Positive fixed points of cubic operators on $\mathbb{R}^{2}$ and Gibbs measures

One model with nearest neighbour interactions of spins with values from the set $[0,1]$ on the Cayley tree of order three is considered in the paper. Translation-invariant Gibbs measures for the model are studied. Results are proved by using properties of the positive fixed points of a cubic operator in the cone $\mathbb{R}_+^{2}$.