Acta Mathematica Sinica

, Volume 20, Issue 2, pp 273–282

Positive Solutions for Semipositone m-point Boundary-value Problems

ORIGINAL ARTICLES

Abstract

Let ξi ∈ (0, 1) with 0 < ξ1 < ξ2 < ··· < ξm−2 < 1, ai, bi ∈ [0,∞) with \( 0 < {\sum\nolimits_{i = 1}^{m - 2} {a_{i} < 1} } \) and \( {\sum\nolimits_{i = 1}^{m - 2} {b_{i} < 1} } \). We consider the m-point boundary-value problem
$$ {u}\ifmmode{''}\else$''$\fi + \lambda f{\left( {t,u} \right)} = 0,{\kern 1pt} {\kern 1pt} {\kern 1pt} t \in {\left( {0,1} \right)}, $$
$$ {x}\ifmmode{'}\else$'$\fi{\left( 0 \right)} = {\sum\limits_{i = 1}^{m - 2} {b_{i} {x}\ifmmode{'}\else$'$\fi{\left( {\xi _{i} } \right)},{\kern 1pt} {\kern 1pt} {\kern 1pt} x{\left( 1 \right)} = {\sum\limits_{i = 1}^{m - 2} {a_{i} x{\left( {\xi _{i} } \right)},} }} } $$
where f(x, y) ≥ −M, and M is a positive constant. We show the existence and multiplicity of positive solutions by applying the fixed point theorem in cones.

Keywords

Ordinary differential equation Existence of solutions Multi-point boundary value problems Fixed point theorem in cones 

MR (2000) Subject Classification

34B10 

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References

  1. 1.
    Moshinsky, M.: Sobre los problemas de condiciones a la frontiera en una dimension de caracteristicas discontinuas. Bol. Soc. Mat. Mexicana, 7, 1–25 (1950)MathSciNetGoogle Scholar
  2. 2.
    Timoshenko, S.: Theory of Elastic Stability, McGraw-Hill, New York (1961)Google Scholar
  3. 3.
    Il'in, V. A. and Moiseev, E. I.: Nonlocal boundary value problem of the second kind for a Sturm-Liouville operator. Differential Equations, 23(8), 979–987 (1987)Google Scholar
  4. 4.
    Gupta, C. P.: A generalized multi-point boundary value problem for second order ordinary differential equations. Appl. Math. Comput., (89), 133–146 (1998)Google Scholar
  5. 5.
    Ma, R.: Positive solutions of a nonlinear three-point boundary value problem. Electronic Journal of Differential Equations, 34, 1–8 (1999)Google Scholar
  6. 6.
    Feng, W. and Webb, J. R. L.: Solvability of a m-point boundary value problems with nonlinear growth. J. Math. Anal. Appl., 212, 467–480 (1997)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Feng, W.: On a m-point nonlinear boundary value problem. Nonlinear Analysis TMA, 30(6), 5369–5374 (1997)CrossRefGoogle Scholar
  8. 8.
    Ma, R.: Existence theorems for a second order m-point boundary value problem. J. Math. Anal. Appl., 211, 545–555 (1997)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Staněk, S.: On some boundary value problems for second order functional differential equations. Nonlinear Analysis TMA, 28(3), 539–546 (1997)CrossRefGoogle Scholar
  10. 10.
    Guo, D. J. and Lakshmikantham, V.: Nonlinear Problems in Abstract Cones, Academic Press, San Diego (1988)Google Scholar
  11. 11.
    Anuradha, V., Hai, D. D. and Shivaji, R.: Existence results for superlinear semipositone boundary value problems. Proc. Amer. Math. Soc., 124(3), 757–763 (1996)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Mawhin, J.: “Topological Degree Methods on Nonlinear Boundary Value Problems”. NSF-CBMS Regional Conference Series in Math., Vol. 40, Amer. Math. Soc., Providence, RI (1979)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of MathematicsNorthwest Normal UniversityLanzhouP. R. China

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