  In this paper we consider four competing interactions (external field,
nearest neighbor, second neighbors and triples of neighbors) of models with
uncountable (i.e. $[0,1]$) set of spin values on the Cayley tree of order two.
We reduce the problem of describing the "splitting Gibbs measures" of the model
to the analysis of solutions to some nonlinear integral equation and study some
particular cases for Ising and Potts models. Also we show that periodic Gibbs
measures for given models are either translation-invariant or periodic with
period two and we give examples of the non-uniqueness of translation-invariant
Gibbs measures.
