On the Stability, Integrability and Boundedness Analysis of Systems of Integro-Differential Equations With Time-Delay

Abstract

In the paper "AIMS Math. 5 (2020), no. 6, 6448-6456", Tian et al. [15, Theorem 1] considered a linear system of integro-time delay differential equations (IDDEs) with constant time retardation. In [15], firstly, a generalized double integral inequality was obtained and then less conservative asymptotically stability criteria were proposed by using that double integral inequality and choosing a new Lyapunov-Krasovskii functional (LKF). To the best of the information, we would like to note that the asymptotically stability criteria of Tian et al. [15, Theorem 1] consist of very interesting but strong conditions. However, in this paper, we define a more suitable LKF, then we obtain the result of Tian et al. [15, Theorem 1] for uniformly asymptotically stability under very weaker conditions using the LKF and also investigate the integrability of the norm and boundedness of solutions. To show the effectiveness of our results, two numerical examples are proposed for the uniformly asymptotically stability as well as integrability and boundedness of solutions. By this work, we do contributions to the work of Tian et al. [15, Theorem 1] under very weaker conditions and those in the previous relevant literature, and we obtain two more new results on the integrability of the norm of solutions and boundedness of solutions. The results of this paper are new and they may be useful for researchers working on the topics of this paper.