On variation of constant formulae for linear fractional delayed system with Lebesgue integrable initial conditions

In the present work is considered the Cauchy (initial) problem (IP) for a linear delayed system with derivatives in Caputo's sense of incommensurate order, distributed delays and locally Lebesgue integrable initial functions. For this IP is studied the important problem of existence a formula of variation of constants. The obtained results extend the corresponding ones in the particular cases of fractional systems with constant and variable delays. The proposed conditions are almost the same as the conditions which guarantee the same result in the case of linear differential equations with distributed delays with integer order of differentiation.