一类三阶常微分方程边值问题的可解性

Solvability of a Class of Boundary Value Problems for Third-order Ordinary Differential Equations

利用等价范数、积分方程组和Leray-Schauder不动点定理考察了半线性三阶两点边值问题u(t)+f(t,u,u′)+g(t,u,u′)=0,u(0)=A,u′(0)=B,u′(1)=C的解和正解的存在性.主要条件都是局部的,换句话说,只要非线性项的主部f(t,u,v)在其定义域的某个有界子集上的“高度”是适当的,该问题必然存在解或者正解.

By making use of the equivalent norm and system of integral equations and Leray-Schauder fixed point theorem,the existence of solution and positive solution is considered for the sublinear third-order two-point boundary value problem u+f(t,u,u′)+g(t,u,u′)=0,u(0)=A,u′(0)=B,u′(1)=C.The main conditions are local.In other words,the problem may possess a solution or positive solution provided the 'height' of major part f(t,u,v) of nonlinear term is appropriate on a bounded subset of its domain.