On the dichotomic behavior of discrete dynamical systems on the half-line

Bogdan Sasu; Adina Luminiţa Sasu

doi:10.3934/dcds.2013.33.3057

Article Contents

2013, Volume 33, Issue 7: 3057-3084. Doi: 10.3934/dcds.2013.33.3057

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On the dichotomic behavior of discrete dynamical systems on the half-line

Bogdan Sasu^1, and
Adina Luminiţa Sasu^1,

1.
Faculty of Mathematics and Computer Science, West University of Timişoara, V. Pârvan Blvd. No. 4, 300223 Timişoara

Received: March 2012

Revised: November 2012

Published online: January 2013

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Abstract

The aim of this paper is to obtain new criteria for the existence of the dichotomies of dynamical systems on the half-line. We associate to a discrete dynamical system an input-output system between two abstract sequence spaces. We deduce conditions for the existence of ordinary dichotomy and exponential dichotomy of the initial discrete system, by using certain admissibility properties of the associated input-output system. We establish the axiomatic structures of the input and output spaces, in each case, clarifying the underlying hypotheses as well as the generality of the proposed method. Next, we present a new and direct proof for the equivalence between the exponential dichotomy of an evolution family on the half-line and the exponential dichotomy of the associated discrete dynamical system. Finally, we apply our main results to the study of the exponential dichotomy of evolution families on the half-line.

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Mathematics Subject Classification: Primary: 34D09, 37B55, 37N35; Secondary: 93C55.

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