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The aim of the present paper is to contribute to the development of the studyof Cauchy problems involving Riemann-Liouville and Caputo fractionalderivatives. Firstly existence-uniqueness results for solutions of non-linearCauchy problems with vector fractional multi-order are addressed. A qualitativeresult about the behavior of local but non-global solutions is also provided.Finally the major aim of this paper is to introduce notions of fractionalstate-transition matrices and to derive fractional versions of the classicalDuhamel formula. We also prove duality theorems relying left state-transitionmatrices with right state-transition matrices.

Cauchy-Lipschitz theory for fractional multi-order dynamics -- State-transition matrices, Duhamel formulas and duality theorems