On Dynamical Systems with Nabla Half Derivative on Time Scales


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.

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The research has been supported by the Grant GA17-03224S of the Czech Science Foundation. The preparation of the final version was supported by the Grant GA20-11846S of the Czech Science Foundation.

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Correspondence to Tomáš Kisela.

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Kisela, T. On Dynamical Systems with Nabla Half Derivative on Time Scales. Mediterr. J. Math. 17, 187 (2020). https://doi.org/10.1007/s00009-020-01629-w

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  • Fractional calculus
  • time scales
  • Nabla half derivative
  • dynamical systems
  • Mittag–Leffler function
  • existence and uniqueness

Mathematics Subject Classification

  • Primary 26A33
  • 26E70
  • Secondary 39A12
  • 39A13