The aim of the present work is to establish explicit conditions for global asymptotic stability of neutral linear fractional differential system with distributed delays. The derivatives in the system can be of Riemann-Liouville or Caputo type. We consider basically the incommensurate case but some results are given for the commensurate case too. The proposed conditions are based on the logarithmic norm technique and part of the results are new even in the particular case of system with constant delays.

The aim of the present work is to study the initial value problem for neutral linear fractional differential system with distributed delays in incommensurate case. Furthermore, in the autonomous case with derivatives in the Riemann-Liouville or Caputo sense we establish that if all roots of the introduced characteristic equation have negative real parts, then the zero solution is globally asymptotically stable. The proposed condition coincides with the conditions which guaranty the same result in the particular case of system with constant delays.

In the present work we study the Cauchy problem for linear incommensurate fractional differential system with distributed delays. For the autonomous case with distributed delays with derivatives in Riemann-Liouville or Caputo sense, we establish sufficient conditions under which the zero solution is globally asymptotic stable. The established conditions coincide with the conditions which guaranty the same result in the particular case of system with constant delays and for the case of system without delays in the commensurate case too.

The goal of the present paper is to obtain sufficient conditions that guaranty the existence and uniqueness of an absolutely continuous fundamental matrix for a retarded linear fractional differential system with Caputo type derivatives and distributed delays. Some applications of the obtained result concerning the integral representation of the solutions are given too.

In the present paper, first we obtain sufficient conditions for the existence and uniqueness of the solution of the Cauchy problem for an inhomogeneous neutral linear fractional differential system with distributed delays (even in the neutral part) and Caputo type derivatives, in the case of initial functions with first kind discontinuities. This result allows to prove that the corresponding homogeneous system possesses a fundamental matrix $C(t,s)$ continuous in $t,t\in [a,\infty ),a\in \mathbb{R}$. As an application, integral representations of the solutions of the Cauchy problem for the considered inhomogeneous systems are obtained.