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In the first part of the paper, we prove a fractional fundamental (duBois-Reymond) lemma and a fractional variant of the integration by partsformula. The proof of the second result is based on an integral representationof functions possessing Riemann-Liouville fractional derivatives, derived inthis paper too.  In the second part of the paper, we use the previous results to givenecessary optimality conditions of Euler-Lagrange type (with boundaryconditions) for fractional Bolza functionals and to prove an existence resultfor solutions of linear fractional boundary value problems. In the last case weuse a Hilbert structure and the Stampacchia theorem.

Fractional fundamental lemma and fractional integration by parts formula  -- Applications to critical points of Bolza functionals and to linear  boundary value problems