Fixed points of Lyapunov integral operators and Gibbs measures

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