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The Poisson distribution, abstract fractional difference equations, and stability


Author: Carlos Lizama
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
Published electronically: May 24, 2017
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Abstract: We study the initial value problem

$\displaystyle \left \{\begin {array}{rll} _C\Delta ^{\alpha } u(n) &= Au(n+1), \quad n \in \mathbb{N}_0; \\ u(0) &= u_0 \in X, \end{array}\right .$ ($ *$)

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.

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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
Email: carlos.lizama@usach.cl

DOI: https://doi.org/10.1090/proc/12895
Keywords: Fractional differences, abstract fractional difference equations, abstract fractional differential equations, Poisson distribution, $\alpha$-resolvent families
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
Article copyright: © Copyright 2017 American Mathematical Society

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