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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 {\it 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
Full Text:
##### References:
 [1] Il’in, V. A.; Moiseev, E. I.: Nonlocal boundary value problem of the first kind for a Sturm-Liouville operator in its differential and finite difference aspects. Differential equations 23, No. 7, 803-810 (1987) [2] Gupta, C. P.: Solvability of a three-point nonlinear boundary value problem for a second order ordinary differential equation. J. math. Anal. appl. 168, 540-551 (1992) · Zbl 0763.34009 [3] Gupta, C. P.: A sharper condition for the solvability of a three-point second order bounder value problem. J. math. Anal. appl. 205, 586 (1997) · Zbl 0874.34014 [4] Ma, R.: Positive solutions of a nonlinear three-point boundary-value problem. Eletron. J. Diff. eqns. 34, 1-8 (1999) · Zbl 0926.34009 [5] Hai, D. D.: Positive solutions for semilinear elliptic equations in annular. Nonlinear analysis 37, 1051-1058 (1999) · Zbl 1034.35044