Skip to main content
SpringerLink
Log in

Multiple positive solutions for nonlinear third order general three-point boundary value problems

  • Published:
Differential Equations and Dynamical Systems Aims and scope Submit manuscript

Abstract

We consider the third order nonlinear three point boundary value problem,

$$ y'''(t) + f(t,y(t)) = 0,t \in [t_1 ,t_3 ], $$

subject to the general boundary conditions

$$ \sum\limits_{j = 1}^3 {\alpha _{ij} y^{(j - 1)} (t_i ) = 0,i = 1,2,3,} $$

where f: [t 1, t 3] × IR → IR is continuous, t 1 < t 2 < t 3 and α i1, α i2, α i3, i = 1, 2, 3, are real constants. We establish the existence of at least three positive solutions by using well-known Leggett-Williams fixed point theorem and an example is presented.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. D. R. Anderson, Green’s function for third order generaliged right focal problem, J. Math. Anal. Appl., 288(2003), 1–14.

    Article  MATH  MathSciNet  Google Scholar 

  2. J. M. Davis, J. Henderson, K. R. Prasad and W. Yin, Eigenvalue intervals for nonlinear right focal problems, Appl. Anal., 74(2000), 215–231.

    Article  MATH  MathSciNet  Google Scholar 

  3. P.W. Eloe and J. Henderson, Positive solutions for (n-1,1) conjugate boundary value problems, Nonlinear. Anal., 28(1997), 1669–1680.

    Article  MATH  MathSciNet  Google Scholar 

  4. P. W. Eloe and J. Henderson, Positive solutions for nonlinear (k,n-k) conjugate eigenvalue problems, Diff. Eqns. Dyn. Sys., 6(1998), 309–317.

    MATH  MathSciNet  Google Scholar 

  5. P. W. Eloe and J. Henderson, Positive solutions for nonlinear multipoint conjugate eigenvalue problems, Elec. J. Diff. Eqns., 1997(1997), No. 3, 1–11.

    MathSciNet  Google Scholar 

  6. L. H. Erbe, S. Hu and H. Wang, Multiple positive solutions of some boundary value problems, J. Math. Anal. Appl., 184(1994), 640–648.

    Article  MATH  MathSciNet  Google Scholar 

  7. L. H. Erbe and H. Wang, On the existence of positive solutions of ordinary differential equations, Proc. Amer. Math. Soc., 120(1994), 743–748.

    Article  MATH  MathSciNet  Google Scholar 

  8. J. R. Graef, C. Qian and B. Yang, Multiple symmetric positive solutions of a class of boundary value problems for higher order differential equations, Proc. Amer. Math. Soc., 131(2003), 577–585.

    Article  MATH  MathSciNet  Google Scholar 

  9. M. Gregus, “Third Order Linear Differential Equations and Mathematical Applications”, Reidel, Dordrecht, 1987.

    Google Scholar 

  10. L. J. Guo, J. P. Sum and Y. H. Zhao, Existence of Positive Solutions for nonlinear third order three point Boundary Value Problems, Nonlinear Anal., (2007), doi,10.1016 J.na. 2007.03.008.

  11. J. Henderson and E. R. Kaufmann, Multiple positive solutions for focal boundary value problems, Comm. Appl. Anal., 1(1997), 53–60.

    MATH  MathSciNet  Google Scholar 

  12. E. R. Kaufmann, Multiple positive solutionsfor higer order boundary value problems, Rocky Mtn. J. Math., 28(1998), 1017–1028.

    Article  MATH  MathSciNet  Google Scholar 

  13. W. C. Lian, F. H. Wang and C. C. Yeh, On the existence of positive solution of nonlinear second order differential equations, Proc. Amer. Math. Soc., 124(1996), 1111–1126.

    Article  MathSciNet  Google Scholar 

  14. K. R. Prasad and P. Murali, Eigenvalue intervals for n th order two-point boundary value problems, Nonlinear Stud., 2008, in press.

  15. K. R. Prasad and P. Murali, Eigenvalue intervals for n th order differential equations on time scales, International J. Pure and Appl. Math., 44(2008), 737–753.

    MATH  MathSciNet  Google Scholar 

  16. K. R. Prasad and P. Murali, Existence of positive solutions for n th order two point boundary value problems on time scales, Elec. J. Diff. Eqn., (2008).

  17. Y. Sun, Positive solutions of singular third order three point boundary value problems, J. Math. Anal. Appl., 306(2005), 589–603.

    Article  MATH  MathSciNet  Google Scholar 

  18. H. R. Sun and W. T. Li, Positive solutions for nonlinear three-point boundary value problems on time scales, J. Math. Anal. Appl., 299(2004), 508–524.

    Article  MATH  MathSciNet  Google Scholar 

  19. Q. Yao, The existence and multiplicity of positive solutions for a third-order three-point boundary value problem, Acta. Math. Appl. Sinica, 19(2003), 117–122.

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Applied Mathematics, Andhra University, Visakhapatnam, 530003, India

    K. R. Prasad & P. Murali

Authors
  1. K. R. Prasad
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. P. Murali
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to K. R. Prasad.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Prasad, K.R., Murali, P. Multiple positive solutions for nonlinear third order general three-point boundary value problems. Differ Equ Dyn Syst 16, 63–75 (2008). https://doi.org/10.1007/s12591-008-0005-3

Download citation

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12591-008-0005-3

MSC

Keywords

Search

Navigation