On well-posedness of vector-valued fractional differential-difference equations
Abstract
We develop an operator-theoretical method for the analysis on well posedness of partial differential 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 is an closed linear operator defined on a Banach space . 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 strongly continuous sequences of bounded operators generated by and natural restrictions on the nonlinearity . Finally we present some original examples to illustrate our results.
Cite
@article{arxiv.1606.05237,
title = {On well-posedness of vector-valued fractional differential-difference equations},
author = {Luciano Abadias and Carlos Lizama and Pedro J. Miana and M. Pilar Velasco},
journal= {arXiv preprint arXiv:1606.05237},
year = {2016}
}