Stability of neutral Volterra integro-differential equations with infinite delay

Improved Stability and Instability Results for Neutral Integro-Differential Equations including Infinite Delay

“…From these information, we see that in despite of the existence of numerous results with regard to IDEs with and without delays, there exists a number of limited results on the qualitative results of IDEs in neutral forms such that in which the LKF method used as basic tool in the proofs. In addition, as for qualitative results with regard to systems of NVIDEs, in which the LKF method used as basic tool in the proofs, we found only the above results, see Li and Jiang [20]. The proper reason for this case is that because of the difficulty to find a suitable LK F which allows meaning results for nonlinear systems of NVIDEs.…”

“…Proof. As similar to Li and Jiang [20], from the definition of Dx t and system of NVIDEs (2), we obtain D + Dx t , which denotes the upper right derivative , as the below:…”

“…In [23], the authors utilized using Sadovskii's fixed point theorem to fulfill the proofs. Li and Jiang [20] investigated the stability, uniform stability and instability of trivial solution of the system of linear NVIDEs of first order with infinite delay via the LKF method:…”

“…Li and Jiang [20] obtained four new qualitative results with regard to zero solution of the system of linear NVIDEs (1) of first order with infinite delay.…”

“…We establish three new outcomes as new theorems with regard to these concepts defining and using two new LKFs. The motivation of this study has been inspired from the results of the papers, the books above and in particular from the results of Li and Jiang [20]. We aim to obtain new outcomes for the concepts mentioned and do new contributions to qualitative theory of NVIDEs via the LKF method.…”