×

Positive solutions of a nonlinear \(n\)th order boundary value problem with nonlocal conditions. (English) Zbl 1074.34022

Summary: We discuss the existence of positive solutions of a nonlinear \(n\)th-order boundary value problem \[ u^{(n)}+ a(t) f(u)= 0,\quad t\in (0,1), \]
\[ u(0)= 0,\quad u'(0)= 0,\;u'(0)= 0,\dots, u^{(n-2)}(0)= 0,\quad\alpha u(\eta)= u(1), \] with \(0< \eta< 1\), \(0< \alpha\eta^{n-1}< 1\). In particular, we establish the existence of at least one positive solution if \(f\) is either superlinear or sublinear by applying the fixed-point theorem in cones due to Krasnoselskij and Guo.

MSC:

34B18 Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15 Nonlinear boundary value problems for ordinary differential equations
PDFBibTeX XMLCite
Full Text: DOI

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, Differ. Equ., 23, 7, 803-810 (1987) · Zbl 0668.34025
[2] Il’in, V. A.; Moiseev, E. I., Nonlocal boundary value problem of the second kind for a Sturm Liouville operator, Differ. Equ., 23, 8, 979-987 (1987) · Zbl 0668.34024
[3] Gupta, G. 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
[4] Eloe, P. W.; Henderson, J., Positive solutions for higher order ordinary differential equations, Electron. J. Differential Equations, 03, 1-8 (1995) · Zbl 0814.34017
[5] Eloe, P. W.; Henderson, J.; Wong, P. J.Y., Positive solutions for two point boundary value problems, Dynam. Systems Appl., 2, 135-144 (1996) · Zbl 0876.34016
[6] Lian, W. C.; Wong, F. H.; Yeh, C. C., On existence of positive solutions of nonlinear second order differential equations, Proc. Amer. Math. Soc., 124, 1111-1126 (1996) · Zbl 0857.34036
[7] Ma, R., Positive solutions for a nonlinear three-point boundary value problem, Electron. J. Differential Equations, 34, 1-8 (1998)
[8] Ma, R., Positive solutions of a nonlinear m-point boundary value problem, Comput. Math. Appl., 42, 755-765 (2001) · Zbl 0987.34018
[9] Coppel, W., Disconjugacy, (Lecture Notes in Mathematics, vol. 220 (1971), Springer-Verlag: Springer-Verlag New York, Berlin) · Zbl 0224.34003
[10] Guo, D.; Lakshmikantham, V., Nonlinear Problems in Abstract Cones (1988), Academic Press: Academic Press San Diego · Zbl 0661.47045
[11] 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
[12] Advances in Dynamic Equations on Time Scales, (Bohner, M.; Peterson, A. (2003), Birkhäuser: Birkhäuser Boston) · Zbl 1025.34001
[13] P.W. Eloe, Positive operators and maximum principles, Cubo (in press); P.W. Eloe, Positive operators and maximum principles, Cubo (in press)
[14] Eloe, P. W.; Henderson, J., Inequalities based on a generalization of concavity, Proc. Amer. Math. Soc., 125, 2103-2108 (1997) · Zbl 0868.34008
[15] Eloe, P. W., Maximum principles for a family of nonlocal boundary value problems, Adv. Difference Equations, 3, 201-210 (2004) · Zbl 1083.39018
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.