变系数四阶周期边值问题正解的局部存在与多解性

Local Existence and Multiplicity of Positive Solutions to a Fourth-order Periodic Boundary Value Problem with Variable Coefficient

考察了一类变系数非线性四阶周期边值问题的正解存在性与多解性.主要工具是相关线性问题的Green函数,积分方程以及锥上的不动点指数定理.结论表明,这个问题可以具有n个正解,只要非线性项在某些有界集上的增长速度函数受到系数函数控制,其中n是一个任意的自然数.

The existence and multiplicity of positive solutions are considered to a class of nonlinear fourth-order periodic boundary value problem with variable coefficient. Main tools are the Green function of associated linear problem, integral equation and fixed point index theorem on cone. The results show that the problem can have n positive solutions provided that the growth rates functions of nonlinear term on some bounded sets are dominated by the coefficient function,where n is an arbitrary natural number.