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.
Similar content being viewed by others
References
V. Doležal, Monotone Operators and Applications in Control and Network Theory (Studies in Automation and Control, Vol. 2), Elsevier, New York (1979).
A. A. Richard, A. de Korvin, and V. Van Tho, “Integration theory for Hammerstein operators,” J. Math. Anal. Appl., 61, 72–96 (1977).
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).
E. Zeidler, Nonlinear Functional Analyses and Its Applications, Vol. I: Fixed-Point Theorem, Springer, New York (1986).
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).
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).
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).
H. Amann, “On the number of solutions of nonlinear equations in ordered Banach spaces,” J. Funct. Anal., 11, 346–384 (1972).
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).
A. Ruiz-Herrera, “Permanence of two species and fixed point index,” Nonlinear Anal., 74, 146–153 (2011).
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).
U. A. Rozikov, Gibbs Measures on Cayley Trees, World Sci., Singapore (2013).
F. H. Haydarov, “Fixed points of Lyapunov integral operators and Gibbs measures,” Positivity, 22, 1165–1172 (2018).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Teoreticheskaya i Matematicheskaya Fizika, 2021, Vol. 208, pp. 440–451 https://doi.org/10.4213/tmf10083.
Rights and permissions
About this article
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
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0040577921090051