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

Consider the homogeneous equation $$u'(t) = l(u)(t){rm{ for a}}{rm{.e}}{rm{. }}t in [a,b]$$ where ?: C([a, b];?) → L([a, b];?) 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.