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

Anderson, D.

Math. Comput. Modelling 27, No. 6, 49-57 (1998).

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).

Reviewer: I.Rachůnková (Olomouc)

MSC:

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

Keywords:

multipoint BVP; third-order ODE; Green function; fixed point; cone

Citations:

Zbl 0421.47033

References:

[1]	Agarwal, R. P.; Wong, P. J.Y., Double positive solutions of \((n,p)\) boundary value problems for higher order difference equations, Computers Math. Applic., 32, 8, 1-21 (1996) · Zbl 0873.39008
[2]	Anderson, D.; Avery, R.; Peterson, A., Three positive solutions to a discrete focal boundary value problem, (Agarwal, R., Positive Solutions of Nonlinear Problems. Positive Solutions of Nonlinear Problems, Journal of Computational and Applied Mathematics (1998)) · Zbl 1001.39021
[3]	Avery, R., Multiple positive solutions of an \(n\) th order focal boundary value problem, PanAmerican Mathematical Journal, 8 (1998) · Zbl 0960.34503
[4]	R. Avery, Three positive solutions of a discrete second order conjugate problem, PanAmerican Mathematical Journal; R. Avery, Three positive solutions of a discrete second order conjugate problem, PanAmerican Mathematical Journal · Zbl 0958.39024
[5]	Eloe, P. W.; Henderson, J., Singular nonlinear boundary value problems for higher order ordinary differential equations, Nonlinear Analysis, 1, 709-729 (1970)
[6]	Erbe, L. H., Boundary value problems for ordinary differential equations, Rocky Mountain J. Math, 17, 1-10 (1991) · Zbl 0731.34015
[7]	Erbe, L. H.; Hu, S.; Wang, H., Multiple positive solutions of some boundary value problems, Journal of Mathematical Analysis and Applications, 184, 640-648 (1994) · Zbl 0805.34021
[8]	Erbe, L. H.; Wang, H., On the existence of positive solutions of ordinary differential equations, (Proceedings of the American Mathematical Society, 120 (1994)), 743-748 · Zbl 0802.34018
[9]	Hartman, P., Ordinary Differential Equations (1964), John Wiley and Sons · Zbl 0125.32102
[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