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In this article, we establish the LaSalle-type theorem for stochastic functional differential equations with Markovian switching (SFDEwMSs) under much weaken conditions. We would like to emphasize that we do not require the linear growth condition and the bounded moment condition on the solutions. Indeed, we allow the Lyapunov function operator could be dependent of time to cover a much wider class of SFDEwMSs. As a bonus, we obtain the criterion on the asymptotical stability and asymptotical boundedness for SFDEwMSs. For comparison and verification, we also present a specific example with much general coefficients.

LaSalle-type theorems for stochastic functional differential equations with Markovian switching