Abstract:

In this paper, the existence, uniqueness and $ \lambda $ stability of the global solutions to the pantogragh stochastic functional differential equations with Markovian switching are investigated. By applying the vector Lyapunov function method, we obtain some theorems ensuring the existence, uniqueness, moment $ \lambda $ stability and almost sure $ \lambda $ stability of the global solutions to the equations. Furthermore, impulsive effects are introduced to the systems.According to the different characteristics of the impulses and different decay rates of the $ \lambda $ functions we prove that similar results still hold under suitable conditions.Our results generalize some existing results.Finally, an example and numerical simulation is given to illustrate the effectiveness of the results.

Key words: existence and uniqueness, stability, pantogragh stochastic functional differential equation, impulse, Itô, formula