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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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[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, 7, 803-810, (1987) · Zbl 0668.34025
[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)
[5] Hai, D.D., Positive solutions for semilinear elliptic equations in annular, Nonlinear analysis, 37, 1051-1058, (1999) · Zbl 1034.35044
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