一般Lidstone边值问题的n个正解的存在性

Existence of n Positive Solutions to General Lidstone Boundary Value Problems

考察了含所有偶数阶导数的一般Lidstone边值问题的正解和对称正解.通过 选择合适的Banach空间和锥,对该问题建立了n个正解或者对称正解的存在性,其 中n是一个任意的自然数.基本工具是等价范数和锥拉伸与锥压缩型的Krasnosel'skii 不动点定理. 结论的主要条件是局部的.换言之,如果非线性项f在某些有界集上的 "高度"是适当的,则该问题可以具有n个正解.

The positive solutions and symmetric positive solutions of the general Lidstone boundary value problem with all even-order derivatives are considered. By choosing suitable Banach space and cone, the existence of n positive solutions or symmetric positive solutions is established for the problem, where n is an arbitrary natural number. The basic tools used are equivalent norm and Krasnosel'skii fixed point theorem of cone expansion-compression type. The main conditions of the results are local. In other words, the problem may possess n positive solutions provided the 'height' of the nonlinear term f on some bounded sets are appropriate.