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Zbl 0989.34009
Ma, Ruyun
Positive solutions for second-order three-point boundary value problems.
(English)
[J] Appl. Math. Lett. 14, No.1, 1-5 (2001). ISSN 0893-9659

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 {\it D. D. Hai} [Nonlinear Anal., Theory Methods Appl. 37A, No. 8, 1051-1058 (1999; Zbl 1034.35044)].
MSC 2000:
*34B18 Positive solutions of nonlinear boundary value problems

Keywords: three-point boundary value problems; positive solution; Schauder fixed-point theorem

Citations: Zbl 0926.34009; Zbl 1034.35044

Cited in: Zbl 1213.34043

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