Phase Transitions for a model with uncountable set of spin values on a Cayley tree

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In this paper we consider a model with nearest-neighbor interactions and with the set [0,1] of spin values, on a Cayley tree of order k ≥ 2. To study translation-invariant Gibbs measures of the model we drive an nonlinear functional equation. For k=2 and 3 under some conditions on parameters of the model we prove non-uniqueness of translation-invariant Gibbs measures (i.e. there are phase transitions).