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On the Exponential Stability of Two-Dimensional Nonautonomous Difference Systems Which Have a Weighted Homogeneity of the Solution

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Advances in Difference Equations and Discrete Dynamical Systems (ICDEA 2016)

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Abstract

The present paper is considered a two-dimensional difference system:

$$\begin{aligned} \varDelta x(n) = a(n)x(n)+b(n)\phi _{p^*\!}(y(n)), \quad \varDelta y(n) = c(n)\phi _p(x(n))+d(n)y(n), \end{aligned}$$

where all coefficients are real-valued sequences; p and \(p^*\) are positive numbers satisfying \(1/p + 1/p^* = 1\); and \(\phi _p(x) = |x|^{p-2}x\) for \(x \ne 0\), and \(\phi _p(0) = 0\). The aim of this paper is to clarify that uniform asymptotic stability and exponential stability are equivalent for the above system. To illustrate the obtained results, an example is given. In addition, a figure of a solution orbit which is drawn by a computer is also attached for a deeper understanding.

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Authors and Affiliations

  1. Department of Applied Mathematics, Okayama University of Science, 700-0005, Okayama, Japan

    Masakazu Onitsuka

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  1. Masakazu Onitsuka
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Correspondence to Masakazu Onitsuka .

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Editors and Affiliations

  1. Department of Mathematics, Trinity University, San Antonio, Texas, USA

    Saber Elaydi

  2. Department of Information Science, Okayama University of Science, Okayama, Japan

    Yoshihiro Hamaya

  3. Department of Mathematical Sciences, Osaka Prefecture University, Sakai, Osaka, Japan

    Hideaki Matsunaga

  4. Institut für Mathematik, Alpen-Adria-Universität Klagenfurt, Klagenfurt, Austria

    Christian Pötzsche

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Onitsuka, M. (2017). On the Exponential Stability of Two-Dimensional Nonautonomous Difference Systems Which Have a Weighted Homogeneity of the Solution. In: Elaydi, S., Hamaya, Y., Matsunaga, H., Pötzsche, C. (eds) Advances in Difference Equations and Discrete Dynamical Systems. ICDEA 2016. Springer Proceedings in Mathematics & Statistics, vol 212. Springer, Singapore. https://doi.org/10.1007/978-981-10-6409-8_11

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