On well-posedness of vector-valued fractional differential-difference equations

Luciano Abadías; Carlos Lizama; Pedro J. Miana; M. Pilar Velasco

doi:10.3934/dcds.2019112

Article Contents

2019, Volume 39, Issue 5: 2679-2708. Doi: 10.3934/dcds.2019112

This issue Previous Article Local well-posedness and blow-up for the half Ginzburg-Landau-Kuramoto equation with rough coefficients and potential Next Article Global well-posedness for the 2D Boussinesq equations with a velocity damping term

On well-posedness of vector-valued fractional differential-difference equations

1.
Departamento de Matemáticas, Instituto Universitario de Matemáticas y Aplicaciones, Universidad de Zaragoza, 50009 Zaragoza, Spain
2.
Departamento de Matemática y Ciencia de la Computación, Universidad de Santiago de Chile, Las Sophoras 173, Estación Central, Santiago, Chile
3.
Escuela Técnica Superior de Ingeniería y Sistemas de Teleomunicación, Universidad Politécnica de Madrid, C/Nikola Tesla, s/n 28031 Madrid, Spain

^* Corresponding author: Carlos Lizama
^* Corresponding author: Carlos Lizama

Received: April 30, 2018

Revised: June 30, 2018

Published: January 2019

C. Lizama has been partially supported by DICYT, Universidad de Santiago de Chile and FONDECYT 1180041. L. Abadias and P. J. Miana have been partially supported by Project MTM2016-77710-P, DGI-FEDER, of the MCYTS, and Project E-64 D.G. Aragón. M. P. Velasco has been partially supported by Project ESP2016-79135-R of the MCYTS

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Abstract

We develop an operator-theoretical method for the analysis on well posedness of partial differential-difference equations that can be modeled in the form

$ \begin{equation*} (*) \left\{\begin{array}{rll} \Delta^{\alpha} u(n) & = & Au(n+2) + f(n,u(n)), \quad n \in \mathbb{N}_0, \,\, 1< \alpha \leq 2; \\ u(0) & = & u_0;\\ u(1) & = & u_1, \end{array}\right. \end{equation*} $

where $ A $ is a closed linear operator defined on a Banach space $ X $. Our ideas are inspired on the Poisson distribution as a tool to sampling fractional differential operators into fractional differences. Using our abstract approach, we are able to show existence and uniqueness of solutions for the problem (^*) on a distinguished class of weighted Lebesgue spaces of sequences, under mild conditions on sequences of strongly continuous families of bounded operators generated by $ A, $ and natural restrictions on the nonlinearity $ f $. Finally we present some original examples to illustrate our results.

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Mathematics Subject Classification: Primary: 35R11; Secondary: 39A14, 47D06.

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