This paper is devoted to study of dynamical systems involving nabla half derivative on an arbitrary time scale. We prove existence and uniqueness of the solution of such system supplied with a suitable initial condition. Both Riemann–Liouville and Caputo approaches to noninteger-order derivatives are covered. Under special conditions we present an explicit form of the solution involving a time scales analogue of Mittag–Leffler function. Also an algorithm for solving of such problems on isolated time scales is established. Moreover, we show that half power functions are positive and decreasing with respect to \(t-s\) on an arbitrary time scale.

本文致力于研究在任意时间尺度上涉及纳布拉半导数的动力学系统。我们证明了具有合适初始条件的此类系统解决方案的存在性和唯一性。涉及非整数阶导数的Riemann-Liouville和Caputo方法都包括在内。在特殊条件下，我们提出了一种解决方案的显式形式，其中包括一个与Mittag–Leffler函数类似的时标。还建立了一种在孤立的时间尺度上解决此类问题的算法。此外，我们表明半幂函数是正的，并且在任意时间尺度上相对于\（ts \）递减。