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Existence of positive solutions for nonlinear third-order three-point boundary value problems. (English) Zbl 1141.34310
Summary: This paper is concerned with the following nonlinear third-order three-point boundary value problem:
$u'''(t)+a(t)f(u(t))=0,\quad t\in(0,1),\;u(0)=u'(0)=0,\;u'(1)=\alpha u'(\eta),$
where $$0<\eta<1$$ and $$1\leq\alpha<\frac 1\eta$$. First, the Green’s function for the associated linear boundary value problem is constructed, and then, some useful properties of the Green’s function are obtained by a new method. Finally, existence results for at least one positive solution for the above problem are established when $$f$$ is superlinear or sublinear.

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations 34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
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