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

TL;DR

This paper investigates positive fixed points of Hammerstein integral operators with degenerate kernels and applies the findings to prove the uniqueness of Gibbs measures in a spin model on a Cayley tree.

Contribution

It introduces a method to analyze positive fixed points of degenerate kernel operators and applies it to establish uniqueness of Gibbs measures for a specific spin model.

Findings

Positive fixed points correspond to roots of certain polynomials.

Uniqueness of translational-invariant Gibbs measures is proved.

The approach links integral operator fixed points to polynomial root analysis.

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.