On solutions to nonlinear integral equation of the Hammerstein type and its applications to Gibbs measures for continuous spin systems

The paper deals with the problem of constructing kernels of Hammerstein-type equations whose positive solutions are not unique. This problem arises from the theory of Gibbs measures, and each positive solution of the equation corresponds to one translation-invariant Gibbs measure. Also, the problem of finding kernels for which the number of positive solutions to the equation is greater than one is equivalent to the problem of finding models which has phase transition. In these articles, the number of solutions corresponding to the constructed kernels does not exceed 3, and in turn, it only gives us a chance to check the existence of phase transitions. The constructed kernels in this paper are more general than the kernels in the abovementioned papers and except for checking phase transitions, it allows us to classify the set of Gibbs measures.