Fundamental Matrix, Measure Resolvent Kernel and Stability Properties of Fractional Linear Delayed System with Discontinuous Initial Conditions

In the present work, a Cauchy (initial) problem for a fractional linear system with distributed delays and Caputo-type derivatives of incommensurate order is considered. As the main result, a new straightforward approach to study the considered initial problem via an equivalent Volterra–Stieltjes integral system is introduced. This approach is based on the existence and uniqueness of a global fundamental matrix for the corresponding homogeneous system, which allows us to prove that the corresponding resolvent system possesses a unique measure resolvent kernel. As a consequence, an integral representation of the solutions of the studied system is obtained. Then, using the obtained results, relations between the stability of the zero solution of the homogeneous system and different kinds of boundedness of its other solutions are established.