Journal
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS
Volume 39, Issue 5, Pages 2679-2708Publisher
AMER INST MATHEMATICAL SCIENCES-AIMS
DOI: 10.3934/dcds.2019112
Keywords
Difference equations; fractional and nonlinear PDE; Poisson distribution; weighted Lebesgue space; well posedness
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Funding
- DICYT, Universidad de Santiago de Chile
- FONDECYT [1180041]
- DGI-FEDER, of the MCYTS [MTM2016-77710-P]
- MCYTS [ESP2016-79135-R]
- [E-64]
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We develop an operator-theoretical method for the analysis on well posedness of partial differential-difference equations that can be modeled in the form (*) {Delta(alpha) u(n) = Au(n + 2) + f(n, u(n)), n is an element of N-0, 1 < alpha <= 2; u(0) = u(0); u(1) = u(1); 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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