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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 $u^{\prime \prime }+a(t)f(u)=0,\quad u(0)=0,\quad u(1)-\alpha u(\eta)=b,$ 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)].

##### MSC:
 34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
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##### References:
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