一类四阶微分方程m点边值问题的正解

Positive Solutions of m-point Boundary Value Problem for One Class of Fourth-order Differential Equation

讨论一类四阶微分方程m点边值问题{u^((4))(t)+h(t)f(u)=0,u(0)=u'(0)=u″(0)=0,u″(1)=∑m=2i=1β_iu″(η_i),其中,η_i∈(0,1),0<η_1<η_2<…<η_(m-2)<1,β_i∈[0,∞)且m=2∑i=1β_iη_i<1.通过与一个线性算子相关的第一特征值的讨论,运用不动点指数定理,得到正解存在的结果,最后给出一个例子用以说明定理的应用.

The existence of positive solutions for the fourth-order m-point boundary value problem {u^（（4））（t）＋h（t）f（u）=0,u（0）=u＇（0）=u″（0）=0,u″（1）=∑m=2i=1β_iu″（η_i） is considered under some conditions concerning the first eigenvalue of the relevant linear operator,where η_i∈（0,1）,0η_1η_2…η_（m-2）1,β_i∈[0,∞） with m=2∑i=1β_iη_i1.The existence of positive solutions is proved by means of fixed point index theory,and an example is given to show the application of the result.