Unique existence of periodic solutions of neutral Volterra integrodifferential equations (Q753075)

The paper is concerned with the neutral Volterra-type integrodifferential equation \((I)\quad x'(t)=f(t)+B(t)x(t)+\int^{t}_{- \infty}C(t,s)ds+\int^{t}_{-\infty}D(t,s)x'(s)ds,\) where \(t\in {\mathbb{R}}\), \(x\in {\mathbb{R}}^ n\); \(f\in C({\mathbb{R}},{\mathbb{R}}^ n)\); B, C and D are \(n\times n\) continuous matrices; \(f(t+T)\equiv f(t)\), \(B(t+T)\equiv B(t)\), \(C(t+T,s+T)\equiv C(t,s)\) and \(D(t+T,s+T)\equiv D(t,s)\) for some number \(T>0\). Assuming that \(\int^{t}_{-\infty}| C(t,s)| ds<\infty\) and \(\int^{t}_{-\infty}| D(t,s)| ds<\infty,\) the author obtains some sufficient conditions for the unique existence of a T-periodic solution to equation (I). The method of proof used in the paper is transposed from the theory of ordinary differential equation [cf. \textit{J. K. Hale}, Ordinary differential equations (1969; Zbl 0186.409)]. The special case when all elements of the matrix B being zero is discussed. Two illustrating examples are also indicated.