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June  2009, 24(2): 589-611. doi: 10.3934/dcds.2009.24.589

Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications

Wan-Tong Li 1, and  Bin-Guo Wang 1,

1. 

School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China

Received  February 2008 Revised  November 2008 Published  March 2009

Skew-product semiflows induced by semi-convex and type-K competitive almost periodic delay differential equations are studied. If $M$ is a compact positively invariant subset of the skew-product semiflow, then continuous separation of the skew-product semiflow on $M$ holds. Furthermore, if two minimal subsets $M_{1}$ and $M_{2}$ of the skew-product semiflow satisfying completely strongly type-K ordering $M_{1}$«$^C_K M_{2}$, then $M_{1}$ is an attractor. Finally, these results are applied to a nonautonomous delayed Hopfield-type neural networks with the diagonal-nonnegative type-K monotone interconnection matrix and sufficient conditions are obtained for the existence of global or partial attractors.
Keywords: Type-K monotone; skew-product semiflows; almost periodic delay differential equations; type-K convex (semi-convex); neural networks..
Mathematics Subject Classification: Primary: 37B55; 34K14; Secondary: 92C2.
Citation: Wan-Tong Li, Bin-Guo Wang. Attractor minimal sets for nonautonomous type-K competitive and semi-convex delay differential equations with applications. Discrete & Continuous Dynamical Systems - A, 2009, 24 (2) : 589-611. doi: 10.3934/dcds.2009.24.589
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