### Abstract:

In this paper there are established effective criteria for the existence and uniqueness of the solution of one two-point boutndary value problem for the system of differential equations with delays dx(t)=f (t,x(T1 (t)),....,x (Tm(t))) ----- dt x(0) = (1 - u)x(l) + uc, x(0) =(t) for tˇ0, where for a natural number n, an integer m, u ţ[0, 1], and the interval I = [0, l]ţR, the function f: I x R n(1+m) »Rn is a vector-valued function satisfying the local Carathočodory conditions, Tj : I»R (j = l,....m) are measurable functions such that Tj(t)ˇt (j=l,...m) for almost all tţI, cţRn, is a continuous and bounded vector-valued function.