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The upper and lower solution method for some fourth-order boundary value problems. (English) Zbl 1122.34010

Summary: We are concerned with the fourth-order two-point boundary value problem
\[ \begin{aligned} & u^{(iv)}(t) = f(t, u(t), u'(t), u''(t), u'''(t)),\quad 0 <t < 1,\\ & u(0) = u'(1) = u''(0) = u'''(1) = 0.\end{aligned} \]
By implying certain restrictions on the nonlinear term \(f\), we obtain the existence results for the fourth-order two-point boundary value problem via the lower and upper solution method. In particular, a new truncating technique and an appropriate Nagumo-type condition are introduced and employed.

MSC:

34B15 Nonlinear boundary value problems for ordinary differential equations
34A45 Theoretical approximation of solutions to ordinary differential equations
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References:

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