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In this paper we study the well-known Khasminskii-Type Theorem, i.e. theexistence and uniqueness of solutions of stochastic evolution delay equations,under local Lipschitz condition, but without linear growth condition. We thenestablish one stochastic LaSalle-type theorem for asymptotic stability analysisof strong solutions. Moreover, several examples are established to illustratethe power of our theories.

Khasminskii-Type Theorem and LaSalle-Type Theorem for Stochastic Evolution Delay Equations