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Journal of Mathematics
Volume 2018, Article ID 7490936, 9 pages
https://doi.org/10.1155/2018/7490936
Research Article

Lyapunov Stability of the Generalized Stochastic Pantograph Equation

1Dagestan Research Center of the Russian Academy of Sciences and Department of Mathematics, Dagestan State University, Makhachkala 367005, Russia
2Norwegian University of Life Sciences, Faculty of Sciences and Technology, P.O. Box 5003, N-1432 Ås, Norway

Correspondence should be addressed to Arcady Ponosov; on.ubmn@idakra

Received 31 January 2018; Accepted 14 May 2018; Published 19 June 2018

Academic Editor: Qamar Din

Copyright © 2018 Ramazan Kadiev and Arcady Ponosov. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The purpose of the paper is to study stability properties of the generalized stochastic pantograph equation, the main feature of which is the presence of unbounded delay functions. This makes the stability analysis rather different from the classical one. Our approach consists in linking different kinds of stochastic Lyapunov stability to specially chosen functional spaces. To prove stability, we check that the solutions of the equation belong to a suitable space of stochastic processes, instead of searching for an appropriate Lyapunov functional. This gives us possibilities to study moment stability, stability with probability 1, and many other stability properties in an efficient way. We show by examples how this approach works in practice, putting emphasis on delay-independent stability conditions for the generalized stochastic pantograph equation. The framework can be applied to any stochastic functional differential equation with finite dimensional initial conditions.