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Positive solutions of a fourth order boundary value problem. (English) Zbl 1042.34051
Summary: The existence of positive solutions of the nonlinear fourth-order problem \[ u^{(4)}(x)=\lambda a(x) f(u(x)),\quad u(0)= u'(0)= u'(1)= u'''(1)= 0, \] is studied, where \(a:[0,1]\to\mathbb{R}\) may change sign, \(f(0)> 0\), and \(\lambda> 0\) is sufficiently small. Our approach is based on the Leray-Schauder fixed-point theorem.
MSC:
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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