Projective surjectivity of quadratic stochastic operators $L_1$ and its application

TL;DR

This paper investigates the projective surjectivity of quadratic stochastic operators on probability measures, extending to infinite-dimensional nonlinear Markov operators and applying results to solve certain integral equations.

Contribution

It introduces new results on the surjectivity of quadratic stochastic operators on $L^1$ spaces and connects these findings to the existence of solutions for Hammerstein integral equations.

Findings

Proved projective surjectivity of quadratic stochastic operators.

Established surjectivity of infinite dimensional nonlinear Markov operators.

Applied results to demonstrate existence of solutions for specific integral equations.

Abstract

A nonlinear Markov chain is a discrete time stochastic process whose transitions depend on both the current state and the current distribution of the process. The nonlinear Markov chain over a infinite state space can be identified by a continuous mapping (the so-called nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex). In the present paper, we consider a continuous analogue of the mentioned mapping acting on $L^1$-spaces. Main aim of the current paper is to investigate projective surjectivity of quadratic stochastic operators (QSO) acting on the set of all probability measures. To prove the main result, we study the surjectivity of infinite dimensional nonlinear Markov operators and apply them to the projective surjectivity of a QSO. Furthermore, the obtained result has been applied for the existence of positive solution of some Hammerstein integral equations.