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Localized derivatives in spaces of functions representable by localized fractional integrals

ABSTRACT The form of a localized fractional derivative of the Marchaud type is obtained. The compositions of localized fractional Riemann-Liouville and Marchaud type derivatives and localized fractional integrals are considered. For homogeneous differential equations with such compositions, non-local uniqueness conditions for the solution are obtained. A physical interpretation of such initial conditions is given.