On positive solutions of the homogeneous Hammerstein integral equation

TL;DR

This paper investigates the existence and uniqueness of positive solutions for a nonlinear Hammerstein integral equation, with applications to Gibbs measures on Cayley trees, providing new theoretical insights.

Contribution

It establishes conditions for finite positive solutions and applies these results to models on Cayley trees, advancing understanding of such integral equations.

Findings

Proved existence of finite positive solutions.

Established uniqueness conditions.

Applied results to Gibbs measures on Cayley trees.

Abstract

In this paper the existence and uniqueness positive fixed points of the one nonlinear integral operator are discussed. We prove that existence finite positive solutions of the integral equation of Hammerstein type. Obtained results applied to study Gibbs measures for models on a Cayley tree.