Fixed points of Lyapunov integral operators and Gibbs measures

Abstract

In this paper we shall consider the connections between Lyapunov integral operators and Gibbs measures for models with four competing interactions and uncountable (i.e. [0, 1]) set of spin values on a Cayley tree. We prove the existence of fixed points of Lyapunov integral operators and give a condition of uniqueness of a fixed point.

On solutions to nonlinear integral equation of the Hammerstein type and its applications to Gibbs measures for continuous spin systems

“…For the case k ≥ 2, models in which phase transitions exist are constructed in [8] and the generalization of the constructed model is studied in [7,9]. In [6], the problem of finding translation-invariant (periodic with period two) Gibbs measures of the model (2.6) was reduced to finding positive fixed points of the nonlinear operator of Hammerstein type and the problem of the existence of fixed points of this operator considered in [10,15,16] ( [13,14]). But, from the Gibbs measure theory, it is interesting to study fixed points of the nonlinear operator of the Hammerstein type with degenerate kernel and all the above works corresponding to non-degenerate kernels.…”

“…Also, in [7][8][9][10], results on non-uniqueness of positive solutions to the equation with non-degenerate kernels are given and for the case of degenerate kernel can be seen in [11,12]. Finally, in [13,14] translation-invariant Gibbs measures for models with four competing interactions are described by solutions to the Lyapunov integral equation. In this paper, we construct more general kernels which gives us a chance to check the existence of phase transitions and to classify the set of Gibbs measures.…”

Positive fixed points of Lyapunov operator

“…The existence of fixed points of Lyapunov's operator A is proved in [16]. A sufficient condition of uniqueness of fixed points of Lyapunov operator A s given (see [8]).…”