Skip to main content
Log in

Existence and uniqueness of fixed points of an integral operator of Hammerstein type

  • Research Articles
  • Published:
Theoretical and Mathematical Physics Aims and scope Submit manuscript

Abstract

We study the existence and uniqueness of positive fixed points of an integral operator of the Hammerstein type in the space of continuous functions. Under certain conditions on the integral operator, we establish a sufficient condition for the the integral operator to have exactly one positive fixed point.

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

Access this article

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

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. V. Doležal, Monotone Operators and Applications in Control and Network Theory (Studies in Automation and Control, Vol. 2), Elsevier, New York (1979).

    MATH  Google Scholar 

  2. A. A. Richard, A. de Korvin, and V. Van Tho, “Integration theory for Hammerstein operators,” J. Math. Anal. Appl., 61, 72–96 (1977).

    Article  MathSciNet  Google Scholar 

  3. A. Cabada, J. Á. Cid, and G. Infante, “A positive fixed point theorem with applications to systems of Hammerstein integral equations,” Bound. Value Probl., 2014, 254, 10 pp. (2014).

    Article  MathSciNet  Google Scholar 

  4. E. Zeidler, Nonlinear Functional Analyses and Its Applications, Vol. I: Fixed-Point Theorem, Springer, New York (1986).

    Book  Google Scholar 

  5. F. Li, Y. Li, and Z. Liang, “Existence of solutions to nonlinear Hammerstein integral equations and applications,” J. Math. Anal. Appl., 323, 209–227 (2006).

    Article  MathSciNet  Google Scholar 

  6. W.-Q. Deng, “An iterative solution to a system of nonlinear Hammerstein type equations and a system of generalized mixed equilibrium problems,” J. Fixed Point Theor. Appl., 19, 2051–2068 (2017).

    Article  MathSciNet  Google Scholar 

  7. D. Guo and V. Lakshmicantham, Nonlinear Problems in Abstract Cones (Notes and Reports in Mathematics in Science and Engineering, Vol. 5), Academic Press, New York (1988).

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  9. U. A. Rozikov and F. Kh. Khaidarov, “Four competing interactions for models with an uncountable set of spin values on a Cayley tree,” Theoret. and Math. Phys., 191, 910–923 (2017).

    Article  ADS  MathSciNet  Google Scholar 

  10. A. Ruiz-Herrera, “Permanence of two species and fixed point index,” Nonlinear Anal., 74, 146–153 (2011).

    Article  MathSciNet  Google Scholar 

  11. Yu. Kh. Eshkabilov, F. H. Haydarov, and U. A. Rozikov, “Uniqueness of Gibbs measure for models with uncountable set of spin values on a Cayley tree,” Math. Phys. Anal. Geom., 16, 1–17 (2013).

    Article  MathSciNet  Google Scholar 

  12. U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore (2013).

    Book  Google Scholar 

  13. F. H. Haydarov, “Fixed points of Lyapunov integral operators and Gibbs measures,” Positivity, 22, 1165–1172 (2018).

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to F. H. Haydarov.

Additional information

Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 208, pp. 440–451 https://doi.org/10.4213/tmf10083.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Haydarov, F.H. Existence and uniqueness of fixed points of an integral operator of Hammerstein type. Theor Math Phys 208, 1228–1238 (2021). https://doi.org/10.1134/S0040577921090051

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0040577921090051

Keywords

Navigation