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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

${u}^{\text{'}\text{'}}+a\left(t\right)f\left(u\right)=0,\phantom{\rule{1.em}{0ex}}u\left(0\right)=0,\phantom{\rule{1.em}{0ex}}u\left(1\right)-\alpha u\left(\eta \right)=b,$

where (A1) $\eta \in \left(0,1\right)$ and $0<\alpha \eta <1$, (A2) $f:\left[0,$ $\infty \right)\to \left[0,\infty \right)$ 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) $a:\left[0,1\right]\to \left[0,\infty \right)$ is continuous and $a\equiv 0$ does not hold on any subinterval of $\left[\eta ,1\right]·$ It is proved that there exists a positive number ${b}^{*}$ such that the problem above has at least one positive solution for $b:0 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