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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 $f:[0,1]\times \bbfR^{2}\rightarrow \bbfR$ 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.

MSC:
34B15Nonlinear boundary value problems for ODE
34B27Green functions
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References:
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