Positive fixed points of Hammerstein integral operators with degenerate kernel

Positive fixed points of the Hammerstein integral operators with a degenerate kernel in the space of continuous functions $C[0,1]$ were explored. The problem of determining the number of positive fixed points of the Hammerstein integral operator was reduced to analyzing the positive roots of polynomials with real coefficients. A model on a Cayley tree with nearest-neighbor interactions and with the set $[0,1]$ of spin values was considered. It was proved that a unique translation-invariant Gibbs measure exists for this model.