Some Applications of Arutyunov Mordukhovich Zhukovskiy Theorem to Stochastic Integral Equations

TL;DR

This paper applies Mordukhovich derivatives and the AMZ Theorem to establish existence results for solutions to stochastic linear and integral equations in infinite-dimensional Banach spaces.

Contribution

It introduces a method to compute covering constants for linear mappings in lp spaces and applies the AMZ Theorem to prove solution existence for stochastic systems.

Findings

Derived explicit covering constants for linear mappings in lp spaces.

Proved existence of solutions for stochastic linear functional equations.

Established solution existence for stochastic integral equations.

Abstract

Mordukhovich derivatives (Mordukhovich coderivatives) of set-valued mappings in Banach spaces have firmly laid the foundation of the theory of generalized differentiation in set-valued analysis, which has been widely applied to optimization theory, equilibrium theory, variational analysis, and so forth, with respect to set-valued mappings. One of the most important applications of Mordukhovich derivatives is to define the covering constants for set-valued mappings in Banach spaces, which play an important role in the well-known Arutyunov Mordukhovich Zhukovskiy Parameterized Coincidence Point Theorem (Theorem 3.1 in [1]). In [15], this theorem is simply named as AMZ Theorem. In this paper, we consider locally or globally stochastic infinitely dimensional systems of linear equations in lp space. We use the Mordukhovich derivatives to precisely find the covering constants for linear and continuous mappings in lp spaces. Then, by using the AMZ Theorem, we prove an existence theorem for solutions to some locally or globally stochastic infinitely dimensional systems of linear functional equations in lp spaces and an existence theorem for solutions to some stochastic integral equations