一类非线性二阶三点边值问题的局部正解

当α0或者αη1时考察了非线性二阶三点边值问题u″(t)+h(t)f(t,u(t))=0,0t1,u(0)=0,αu(η)=u(1)(1)的局部正解,此时相应的Green函数不是非负的,传统的正函数锥不再适用。通过引入局部正函数锥,该问题被转化为此锥上的一个Hammerstein积分方程。根据局部正函数锥的性质构造了两个控制函数以便控制非线性的增长变化。在这些锥和控制函数的基础上,使用锥上的不动点指数定理获得了一、二个局部正解的存在性。

Partial positive solutions of a class of nonlinear second-order three-point boundary value problems

The existence of partial positive solutions is considered for the nonlinear second-order three-point boundary value problem u'(t)+h(t)f(t,u(t))=0,0＜t＜1,u(0)=0,αu(η)=u(1)(1)when α＜0 or αη＞1.In the case,the related Green function is not nonnegative.The usual cone of positive functions is inapplicable.By introducing cone of partial positive functions,the problem is transformed into a Hammerstenin integral equation on the cone.By the properties of cone of partial positive functions,two control functions are constructed for controlling the growth changes of nonlinear term.On the basis of cone and control functions,the existence of one and two partial positive solutions is obtained by applying the fixed point index theorem on cone.