Solvability of nonlinear integral equations and surjectivity of nonlinear Markov operators

Abstract

In the present paper, we consider integral equations, which are associated with nonlinear Markov operators acting on an infinite‐dimensional space. The solvability of these equations is examined by investigating nonlinear Markov operators. Notions of orthogonal preserving and surjective nonlinear Markov operators defined on infinite dimension are introduced, and their relations are studied, which will be used to prove the main results. We show that orthogonal preserving nonlinear Markov operators are not necessarily satisfied surjective property (unlike finite case). Thus, sufficient conditions for the operators to be surjective are described. Using these notions and results, we prove the solvability of Hammerstein equations in terms of surjective nonlinear Markov operators.

A class of bijective Lotka–Volterra operators and its application

“…It turns out that the surjectivity of multilinear stochastic operators and their orthogonal preserving properties are equivalent 20 . Applications of such kind of results to the solutions of nonlinear Hammerstein equations have been carried out in Mukhamedov et al 21,22 Dynamical behavior of such kind of operators has been explored in Mukhamedov et al 23 …”

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

“…We refer the reader to [11,18,27] as the exposition of the recent achievements and open problems in the theory of the QSO can be further researched. In [22,23] the surjectivity of DQSO and its relation with orthogonal preserving property of V have been investigated.…”