On the solvability and estimates of solutions to functional-differential equations in Sobolev spaces (Q2714709)

The author of this interesting paper presents results concerning the unique solvability and asymptotic behavior of solutions to initial-boundary value problems for a class of neutral linear differential-difference equations whose coefficients are operator functions taking values in the set of unbounded operators in a Hilbert space. The case of variable delays is investigated. The considered problem has the form: NEWLINE\[NEWLINE\begin{multlined} du/dt+ Au(t)+ B_0(t)CAu(t)+ \sum_{j=1}^\infty (B_j(t) S_{g_j}(Au)(t)+ D_j(t) S_{g_j}(du/dt)(t))+\\ +\int_0^{+\infty} (K(t-s)Au(s)+ Q(t-s)u^{(1)}(s)) ds= f(t), \quad t\in \mathbb{R}_+, \end{multlined}NEWLINE\]NEWLINE with initial condition \(u(+0)= \varphi_0\). Here \(B_0(t)\), \(B_j(t)\) and \(D_j(t)\) \((j=1,2,\dots)\) are strongly continuous operator functions taking values in the ring of bounded operators in a separable Hilbert space, \(K(t)\) and \(Q(t)\) are operator functions taking values in the ring of bounded operators in a separable Hilbert space and such that the operator functions \(\exp(-\kappa t)K(t)\) and \(\exp(-\kappa t)Q(t)\) are Bochner integrable on \(\mathbb{R}_+\) for a certain \(\kappa>0\), \(C\) is a compact operator in a separable Hilbert space, \(\varphi_0\) is a proper vector and \(S_{g_j}\) are some operators.NEWLINENEWLINEFor the entire collection see [Zbl 0952.00006].