Here, the author considers the three-point boundary value problem
where (A1) and , (A2) is continuous and satisfies and , (A3) is continuous and does not hold on any subinterval of It is proved that there exists a positive number such that the problem above has at least one positive solution for and no solution for . The particular case where 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)].