Abstract: | We consider the following singularly perturbed boundary-value problem: on the interval 0 ≤x ≤ 1. We study the existence and uniqueness of its solutionu(x, ε) having the following properties:u(x, ε) →u
0(x) asε → 0 uniformly inx ε 0, 1], whereu
0(x) εC
∞ 0, 1] is a solution of the degenerate equationf(x, u, u′)=0; there exists a pointx
0 ε (0, 1) such thata(x
0)=0,a′(x
0) > 0,a(x) < 0 for 0 ≤x <x
0, anda(x) > 0 forx
0 <x ≤ 1, wherea(x)=f′
v(x,u
0(x),u′
0(x)).
Translated fromMatematicheskie Zametki, Vol. 67, No. 4, pp. 520–524, April, 2000. |