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