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**Positive solutions for second-order three-point eigenvalue problems.**
*(English)*
Zbl 1204.34029

Summary: By means of the fixed point index theorem in cones, we get an existence theorem concerning the existence of a positive solution for the second-order three-point eigenvalue problem

\[ x''(t)+\lambda f(t,x(t))=0,\quad 0\leq t\leq 1,\;x(0)=0,\;x(1)=x(\eta), \]

where \(\lambda\) is a parameter. An illustrative example is given to demonstrate the effectiveness of the obtained result.

\[ x''(t)+\lambda f(t,x(t))=0,\quad 0\leq t\leq 1,\;x(0)=0,\;x(1)=x(\eta), \]

where \(\lambda\) is a parameter. An illustrative example is given to demonstrate the effectiveness of the obtained result.

### MSC:

34B18 | Positive solutions to nonlinear boundary value problems for ordinary differential equations |

34B09 | Boundary eigenvalue problems for ordinary differential equations |

34B10 | Nonlocal and multipoint boundary value problems for ordinary differential equations |

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\textit{C. Bai}, Abstr. Appl. Anal. 2010, Article ID 236826, 8 p. (2010; Zbl 1204.34029)

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