The aim of this paper is to consider a fully cantilever beam equation with one end fixed and the other connected to a resilient supporting device, that is,

$$ \textstyle\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)), \quad t\in [0,1], \\ u(0)=u'(0)=0, \\ u''(1)=0,\qquad u'''(1)=g(u(1)), \end{cases} $$

where \(f:[0,1]\times \mathbb{R}^{4}\rightarrow \mathbb{R}\), \(g: \mathbb{R}\rightarrow \mathbb{R}\) are continuous functions. Under the assumption of monotonicity, two existence results for solutions are acquired with the monotone iterative technique and the auxiliary truncated function method.

中文翻译：

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带支撑的全弹性悬臂梁方程的上下解法

本文的目的是考虑一个一端固定另一端连接到弹性支撑装置的全悬臂梁方程，即，

$$ \textstyle\begin{cases} u^{(4)}(t)=f(t,u(t),u'(t),u''(t),u'''(t)) , \quad t\in [0,1], \\ u(0)=u'(0)=0, \\ u''(1)=0,\qquad u'''(1)=g( u(1)), \end{cases} $$

其中\(f:[0,1]\times \mathbb{R}^{4}\rightarrow \mathbb{R}\) , \(g: \mathbb{R}\rightarrow \mathbb{R}\)是连续函数。在单调性假设下，利用单调迭代法和辅助截断函数法得到解的两个存在性结果。