一类非线性二阶常微分方程m+1点边值问题的可解性

The Solvability on a Nonlinear Second-order Ordinary Differential Equation(m+1)-point Boundary Value Problem

设e∈C[0,1].设η∈[0,1],α∈R,iξ∈[0,1],ai∈R(i=1,2,…,m-2)为给定常数,满足α≠1,0<1ξ<2ξ<…<mξ-2<1,所有ai具有相同符号且∑m-2Carathéodory条件和一些符号条件的前提下考虑非线性二阶常微分方程m+1点边值问题x″=f(t,x(t),x′(t))+e(t),0<t<1,(1)x′(0)=αx′(η),x(1)=∑m-2i=1aix(ξi)(2)的可解性,允许非线性项在无穷远处的增长不受限制.研究工具是Leray-Schauder非线性抉择.

Let f：[0,1]×R^2→R satisfy the Caratheodory condition and e∈C[0,1],Letη∈[0,1],a∈R, and a≠1 ai∈R（i=1,2,…,m-2） all of the ai＇s having the same sign and ∑m-2 i=1 ai≠1,ξi∈[0,1]and 0〈ξ1〈ξ2〈…〈ξm-2〈1,be given. The existence of solutions for second-order ordinary differential equation （m＋ 1）-point boundary is studied. x^″=f（t,x（t））＋e（t）,0〈t〈1, （1） x′（0）=ax′（η）,x（1）=∑m-2 i=1 aox（ξi） （2） Under some sign conditions of f about the origin, an existence theorem for the problem （1）-- （2） is obtained. Analysis is based on a nonlinear alternative of Leray-Schauder.