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**On the existence of positive solution for a kind of multi-point boundary value problem at resonance.**
*(English)*
Zbl 1200.34018

The paper deals with the existence of positive solutions for a second-order multi-point boundary value problem at resonance

\[ x''(t)+f(t,x(t))=0, \quad t\in (0,1), \]

\[ x(0)=\sum_{i=1}^{m-2}\alpha_{i}x(\xi_{i}), \quad x(1)=\sum_{i=1}^{m-2}\beta_{i}x(\xi_{i}). \]

The key tool is the Leggett-Williams norm-type theorem for coincidence equations due to D. O’Regan and M. Zima [Arch. Math. 87, No. 3, 233–244 (2006; Zbl 1109.47051)].

Reviewer’s remark: There is a gap in the proof of one of the main results of the paper. Namely, the assumptions of Theorem 2 do not imply the condition (C2) of Lemma 2.1.

\[ x''(t)+f(t,x(t))=0, \quad t\in (0,1), \]

\[ x(0)=\sum_{i=1}^{m-2}\alpha_{i}x(\xi_{i}), \quad x(1)=\sum_{i=1}^{m-2}\beta_{i}x(\xi_{i}). \]

The key tool is the Leggett-Williams norm-type theorem for coincidence equations due to D. O’Regan and M. Zima [Arch. Math. 87, No. 3, 233–244 (2006; Zbl 1109.47051)].

Reviewer’s remark: There is a gap in the proof of one of the main results of the paper. Namely, the assumptions of Theorem 2 do not imply the condition (C2) of Lemma 2.1.

Reviewer: Mirosława Zima (Rzeszow)

### MSC:

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

34B15 | Nonlinear boundary value problems for ordinary differential equations |

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

47N20 | Applications of operator theory to differential and integral equations |

### Citations:

Zbl 1109.47051
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\textit{L. Yang} and \textit{C. Shen}, Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 72, No. 11, 4211--4220 (2010; Zbl 1200.34018)

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### References:

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