Attractor minimal sets for nonautonomous type-Kcompetitive and semi-convex delay differential equations withapplications

Wan-Tong Li; Bin-Guo Wang

doi:10.3934/dcds.2009.24.589

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2009, Volume 24, Issue 2: 589-611. Doi: 10.3934/dcds.2009.24.589

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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

Received: January 31, 2008

Revised: October 31, 2008

Published: May 2009

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Abstract

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: 92C20.

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

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