On a Family of Solutions of a Linear Second-OrderV Differential Equation with Constant Unbounded Operator Coefficients in a Banach Space

摘要

In a Banach space E, we study the equation u(t) + Bu(t) + Cu(t) = f(t), 0 ≤ t < ∞, (1) where f(t) ∈ C([0,∞);E), B,C ∈ N(E), and N(E) is the set of closed unbounded linear operators from E to E with dense domain in E. We find a two-parameter family of solutions of Eq. (1) in two cases: (a) the operator discriminant D = B2 - 4C of Eq. 1 is zero; (b) D = F2, 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 x1, x2 ∈ E.