*(English)*Zbl 0989.34009

Here, the author considers the three-point boundary value problem

where (A1) $\eta \in (0,1)$ and $0<\alpha \eta <1$, (A2) $f:[0,$ $\infty )\to [0,\infty )$ is continuous and satisfies ${lim}_{u\to {0}^{+}}f\left(u\right)/u=0$ and ${lim}_{u\to \infty}f\left(u\right)/u=\infty $, (A3) $a:[0,1]\to [0,\infty )$ is continuous and $a\equiv 0$ does not hold on any subinterval of $[\eta ,1]\xb7$ 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)].

##### MSC:

34B18 | Positive solutions of nonlinear boundary value problems for ODE |