Uniqueness of solutions of boundary-value problems for operator-differential equations on a finite segment and on a semiaxis

Abstract

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.