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Fractional \(h\)-difference equations arising from the calculus of variations. (English) Zbl 1289.39007

The authors deal with the definitions and properties of the left and right fractional \(h\)-differences. Left and right fractional \(h\)-differences represent the discrete versions of the Riemann-Liouville left and right fractional derivatives, while the fractional \(h\)-sum represents the discrete version the fractional integral. The properties of the left and right fractional \(h\)-differences prove to be analogous to the properties of the left and right Riemann-Liouville derivatives. The central role plays the fractional \(h\)-difference of the power-type function, as well as the exponent law. The authors also obtain the function whose left (right) fractional \(h\)-difference is zero. These results are used in order to formally solve Euler-Lagrange equations in two simple cases of given Lagrangians.

MSC:

39A12 Discrete version of topics in analysis
49J05 Existence theories for free problems in one independent variable
49K05 Optimality conditions for free problems in one independent variable
26A33 Fractional derivatives and integrals
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