Abstract
In this paper we consider a model with nearest-neighbor interactions and with the set [0, 1] of spin values, on the Cayley tree of order two. Translational-invariant Gibbs measures for the model investigated by properties of positive fixed points of quadratic operators in the cone of \(\mathbb {R}^{2}\). In the last section it is constructed kernels for the Hammerstien’s integral operator, which there exists two and three positive fixed points.
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