一类高阶多点边值问题在共振条件下的可解性

Solvability of a Higher Order Multi-Point Boundary Value Problem at Resonance

考虑非线性高阶多点边值问题x（n）（t）=f（t,x（t）,x＇（t）,…,x（n-1）（t））＋e（t）,t∈（0,1）,x（i）（0）=0,i=0,1,…,n-2,x（n-2）（1）=∑m-2j=1βjx（n-2）（ηj{）解的存在性,这里f：[0,1]×n→是连续函数,e（t）∈L1[0,1],βj（j=1,2,…,m-2）为符号不全相同的实数,0〈η1〈η2〈…〈ηm-2〈1.利用Mawhin连续性定理对于上述共振条件下的非线性n阶多点边值问题建立了解的存在性结果.

The existence of nonlinear higher order multi-point boundary value is considered, x（n）（t）=f（t,x（t）,x′（t）,…,x（n-1）（t））＋e（t）,t∈（0,1）, x（i）（0）=0,i=0,1,…,n-2, x（n-2）（1）=∑m-2j=1βjx（n-2）（ηj）, where f：×Rn→R is a continuous function,e（t）∈L1, all the βj（j=1,2,…,m-2） are real numbers and have not the same sign,and 0η1η2…ηm-21. By using the Mawhin＇s continuation theorem,an existence result of solutions for the above nonlinear higher order multi-point boundary value at resonance is obtained.