Abstract: In this paper, we study the following perturbed nonlinear boundary value problemof the form:εx’=f(t, x,y, ε)εy’ =g(t, x,y, ε)x(0)= A(ξ₁,ξ₂ ,x(1) -x(0), y(1)- y(0), ε)y(0)=B(ξ₁, ξ₂,x(1)-x(0),y(1)-g(0),ε)where ξ₁, ξ₂ are functions of ε. 0<ε<<1. Under some suitable conditions, we give the asymptotic expansion of solution of any order, and obtain the estimation of remaindet term by using the comparison theorem.

Key words: nonlinear boundary value, singular perturbation, comparisontheorem, asymptotic expansion