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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.

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