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From \cite{re} From \cite{re} it is known that ``translation-invariant Gibbsmeasures" of the model with an uncountable set of spin values can be describedby positive fixed points of a nonlinear integral operator of Hammerstein type.In \cite{enh2015, MSSX} there are main results on positive fixed points of theoperator of Hammerstein type with degenerate kernels, but it was not solved theexistence of Gibbs measures corresponding to the founded fixed points forconstructed kernels. This paper is an investigation of the papers\cite{enh2015} and \cite{MSSX}. In this paper we construct new degeneratekernels of the Hammerstein operator by taking into account problems in thetheory of Gibbs measure, i.e. each positive fixed point of the operator givestranslational-invariant Gibbs measure.

On positive fixed points of operator of Hammerstein type with degenerate kernel and Gibbs Measures