一类奇异常微分方程边值问题解的存在性

Solvability of a Singular Boundary Value Problems for Ordinary Differential Equation

设f:[0,1]×R2→R满足Caratheodory,a(.)∈L1[0,1],a(.)≥0∫,0t0a(t)dt-∫1t0a(t)dt≠1,t0∈(0,1),(1-t)e(t)∈L1(0,1).运用Leray-Schauder原理考虑了二阶奇异边值问题:x″(t)=f(t,x(t),x′(t))+e(t),0<t<1x′(0)=0,x(1)=∫0t0a(t)x(t)dt-∫1t0a(t)x(t)dt在C1[0,1)上解的存在性.

Let f： [0,1]×R^2→R be a function satisfying Carathéodory＇s conditions and ,（1-t）e（t）∈L^1（0,1）.Let∫0^t0 α（t）dt-∫t0^1 α（t）dt≠1,t0∈（0,1）,α（t）∈L^1[0,1] be given. This paper is concerned with the problem of solvability of a C^1 [0,1 ） solution to the m - point boundary value problem {x^n （t）=f（t,x（t）,x′（t））＋e（t）,0〈t〈1 x′（0）=0,x（1）=∫0^t0 α（t）x（t）dt-∫t0^1 α（t）x（t）dt The proof of our main result is based upon the Leray Schauder continuation theorem.