Globally stable periodic solutions of linear neutral Volterra integrodifferential equations (Q1103151)

Consider the linear neutral Volterra integrodifferential equation (all functions are continuous) \[ d/dt[x(t)-\int^{t}_{0}C(t-s)x(s)ds- g(t)]=Ax(t)+\int^{t}_{0}G(t-\quad s)x(s)ds+f(t), \] where A, C and G are matrices (A is constant); g, f are periodic vectors and x(0) is given. One can show that \(x(t)=Z(t)[x(0)-g(0)]+g(t)+\int^{t}_{0}Z'(t- s)g(s)ds+\int \quad^{t}_{0}Z(t-s)f(s)ds,\) where Z(t) satisfies the first equation with \(g(t)=f(t)=0\) and \(Z(0)=1\). If Z,Ż\(\in L\) 1(R \(+)\), then conclusions about the asymptotic behavior of x can be drawn. If A is a stable matrix and sufficiently large compared to the matrices C and G, then Z,Ż\(\in L\) 1(R \(+)\). These are the results of the paper.