Hyers-Ulam stability of abstract second order linear dynamic equations on time scales

Abstract

In this paper we investigate the Hyers-Ulam Stability of the abstract dynamic equation of the form ∆∆ () + () ∆ () + () () = (), ∈ , where , : → L(), the space of all bounded linear operators from a Banach space into itself, and is rd-continuous from a time scale to. Some examples illustrate the applicability of the main result.

Hyers-Ulam-Rassias stability of abstract second-order linear dynamic equations on time scales

“…In this paper, we extend the results of [14] concerning Hyers-Ulam and Hyers-Ulam-Rassias stability of the equation…”

“…In [4] Anderson, Gates and Heuer extend the work of Li and Shen [19,20] to prove Hyers-Ulam stability of the scalar second-order linear nonhomogeneous dynamic equation on bounded time scales. Hamza and Yaseen [13] generalized these results for unbounded time scales.…”

“…New sufficient conditions for Hyers-Ulam and Hyers-Ulam-Rassias stability of equation (1.1) are obtained. For the concepts of Hyers-Ulam and Hyers-Ulam-Rassias stability, see [4,14].…”

Hyers-Ulam-Rassias Stability of Abstract Dynamic Equations on Time Scales

“…Ten years after the publication of Hyers's theorem, D. G. Bourgin [5] extended the theorem of Hyers and stated it in his paper [5] without proof. Hamza and Yassen extended the work of Douglas, Gates and Heuer, and investigated Hyers-Ulam stability of abstract second order linear dynamic equations on time scales [3,4] Unfortunately, it seems that this result of Bourgin failed to receive attention from mathematicians at that time. No one has made use of this result for a long time.…”

Hyers-Ulam-Rassias Stability For Volterra Integral Equations on Time Scales

“…Many papers introduced the Hyers-Ulam and Hyers-Ulam-Rassias stability for differential equations and integral equations [10,18]. On the other hand side the papers which were presented the Hyers-Ulam and Hyers-Ulam-Rassias stability of dynamic equations are still very few, may be except the studies which were presented in the papers [3,4,5].…”