## The method of lower and upper solutions for fourth-order two-point boundary value problems.(English)Zbl 0892.34009

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.
Reviewer: Eduardo Liz (Vigo)

### MSC:

 34B15 Nonlinear boundary value problems for ordinary differential equations 34B27 Green’s functions for ordinary differential equations
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### References:

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