The generalized Bohl-Perron principle for the neutral type vector functional differential equations (Q2376960)

Consider the linear neutral functional differential equation \[ [y(t)-(E_1y)(t)]' = (E_0y)(t), \] where \(E_0\) and \(E_1\) are vector-valued functionals of some special form, and the nonhomogeneous equation \[ [y(t)-(E_1y)(t)]' = (E_0y)(t) +f(t)\quad (t\geq 0). \] It is shown that, under certain conditions on the norms of \(E_0\) and \(E_1\), if the nonhomogeneous equation with initial value \(y(t) = 0\) for \(t\leq 0\) has a bounded solution for any \(f\in C(0, \infty)\), then the homogeneous equation is exponentially stable. This is a generalisation of the Bohl-Perron principle from ordinary differential equations to neutral functional differential equations.