The Poisson distribution, abstract fractional difference equations, and stability
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- by Carlos Lizama PDF
- Proc. Amer. Math. Soc. 145 (2017), 3809-3827 Request permission
Abstract:
We study the initial value problem \begin{equation*} \tag {$*$} \left \{\begin {array}{rll} _C\Delta ^{\alpha } u(n) &= Au(n+1), \quad n \in \mathbb {N}_0; \\ u(0) &= u_0 \in X, \end{array}\right . \end{equation*} when $A$ is a closed linear operator with domain $D(A)$ defined on a Banach space $X$. We introduce a method based on the Poisson distribution to show existence and qualitative properties of solutions for the problem $(*)$, using operator-theoretical conditions on $A$. We show how several properties for fractional differences, including their own definition, are connected with the continuous case by means of sampling using the Poisson distribution. We prove necessary conditions for stability of solutions, that are only based on the spectral properties of the operator $A$ in the case of Hilbert spaces.References
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Additional Information
- Carlos Lizama
- Affiliation: Departamento de Matemática y Ciencia de la Computación, Facultad de Ciencias, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile
- MR Author ID: 114975
- Email: carlos.lizama@usach.cl
- Received by editor(s): June 30, 2014
- Received by editor(s) in revised form: June 27, 2015, July 8, 2015, and July 13, 2015
- Published electronically: May 24, 2017
- Additional Notes: The author was partially supported by FONDECYT grant number 1140258
- Communicated by: Mark M. Meerschaert
- © Copyright 2017 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 3809-3827
- MSC (2010): Primary 39A13, 39A14, 39A06, 39A60, 47D06, 47D99
- DOI: https://doi.org/10.1090/proc/12895
- MathSciNet review: 3665035