Boundary value problems for systems of ordinary differential equations with a small parameter $\varepsilon$ and with a finite number of measurable delays of the argument are considered. Under the assumption that the number $m$ of boundary conditions does not exceed the dimension $n$ of the differential system, it is proved that the point $\varepsilon =0$ generates $\rho$ -parametric families (where $\rho =n-m$ ) of solutions of the initial problem. Bifurcation conditions of such solutions are established. Also, it is shown that the index of the operator, which is determined by the initial boundary value problem, is equal to $\rho$ and coincides with the index of the unperturbed problem. Finally, an algorithm for construction of solutions (in the form of Laurent series with a finite number terms of negative power of $\varepsilon$ ) of the boundary value problem under consideration is suggested.