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Sequence Spaces with Variable Exponents for Lattice Systems with Nonlinear Diffusion

  • Xiaoying Han
  • Peter E. KloedenEmail author
  • Jacson Simsen
Chapter
Part of the Understanding Complex Systems book series (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 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 (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.

Notes

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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Copyright information

© Springer International Publishing AG, part of Springer Nature 2019

Authors and Affiliations

  • Xiaoying Han
    • 1
  • Peter E. Kloeden
    • 2
    Email author
  • Jacson Simsen
    • 3
  1. 1.Department of Mathematics and StatisticsAuburn UniversityAuburnUSA
  2. 2.School of Mathematics and StatisticsHuazhong University of Science & TechnologyWuhanChina
  3. 3.Instituto de Matemática e ComputaçãoUniversidade Federal de ItajubáBairro PinheirinhoBrazil

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