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Sequence Spaces with Variable Exponents for Lattice Systems with Nonlinear Diffusion

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Modern Mathematics and Mechanics

Part of the book series: Understanding Complex Systems ((UCS))

Abstract

Motivated by the study of lattice dynamical systems, i.e., infinite dimensional systems of ordinary differential equations, with nonlinear and state dependent diffusion, a new sequence space with variable exponents is introduced. In particular, given an exponent sequence \({\boldsymbol p} = (p_i)_{i \in \mathbb {Z}}\), a discrete Musielak-Orlicz space of real valued bi-infinite sequences ℓ p is defined and equipped with a norm ∥⋅∥p induced by a semi-modular ρ(⋅). Properties of ∥⋅∥p and ρ(⋅), as well as properties of the space (ℓ p, ∥⋅∥p) are discussed in greater detail. While these properties largely facilitate dynamical analysis of a much wider class of lattice systems, this work is a step towards the construction of an integral mathematical framework for the study of lattice models with complicated diffusion structures.

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Acknowledgements

This work has been partially supported by the National Science Foundation of China grant number 11571125 (PEK) and FAPEMIG process CEX-PPM-00329-16 (JS).

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Auburn University, Auburn, AL, USA

    Xiaoying Han

  2. School of Mathematics and Statistics, Huazhong University of Science & Technology, Wuhan, Hubei, China

    Peter E. Kloeden

  3. Instituto de Matemática e Computação, Universidade Federal de Itajubá, Bairro Pinheirinho, Itajubá - MG, Brazil

    Jacson Simsen

Authors
  1. Xiaoying Han
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  2. Peter E. Kloeden
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  3. Jacson Simsen
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Corresponding author

Correspondence to Peter E. Kloeden .

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Editors and Affiliations

  1. Lomonosov Moscow State University, Moscow, Russia

    Victor A. Sadovnichiy

  2. National Technical University of Ukraine, “Igor Sikorsky Kyiv Polytechnic Institute”, Kyiv, Ukraine

    Michael Z. Zgurovsky

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Han, X., Kloeden, P.E., Simsen, J. (2019). Sequence Spaces with Variable Exponents for Lattice Systems with Nonlinear Diffusion. In: Sadovnichiy, V., Zgurovsky, M. (eds) Modern Mathematics and Mechanics. Understanding Complex Systems. Springer, Cham. https://doi.org/10.1007/978-3-319-96755-4_12

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  • DOI: https://doi.org/10.1007/978-3-319-96755-4_12

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-96754-7

  • Online ISBN: 978-3-319-96755-4

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