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Nonuniform Stability of Arbitrary Difference Equations

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Abstract

We describe two approaches to study the robustness of the exponential stability of an arbitrary difference equation with finite delay on a Banach space. The first approach is based on a characterization of Perron-type of the exponential stability in terms of the invertibility of a certain linear operator between spaces of bounded sequences. The second approach is based on looking at the dynamics on a higher-dimensional space without delay and so for which one has the cocycle property. We emphasize that none of the results obtained with the two approaches implies the other. More precisely, the second approach allows to obtain the robustness of a weaker notion although also with a stronger hypothesis. We consider the general case of nonuniform exponential stability.

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Authors and Affiliations

  1. Departamento de Matemática, Instituto Superior Técnico, Universidade de Lisboa, 1049-001, Lisbon, Portugal

    Luis Barreira & Claudia Valls

  2. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW 2052, Australia

    Davor Dragičević

Authors
  1. Luis Barreira
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  2. Davor Dragičević
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  3. Claudia Valls
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Corresponding author

Correspondence to Luis Barreira.

Additional information

L.B. and C.V. were supported by FCT/Portugal through UID/MAT/04459/2013. D.D. was supported in part by an Australian Research Council Discovery Project DP150100017, Croatian Science Foundation under the Project IP-2014-09-2285 and by the University of Rijeka research Grant 13.14.1.2.02.

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Barreira, L., Dragičević, D. & Valls, C. Nonuniform Stability of Arbitrary Difference Equations. Results Math 71, 333–346 (2017). https://doi.org/10.1007/s00025-015-0499-2

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  • DOI: https://doi.org/10.1007/s00025-015-0499-2

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