Remarks on “Periodic Solutions of Linear Volterra Equations”

Consider the Volterra equationwhere x is an mathrmn -vector, the function $f(t)$ is continuous on $(-infty, infty)$ with the value in Rⁿ , D is an $ntimes n$ matrix of functions continuous on $(-infty, infty)$ with $D(t+T)=$ $D(t)$ for a fixed positive number T , $E(t, s)$ is an $ntimes n$ matrix of functions continuous for $-infty<sleq t<infty$ and int-inftyᵗ||E(t, s)|ds is continuous and bounded on $(-infty, infty)$ .In [1], T. A. Burton has proved the following theorems (see [1], Theorems 1 and 6).