LAUSR

Recently, a new class of stochastic systems induced by linear discrete time noises was proposed and studied. Up to now, the existing literatures mainly investigated the exponential stability of such stochastic systems under the global Lipschitz condition. Our aim here is to weaken the strictly global Lipschitz condition and explore new stability theory for a new class of Markov switched stochastic Hopfield neural networks induced by nonlinear discrete time noises. In the present paper, we propose such Markov switched stochastic Hopfield neural networks, and creatively introduce a new class of Lyapunov functionals to investigate the $$H_{\infty }$$H∞ stability, asymptotic stability and exponential stability for such systems under the local Lipschitz condition using some novel skills. Furthermore, we specially study the case induced by linear discrete time noises.