Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay

Ya Wang; Fuke Wu; Xuerong Mao; Enwen Zhu

doi:10.3934/dcdsb.2019182

Article Contents

2020, Volume 25, Issue 1: 287-300. Doi: 10.3934/dcdsb.2019182

This issue Previous Article Analysis of time-domain Maxwell's equations in biperiodic structures Next Article Pullback exponential attractors for differential equations with variable delays

Advances in the LaSalle-type theorems for stochastic functional differential equations with infinite delay

1.
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China
2.
Department of Mathematics and Statistics, University of Strathclyde, Glasgow G1 1XH, UK
3.
School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410004, China

¹Corresponding author
¹Corresponding author

Received: December 31, 2018

Early access: July 2019

Published: January 2020

The research was supported in part by the National Natural Science Foundations of China (Grant Nos. 1161101211 and 61873320), and the Royal Society and the Newton Fund (NA160317, Royal Society-Newton Advanced Fellowship)

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Abstract

This paper considers stochastic functional differential equations (SFDEs) with infinite delay. The main aim is to establish the LaSalle-type theorems to locate limit sets for this class of SFDEs. In comparison with the existing results, this paper gives more general results under the weaker conditions imposed on the Lyapunov function. These results can be used to discuss the asymptotic stability and asymptotic boundedness for SFDEs with infinite delay. In the end, two examples will be given to illustrate applications of our new results established.

Keywords:

Stochastic functional differential equations,
the LaSalle-type theorem,
infinite delay,
asymptotic stability,
nonnegative semimartingale convergence theorem.

Mathematics Subject Classification: Primary: 34K50, 60H10; Secondary: 34D45, 37C75.

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