The method of lower and upper solutions for fourth-order two-point boundary value problems
The method of upper and lower solutions coupled with the monotone iterative technique is used to guarantee the existence of a couple of monotone sequences converging to the extremal solutions in a sector of the following fourth-order boundary value problem: \[ u^{(IV)}(x)=f(x,u(x),u''(x)), \qquad u(0)=u(1)=u''(0)=u''(1)=0, \] where \(f:[0,1]\times \mathbb{R}^{2}\rightarrow \mathbb{R}\) is continuous. In order to demonstrate the main result, the existence of a lower solution \(\beta\) and an upper solution \(\alpha\) with \(\beta\leq\alpha\) and \(\beta''\geq\alpha''\) on \([0,1]\) is assumed. As a previous result, the authors also prove a maximum principle.
scientific article; zbMATH DE number 1115423
scientific article