On the stability, integrability and boundedness analyses of systems of integro-differential equations with time-delay retardation

In the paper, Tian and Wang (Appl Math Lett 105:106325, 8 pp, 2020, Theorem 1) took into consideration a linear system of integro-delay differential equations (IDDEs) with constant time retardation. In Tian and Wang (2020), the authors proved a new and interesting theorem concerning asymptotically stability of zero solution of that linear system of IDDEs with constant time retardation. Tian and Wang (2020) constructed a new Lyapunov–Krasovskiĭ functional (LKF) and used that LKF to prove the related theorem on the asymptotically stability. To the best of the information, we would like to note that the asymptotically stability result of Tian and Wang (2020, Theorem 1) consists of very interesting and strong conditions. However, in this paper, we construct a more suitable LKF, then we obtain the result of Tian and Wang (2020, Theorem 1) for uniformly asymptotically stability of zero solution under very weaker condition using that LKF as well as we investigate the integrability of the norm and boundedness of solutions. For illustrative aims, in particular cases, two numerical examples are provided for the uniformly asymptotically stability of zero solution as well as integrability and boundedness of solutions. By this work, we do a contribution to the topic of the paper and relevant literature. The results of this paper have also new contributions to the former literature and they may useful for researchers working on the topics of this paper.