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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 '' +a(t)f(u)=0,u(0)=0,u(1)-αu(η)=b,

where (A1) η(0,1) and 0<αη<1, (A2) f:[0, )[0,) is continuous and satisfies lim u0 + f(u)/u=0 and lim u f(u)/u=, (A3) a:[0,1][0,) is continuous and a0 does not hold on any subinterval of [η,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)].

34B18Positive solutions of nonlinear boundary value problems for ODE