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

Abstract

For the equationL ₀ x(t)+L ₁x′(t)+...+L _n x ⁽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 semiaxist≥a with its firstn derivatives.