Mean square stability with general decay rate of nonlinear neutral stochastic function differential equations in the $ G $-framework

Guangjie Li; Guangjie Li

doi:10.3934/math.2022318

AIMS Mathematics

2022, Volume 7, Issue 4: 5752-5767. doi: 10.3934/math.2022318

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Research article

Mean square stability with general decay rate of nonlinear neutral stochastic function differential equations in the $ G $-framework

Guangjie Li ^,

School of Mathematics and Statistics, Guangdong University of Foreign Studies, Guangzhou 510006, China

Received: 08 October 2021 Revised: 06 December 2021 Accepted: 27 December 2021 Published: 11 January 2022
MSC : 34K20, 60H10, 34K50

Few results seem to be known about the stability with general decay rate of nonlinear neutral stochastic function differential equations driven by $ G $-Brownain motion ($ G $-NSFDEs in short). This paper focuses on the $ G $-NSFDEs, and the coefficients of these considered $ G $-NSFDEs can be allowed to be nonlinear. It is first proved the existence and uniqueness of the global solution of a $ G $-NSFDE. It is then obtained the trivial solution of the $ G $-NSFDE is mean square stable with general decay rate (including the trivial solution of the $ G $-NSFDE is mean square exponentially stable and the trivial solution of the $ G $-NSFDE is mean square polynomially stable) by $ G $-Lyapunov functions technique. In this paper, auxiliary functions are used to dominate the Lyapunov function and the diffusion operator. Finally, an example is presented to illustrate the obtained theory.
- mean square stability,
- neutral stochastic functional differential equations,
- $ G $-Brownain motion,
- general decay,
- $ \lambda $-type function
Citation: Guangjie Li. Mean square stability with general decay rate of nonlinear neutral stochastic function differential equations in the $ G $-framework[J]. AIMS Mathematics, 2022, 7(4): 5752-5767. doi: 10.3934/math.2022318

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Abstract

Few results seem to be known about the stability with general decay rate of nonlinear neutral stochastic function differential equations driven by $ G $-Brownain motion ($ G $-NSFDEs in short). This paper focuses on the $ G $-NSFDEs, and the coefficients of these considered $ G $-NSFDEs can be allowed to be nonlinear. It is first proved the existence and uniqueness of the global solution of a $ G $-NSFDE. It is then obtained the trivial solution of the $ G $-NSFDE is mean square stable with general decay rate (including the trivial solution of the $ G $-NSFDE is mean square exponentially stable and the trivial solution of the $ G $-NSFDE is mean square polynomially stable) by $ G $-Lyapunov functions technique. In this paper, auxiliary functions are used to dominate the Lyapunov function and the diffusion operator. Finally, an example is presented to illustrate the obtained theory.

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