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Duality for the left and right fractional derivatives

机译：左右分数阶导数的对偶

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摘要

We prove duality between the left and right fractional derivatives, independently on the type of fractional operator. Main result asserts that the right derivative of a function is the dual of the left derivative of the dual function or, equivalently, the left derivative of a function is the dual of the right derivative of the dual function. Such duality between left and right fractional operators is useful to obtain results for the left operators from analogous results on the right operators and vice versa. We illustrate the usefulness of our duality theory by proving a fractional integration by parts formula for the right Caputo derivative and by proving a Tonelli-type theorem that ensures the existence of minimizer for fractional variational problems with right fractional operators.
机译：我们证明了左和右分数导数之间的对偶性，独立于分数算符的类型。主要结果断言，函数的右导数是对偶函数的左导数的对偶，或者等效地，函数的左导数是对偶函数的右导数的对偶。左和右小数运算符之间的这种对偶性可用于从右运算符上的相似结果获得左运算符的结果，反之亦然。我们通过证明正确的Caputo导数的分式公式进行分数积分并证明Tonelli型定理来证明对偶理论的有用性，该定理可确保使用正确的分数算子来确保分数变分问题的最小化子的存在。

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