On a family of solutions of a linear second-order differential equation with constant unbounded operator coefficients in a Banach space

In a Banach space E, we study the equation u"(t) + Bu'(t) + Cu(t) = f(t), 0 <= t < infinity, where f(t) epsilon C([0, infinity); E), B, C epsilon N(E), and N(E) is the set of closed unbounded linear operators from E to E with dense domain in E. We. nd a two-parameter family of solutions of Eq. (1) in two cases: (a) the operator discriminant D = B-2 - 4C of Eq. 1 is zero; (b) D = F-2, where F is some operator in N(E). We suggest a method for increasing the smoothness of such solutions by imposing more restrictive conditions on the input data W = (B, C, f(t)) and the parameters x(1), x(2) epsilon E.