##
**Positive solutions for second-order three-point boundary value problems.**
*(English)*
Zbl 0989.34009

Here, the author considers the three-point boundary value problem
\[
u^{\prime \prime }+a(t)f(u)=0,\quad u(0)=0,\quad u(1)-\alpha u(\eta)=b,
\]
where (A1) \(\eta \in (0,1)\) and \(0<\alpha \eta <1\), (A2) \(f:[0,\) \(\infty)\rightarrow [0,\infty)\) is continuous and satisfies \(\lim_{u\rightarrow 0^{+}}f(u)/u=0\) and \(\lim_{u\rightarrow \infty }f(u)/u=\infty \), (A3) \(a:[0,1]\rightarrow [0,\infty)\) is continuous and \(a\equiv 0\) does not hold on any subinterval of \([\eta ,1].\) It is proved that there exists a positive number \(b^{*}\) such that the problem above has at least one positive solution for \(b:0<b<b^{*}\) and no solution for \(b>b^{*}\). The particular case where \(b=0\) was previously studied by the same author [Electron. J. Differ. Equ. 1999, Paper. No. 34 (1999; Zbl 0926.34009)]. The proof is based upon the Schauder fixed-point theorem and motivated by D. D. Hai [Nonlinear Anal., Theory Methods Appl. 37A, No. 8, 1051-1058 (1999; Zbl 1034.35044)].

Reviewer: Idris Addou (Montréal)

### MSC:

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

Full Text:
DOI

### References:

[1] | Il’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects, Differential Equations, 23, 7, 803-810 (1987) · Zbl 0668.34025 |

[2] | Gupta, C. P., Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation, J. Math. Anal. Appl., 168, 540-551 (1992) · Zbl 0763.34009 |

[3] | Gupta, C. P., A sharper condition for the solvability of a three-point second order bounder value problem, J. Math. Anal. Appl., 205, 586 (1997) · Zbl 0874.34014 |

[4] | Ma, R., Positive solutions of a nonlinear three-point boundary-value problem, Eletron. J. Diff. Eqns., 34, 1-8 (1999) |

[5] | Hai, D. D., Positive solutions for semilinear elliptic equations in annular, Nonlinear Analysis, 37, 1051-1058 (1999) · Zbl 1034.35044 |

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