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Fractional differences and integration by parts. (English) Zbl 1225.39008
Inspired by the results of F. M. Atici and P. W. Eloe [Int. J. Difference Equ. 2, No. 2, 165–176 (2007), Proc. Am. Math. Soc. 137, No. 3, 981–989 (2009; Zbl 1166.39005)] and K. S. Miller and B. Ross [An introduction to the fractional calculus and fractional differential equations. New York: John Wiley & Sons (1993; Zbl 0789.26002)], the authors introduce right fractional sum and difference operators. Based upon the provided theory, a by-part formula is given analogous to the one in usual fractional calculus. Towards the end of the paper, the obtained results are implemented to derive Euler-Lagrange equations for a discrete variational problem in fractional calculus.
MSC:
39A12Discrete version of topics in analysis
26A33Fractional derivatives and integrals (real functions)
39A10Additive difference equations
49J15Optimal control problems with ODE (existence)
49M25Discrete approximations in calculus of variations
39A70Difference operators