Abstract
We formulate fractional difference equations of Riemann–Liouville and Caputo type in a functional analytical framework. Main results are existence of solutions on Hilbert space-valued weighted sequence spaces and a condition for stability of linear fractional difference equations. Using a functional calculus, we relate the fractional sum to fractional powers of the operator \(1 - \tau ^{-1}\) with the right shift \(\tau ^{-1}\) on weighted sequence spaces. Causality of the solution operator plays a crucial role for the description of initial value problems.
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Acknowledgements
The research of A.B. was funded by the National Science Centre in Poland granted according to decision DEC-2015/19/D/ST7/03679. The research of A.C. was supported by the Statutory Funding of the Faculty of Automatic Control, Electronics and Computer Science, Silesian University of Technology, Gliwice, Poland, 02/990/BK_19/0121. The research of M.N. was supported by the Polish National Agency for Academic Exchange according to the decision PPN/BEK/2018/1/00312/DEC/1. The research of S.S. was partially supported by an Alexander von Humboldt Polish Honorary Research Fellowship. The work of H.T. Tuan was supported by the joint research project from RAS and VAST QTRU03.02/437 18-19.
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Anh, P.T. et al. (2020). A Hilbert Space Approach to Fractional Difference Equations. In: Bohner, M., Siegmund, S., Šimon Hilscher, R., Stehlík, P. (eds) Difference Equations and Discrete Dynamical Systems with Applications. ICDEA 2018. Springer Proceedings in Mathematics & Statistics, vol 312. Springer, Cham. https://doi.org/10.1007/978-3-030-35502-9_4
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