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We prove optimality conditions for different variational functionalscontaining left and right Caputo fractional derivatives. A sufficient conditionof minimization under an appropriate convexity assumption is given. AnEuler-Lagrange equation for functionals where the lower and upper bounds of theintegral are distinct of the bounds of the Caputo derivative is also proved.Then, the fractional isoperimetric problem is formulated with an integralconstraint also containing Caputo derivatives. Normal and abnormal extremalsare considered.

Necessary and sufficient conditions for the fractional calculus of variations with Caputo derivatives