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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 $1\le\alpha<\frac 1\eta$. 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.

34B18Positive solutions of nonlinear boundary value problems for ODE
34B10Nonlocal and multipoint boundary value problems for ODE
Full Text: DOI
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