Translation-invariant Gibbs measures for a model with logarithmic potential on a Cayley tree

In this paper, we consider a model with logarithmical potential and with the set [0, 1] of spin values, on a Cayley tree $\Gamma^k$ of the order $k$. In the case $k= 2;3$, we shall prove that, there is a unique translation-invariant splitting Gibbs measure for this model. For the case $k=4$, we show that there are three translation-invariant Gibbs measures for this model.