About the Resolvent Kernel of Neutral Linear Fractional System with Distributed Delays

The present work considers the initial problem (IP) for a linear neutral system with derivatives in Caputo's sense of incommensurate order, distributed delay and various kinds of initial functions. For the considered IP, the studied problem of existence and uniqueness of a resolvent kernel under some natural assumptions of boundedness type. In the case when, in the system, the term which describes the outer forces is a locally Lebesgue integrable function and the initial function is continuous, it is proved that the studied IP has a unique solution, which has an integral representation via the corresponding resolvent kernel. Applying the obtained results, we establish that, from the existence and uniqueness of a resolvent kernel, the existence and uniqueness of a fundamental matrix of the homogeneous system and vice versa follows. An explicit formula describing the relationship between the resolvent kernel and the fundamental matrix is proved as well.