Uniqueness of solutions of boundary-value problems for operator-differential equations on a finite segment and on a semiaxis - Radzievskii, G. V. (Ukrains’kyi Matematychnyi Zhurnal, 1994)
For the equation L_0x(t)+L_1x′(t) + ... + L_nx^(n)(t) = O, where L_k, k = 0,1,...,n, are operators acting in a Banach space, we establish criteria for an arbitrary solution x(t) to be zero provided that the following conditions are satisfied: x^(1−1) (a) = 0, 1 = 1, ..., p, and x^(1−1) (b) = 0, 1 = 1,...,q, for -∞ < a < b < ∞ (the case of a finite segment) or x^(1−1) (a) = 0, 1 = 1,...,p, under the assumption that a solution x(t) is summable on the semiaxis t ≥ a with its first n derivatives.