Skip to main content
SpringerLink
Log in

A multiplicity result of positive solutions for a class of multi-parameter ordinary differential systems

  • Published:
Acta Mathematicae Applicatae Sinica, English Series Aims and scope Submit manuscript

Abstract

In this paper, we establish a result of Leray-Schauder degree on the order interval which is induced by a pair of strict lower and upper solutions for a system of second-order ordinary differential equations. As applications, we prove the global existence of positive solutions for a multi-parameter system of second-order ordinary differential equations with respect to parameters. The discussion is based on the result of Leray-Schauder degree on the order interval and the fixed point index theory in cones.

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. Amann, H. Existence of multiple solutions for nonlinear elliptic boundary value problems. Indiana Univ. Math. J., 21: 925–935 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  2. Amann, H. On the number of solutions of nonlinear equations in ordered Banach spaces. J. Funct. Anal., 11: 346–384 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amann, H. Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces. SIAM Review, 18: 620–709 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bernfeld, S.R., Lakshmikantham, V. An Introduction to Nonlinear Boundary Value Problems. Academic Press, New York, 1974

    MATH  Google Scholar 

  5. Cheng, X., Zhong, C. Existence of positive solutions for a second-order ordinary differential system. J. Math. Anal. Appl., 312: 14–23 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheng, X. Existence of positive solutions for a class of second-order ordinary differential systems. Nonlinear Anal., 69: 3042–3049 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  7. Cheng, X.Y., Zhang, Z.T. Existence of positive solutions for a general nonlinear eigenvalue problem. Acta Math. Appl. Sinica, English Series, 27: 367–372 (2011)

    Article  MathSciNet  Google Scholar 

  8. Cheng, X.Y., Zhang, Z.T. Positive solutions for a multi-parameter system of second-order ordinary differential equations. Sci China Math, 54: 959–972 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  9. Correa, F.J.S.A. On pairs of positive solutions for a class of sub-superlinear elliptic problems. Diff. and Int. Equations, 5: 387–392 (1992)

    MathSciNet  MATH  Google Scholar 

  10. Deimling, K. Nonlinear Functional Analysis. Springer-Verlag, New York, 1985

    Book  MATH  Google Scholar 

  11. Dunninger, D.R., Wang, H. Existence and multiplicity of positive solutions for elliptic systems. Nonlinear Anal., 29: 1051–1060 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Dunninger, D.R., Wang, H. Multiplicity of positive radial solutions for an elliptic system on an annulus. Nonlinear Anal., 42: 803–811 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Henderson, J., Wang, H. Positive solutions for nonlinear eigenvalue problems. J. Math. Anal. Appl., 208: 252–259 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo, D., Lakshmikantham, V. Nonlinear Problems in Abstract Cones. Academic Press, New York, 1988

    MATH  Google Scholar 

  15. Lan, K., Webb, R.L. Positive solutions of semilinear equations with singularities. J. Diff. Equations, 148: 407–421 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lee, Y.H. A multiplicity result of positive radial solutions for a multiparameter elliptic system on an exterior domain. Nonlinear Anal., 45: 597–611 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lee, Y.H. Multiplicity of positive radial solutions for multiparameter semilinear elliptic systems on an annulus. J. Diff. Equations, 174: 420–441 (2001)

    Article  MATH  Google Scholar 

  18. Liu, X.L., Wu, J.H. Positive solutions for a Hammerstein integral equation with a parameter. Appl. Math. Lett., 22: 490–494 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ma, R. Existence of positive radial solutions for elliptic systems. J. Math. Anal. Appl., 201: 375–386 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  20. do Ó, J.M., Lorca, S., Ubilla, P. Local superlinearity for elliptic systems involving parameters. J. Diff. Equations 211: 1–19 (2005)

    Google Scholar 

  21. do Ó, J.M., Lorca, S., Sánchez, J., Ubilla, P. Positive solutions for a class of multiparameter ordinary elliptic systems. J. Math. Anal. Appl., 332: 1249–1266 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Whyburn, G.T. Topological Analysis. Princeton University Press, Princeton, 1958

    MATH  Google Scholar 

  23. Zhong, C., Fan, X., Chen, W. An Introduction to Nonlinear Functional Analysis. Lanzhou University Press, Lanzhou, 1998 (in Chinese)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Xi-you Cheng

  2. School of Science, Tianjin Polytechnic University, Tianjin, 300387, China

    Xiao-lin Yan

Authors
  1. Xi-you Cheng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xiao-lin Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xi-you Cheng.

Additional information

Supported in part by the National Natural Science Foundation (No. 11101404) of China and the Fundamental Research Funds for the Central Universities (No. lzujbky-2012-11).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Cheng, Xy., Yan, Xl. A multiplicity result of positive solutions for a class of multi-parameter ordinary differential systems. Acta Math. Appl. Sin. Engl. Ser. 28, 653–662 (2012). https://doi.org/10.1007/s10255-012-0180-4

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10255-012-0180-4

Keywords

2000 MR Subject Classification

Search

Navigation