Y-H Xia | Searcharxiv

arXiv2022

Higher regularity of homeomorphisms in the Hartman-Grobman theorem and a conjecture on its sharpness

Hartman-Grobman theorem states that there is a homeomorphism H sending the solutions of the nonlinear system onto those of its linearization under suitable assumptions. Many mathematicians have made contributions to prove Hölder continuity of the homeomorphisms. However, is it possible to improve the Hölder continuity to Lipschitzian continuity? This paper gives a positive answer. We formulate the first result that the homeomorphism is Lipschitzian, but not $C^1$, while its inverse is merely Hölder continuous, but not Lipschitzian. It is interesting that the regularity of the homeomorphism is different from its inverse. Moreover, some illustrative examples are presented to show the effectiveness of our results. Further, motivated by our example, we also propose a conjecture, saying, the regularity of the homeomorphisms is sharp and it could not be improved any more.

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arXiv2018

Solve the linear quaternion-valued differential equations having multiple eigenvalues

The theory of two-dimensional linear quaternion-valued differential equations (QDEs) was recently established (see Kou and Xia, SAPM). Some profound differences between QDEs and ODEs were observed. Also, an algorithm to evaluate the fundamental matrix by employing the eigenvalues and eigenvectors was presented in [Kou and Xia, SAPM]. However, the fundamental matrix can be constructed providing that the eigenvalues are simple. If the linear system has multiple eigenvalues, how to construct the fundamental matrix? In particular, if the number of independent eigenvectors might be less than the dimension of the system. That is, the numbers of the eigenvectors is not enough to construct a fundamental matrix. How to find the "missing solutions"? The main purpose of this paper is to answer this question. Furthermore, Caley determinant for Quaternion-valued matrix was adopted to proceed the theory of QDEs in [Kou and Xia, SAPM]. One big disadvantage of Caley determinant is that it can be expanded along the different rows and columns. This may lead to different results due to non-commutativity of the quaternions. This approach is not convenient to be used. The novel definition of determinant for Quaternion-valued matrix based on permutation is introduced to analyze the theory. This newly definition of determinant has great advantage compare to Calay determinant.

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