四阶微分方程的迭代解

利用一个构造性的方法,在假设边值问题存在上解α和下解β,满足β≤α的前提下,给出了两个单调序列它们一致收敛于如下两类边值问题的极值解u(4)(x)-Mu″(x)=f(x,u(x),u'(x),u″(x),u″'(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0;u(4)(x)-Mu″(x)=g(x,u(x),u'(x),u″(x)),0＜x＜1,u(0)=u'(1)=u″(0)=u″'(1)=0.

The Iterative Solution for some Fourth-Order DifferentialEquations

In this paper we describe a constructive method which yields two monotone sequences that converge uniformly to extremal solutions to the following boundary value problems $$ u^{(4)}(x)-Mu'(x) =f(x,u(x),u'(x),u'(x), u'(x)) , 0<x<1 ,$$ $$ u(0)=u'(1)=u'(0)=u'(1) = 0 ,$$ and $$ u^{(4)}(x)-Mu'(x) =g(x,u(x),u'(x),u'(x)) , \ \ 0<x<1 , $$ $$ u(0)=u'(1)=u'(0)=u'(1) = 0,$$ if there exist an upper solution $\beta$ and a lower solution $\alpha$ with $\beta \leq \alpha$.