Singular integro-differential equations and product spaces (Q1310877)

The authors study the well-posedness, in a state space with product space structure, of neutral functional differential equations of the form (1) \[ {d\over {dt}} \bigl( x(t)-\sum_{j=1}^ p B_ j x(t-h_ j)- \int_{-h}^ 0 B(s)x(t+s)ds\bigr)= \sum_{j=0}^ p A_ j x(t-h_ j)- \int_{-h}^ 0 A(s)x(t+s)ds, \] where \(0=h_ 0< h_ 1<\dots< h_ p=h\), \(A_ j,B_ j\in \mathbb{R}^{n\times n}\), and \(A(.),B(.)\in L^ 2(- h,0; \mathbb{R}^{n\times n})\). An affirmative answer is proved to the question: Is it possible to find a state space with product space structure for equation (1)? It is shown in the paper that such a space can be found and it is the extrapolation space corresponding to the solution semigroup of the problem on the space \(L_ g^ 2\). The extended state spaces are also discussed in details.