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Positive Solutions of Fourth-Order Problem Subject to Nonlocal Boundary Conditions

  • Lin Han
  • Guowei ZhangEmail author
  • Hongyu Li
Article
  • 14 Downloads

Abstract

In this paper, we study the fourth-order problem with the second derivative in nonlinearity under nonlocal boundary value conditions involving Stieltjes integrals. Some inequality conditions on nonlinearity and the spectral radius conditions of linear operators are presented that guarantee the existence of positive solutions to the problem by the theory of fixed point index on special cone. The conditions allow that the nonlinearity has superlinear or sublinear growth. Two examples are provided to support the main results under mixed boundary conditions involving multi-point with sign-changing coefficients and integral with sign-changing kernel.

Keywords

Positive solution Fixed point index Cone Spectral radius 

Mathematics Subject Classification

34B18 34B10 34B15 

Notes

Acknowledgements

The authors express their sincere gratitude to the referees for their careful reading of the original manuscript and thoughtful comments.

References

  1. 1.
    Alves, E., Ma, T.F., Pelicer, M.L.: Monotone positive solutions for a fourth order equation with nonlinear boundary conditions. Nonlinear Anal. 71, 3834–3841 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Bai, Z.: Positive solutions of some nonlocal fourth-order boundary value problem. Appl. Math. Comput. 215, 4191–4197 (2010)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Deimling, K.: Nonlinear Functional Analysis. Springer, Berlin (1985)CrossRefGoogle Scholar
  4. 4.
    Guo, D., Lakshmikantham, V.: Nonlinear Problems in Abstract Cones. Academic Press, San Diego (1988)zbMATHGoogle Scholar
  5. 5.
    Guo, Y., Yang, F., Liang, Y.: Positive solutions for nonlocal fourth-order boundary value problems with all order derivatives. Bound. Value Probl. (2012).  https://doi.org/10.1186/1687-2770-2012-29 MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Infante, G., Pietramala, P.: A cantilever equation with nonlinear boundary conditions. Electron. J. Qual. Theory Differ. Equ. (2009).  https://doi.org/10.14232/ejqtde.2009.4.15 MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Krasnosel’skiĭ, M.A.: Positive Solutions of Operator Equations (English translation). P. Noordhoff Ltd., Groningen (1964)Google Scholar
  8. 8.
    Kaufmann, E.R., Kosmatov, N.: Elastic beam problem with higher order derivatives. Nonlinear Anal. Real World Appl. 8, 811–821 (2007)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Li, Y.: Existence of positive solutions for the cantilever beam equations with fully nonlinear terms. Nonlinear Anal. Real World Appl. 27, 221–237 (2016)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Li, Y.: On the existence of positive solutions for the bending elastic beam equations. Appl. Math. Comput. 189, 821–827 (2007)MathSciNetzbMATHGoogle Scholar
  11. 11.
    Li, Y.: Positive solutions of fourth-order boundary value problems with two parameters. J. Math. Anal. Appl. 281, 477–484 (2003)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Ma, R.: Existence of positive solutions of a fourth-order boundary value problem. Appl. Math. Comput. 168, 1219–1231 (2005)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Minhós, F., Gyulov, T., Santos, A.I.: Lower and upper solutions for a fully nonlinear beam equation. Nonlinear Anal. 71, 281–292 (2009)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Ma, Y., Yin, C., Zhang, G.: Positive solutions of fourth-order problems with dependence on all derivatives in nonlinearity under Stieltjes integral boundary conditions. Bound. Value Probl. (2019).  https://doi.org/10.1186/s13661-019-1155-7 MathSciNetCrossRefGoogle Scholar
  15. 15.
    Webb, J.R.L., Infante, G.: Non-local boundary value problems of arbitrary order. J. London Math. Soc. 79, 238–259 (2009)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Webb, J.R.L., Infante, G., Franco, D.: Positive solutions of nonlinear fourth-order boundary-value problems with local and non-local boundary conditions. Proc. R. Soc. Edinburgh 138A, 427–446 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Yao, Q.: Local existence of multiple positive solutions to a singular cantilever beam equation. J. Math. Anal. Appl. 363, 138–154 (2010)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Yao, Q.: Monotonically iterative method of nonlinear cantilever beam equations. Appl. Math. Comput. 205, 432–437 (2008)MathSciNetzbMATHGoogle Scholar
  19. 19.
    Zhang, J., Zhang, G., Li, H.: Positive solutions of second-order problem with dependence on derivative in nonlinearity under Stieltjes integral boundary condition. Electron. J. Qual. Theory Differ. Equ. (2018).  https://doi.org/10.14232/ejqtde.2018.1.4 MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2020

Authors and Affiliations

  1. 1.Department of MathematicsNortheastern UniversityShenyangChina
  2. 2.College of Mathematics and Systems ScienceShandong University of Science and TechnologyQingdaoChina

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