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We apply a new definition of periodicity on isolated time scales introduced by Bohner, Mesquita, and Streipert to the study of Ulam stability. If the graininess (step size) of an isolated time scale is bounded by a finite constant, then the linear \(\omega \) -periodic dynamic equations are Ulam stable if and only if the exponential function has modulus different from unity. If the graininess increases at least linearly to infinity, the \(\omega \) -periodic dynamic equations are not Ulam stable. Several examples of specific isolated time scales are also supplied.

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Ulam Stability and Instability of First-Order Linear \(\omega \) -Periodic Dynamic Equations on Isolated Time Scales

  • 章节

  • 发表日期：

    2025年

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  • Douglas R. Anderson,
  • Masakazu Onitsuka

摘要

We apply a new definition of periodicity on isolated time scales introduced by Bohner, Mesquita, and Streipert to the study of Ulam stability. If the graininess (step size) of an isolated time scale is bounded by a finite constant, then the linear \(\omega \) -periodic dynamic equations are Ulam stable if and only if the exponential function has modulus different from unity. If the graininess increases at least linearly to infinity, the \(\omega \) -periodic dynamic equations are not Ulam stable. Several examples of specific isolated time scales are also supplied.