  In this paper we study positive fixed points of Hammerstein integral
operators with degenerate kernel in the cone of C[0, 1]. Problem on a number of
positive fixed points of the Hammerstein integral operator leads to the study
positive roots of polynomials with real coefficients. Consider a model on a
Cayley tree with nearest-neighbor interactions and with the set [0, 1] of spin
values. The uniqueness translational-invariant Gibbs measures for the given
model is proved.
