Stability Analysis of the First Order Non-linear Impulsive Time Varying Delay Dynamic System on Time Scales

• Published in: Qualitative theory of dynamical systems Vol. 18; no. 3; pp. 825 - 840
• Main Authors: Shah, Syed Omar, Zada, Akbar, Hamza, Alaa E.
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 01.12.2019; Springer Nature B.V
• Subjects: Delay; Difference and Functional Equations; Dynamic stability; Dynamical systems; Dynamical Systems and Ergodic Theory; Mathematics; Mathematics and Statistics; Stability analysis; Time; Impulses; 34G20; 34A37; Delay dynamic system; 34N05; 35B35; Time scale; Hyers–Ulam stability
• ISSN: 1575-5460; 1662-3592
• DOI: 10.1007/s12346-019-00315-x

In this paper, we study Hyers–Ulam stability and Hyers–Ulam–Rassias stability of first order non-linear impulsive time varying delay dynamic system on time scales, via a fixed point approach. We obtain some results of existence and uniqueness of solutions by using Picard operator. The main tools for our results are the Grönwall’s inequality on time scales, abstract Grönwall lemma and Banach contraction principle. In order to overcome difficulties arises in our considered model, we pose some conditions along with Lipchitz condition. At the end, an example is given that shows the validity of our main results.