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In this brief, we discuss some variants of generalized Halanay inequalities that are useful in the discussion of dissipativity and stability of delayed neural networks, integro-differential systems, and Volterra functional differential equations. We provide some generalizations of the Halanay inequality, which is more accurate than the existing results. As applications, we discuss invariant set, dissipative synchronization, and global asymptotic stability for the Hopfield neural networks with infinite delays. We also prove that the dynamical systems with unbounded time-varying delays are globally asymptotically stable.