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A Hilbert space approach to fractional difference equations. (English) Zbl 1442.39006

Bohner, Martin (ed.) et al., Difference equations and discrete dynamical systems with applications. ICDEA 24, Dresden, Germany, May 21–25, 2018. Proceedings of the 24th international conference on difference equations and applications. Cham: Springer. Springer Proc. Math. Stat. 312, 115-131 (2020).
Summary: 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.
For the entire collection see [Zbl 1443.39002].

MSC:

39A13 Difference equations, scaling (\(q\)-differences)
39A12 Discrete version of topics in analysis
26A33 Fractional derivatives and integrals
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