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.


39A12 Discrete version of topics in analysis
26A33 Fractional derivatives and integrals
39A10 Additive difference equations
49J15 Existence theories for optimal control problems involving ordinary differential equations
49M25 Discrete approximations in optimal control
39A70 Difference operators