Open Access

Existence and multiplicity of solutions for some three-point nonlinear boundary value problems

Journal of Inequalities and Applications20062006:79653

DOI: 10.1155/JIA/2006/79653

Received: 30 September 2004

Accepted: 20 October 2004

Published: 11 April 2006

Abstract

We study the existence and multiplicity of solutions for the three-point nonlinear boundary value problem , ; , where , , and are assumed to be positive and have some singularities, and is a positive parameter. Under certain conditions, we prove that there exists such that the three-point nonlinear boundary value problem has at least two positive solutions for , at least one solution for , and no solution for .

[12345678910111213]

Authors’ Affiliations

(1)
Department of Mathematics, Xuzhou Normal University
(2)
Department of Mathematics, National University of Ireland, Galway

References

  1. Choi YS: A singular boundary value problem arising from near-ignition analysis of flame structure. Differential and Integral Equations. An International Journal for Theory and Applications 1991,4(4):891–895.MathSciNetMATHGoogle Scholar
  2. Dalmasso R: Positive solutions of singular boundary value problems. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1996,27(6):645–652.MathSciNetView ArticleMATHGoogle Scholar
  3. Feng W, Webb JRL: Solvability of m-point boundary value problems with nonlinear growth. Journal of Mathematical Analysis and Applications 1997,212(2):467–480. 10.1006/jmaa.1997.5520MathSciNetView ArticleMATHGoogle Scholar
  4. Gupta CP, Trofimchuk SI: Existence of a solution of a three-point boundary value problem and the spectral radius of a related linear operator. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1998,34(4):489–507.MathSciNetView ArticleMATHGoogle Scholar
  5. Ha KS, Lee Y-H: Existence of multiple positive solutions of singular boundary value problems. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1997,28(8):1429–1438.MathSciNetView ArticleMATHGoogle Scholar
  6. Liu X: Nontrivial solutions of singular nonlinear m-point boundary value problems. Journal of Mathematical Analysis and Applications 2003,284(2):576–590. 10.1016/S0022-247X(03)00365-2MathSciNetView ArticleMATHGoogle Scholar
  7. Ma R, Castaneda N: Existence of solutions of nonlinear-point boundary-value problems. Journal of Mathematical Analysis and Applications 2001,256(2):556–567. 10.1006/jmaa.2000.7320MathSciNetView ArticleMATHGoogle Scholar
  8. Webb JR: Positive solutions of some three point boundary value problems via fixed point index theory. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 2001,47(7):4319–4332.MathSciNetView ArticleMATHGoogle Scholar
  9. Wong FH: Existence of positive solutions of singular boundary value problems. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1993,21(5):397–406.MathSciNetView ArticleMATHGoogle Scholar
  10. Xu X: Multiplicity results for positive solutions of some semi-positone three-point boundary value problems. Journal of Mathematical Analysis and Applications 2004,291(2):673–689. 10.1016/j.jmaa.2003.11.037MathSciNetView ArticleMATHGoogle Scholar
  11. Xu X: Positive solutions for singular-point boundary value problems with positive parameter. Journal of Mathematical Analysis and Applications 2004,291(1):352–367. 10.1016/j.jmaa.2003.11.009MathSciNetView ArticleMATHGoogle Scholar
  12. Xu X, Ma J: A note on singular nonlinear boundary value problems. Journal of Mathematical Analysis and Applications 2004,293(1):108–124. 10.1016/j.jmaa.2003.12.017MathSciNetView ArticleMATHGoogle Scholar
  13. Zhang Z, Wang J: The upper and lower solution method for a class of singular nonlinear second order three-point boundary value problems. Journal of Computational and Applied Mathematics 2002,147(1):41–52. 10.1016/S0377-0427(02)00390-4MathSciNetView ArticleMATHGoogle Scholar

Copyright

© X.Xian and D. O'Regan. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.