On the dimension of the solution set to the homogeneous linear functional differential equation of the first order

Consider the homogeneous equation u'(t) = l(u)(t) for a,e, t is an element of[a, b] where a"": C([a, b];a"e) -> L([a, b];a"e) is a linear bounded operator. The efficient conditions guaranteeing that the solution set to the equation considered is one-dimensional, generated by a positive monotone function, are established. The results obtained are applied to get new efficient conditions sufficient for the solvability of a class of boundary value problems for first order linear functional differential equations.