An estimate for the spectral radius of an operator associated with an equation of neutral type

We obtain new bounds for the spectral radius of the operator $(Ax)(t)=a(t)x(t-h)$ in spaces of functions which are $\omega$-periodic, almost periodic, and continuous and bounded on the whole axis. The results are used to prove a theorem on the existence ofohgr-periodic, bounded, and almost periodic solutions for linear functional-differential equations of neutral type.