On solutions to nonlinear integral equation of the Hammerstein type and its applications to Gibbs measures for continuous spin systems
https://doi.org/10.17586/2220-8054-2024-15-1-23-30
Abstract
The paper deals with the problem of constructing kernels of Hammerstein-type equations whose positive solutions are not unique. This problem arises from the theory of Gibbs measures, and each positive solution of the equation corresponds to one translation-invariant Gibbs measure. Also, the problem of finding kernels for which the number of positive solutions to the equation is greater than one is equivalent to the problem of finding models which has phase transition. In these articles, the number of solutions corresponding to the constructed kernels does not exceed 3, and in turn, it only gives us a chance to check the existence of phase transitions. The constructed kernels in this paper are more general than the kernels in the abovementioned papers and except for checking phase transitions, it allows us to classify the set of Gibbs measures.
Keywords
Supporting agencies: We thank the referee for careful reading of the manuscript; in particular, for some comments, useful discussions, and suggestions which have improved the paper
About the Authors
I. M. Mavlonov
National University of Uzbekistan
Uzbekistan
Uzbekistan
Ismoil M. Mavlonov
University str., 4 Olmazor district, Tashkent, 100174
A. M. Sattarov
University of Business and Science
Uzbekistan
Uzbekistan
Aloberdi M. Sattarov
Namangan region, Namangan city, 111 Beshkapa str., Namangan, 160100
S. A. Karimova
National University of Uzbekistan
Uzbekistan
Uzbekistan
Sevinchbonu A. Karimova
University str., 4 Olmazor district, Tashkent, 100174
F. H. Haydarov
Institute of Mathematics; New Uzbekistan University; National University of Uzbekistan
Uzbekistan
Uzbekistan
Farhod H. Haydarov
54 Mustaqillik Ave., Tashkent, 100007; University str., 4 Olmazor district, Tashkent, 100174; 9, University str., Tashkent, 100174
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Review
For citations:
Mavlonov I.M., Sattarov A.M., Karimova S.A., Haydarov F.H. On solutions to nonlinear integral equation of the Hammerstein type and its applications to Gibbs measures for continuous spin systems. Nanosystems: Physics, Chemistry, Mathematics. 2024;15(1):23-30. https://doi.org/10.17586/2220-8054-2024-15-1-23-30
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