A Comparison of a Priori Estimates of the Solutions of a Linear Fractional System with Distributed Delays and Application to the Stability Analysis

Abstract

In this article, we consider a retarded linear fractional differential system with distributed delays and Caputo type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches—by the use of the Gronwall’s inequality and by the use of integral representations of the solutions are obtained. As application of the obtained estimates, different sufficient conditions which guaranty finite-time stability of the solutions are established. A comparison of the obtained different conditions in respect to the used estimates and norms is made.

Continuous Dependence on the Initial Functions and Stability Properties in Hyers–Ulam–Rassias Sense for Neutral Fractional Systems with Distributed Delays

“…For the IP ( 15), (3) with inhomogeneous term F R (t) = R(t) + F(t), F R (t) ∈ BL loc 1 (J, R n ) on the right side of ( 15) and initial function Y(t)| [a−h,a] = Φ Y (t − a) denote the corresponding unique solution by X R (t) ∈ C(J, R n ), existing according to Theorem 1. Then, from (20), (23) and Theorem 1, it follows that X R (t) ≡ Y(t)| J for t ∈ J and hence possess the following integral representation (see Theorem 4 in [9])…”

Fractional Calculus—Theory and Applications

“…In article [9], the authors considered a retarded linear fractional differential system with distributed delays and Caputo-type derivatives of incommensurate orders. For this system, several a priori estimates for the solutions, applying the two traditional approaches (Gronwall's inequality and integral representations of the solutions) were obtained.…”