一类四阶边值问题解的存在唯一性及迭代算法

On the Existence, Uniqueness and Iterative Algorithm of Solution to a Fourth-Order Boundary Value Problem

摘要: 文章考虑如下四阶隐式微分方程边值问题：$\{\begin{array}{c}f\left(t,u\left(t\right),u^{″}\left(t\right),u^{\left(4\right)}\left(t\right)\right)=0,t\in \left[0,1\right]\\ u\left(0\right)=u\left(1\right)=u^{″}\left(0\right)=u^{″}\left(1\right)=0\end{array}$ $\begin{array}{c}(1)\\ (2)\end{array}$ 。以格林函数为工具，利用泛函分析中的压缩映象原理，直观给出该问题解的存在性和唯一性条件，及唯一解的迭代序列，并通过具体实例来实现结果的应用。

Abstract: In this paper, we consider the following fourth-order implicit differential equation boundary value problem: $\{\begin{array}{c}f\left(t,u\left(t\right),u^{″}\left(t\right),u^{\left(4\right)}\left(t\right)\right)=0,t\in \left[0,1\right]\\ u\left(0\right)=u\left(1\right)=u^{″}\left(0\right)=u^{″}\left(1\right)=0\end{array}$ $\begin{array}{c}(1)\\ (2)\end{array}$ . By using the Green function as a tool and the principle of compression mapping in functional analysis, the existence and uniqueness conditions of the solution to this problem are intuitively, as well as the iterative sequence of the unique solution. The results are implemented by specific examples.