一类四阶非线性常微分方程两点边值问题三重正解的存在性

Existence of Triple Positive Solutions to A Class of Nonlinear Fourth-Order Two-Point Boundary Value Problem

考虑一类四阶非线性两点边值问题三重正解的存在性问题,这里f:[0,1]×[0,+∞)×(-∞,0]→[0,+∞).通过适当变换可将上述四阶边值问题转化为与其等价的二阶微分-积分方程的两点边值问题,适当定义半序巴拿赫空间及其上的锥,运用Legget-Williams不动点定理,得到二阶微分-积分方程的两点边值问题的三重正解的存在性,再由等价性,得到上述四阶非线性两点边值问题三重正解的存在性.

In this paper, the existence of at least three positive solutions to a class of nonlinear fourth order two- point boundary value problem is considered： Where is continuous. We exploit the fact that the above boundary value problem can be translated to a second order integral-differential boundary value problem. The Legget-Williams fixed point theorem is applyed in cone to the equivalent second order integral-differential boundary value proble, and get the existence of at least three positive solutions for the second order integral- differential boundary value problem. So the existence of at least three positive solutions for the above nonlinear fourth order two-point boundary value problem is proved by the equivalence property.