Positive fixed points of Hammerstein's integral operators with degenerate kernel

Abstract

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.