Duality of singular linear systems of fractional nabla difference equations

Abstract

Please cite this article as: I.K. Dassios, D.I. Baleanu, Duality of singular linear systems of fractional nabla difference equations, Appl. Math. Modelling (2014), doi: http://dx. Abstract:The main objective of this article is to provide a link between the solutions of an initial value problem of a linear singular system of fractional nabla difference equations and its proper (and transposed) dual systems. By taking into consideration the case that the coefficients are constant matrices with the leading coefficient singular, we study the prime system and by using the invariants of its pencil we give necessary and sufficient conditions for existence and uniqueness of solutions. After we prove that by using the pencil of the prime system we can study the existence and uniqueness of solutions of the proper and transposed dual systems. Moreover their solutions, when they exist, can be explicitly represented without resorting to further processes of computations for each one separately. Finally, numerical examples are given based on a singular fractional nabla real dynamical system to justify our theory.

A macroeconomic mathematical model for the national income of a union of countries with interaction and trade

“…The fractional nabla operator is a very interesting tool for this, since it succeeds to provide information from a specific year in the past until the current year, see Dassios and Baleanu (2013), , Ŵ(k−j+1) . Results on stability, robustness, duality, etc., of this operator, see Dassios (2016), Dassios and Baleanu (2015), , may also be used successfully on the suggested macroeconomic models.…”

Lebesgue regularity for differential difference equations with fractional damping

“…Studies on qualitative properties, as for example, the existence of positive solutions for discrete fractional systems, have been provided by Goodrich. () Other interesting contributions are due to Ferreira, Holm, Kovács, Li, and Lubich, Dassios,() Wu, Baleanu et al,() and Tarasov et al()…”

“…Studies on qualitative properties, as for example, the existence of positive solutions for discrete fractional systems, have been provided by Goodrich. [14][15][16] Other interesting contributions are due to Ferreira, 17 Holm, 18 Kovács, Li, and Lubich, 19 Dassios,20,21 Wu, Baleanu et al, [22][23][24][25] and Tarasov et al [26][27][28] Starting with the work of Blunck, 29 the existence and uniqueness of solutions for discrete systems that belong to the Lebesgue space of vector-valued sequences began to be considered by many authors. [30][31][32][33] Some of the studies correspond to a numerical point of view.…”

Sufficient conditions for oscillation of a nonlinear fractional nabla difference system

“…For example, we refer the reader to Atici and Eloe (2007, 2009a, 2009b, 2011, 2012), Abdeljawad and Atici (2012), Atici and Wu (2014), Goodrich (2011, 2012), Anastassiou (2010, 2011), Čermák et al. (2015), Dassios and Baleanu (2013, 2015), Dassios et al. (2014), Hein et al.…”