Variationally Stable Difference Systems

Abstract

We characterize the h-stability in variation for nonlinear difference systems via n -similarity and Lyapunov functions. Furthermore, using Lyapunov's method and ϱ comparison principle, we obtain some results related to stability for the perturbations of nonlinear difference systems. ᮊ

Variationally Asymptotically Stable Difference Systems

“…Then, this theorem can be easily proved by following the proof of Theorem 2.1 in [6] and and Theorem 3.2 in [12].…”

“…Note that Theorem 3.2 in [12] was improved by Theorem 2.1 in [6] and our Theorem 3.7 as we replace the fundamental matrix Φ(n + 1,n 0 ,x 0 ) by Φ(n,n 0 ,x 0 ) in [12 …”

“…Also, Choi and Koo [5] introduced the notion of n ∞ -similarity in the set of all m × m invertible matrices and showed that two concepts of global h-stability and global h-stability in variation are equivalent by using the concept of n ∞ -similarity and Lyapunov functions. Furthermore, they showed that h-stability of the perturbed system can be derived from h-stability in variation of the nonlinear system in [6]. Note that the n ∞ -similarity is not symmetric or transitive relation but still preserves h-stability which included the most types of stability.…”

“…Note that the n ∞ -similarity is not symmetric or transitive relation but still preserves h-stability which included the most types of stability. For the variational stability in difference systems, see [6]. Also, see [7][8][9] for the asymptotic property of difference systems and Volterra difference systems, respectively.…”

h-Stability for linear dynamic equations on time scales

“…In the study of stability properties of differential and difference systems, the notion of h-stability is very useful because, when we study the asymptotic stability, it is not easy to work with nonexponential types of stability. For the detailed results about h-stability for differential and difference systems, we refer to the papers [4][5][6] and [11][12][13][14][15]. Now, we mention without proof several foundational definitions and results from the calculus on time scales in an excellent introductory text by Bohner and Peterson [3].…”