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Linear systems of fractional nabla difference equations. (English) Zbl 1218.39003
The authors consider a system of linear fractional nabla difference equations with constant coefficients of the form
$\nabla_0^\nu y(t)=Ay(t)+f(t),\quad t=1,2,\dots,$
where $$A$$ denotes an $$n\times n$$ matrix with constant entries, $$0<\nu<1$$ and $$\nabla_0^\nu$$ is the Riemann-Liouville fractional difference operator. They construct the fundamental matrix for the homogeneous system and the causal Green’s function for the nonhomogeneous system. Applications to asymptotic behavior are given.

##### MSC:
 39A06 Linear difference equations 39A12 Discrete version of topics in analysis 26A33 Fractional derivatives and integrals 34A08 Fractional ordinary differential equations
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##### References:
 [1] M. Abramowitz and I. Stegun, Handbook of mathematical functions, with formulas, graphs and mathematical tables , Sixth printing, with corrections, in National Bureau of Standards Applied Mathematics Series 55 National Bureau of Standards, Washington, D.C., 1967. · Zbl 0543.33001 [2] R.P. Agarwal, A propos d’une note de M. Pierre Humbert , C.R. Acad. Sci. 236 (1953), 2031-2032. · Zbl 0051.30801 [3] G. Anastassiou, Nabla discrete fractional calculus and nabla inequalities , Math. Comput. Modelling 51 (2010), 562-571. · Zbl 1190.26001 [4] G.E. Andrews, R. Askey and R. Roy, Special functions , in Encyclopedia of mathematics and its applications 71 , Cambridge University Press, Cambridge, 1999. [5] F.M. AtP.W. Eloe, Discrete fractional calculus with the nabla operator , Electron. J. Qual. Theory Differ. Equat. 2009 Special Edition I, No. 3, 12 pages (electronic). · Zbl 1189.39004 [6] —, A transform method in discrete fractional calculus , Inter. J. Difference Equations 2 (2007), 165-176. [7] —, Fractional $$q$$-calculus on a time scale , J. Nonlinear Math. Phys. 14 (2007), 333-344. [8] F.M. AtP.W. Eloe, Initial value problems in discrete fractional calculus , Proc. Amer. Math. Soc. 137 (2009), 981-989. · Zbl 1166.39005 [9] F.M. AtS. Şengül, Modeling with fractional difference equations , J. Math. Anal. Appl. 369 (2010), 1-9. · Zbl 1204.39004 [10] B. Bonilla, M. Rivero and J.J. Trujillo, On systems of linear fractional differential equations with constant coefficients , Appl. Math. Comput. 187 (2007), 68-78. · Zbl 1121.34006 [11] H.J. Haubold, A.M. Mathai and R.K. Saxena, Mittag-Leffler functions and their applications , arxiv.org/abs /09090230. · Zbl 1218.33021 [12] R. Magin, Fractional calculus in bioengineering , Begell House Publishers, Redding, CT, 2004. [13] G.M. Mittag-Leffler, Sur la nouvelle fonction $$E_\alpha(x)$$ , C.R. Acad. Sci. Paris 137 (1903), 554-558. · JFM 34.0435.01 [14] A. Nagai, Discrete Mittag-Leffler function and its applications , Publ. Res. Inst. Math. Sci. Kyoto Univ. 1302 (2003), 1-20. [15] I. Podlubny, Fractional differential equations , Academic Press, New York, 1999. · Zbl 0924.34008 [16] G. Samko, A.A. Kilbas and O.I. Marichev, Fractional integrals and derivatives: Theory and applications , Gordon and Breach, Yverdon, 1993. · Zbl 0818.26003 [17] H.M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leffler function in the kernel , Appl. Math. Comput. 211 (2009), 198-210. · Zbl 1432.30022
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