一类完全三阶常微分方程边值问题的正解

Positive Solutions for a Class of Fully Third-Order Ordinary Differential Equation Boundary Value Problem

用锥上的不动点指数理论与导数估计技巧,研究完全三阶边值问题{-u′′′(t)=f(t,u(t),u′(t),u″(t)),t∈[0,1],u(0)=u′(0)=u″(1)=0正解的存在性,其中f:[0,1]×R_+~3→R_+连续.在f(t,x,y,z)满足|(x,y,z)|充分小或充分大时的一些不等式条件下,得到该方程正解的存在性结果,这些不等式条件允许f(t,x,y,z)关于x,y,z超线性或次线性增长.

By using the fixed point index theory in cones and derivative estimates technique,we studied the existence of positive solutions for the fully third-order boundary value problem {-u′′′(t)=f(t,u(t),u′(t),u″(t)),t∈[0,1],u(0)=u′(0)=u″(1)=0 where f:[0,1]×R_+~3→R_+ was continuous.The existence results of positive solutions were obtained under the condition that f(t,x,y,z)satisfied some inequalities when|(x,y,z)| was small or large enough.These inequality conditions allowed that f(t,x,y,z)was superlinear or sublinear growth on x,y,z.