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Anderson, Douglas R.

Green’s function for a third-order generalized right focal problem. (English) Zbl 1045.34008

J. Math. Anal. Appl. 288, No. 1, 1-14 (2003).
The first part is concerned with the third-order three-point generalized right focal boundary value problem \(x'''(t)=0,\;t_1\leq t\leq t_3,\;x(t_1)=x'(t_2)=0,\;\gamma x(t_3)+\delta x''(t_3)=0,\) with \(\gamma\geq0\) and \(\delta>0\). Green’s function for this homogeneous problem is determined. Then sufficient conditions in terms of coefficients and boundary points are given, which guarantee its positivity. The second part of this paper is concerned with proving the existence of positive solutions of the boundary value problem (BVP) \(x'''(t)=f(t,x(t)),\;x(t_1)=x'(t_2)=0,\;\gamma x(t_3)+\delta x''(t_3)=0,\) (*), where \(f\) is nonnegative for \(x\geq0\). More precisely, Krasnoselskii’s fixed-point theorem is used to find sufficient conditions for the existence of a positive solution of BVP (*) whose norm lies between given numbers. Further, the fixed-theorem of Legget and Williams is used to find conditions for the existence of at least three positive solutions of BVP (*).
Reviewer: Pavel Rehak (Brno)

Cited in 67 Documents

MSC:

34B27 Green’s functions for ordinary differential equations
34B10 Nonlocal and multipoint boundary value problems for ordinary differential equations
34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations

Keywords:

third-order differential equations; boundary value problem; Green’s function; right focal
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\textit{D. R. Anderson}, J. Math. Anal. Appl. 288, No. 1, 1--14 (2003; Zbl 1045.34008)
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References:

[1] Agarwal, R. P., Focal Boundary Value Problems for Differential and Difference Equations (1998), Kluwer Academic: Kluwer Academic Boston · Zbl 0914.34001
[2] Agarwal, R. P.; Wong, P. J.Y.; O’Regan, D., Positive Solutions of Differential, Difference, and Integral Equations (1999), Kluwer Academic: Kluwer Academic Boston · Zbl 0923.39002
[3] Agarwal, R. P.; Bohner, M.; Wong, P. J.Y., Positive solutions and eigenvalues of conjugate boundary value problems, Proc. Edinburgh Math. Soc., 42, 349-374 (1999) · Zbl 0934.34008
[4] Agarwal, R. P.; O’Regan, D., Twin solutions to singular boundary value problems, Proc. Amer. Math. Soc., 128, 2085-2094 (2000) · Zbl 0946.34020
[5] Anderson, D., Green’s function for an \(n\)-point discrete right focal boundary value problem, Panamer. Math. J., 8, 45-70 (1998) · Zbl 0958.39006
[6] Anderson, D., Multiple positive solutions for a three point boundary value problem, Math. Comput. Modelling, 27, 49-57 (1998) · Zbl 0906.34014
[7] Anderson, D., Positivity of Green’s function for an \(n\) point right focal boundary value problem on measure chains, Math. Comput. Modelling, 31, 29-50 (2000) · Zbl 1042.39504
[8] Anderson, D.; Davis, J., Multiple solutions and eigenvalues for third order right focal boundary value problems, J. Math. Anal. Appl., 267, 135-157 (2002) · Zbl 1003.34021
[9] Avery, R., Existence of multiple positive solutions to a conjugate boundary value problem, Math. Sci. Res. Hot-Line, 2, 1-6 (1998) · Zbl 0960.34503
[10] Avery, R., Multiple positive solutions of an \(n\) th order focal boundary value problem, Panamer. Math. J., 8, 39-55 (1998) · Zbl 0960.34015
[11] Avery, R., Three positive solutions of a discrete second order conjugate problem, Panamer. Math. J., 8, 79-96 (1998) · Zbl 0958.39024
[12] Avery, R.; Henderson, J., Three symmetric positive solutions for a second order boundary value problem, Appl. Math. Lett., 13, 1-7 (2000) · Zbl 0961.34014
[13] Davis, J. M.; Erbe, L. H.; Henderson, J., Multiplicity of positive solutions for higher order Sturm-Liouville problems, Rocky Mountain J. Math., 31, 169-184 (2001) · Zbl 0989.34012
[14] Erbe, L. H.; Wang, H., On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120, 743-748 (1994) · Zbl 0802.34018
[15] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045
[16] Henderson, J., Multiple solutions for \(2m\) th order Sturm-Liouville boundary value problems on a measure chain, J. Differ. Equations Appl., 6, 417-429 (2000) · Zbl 0965.39008
[17] Krasnoselskii, M. A., Positive Solutions of Operator Equations (1964), Noordhoff: Noordhoff Groningen
[18] Leggett, R. W.; Williams, L. R., Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J., 28, 673-688 (1979) · Zbl 0421.47033
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