高阶P-Laplace方程边值问题的上下解方法

The method of upper and lower solutions for higher order P-Laplace equation boundary value problems

利用上下解构造迭代序列获得边值问题(φ(x(2m-2)(t)))″=f(t,x,x″(t),x(4)(t),…x(2m-2)(t)),t∈[0,1]x(2j)(0)=0,x(2j)(1)=0,j=0,1,…m-1极值解的存在性。主要通过定义上下解构造凸闭集,通过方程定义算子,然后利用上下解构造两个迭代序列,利用算子在所构造的凸闭集中的性质,证明两个序列为单调序列,且他们是一致有界等度连续的,由Arzela定理得到算子的不动点,极值解的存在性得以证明。

The existence of extremal solutions of the boundary value problems （φ（x^（2m-2）（t）））″=f（t,x,x″（t）,x^（4）（t）,…x^（2m-2）（t））,t∈[0,1]x^（2j）（0）=0,x^（2j）（1）=0,j=0,1,…m-1 is obtained by constructing iterative sequence via upper and lower solutions. Mainly Convex closed set is constructed by defining upper and lower solutions. Operator is defined through the equation, then two iterative sequences is constructed via upper and lower solutions. By the property of the operator in the constructed convex closed set, the two sequences is proved that they are monotone and uniformly bounded and equicontinuous. From Arzela theorem a fixed point of the operator is obtained. Thus the existence of extremal solutions is proved.