In this paper it is found fixed points of Lyapunov integral equation 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.
1. Eshkabilov Yu. Kh, Haydarov F. H., Rozikov U. A. Uniqueness of Gibbs measure for models with uncountable set of spin values on a Cayley tree. Math. Phys. Anal. Geom., 2013, V.16, pp.1–17.
2. Georgii H. O. Gibbs Measures and Phase Transitions, 2nd edn. De Gruyter Studies in Mathematics, Vol. 9. Walter de Gruyter, Berlin, 2011. – 545 p.
3. Krasnoselâski M. A. Positive Solutions of Opertor Equations. Fizmatgiz, Moscow, 1962 (Russian). – 394 p.
4. Nirenberg L. Topics in nonlinear functional analysis. AMS, Courant Lec. Notes in Math, 6, N.Y., 2001. – 145 p.
5. 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, V. 191, No. 3, pp.910–923.
6. Rozikov U. A. Gibbs measures on Cayley trees. World Scientific, 2013. – 404 p.
"Positive fixed points of Lyapunov integral operators and Gibbs measures,"
Bulletin of National University of Uzbekistan: Mathematics and Natural Sciences: Vol. 1:
1, Article 9.