau)y'(t-\tau) &=\int^{\eta}_0 dR(\tau)y(t-\tau)\quad(t\geq 0), \\y(t) &= \phi(t) \text{ for } t\in [-\eta, 0]\end{aligned} \] is considered, where \(\phi\in C^1(-\eta, 0)\), \(R\) and \(\tilde{R}\) are \(n\times n\) real matrix-valued functions defined on \([0, \eta]\) with entries having bounded variations, and the integrals are Lebesgue-Stieljes integrals. Estimates for various norms of the fundamental solution are established. As a direct application of these estimates, stability conditions for equations with nonlinear causal mappings are obtained.