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Existence and location result for a fourth order boundary value problem. (English) Zbl 1157.34310
Summary: We prove an existence and location result for the fourth order nonlinear equation $$u^{(i \upsilon)} = f(t, u, u', u'', u'''),\quad 0 < t < 1,$$ with the Lidstone boundary conditions $$u(0) = u''(0) = u(1) = u''(1) = 0,$$ where $f: [0,1] \times \Bbb{R}^{4} \to \Bbb R$ is a continuous function satisfying a Nagumo type condition. The existence of at least one solution lying between a pair of well ordered lower and upper solutions is obtained by using an a priori estimate, lower and upper solutions method and degree theory.

34B15Nonlinear boundary value problems for ODE