On the existence and uniqueness of periodic solutions of linear differential equation with shifted argument in Banach space (Q2767629)

Let \((B,\|\cdot\|)\) be a complex Banach space. Let \(L(B)\) be a space of linear continuous operators acting in space \(B\), let \(T\) be a positive rational number, and let \(\{A_{i}, i=-N,\ldots,N\}\subset L(B)\). The author obtains necessary and sufficient conditions for the existence and uniqueness of a solution \(x\in C^1(R,B)\), \(\|x\|_{\infty}=\sup_{t\in R}\|x(t)\|<\infty\), \(\forall t\in R: x(t+T)=x(t)\) of the equation \(x'(t)=\sum_{k=-N}^{N}A_{k}x(t-k)+y(t)\), \(t\in R\) for any \(y(t)\in C(R,B)\), \(\|y\|_{\infty}<\infty\), \(\forall t\in R: y(t+T)=y(t)\). The explicit form of the solution is found.