On the solvability of an initial value problem for an equation with delay argument and unbounded operator-valued coefficients (Q1902826)

The author considers the initial value problem \[ \Biggl({1\over i} {d\over dt}- \sum^m_{j= 0} A_j(t) S_{h_j(t)}\Biggr) u(t)= f(t),\;t\geq t_0,\tag{1} \] \[ u(t)= g(t),\quad t\leq t_0,\quad u(t_0+ 0)= g(t_0),\tag{2} \] where \(A_j(t): Y\to Y\) are closed operators, \(A_j(t): X\to Y\) are bounded operators, \(t> t_0\), \(j\geq 0\), \(X\) and \(Y\) are Hilbert spaces with \(X\subset Y\), \(|\cdot|_X\geq |\cdot|_Y\), \(S_h u(t)\equiv u(t- h)\), \(h_0(t)\equiv 0\), \(h_j'(t)\leq r< 1\), \(t\in \mathbb{R}\), \(h_j(t)\in \text{AC}(\mathbb{R})\), \(j\geq 1\). In four theorems he presents sufficient conditions for the existence of a unique solution of (1), (2) with certain additional properties.