Stability results for neutral stochastic functional differential equations via fixed point methods

ABSTRACT

In this paper we prove some results on the mean square asymptotic stability of a class of neutral stochastic differential systems with variable delays by using a contraction mapping principle. Namely, a necessary and sufficient condition ensuring the asymptotic stability is proved. The assumption does not require neither boundedness or differentiability of the delay functions, nor do they ask for a fixed sign on the coefficient functions. In particular, the results improve some previous ones proved by Guo, Y., Xu, C., & Wu, J. [(2017). Stability analysis of neutral stochastic delay differential equations by a generalisation of Banach's contraction principle. International Journal of Control, 90, 1555–1560]. Finally, an example is exhibited to illustrate the effectiveness of the proposed results.

Keywords:

• Fixed points theory
• asymptotic stability in mean square
• neutral stochastic differential equations
• variable delays

AMS Subject Classifications:

• 34K20
• 34K13
• 92B20

Acknowledgments

The authors would like to thank the referee for the helpful and interesting suggestions and comments which allowed to improve the presentation of our paper.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of the second author has been partially supported by Ministerio de Economía y Competitividad grant MTM2015-63723-P (MINECO/FEDER, EU), and Consejería de Economía, Innovación, Ciencia y Empleo, Junta de Andalucía (Spain) under the Proyecto de Excelencia P12-FQM-1492. Also, this work was funded by European Mathematical Society.