Existence and uniqueness of fixed points of integral operator of Hammerstein type and its applications

Equations of the Hammerstein type play a crucial role in the theory of optimal control systems and in automation and network theory so there are some works devoted to fixed points of Hammerstein integral operator on cones. On the other hand, we need to find new results on the uniqueness of fixed points of Hammerstein equations in cones. For instance, during last years, an increasing attention was given to models with uncountable many spin values on a Cayley tree. Splitting Gibbs measures on Cayley trees are described by positive fixed points of Hammerstein integral operator. But, it is impossible to use directly from known results devoted to problems of existence and uniqueness of fixed points of Hammerstein integral operator on cones. In this lecture, we give new conditions of existence and uniqueness of fixed points of Hammerstein integral operator, taking into account problems in the theory of Gibbs measure.