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On the Fréchet Differentiability of Invariant Tori of Countable Systems of Difference Equations Defined on Infinite-Dimensional Tori

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Ukrainian Mathematical Journal Aims and scope

Abstract

By using the method of Green–Samoilenko functions, in the space of bounded number sequences we construct invariant tori of linear and nonlinear systems of discrete equations defined on infinite-dimensional tori. We establish sufficient conditions for the Fréchet differentiability of invariant tori.

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REFERENCES

  1. A. M. Samoilenko and Yu. V. Teplinskii, “Invariant tori of linear countable systems of discrete equations given on an infinite-dimensional torus,” Ukr. Mat. Zh., 50, No. 2, 244–251 (1998).

    Google Scholar 

  2. A. M. Samoilenko and Yu. V. Teplinskii, Limit Theorems in the Theory of Systems of Difference Equations [in Russian], Preprint No. 98.3, Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1998).

  3. Yu. V. Teplins'kyi and N. A. Marchuk, “Truncation method in studying the smoothness of invariant tori of countable systems of difference equations with parameters,” Zb. Nauk. Prats' Kam'yanets'-Podil's'k. Ped. Univ., Ser. Mat., No. 5, 117–126 (2000).

    Google Scholar 

  4. D. I. Martynyuk and H. V. Ver'ovkina, “Invariant sets of countable systems of difference equations,” Visn. Kyiv Univ., Ser. Fizs.-Mat., No. 1, 117–127 (1997).

    Google Scholar 

  5. H. V. Ver'ovkina, “On the introduction of local coordinates for a countable discrete system in the neighborhood of an invariant torus,” Visn. Kyiv Univ., Ser. Fizs.-Mat., No. 4, 23–29 (1997).

    Google Scholar 

  6. A. M. Samoilenko and Yu. V. Teplinskii, Countable Systems of Differential Equations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1993).

    Google Scholar 

  7. A. M. Samoilenko and Yu. V. Teplinskii, “On the smoothness of the invariant torus of a countable linear extension of a dynamical system on an m-dimensional torus,” Differents. Uravn., 30, No. 5, 781–790 (1994).

    Google Scholar 

  8. L. Schwartz, Analysis [Russian translation], Vol. 1, Mir, Moscow (1972).

    Google Scholar 

  9. Yu. D. Latushkin and A. M. Stepin, “Operators of weighted shift and linear extensions of dynamical systems,” Usp. Mat. Nauk, 48, No. 2, 85–143 (1991).

    Google Scholar 

  10. A. G. Ilyukhin, “An approximate method for the solution of a mixed problem for a nonlinear partial differential equation of hyperbolic type with small parameter,” Ukr. Mat. Zh., 14, No. 3, 250–259 (1962).

    Google Scholar 

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Authors and Affiliations

  1. Kam'yanets'-Podil's'k Pedagogic University, Kam'yanets'-Podil's'k, Russia

    Yu. V. Teplins'kyi & N. A. Marchuk

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  1. Yu. V. Teplins'kyi
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  2. N. A. Marchuk
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Teplins'kyi, Y.V., Marchuk, N.A. On the Fréchet Differentiability of Invariant Tori of Countable Systems of Difference Equations Defined on Infinite-Dimensional Tori. Ukrainian Mathematical Journal 55, 93–111 (2003). https://doi.org/10.1023/A:1025024703214

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