Ulam-Hyers-Rassias Stability of Stochastic Functional Differential Equations via Fixed Point Methods

Abdellatif Ben Makhlouf,Mohamed Rhaima,Lassaad Mchiri,Zoran Mitrovic

doi:10.1155/2021/5544847

Abstract

The Ulam-Hyers-Rassias stability for stochastic systems has been studied by many researchers using the Gronwall-type inequalities, but there is no research paper on the Ulam-Hyers-Rassias stability of stochastic functional differential equations via fixed point methods. The main goal of this paper is to investigate the Ulam-Hyers Stability (HUS) and Ulam-Hyers-Rassias Stability (HURS) of stochastic functional differential equations (SFDEs). Under the fixed point methods and the stochastic analysis techniques, the stability results for SFDE are investigated. We analyze two illustrative examples to show the validity of the results.

Highlights

stochastic functional differential equations (SFDEs) play an important role in different areas such as physics, mechanics, population dynamics, ecology, medicine biology, and other areas of sciences
Stability investigation is conducted for stochastic nonlinear differential equations with constant delay
The Lyapunov method is used for stability investigation of different mathematical models such as predator-prey relationships and inverted controlled pendulum

Summary

Introduction

SFDEs play an important role in different areas such as physics, mechanics, population dynamics, ecology, medicine biology, and other areas of sciences. The HUS problem of functional systems began from a question of S. Ulam, queried in 1940, about the stability of functional differential equations for homomorphism as follows. The question regarding the stability problem of homomorphisms is as follows: Denote by H1 the group, and H2 the metric group with a metric ~δ and a constant θ > 0. Ulam assuming that D1, D2 be two Banach spaces in the case of λ-linear transformations, that is. In 1978, Rassias [22] provided a generalized answer to the Ulam question for approximate λ-linear transformations. There are a few papers about the HUS and the HURS of stochastic systems (see [13, 27,28,29,30]). It is interesting to extend the research results on the deterministic functional systems to the stochastic case. Two numerical examples are presented to illustrate the main results

Preliminary

Stability Results

Examples

Conclusion