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)].


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