Multiple symmetric positive solutions of a class of boundary value problems for higher order ordinary differential equations

Authors:
John R. Graef, Chuanxi Qian and Bo Yang

Journal:
Proc. Amer. Math. Soc. **131** (2003), 577-585

MSC (2000):
Primary 34B15

DOI:
https://doi.org/10.1090/S0002-9939-02-06579-6

Published electronically:
June 18, 2002

MathSciNet review:
1933349

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, the authors consider the boundary value problem

and give sufficient conditions for the existence of any number of symmetric positive solutions of (E)-(B). The relationships between the results in this paper and some recent work by Henderson and Thompson (Proc. Amer. Math. Soc. 128 (2000), 2373-2379) are discussed.

**1.**R. P. Agarwal, Focal Boundary Value Problems for Differential and Difference Equations, Kluwer Academic, Dordrecht, 1998. MR**99h:34036****2.**R. P. Agarwal, D. O'Regan, and P. J. Y. Wong, Positive Solutions of Differential, Difference, and Integral Equations, Kluwer Academic, Dordrecht, 1999. MR**2000a:34046****3.**R. Agarwal and F. H. Wong, Existence of positive solutions for higher order boundary value problems, Nonlinear Studies 5 (1998), 15-24. MR**99e:34024****4.**R. I. Avery, J. M. Davis, and J. Henderson, Three symmetric positive solutions for Lidstone problems by a generalization of the Leggett-Williams theorem, Electron. J. Differential Equations, Vol. 2000 (2000), No. 40, pp 1-15. MR**2001c:34048****5.**J. Baxley and L. J. Haywood, Nonlinear boundary value problems with multiple solutions, Nonlinear Anal. 47 (2001), 1187-1198.**6.**J. Baxley and L. J. Haywood, Multiple positive solutions of nonlinear boundary value problems, Dynam. Contin. Discrete Impuls. Systems, to appear.**7.**P. W. Eloe, Nonlinear eigenvalue problems for higher order Lidstone boundary value problems, Electron. J. Differential Equations 2000 No. 2, 1-8. MR**2001a:34032****8.**P. W. Eloe and J. Henderson, Positive solutions for higher order ordinary differential equations, Electron. J. Differential Equations 1995 (1995), 1-8. MR**96a:34037****9.**L. H. Erbe, S. Hu, and H. Y. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl. 184 (1994), 640-648. MR**95f:34023****10.**J. Henderson and H. B. Thompson, Multiple symmetric positive solutions for a second order boundary value problem, Proc. Amer. Math. Soc. 128 (2000), 2373-2379. MR**2000k:34042****11.**M. A. Krasnosel'skii, Positive Solutions of Operator Equations, Noordhoff, Groningen, 1964. MR**31:6107****12.**R. Ma, J. Zhang, and S. Fu, The method of lower and upper solutions for fourth-order two-point boundary value problems, J. Math. Anal. Appl. 215 (1997), 415-422. MR**98i:34037****13.**P. J. Y. Wong and R. Agarwal, Eigenvalues of Lidstone boundary value problems, Appl. Math. Comput. 104 (1999), 15-31. MR**2000b:34034**

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Additional Information

**John R. Graef**

Affiliation:
Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee 37403

Email:
john-graef@utc.edu

**Chuanxi Qian**

Affiliation:
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi 39762

Email:
qian@math.msstate.edu

**Bo Yang**

Affiliation:
Department of Mathematics and Statistics, Mississippi State University, Mississippi State, Mississippi 39762

Email:
by2@ra.msstate.edu

DOI:
https://doi.org/10.1090/S0002-9939-02-06579-6

Keywords:
Boundary value problems,
existence of positive solutions,
higher order equations,
multiple solutions,
nonlinear equations

Received by editor(s):
April 16, 2001

Received by editor(s) in revised form:
October 2, 2001

Published electronically:
June 18, 2002

Communicated by:
Carmen C. Chicone

Article copyright:
© Copyright 2002
American Mathematical Society