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Necessary optimality conditions for fractional difference problems of the calculus of variations. (English) Zbl 1209.49020
Summary: We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler–Lagrange and Legendre type conditions are given. They show that the solutions of the fractional problems coincide with the solutions of the corresponding non-fractional variational problems when the order of the discrete derivatives is an integer value.

MSC:
49K15 Optimality conditions for problems involving ordinary differential equations
39A12 Discrete version of topics in analysis
26A33 Fractional derivatives and integrals
34A08 Fractional ordinary differential equations
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