Year: 2022
Type: article
Abstract: The W -method was introduced and developed by N.V. Azbelev and his students.The main idea of it is to transform a given functional differential equation to an integral equation, which is easier to handle.Using this transformation, one can study boundary value problems and asymptotic behaviour of solutions.The present review article describes a stochastic version of this method, which can be used in stability analysis of stochastic functional differential equations.The classical setting of the Wmethod, developed primarily for linear equations and based on estimation of the norm of an associated integral operator, is considered first.An alternative version of the method, which, in addition, includes the theory of positive invertible matrices, is described after that.In the latter approach, all equations of the system are transformed in different ways, depending on specific properties of the equations, and the norm of each integral operator component is estimated separately.In the classical framework, all equations are also transformed separately, but the norm of the operator is estimated without taking into account the specific behaviour of each equation.The alternative approach to the W -method can be also applied to nonlinear stochastic functional differential equations.Several verifiable stability conditions are given demonstrating the efficiency of the method.
Authors: R. KADIEV, A. PONOSOV
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4316923567