Stability of abstract dynamic equations on time scales by Lyapunov`s second method

Authors : Alaa HAMZA, Karima ORABY

Pages : 841-861

View : 12 | Download : 3

Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :In this paper, we use the Lyapunov`s second method to obtain new sufficient conditions for many types of stability like exponential stability, uniform exponential stability, $h$-stability, and uniform $h$-stability of the nonlinear dynamic equation \begin{equation*} x^{\Delta}insert ignore into journalissuearticles values(t);=Ainsert ignore into journalissuearticles values(t);xinsert ignore into journalissuearticles values(t);+finsert ignore into journalissuearticles values(t,x);,\;t\in \mathbb{T}^+_\tau:=[\tau,\infty);_{\mathbb T}, \end{equation*} on a time scale $\mathbb T$, where $A\in C_{rd}insert ignore into journalissuearticles values(\mathbb T,Linsert ignore into journalissuearticles values(X););$ and $f:\mathbb T\times X\to X$ is rd-continuous in the first argument with $finsert ignore into journalissuearticles values(t,0);=0.$ Here $X$ is a Banach space. We also establish sufficient conditions for the nonhomogeneous particular dynamic equation \begin{equation*} x^{\Delta}insert ignore into journalissuearticles values(t);=Ainsert ignore into journalissuearticles values(t);xinsert ignore into journalissuearticles values(t);+finsert ignore into journalissuearticles values(t);,\,t\in\mathbb{T}^+_{\tau}, \end{equation*} to be uniformly exponentially stable or uniformly $h$-stable, where $f\in C_{rd}insert ignore into journalissuearticles values(\mathbb T,X);$, the space of rd-continuous functions from $\mathbb T$ to $X$. We construct a Lyapunov function and we make use of this function to obtain our stability results. Finally, we give illustrative examples to show the applicability of the theoretical results.
Keywords : Lyapunov stability theory, dynamic equations, time scales