LaSalle-Type Theorems for General Nonlinear Stochastic Functional Differential Equations by Multiple Lyapunov Functions

Abstract

We investigate LaSalle-type theorems for general nonlinear stochastic functional differential equations. With some preliminaries on lemmas and the derivation techniques, we establish three LaSalle-type theorems for the general nonlinear stochastic functional differential equations via multiple Lyapunov functions. For the typical special case with estimations involving ${|x_t|}^p$ for the derivatives of the Lyapunov functions, a theorem is established as the corollary of the main theorem. At the end of the paper, an example is given to illustrate the usage of the method proposed and show the advantage of the results obtained.