Positive fixed points of Lyapunov operator
https://doi.org/10.17586/2220-8054-2020-11-4-373-378
Abstract
In this paper, fixed points of Lyapunov integral equation are found and considered the connections between Gibbs measures for four competing interactions of models with uncountable (i.e. [0,1]) set of spin values on the Cayley tree of order two.
About the Authors
R. N. Ganikhodjaev
National University of Uzbekistan
Uzbekistan
Uzbekistan
100174, Tashkent
R. R. Kucharov
National University of Uzbekistan
Uzbekistan
Uzbekistan
100174, Tashkent
K. A. Aralova
National University of Uzbekistan
Uzbekistan
Uzbekistan
100174, Tashkent
References
1. Pigorov S.A., Sinai Ya.G. Phase diagrams of classical lattice systems (Russian). Theor. and Math. Phys., 1975, 25), P. 358–369.
2. Pigorov S.A., Sinai Ya.G. Phase diagrams of classical lattice systems. Continuation (Russian). Theor. and Math. Phys., 1976, 26, P. 61–76.
3. Sinai Ya.G. Theory of phase transitions: Rigorous Results, Pergamon, Oxford, 1982.
4. Kotecky R., Shlosman S.B. First-order phase transition in large entropy lattice models. Commun. Math. Phys., 1982, 83, P. 493–515.
5. Mazel A., Suhov Y., Stuhl I. A classical WR model with q particle types. J.Stat.Phys., 2015, 159, P. 1040–1086.
6. Mazel A., Suhov Y., Stuhl I., Zohren S. Dominance of most tolerant species in multi-type lattice Widom-Rowlinson models. Journ. Stat. Mech., 2014, P. 08010.
7. Eshkabilov Yu.Kh., Haydarov F.H. On positive solutions of the homogeneous Hammerstein integral equation. Nanosystems: Physics, Chemistry, Mathematics, 2015, 6(5), P. 618–627.
8. Eshkabilov Yu.Kh., Rozikov U.A,Haydarov F.H. Non-uniqueness of Gibbs measure for models with uncountable set of spin values on a Cayley Tree. Journal Statistical Physics, 2012, 147, P. 779–794.
9. Eshkabilov Yu.Kh., Rozikov U.A, Haydarov F.H. Uniqueness of Gibbs measure for models with uncountable set of spin values on a Cayley tree. Math.Phys.Anal.Geom., 2013, 16, P. 1–17.
10. Eshkabilov Yu.K., Haydarov F.H., Nodirov Sh.D. Positive fixed points of quadratic operators and Gibbs measures. Positivity, 2016, 20(4), P. 929–943.
11. Eshkabilov Yu.Kh., Haydarov F.H. Lyapunov operator L with degenerate kernel and Gibbs measures. Nanosystems: Physics, Chemistry, Mathematics, 2017, 8(5), P. 553–558.
12. Rozikov U.A. Gibbs measures on a Cayley trees, World Sci. Pub, Singapore, 2013.
13. Rozikov U.A., Haydarov F.H. Four competing interactions for models with an uncountable set of spin values on a Cayley tree. Theor. Math. Phys., 2017, 191(2), P. 748–761.
14. Krasnosel’ski M.A. Positive Solutions of Opertor Equations. Gos. Izd. Moscow, 1969, Russian.
15. Nirenberg L. Topics in nonlinear functional analysis. AMS, Courant Lec. Notes in Math, 6, N.Y., 2001.
16. Haydarov F.H. Fixed points of Lyapunov integral operators and Gibbs measures. Positivity, 2018, 22(4), P. 1165–1172.
17. Georgii H.O. Gibbs Measures and Phase Transitions, 2nd edn. de Gruyter Studies in Mathematics, vol. 9. Walter de Gruyter, Berlin, 2011.
Review
For citations:
Ganikhodjaev R.N., Kucharov R.R., Aralova K.A. Positive fixed points of Lyapunov operator. Nanosystems: Physics, Chemistry, Mathematics. 2020;11(4):373–378. https://doi.org/10.17586/2220-8054-2020-11-4-373-378
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