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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)),u(0)=u(1)=u '' (0)=u '' (1)=0,

where f:[0,1]× 2 is continuous.

In order to demonstrate the main result, the existence of a lower solution β and an upper solution α with βα and β '' α '' on [0,1] is assumed. As a previous result, the authors also prove a maximum principle.

34B15Nonlinear boundary value problems for ODE
34B27Green functions