najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2024-15-1-23-30najo-27Research ArticleMATHEMATICSМАТЕМАТИКАOn solutions to nonlinear integral equation of the Hammerstein type and its applications to Gibbs measures for continuous spin systemsО решениях нелинейного интегрального уравнения тип Хаммерштейна и его приложения к мерам гиббса длянепрерывной спиновой системыhttps://orcid.org/0009-0009-4889-1889МавлоновИ. М.MavlonovI. M.

Ismoil M. Mavlonov

University str., 4 Olmazor district, Tashkent, 100174

mavlonovismoil16@gmail.comhttps://orcid.org/0009-0008-0115-3937СаттаровА. М.SattarovA. M.

Aloberdi M. Sattarov

Namangan region, Namangan city, 111 Beshkapa str., Namangan, 160100

saloberdi90@mail.ruhttps://orcid.org/0009-0006-3128-6605КаримоваС. А.KarimovaS. A.

Sevinchbonu A. Karimova

University str., 4 Olmazor district, Tashkent, 100174

https://orcid.org/0000-0001-9388-122XХайдаровФ. Х.HaydarovF. H.

Farhod H. Haydarov

54 Mustaqillik Ave., Tashkent, 100007; University str., 4 Olmazor district, Tashkent, 100174; 9, University str., Tashkent, 100174

haydarovimc@mail.ruNational University of UzbekistanUzbekistanUniversity of Business and ScienceUzbekistanInstitute of Mathematics; New Uzbekistan University; National University of UzbekistanUzbekistan2024310520251512330Copyright © Mavlonov I.M., Sattarov A.M., Karimova S.A., Haydarov F.H., 20252025Мавлонов И.М., Саттаров А.М., Каримова С.А., Хайдаров Ф.Х.Mavlonov I.M., Sattarov A.M., Karimova S.A., Haydarov F.H.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/27

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.

В данной работе рассматривается проблема построения ядер уравнений типа Хаммерштейна, чьи положительные решения не являются уникальными. Эта проблема возникает из теории мер Гиббса, и каждое положительное решение уравнения соответствует одной трансляционно-инвариантной мере Гиббса. Кроме того, проблема поиска ядер, для которых количество положительных решений уравнения превышает одно, эквивалентна проблеме поиска моделей с фазовым переходом. Проблема построения ядер уравнения, для которых есть как минимум два положительных решения, также изучается в [1, 4, 12, 13, 15]. В этих статьях количество решений, соответствующих построенным ядрам, не превышает 3, и в свою очередь, это дает нам возможность проверить наличие фазовых переходов. Построенные в данной статье ядра более общие, чем ядра в упомянутых выше работах, и помимо проверки фазовых переходов, они позволяют классифицировать множество мер Гиббса.

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The authors declare that there are no conflicts of interest present.