四阶两点常微分方程边值问题解的存在性

讨论一类四阶两点常微分方程边值问题x(4)=f(t,x,x′,x″,x)，边界条件的解的存在性,并给出相应的结论。其中边界条件如下:x(0)=A,x(1)=B,x″(0)=,x″(1)=, x(0)=A,x(1)=B,x″(0)=,x(1)=, x(0)=A,x(1)=B,x(0)=,x″(1)=, x(0)=A,x′(1)=B,x″(0)=,x″(1)=, x(0)=A,x′(1)=B,x″(0)=,x(1)=, x(0)=A,x′(1)=B,x(0)=,x″(1)=, x′(0)=A,x(1)=B,x″(0)=,x″(1)=, x′(0)=A,x(1)=B,x″(0)=,x(1)=, x′(0)=A,x(1)=B,x(0)=,x″(1)=。这些结论是在假设f(t,x,y,p,r)在形如［0,1］×Dx×Dy×Dp×I的区域内不变号的条件下给出的,其中Dx、Dy、Dp、I分别为某一区间。

The existence results for fourth-order two-point boundary value problems

x(4)=f(t,x,x′,x″,x) ,boundary value conditionwere considered and results were obtained, here the boundary value condition is one of the followings:x(0)=A,x(1)=B,x″(0)=,x″(1)=, x(0)=A,x(1)=B,x″(0)=,x(1)=,x(0)=A,x(1)=B,x(0)=,x″(1)=,x(0)=A,x′(1)=B,x″(0)=,x″(1)=,x(0)=A,x′(1)=B,x″(0)=,x(1)=,x(0)=A,x′(1)=B,x(0)=,x″(1)=,x′(0)=A,x(1)=B,x″(0)=,x″(1)=,x′(0)=A,x(1)=B,x″(0)=,x(1)=,x′(0)=A,x(1)=B,x(0)=,x″(1)=. These results were given to assume that the function f(t,x,y,p,r) satisfies the following condition. There are pairs (four or eight) of suitable constants such that f(t,x,y,p,r) does not change sign on sets of the form ［0,1］×Dx×Dy×Dp×I, where Dx,　Dy,　Dp are closed bounded intervals, I is a closed set in R and bounded by some pairs of constants, mentioned above.