Korean Journal of Mathematics

The Kangwon-Kyungki Mathematical Society (강원경기수학회)
권호 표지

EXISTENCE OF NONNEGATIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEMS

  • Received : 2008.10.20
  • Published : 2009.12.30
Download PDF
⟨ Previous Next ⟩

Abstract

By means of Green function and fixed point theorem related with cone theoretic method we show that there exist multiple nonnegative solutions of a Dirichlet problem $$\array{-[p(t)x^{\prime}(t)]^{\prime}={\lambda}q(t)f(x(t)),\;t{\in}I=[0,\;T]\\x(0)=0=x(T)}$$, and a mixed problem $$\array{-[p(t)x^{\prime}(t)]^{\prime}={\mu}q(t)f(x(t)),\;t{\in}I=[0,\;T]\\x^{\prime}(0)=0=x(T)}$$, where ${\lambda}$ and ${\mu}$ are positive parameters.

Keywords

Acknowledgement

Supported by : Hallym University

References

  1. F. Atici, Two positive solutions of a boundary value problem for a difference equations, J. Differece Equations Appl. 1, (1995), 263-270. https://doi.org/10.1080/10236199508808026
  2. K. Deimling, Nonlinear Functional Annalysis, Springer-Verlag, New York, 1985.
  3. P. W. Eloe and J. Henderson, Positive Solutions for higher order differential equations, Electron. J. Differential Equations. 3, (1995), 1-8.
  4. P. W. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc. 120, (1994), 743-748. https://doi.org/10.1090/S0002-9939-1994-1204373-9
  5. P. W. Erbe and H Wang, Existence or nonexistence of positive solutions in annular domains, WSSIAA. 3, (1994), 207-217.
  6. K. S. Ha and Y. H. Lee Existence of multiple positive solutions of singular boundary value problems, Nonlinear Analysis, Theory, Methods & Applications. Vol 28, (1997), No 8, 1429-1438. https://doi.org/10.1016/0362-546X(95)00231-J
  7. J. Henderson and H. Wang, Positive Solutions for Nonlinear Eigenvalue Problems, Jour. of Math. Anal. and Appl. Publ. 208, (1997), 252-259. https://doi.org/10.1006/jmaa.1997.5334
  8. E. R. Kaufmann and N. Kosmatov, A multiplicity result for a boundary value problems with infinitely many singularities, Jour. of Math. Anal. and Appl. Publ. 269, (2002), 444-453. https://doi.org/10.1016/S0022-247X(02)00025-2
  9. R. Leggett and L. Wiliams, Multiple positive fixed points of nonlinear operators on ordered Banach spaces, Indiana Univ. Math. J. Publ. 28, (1979), 673-688. https://doi.org/10.1512/iumj.1979.28.28046
  10. J. Wang, Solvability of singular nonlinear two-point boundary problems, Nonlinear Analysis, Theory, Methods & Applications. Vol 24, (1995), No 4, 555-561. https://doi.org/10.1016/0362-546X(95)93091-H
  11. Y. H. Lee, A multiplicity result of positive solutions for the generalized Gelfand type singular boundary problems, Nonlinear Analysis, Theory, Methods & Applications. Vol 30, (1997), No 6, 3829-3835. https://doi.org/10.1016/S0362-546X(97)00077-1
  12. E. Zeidler, Nonlinear Functional Annalysis and its Applications I, Fixed-Point Theorems, Springer-Verlag, New York, 1985.