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

For the equationL₀x(t)+L₁x(t)+...+L_nx⁽ⁿ⁾(t)=O, whereL_k,k=0,1,...,n, are operators acting in a Banach space, we establish criteria for an arbitrary solutionx(t) to be zero provided that the following conditions are satisfied:x^(1–1) (a)=0, 1=1, ..., p, andx^(1–1) (b)=0, 1=1,...,q, for - <a< b< (the case of a finite segment) orx^(1–1) (a)=0, 1=1,...,p, under the assumption that a solutionx(t) is summable on the semiaxista with its firstn derivatives.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 3, pp. 279–292, March, 1994.This research was supported by the Ukrainian State Committee on Science and Technology.