Variationally Stable Difference Systems by n∞-Similarity

Variationally Asymptotically Stable Difference Systems

“…The various notions about h-stability given by Definition 2.3 include several types of known stability properties such as uniform stability, uniform Lipschitz stability, and exponential asymptotic stability. See [5,[11][12][13].…”

“…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].…”

“…Now, for the converse of Theorem 5.1, we examine the asymptotic equilibrium for the perturbed system of linear difference system (2.2) by using the comparison principle. To do this we need the following comparison principle in [5] which is a slight modification of [10].…”

Asymptotic Equivalence Between Two Linear Dynamic Systems on Time Scales

“…The notion of similarity is an effective tool to study the theory of stability for differential systems and difference systems [5,7,10,11,12,17,18,19]. Markus [17] introduced the notion of kinematic similarity in the set of all n × n continuous matrices defined on [t 0 , ∞) and showed that the relationship of kinematic similarity is an equivalence relation preserving the type numbers of the linear differential systems.…”

“…, is uniformly, exponentially or strictly stable or has linear asymptotic equilibrium, then the linear system ∆y(n) = B(n)y(n) has also the same properties. Also, Choi and Koo [7] 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. Their approach included most types of stability.…”