Abstract: It is proved that for ε ≥ 0 and δ ≥ 0 the two -point boundary value problem
-{y'(t)+f(t)+h(t)K_e(t,y(t))}=g(t)z(t),A≤t≤B,
z'(t)=K_e(t,y(t)):={(k(t)+ε)/y(t)}^1/N,A≤t≤B,
y(A)=δ-Pz(A)，y(B)=δ+Qz(B),
has a unique solution (y(t,ε,δ),z(t,ε,δ)) under certain hypotheses with the aid of the appropriate Green's function integral operator.The unique solution (ξ_e,η_e,v_e(s)) of the free boundary problem
{(k(v)+ε)|v'|^N-1v'}'+{sg(v)+f(v)}v'+h(v)=0,ε < s < η,
v|_s=ε=A_e-{(k(v)+ε)|^N-1v'}|_s=ε=Pε,
v|_s=η=B_e=B_e{(k(v)+ε)|v'|^N-1v'}_s=η=Qη,
is constructed utilizing the solution (y(t,ε.0),z(t,ε,0)).The fine boundary problem is shown to be a singular perturbation problem when the function k(t) possesses intervals of degeneracy

Key words: Two-point boundary value problem, free boundary problem, interval of dogeneracy