A new method to prove the nonuniform dichotomy spectrum theorem in $\mathbb {R}^n$
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- by Yonghui Xia, Yuzhen Bai and Donal O’Regan PDF
- Proc. Amer. Math. Soc. 147 (2019), 3905-3917 Request permission
Abstract:
This paper presents a new method to prove the nonuniform dichotomy spectrum theorem. Chu et al. [Bull. Sci. Math. 139 (2015), pp. 538–557] and Zhang [J. Funct. Anal. 267 (2014), pp. 1889–1916] generalized the dichotomy spectrum in Siegmund [J. Dynam. Differential Equations 14 (2002), pp. 243–258] to the nonuniform dichotomy spectrum and the authors in these works employed linear integral manifolds (stable and unstable) to establish the spectral theorem. They then used the spectrum theorem to study reducibility. We prove the nonuniform dichotomy spectrum by way of contradiction. In particular, we employ the nonuniform kinematically similarity (nonuniform reducibility) to reduce the shift system into two blocks and then we get a contradiction based on a technique in mathematical analysis. The method in the proof is completely different from previous works.References
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Additional Information
- Yonghui Xia
- Affiliation: Department of Mathematics, Zhejiang Normal University, Jinhua, 321004, People’s Republic of China
- MR Author ID: 729169
- Email: xiadoc@163.com
- Yuzhen Bai
- Affiliation: School of Mathematical Sciences, Qufu Normal University, Qufu, 273165, People’s Republic of China
- MR Author ID: 654437
- Email: baiyu99@126.com
- Donal O’Regan
- Affiliation: School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland
- MR Author ID: 132880
- Email: donal.oregan@nuigalway.ie
- Received by editor(s): November 8, 2018
- Received by editor(s) in revised form: January 3, 2019, and January 4, 2019
- Published electronically: June 14, 2019
- Additional Notes: This work was supported by the National Natural Science Foundation of China under Grants (No.11671176 and No.11271333), Natural Science Foundation of Zhejiang Province under Grant (LY15A010007), Natural Science Foundation of Fujian Province under Grant (No.2018J01001), and a start-up fund of Huaqiao University (Z16J0039).
The second author is the corresponding author. - Communicated by: Wenxian Shen
- © Copyright 2019 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 3905-3917
- MSC (2010): Primary 34D10, 37B55
- DOI: https://doi.org/10.1090/proc/14535
- MathSciNet review: 3993783