Advertisement

Multiple Solutions of Boundary-Value Problems for Fourth-Order Differential Equations with Deviating Arguments

  • T. Jankowski
  • R. Jankowski
Article
First Online:

Abstract

This paper considers fourth-order differential equations with four-point boundary conditions and deviating arguments. We establish sufficient conditions under which such boundary-value problems have positive solutions. We discuss such problems in the cases when the deviating arguments are delayed or advanced. In order to obtain the existence of at least three positive solutions, we use a fixed-point theorem due to Avery and Peterson. To the authors’ knowledge, this is a first paper where the existence of positive solutions of boundary-value problems for fourth-order differential equations with deviating arguments is discussed.

Keywords

Fourth-order differential equations Existence of positive solutions Boundary conditions Advanced and delayed arguments 
This is a preview of subscription content, log in to check access.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bai, C.: Triple positive solutions of three-point boundary value problems for fourth-order differential equations. Comput. Math. Appl. 56, 1364–1371 (2008) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Bai, Z., Wang, H.: On positive solutions of some nonlinear fourth-order beam equations. J. Math. Anal. Appl. 270, 357–368 (2002) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Hao, Z., Liu, L., Debnath, L.: A necessary and sufficient condition for the existence of positive solutions of fourth-order singular boundary value problems. Appl. Math. Lett. 16, 279–285 (2003) MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Li, F., Zhang, Q., Liang, Z.: Existence and multiplicity of solutions of a kind of fourth-order boundary value problem. Nonlinear Anal. 62, 803–816 (2005) MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Liu, B.: Positive solutions of fourth-order two point boundary value problems. Appl. Math. Comput. 148, 407–420 (2004) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Yang, D., Zhu, H., Bai, C.: Positive solutions for semipositone fourth-order two-point boundary value problems. Electron. J. Differ. Equ. 2007(16), 1–8 (2007) CrossRefMathSciNetGoogle Scholar
  7. 7.
    Zhang, X., Liu, L., Zou, H.: Positive solutions of fourth-order singular three point eigenvalue problems. Appl. Math. Comput. 189, 1359–1367 (2007) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Zhang, M., Wei, Z.: Existence of positive solutions for fourth-order m-point boundary value problem with variable parameters. Appl. Math. Comput. 190, 1417–1431 (2007) MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Jankowski, T.: Positive solutions of three-point boundary value problems for second order impulsive differential equations with advanced arguments. Appl. Math. Comput. 197, 179–189 (2008) MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jankowski, T.: Three positive solutions to second order three-point impulsive differential equations with deviating arguments. Int. J. Comput. Math. 87, 215–225 (2010) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Yang, C., Zhai, C., Yan, J.: Positive solutions of the three-point boundary value problem for second order differential equations with an advanced argument. Nonlinear Anal. 65, 2013–2023 (2006) MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Bai, D., Xu, Y.: Positive solutions of second-order two-delay differential systems with twin parameter. Nonlinear Anal. 63, 601–617 (2005) MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Bai, D., Xu, Y.: Existence of positive solutions for boundary value problem of second-order delay differential equations. Appl. Math. Lett. 18, 621–630 (2005) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Du, B., Hu, X., Ge, W.: Positive solutions to a type of multi-point boundary value problem with delay and one-dimensional p-Laplacian. Appl. Math. Comput. 208, 501–510 (2009) MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Jiang, D.: Multiple positive solutions for boundary value problems of second-order delay differential equations. Appl. Math. Lett. 15, 575–583 (2002) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Wang, W., Sheng, J.: Positive solutions to a multi-point boundary value problem with delay. Appl. Math. Comput. 188, 96–102 (2007) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Wang, Y., Zhao, W., Ge, W.: Multiple positive solutions for boundary value problems of second order delay differential equations with one-dimensional p-Laplacian. J. Math. Anal. Appl. 326, 641–654 (2007) MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Avery, R.I., Peterson, A.C.: Three positive fixed points of nonlinear operators on ordered Banach spaces. Comput. Math. Appl. 42, 313–322 (2001) MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of Differential Equations and Applied MathematicsGdansk University of TechnologyGdańskPoland
  2. 2.Faculty of Civil and Environmental EngineeringGdansk University of TechnologyGdańskPoland

Personalised recommendations