Uniqueness of Gibbs Measure for Models With Uncountable Set of Spin Values on a Cayley Tree

We consider models with nearest-neighbor interactions and with the set $[0,1]$ of spin values, on a Cayley tree of order $k\ge 1$. It is known that the “splitting Gibbs measures” of the model can be described by solutions of a nonlinear integral equation. For arbitrary $k\ge 2$ we find a sufficient condition under which the integral equation has unique solution, hence under the condition the corresponding model has unique splitting Gibbs measure.