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On the number of positive solutions for a nonlinear third order boundary value problem. (English) Zbl 1114.34021
Summary: We consider the nonlinear third-order ordinary differential equation \[ u'''(t)= \lambda a(t)f(u(t)),\;0<t<1 \] with the boundary conditions \[ \alpha u'(0)-\beta u''(0)=0,\;u(1)=u'(1)=0. \] Some sufficient conditions for the nonexistence and existence of at least one, two and \(n\) positive solutions for the boundary value problem are established. In doing so the usual restriction that \(f_0=\lim_{u\to 0^+}\frac{f(u)}{u}\) and \(f_\infty =\lim_{u\to\infty}\frac{f(u)}{u}\) exist is removed. An example is also given to illustrate the main results.

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B24 Sturm-Liouville theory
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