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<p>In this paper, we study discrete dichotomy and Hyers-Ulam (H-U) stability for linear and semilinear quaternion-valued periodic difference systems (hereafter referred to as LQPDS and SQPDS, respectively). Firstly, since the projection for the specific eigenvalues of the matrix extracted by the Cauchy integral formula on the quaternion skew field is not valid, a matrix transformation is constructed to replace the traditional spectral mapping, and it is proved that the transformation has similar properties to the complex spectral projection. Secondly, by the constructed matrix transformation, it is obtained that the first-order LQPDS is H-U stable if and only if the monodromy matrix has a discrete dichotomy. Moreover, in order to overcome the difficulty that the LQPDS can not be directly converted into the characteristic equation, the sufficient condition for H-U stability of the second-order LQPDS is obtained by discrete dichotomy and analyzing the coefficients of the difference equations. Furthermore, the H-U stability of the SQPDS is considered. Finally, three illustrative examples are provided to demonstrate the theoretical results.</p>

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Dichotomy and Hyers-Ulam stability for linear quaternion-valued periodic difference systems

  • 期刊论文

  • 发表日期：

    2026年4月1日

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  • Jiaojiao Lv,
  • JinRong Wang

摘要

In this paper, we study discrete dichotomy and Hyers-Ulam (H-U) stability for linear and semilinear quaternion-valued periodic difference systems (hereafter referred to as LQPDS and SQPDS, respectively). Firstly, since the projection for the specific eigenvalues of the matrix extracted by the Cauchy integral formula on the quaternion skew field is not valid, a matrix transformation is constructed to replace the traditional spectral mapping, and it is proved that the transformation has similar properties to the complex spectral projection. Secondly, by the constructed matrix transformation, it is obtained that the first-order LQPDS is H-U stable if and only if the monodromy matrix has a discrete dichotomy. Moreover, in order to overcome the difficulty that the LQPDS can not be directly converted into the characteristic equation, the sufficient condition for H-U stability of the second-order LQPDS is obtained by discrete dichotomy and analyzing the coefficients of the difference equations. Furthermore, the H-U stability of the SQPDS is considered. Finally, three illustrative examples are provided to demonstrate the theoretical results.