A global attractivity result for maps with invariant boxes

M. R. S. Kulenović; Orlando Merino

doi:10.3934/dcdsb.2006.6.97

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2006, Volume 6, Issue 1: 97-110. Doi: 10.3934/dcdsb.2006.6.97

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A global attractivity result for maps with invariant boxes

M. R. S. Kulenović^1, and
Orlando Merino^1,

1.
Department of Mathematics, University of Rhode Island, Kingston, Rhode Island 02881-0816

Received: April 2005

Revised: September 2005

Published: January 2006

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Abstract

We present a global attractivity result for maps generated by systems of autonomous difference equations. It is assumed that the map of the system leaves invariant a box, is monotone in a coordinate-wise sense (but not necessarily monotone with respect to a standard cone), and satisfies certain algebraic condition. It is shown that there exists a unique equilibrium, and that it is a global attractor. As an application, it is shown that a discretized version of the Lotka-Volterra system of differential equations of order $k$ has a global attractor in the positive orthant for certain range of parameters.

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Mathematics Subject Classification: Primary: 37B25; Secondary: 39A11, 39A20.

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