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The present work was primarily motivated by our findings in the literature ofsome flaws within the proof of the second-order Legendre necessary optimalitycondition for fractional calculus of variations problems. Therefore we wereeager to elaborate a correct proof and it turns out that this goal is highlynontrivial, especially when considering final constraints. This paper is theresult of our reflections on this subject. Precisely we consider here aconstrained minimization problem of a general Bolza functional that depends ona Caputo fractional derivative of order 0 < $\alpha$ $\le$ 1 and on aRiemann-Liouville fractional integral of order $\beta$ > 0, the constraint setdescribing general mixed initial/final constraints. The main contribution ofour work is to derive corresponding first-and second-order necessary optimalityconditions, namely the Euler-Lagrange equation, the transversality conditionsand, of course, the Legendre condition. A detailed discussion is provided onthe obstructions encountered with the classical strategy, while the new proofthat we propose here is based on the Ekeland variational principle. Furthermorewe underline that some subsidiary contributions are provided all along thepaper. In particular we prove an independent and intrinsic result of fractionalcalculus stating that it does not exist a nontrivial function which is,together with its Caputo fractional derivative of order 0 < $\alpha$ < 1,compactly supported. Moreover we also discuss some evidences claiming thatRiemann-Liouville fractional integrals should be considered in the formulationof fractional calculus of variations problems in order to preserve theexistence of solutions.

Legendre's necessary condition for fractional Bolza functionals with mixed initial/final constraints