Abstract: In this paper we study the Robin boundary value problem with a small parameter
εy"=f(t, y, ω(ε)y', ε),
a₀y(0) +b₀y'(0)=ξ(ε), a₁y(1)+b₁y'(1)=η(ε),
where the function ω(ε) is continuous on ε ≥ 0 with ω(0)=0. Assuming all known functions are suitably smooth, f satisfies Nagumo's condition, f_y>0, a_i²-b_i²≠0, (-1)ⁱa_ib_i ≤ 0 (i=0, 1) and the reduced equation 0=f(t, y, 0, 0) has a solution y(t) (0 ≤ t ≤ 1), we prove the existence and the uniqueness of the solution for the boundary value problem and givo an asymptotic expansion of the solution in the power ε^1/2 which is uniformly valid on 0 ≤ t ≤ 1.