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In this article the properties of attractors of dynamical systems in locally compact metric space are discussed. Existing conditions of attractors and related results are obtained by the near isolating block which we present.

In this paper, our aim is to investigate the region stability of stochastic discrete time-delay systems. Firstly, region stabilities of systems are defined by a spectrum operator. Secondly, with the aid of linear matrix inequality, some equivalent conditions and related results for region stabilities of systems are obtained. Finally, the relation between region stabilities and the convergence speed of the trajectories of systems is given.

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