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We consider the stability problem for the difference system x n = Ax n−1 + Bx n−k , where A, B are real matrixes and the delay k is a positive integer. In the case A = −I, the equation is asymptotically stable if and only if all eigenvalues of the matrix B lie inside a special stability oval in the complex plane. If k is odd, then the oval is in the right half-plane, otherwise, in the left half-plane. If A + B < 1, then the equation is asymptotically stable. We derive explicit sufficient stability conditions for A I and A −I.

We prove that if a s $ 0 ð1 # s # kÞ and 0 , X k s¼1 a s 2 sin p 2ð2s21Þ , 1; then the equationa s xðn 2 sÞ is asymptotically stable. As a corollary, we obtain sufficient asymptotic stability conditions: a s > 0 and 0 , X k s¼1 sa s # p 2

a b s t r a c tThis paper further develops a method, originally introduced by Mori et al., for proving local stability of steady states in linear systems of delay differential equations. A nonlinear nonautonomous system of delay differential equations with several delays is considered. Explicit delay-independent sufficient conditions for global attractivity of the solutions with an extremely simple form are provided. The above-mentioned conditions make the stability test quite practical. We illustrate application of this test to the Hopfield neural network models. The results obtained were also applied to a new marine protected areas model with delay that describes the ecological linkage between the reserve and fishing ground.