# On the qualitative study of solutions of a class of nonlinear abstract dynamic equations on time scales and applications

## DOI:

https://doi.org/10.54021/seesv5n2-008## Keywords:

Abstract Dynamic Equation, Time Scale, Integral Inequality, Semi-Group, H-Stability## Abstract

The theory of dynamic equations on time scale was introduced in [7, 8] whose main objective is to provide a unified approach to continuous and discrete analysis. The calculus on time scales and dynamic equations on time scales have applications in any field that requires simultaneous modeling of continuous and discrete processes, because they bridge the divide between continuous and discrete aspects of processes. Perturbation theory is a pertinent discipline for the applications of time scale dynamics which is a compilation of methods systematically used to evaluate the global behavior of solutions of dynamical systems with occurrence on non uniform domains. One of the analytic methods of the perturbation theory was referred to integral inequalities to quest some type of stability. In the last few years, the search on qualitative properties of dynamics was directed to the time scale integral inequalities using diverse techniques and some significant results were obtained. Some of the original references on this approach include [1,2,3,4,5,9,10,11,12,13,14,17,22,24]. In this work, we study the h-stability problems of some classes of dynamic equations as an extension of exponential stability. We derive some sufficient conditions that guarantee h-stability of perturbed dynamic equations using Grönwall- typ integral inequality approach and Lyapunov function approach. We prove under certain conditions on the linear and nonlinear perturbations that the resulting perturbed nonlinear abstract dynamic equation still acquired h stable, if the associated dynamic equation has already owned this property. Finally, an numerical example is introduced to illustrate the applicability of the main results.

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*STUDIES IN ENGINEERING AND EXACT SCIENCES*,

*5*(2), e5408. https://doi.org/10.54021/seesv5n2-008