Lyapunov operator $\mathcal{L}$ with degenerate kernel and Gibbs measures

In this paper, we studied the fixed points of the Lyapunov operator with degenerate kernel, in which each fixed point of the operator is corresponds to a translation-invariant Gibbs measure with four competing interactions of models with uncountable set of spin values on the Cayley tree of order two. Also, it was proved that Lyapunov operator with degenerate kernel has at most three positive fixed points.