共振情形m-点边值问题解的存在性

On the Existence of m-Point Boundary Value Problem at Resonance

研究一类共振情形二阶微分方程m-点边值问题其中m≥3为整数,ai≥0,ξi∈(0,1)(i=1,2,…,m-2)为常数满足∑i=1m=2 ai=1, 0<ξ1<ξ2<…<ξm-2．利用Mawhin重合度拓展定理,作者得到了边值问题解存在的新结果．有意义的是本文允许函数．f(t,x,y)关于变量x和y的次数大于1,特别是允许变量x的次数大于y的次数,这些结果与已有工作是不同的．

By means of Mawhin＇s continuation theorem, we study m-point boundary value problem at resonance in the following form{x＇＇（t） = f（t,x（t）,x＇（t）） ＋ e（t）, t ∈ （0, 1）x＇（0）= 0, x（1） = m-2∑i=1 aix（ξi） where m≥ 3 is an integer, ai≥0,ξi∈（0,1）（i=1,2…,m-2）are constants satisfying∑i=1 m-2ai=1,=1and 0〈ξ1〈ξ2〈…〈ξm-2 m-2 A new result on the existence of solutions is obtained. The significance is that we allow the degree of power with respect to the second variable x and the third variable y of f（t, x, y） to be greater than 1, espeeially, the degree of variable x may be grater then the degree of variable y, which is different from corresponding ones of the past work.