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Existence of solutions with turning points for nonlinear singularly perturbed boundary-value problems

Authors:

A. Yu. Kolesov N. Kh. Rozov

Affiliation:

(1) P. G. Demidov Yaroslavl State University, USSR;(2) M. V. Lomonosov Moscow State University, USSR

Abstract:

We consider the following singularly perturbed boundary-value problem:

$varepsilon u' = fleft( {x,u,u'} right), 0< varepsilon ll 1, g_j left( {uleft| {_{x = 0} ,} right.uleft| {_{x = 1} ,u'left| {_{x = 0} ,} right.} right.u'left| {_{x = 1} } right.} right) = 0, j = 1,2,$

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.

Keywords:

nonlinear boundary-value problem singular perturbation solution with turning points existence and uniqueness of solutions

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