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Multiple positive solutions for a three-point boundary value problem. (English) Zbl 0906.34014

The author deals with the three-point boundary value problem \[ -x'''+ f(x(t))= 0,\quad x(0)= x'(t_2)= x''(1)= 0\tag{1} \] with \(t_2\in\left[{1\over 2},1\right)\), \(f: \mathbb{R}\to \mathbb{R}\) is continuous and nonnegative for \(x\geq 0\).
Using properties of the Green function of the corresponding linear problem and a theorem by R. W. Leggett and L. R. Williams [Indiana Univ. Math. J. 28, 673-688 (1979; Zbl 0421.47033)], the author proves the existence of at least three positive solutions to (1).

MSC:

34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B27 Green’s functions for ordinary differential equations

Citations:

Zbl 0421.47033
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References:

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[10] Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana University Mathematics Journal, 28, 673-688 (1979) · Zbl 0421.47033
[11] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego, CA · Zbl 0661.47045
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