Existence and Uniqueness of Solutions to Nonlinear Functional Integral Itô Equations

A new class of Itô integral equations is considered that contains many classicalproblems, for example, the Cauchy problem for differential equations of integer and fractionalorder with and without stochastic perturbations as well as some less known and little-studiedtypes of equations that have been introduced recently. The purpose of the study is to findsufficiently general conditions that guarantee the existence and uniqueness of solutions to suchequations taking into account their specific features. The article therefore proposes to use aspecial generalized Lipschitz condition, which, due to its flexibility, allows one to obtain effectivesolvability criteria in terms of the right-hand sides of equations. Numerous examples areconsidered, covering in particular Itô differential equations of fractional order with andwithout aftereffect, equations with fractional Wiener processes, and Itô equations withseveral time scales, as well as their generalizations.