Characterizations of stability of abstract dynamic equations on time scales
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In this paper, we investigate many types of stability, like (uniform stability, exponential stability and $h$-stability) of the first order dynamic equations of the form \begin{equation*} \left\lbrace \begin{array}{l l} u^{\Delta}(t)=Au(t)+f(t), & \text{ $t\in\mathbb{T},~t>t_0$}\\ u(t_0)=x\in D(A),& \end{array} \right. \end{equation*} and \begin{equation*} \left\lbrace \begin{array}{l l} u^{\Delta}(t)=Au(t)+f(t,u), & \text{ $t\in\mathbb{T},~t>t_0$}\\ u(t_0)=x\in D(A),& \end{array} \right. \end{equation*} in terms of the stability of the homogeneous equation \begin{equation*} \left\lbrace \begin{array}{l l} u^{\Delta}(t)=Au(t), & \text{ $t\in\mathbb{T},~t>t_0$}\\ u(t_0)=x\in D(A),& \end{array} \right. \end{equation*} where $f$ is rd-continuous in $t\in \mathbb T$ and with values in a Banach space $X$, with $f(t,0)=0$, and $A$ is the generator of a $C_0$-semigroup $\{T(t): t\in\mathbb{T}\}\subset L(X)$, the space of all bounded linear operators from $X$ into itself. Here $D(A)$ is the domain of $A$ and $\mathbb{T}\subseteq \mathbb R^{\geq 0}$ is a time scale which is an additive semigroup with property that $a-b\in\mathbb T$ for any $a,b\in\mathbb T$ such that $a>b$. Finally, we give illustrative examples.
Abstract
In this paper, we investigate many types of stability, like (uniform stability, exponential stability and $h$-stability) of the first order dynamic equations of the form \begin{equation*} \left\lbrace \begin{array}{l l} u^{\Delta}(t)=Au(t)+f(t), & \text{ $t\in\mathbb{T},~t>t_0$}\\ u(t_0)=x\in D(A),& \end{array} \right. \end{equation*} and \begin{equation*} \left\lbrace \begin{array}{l l} u^{\Delta}(t)=Au(t)+f(t,u), & \text{ $t\in\mathbb{T},~t>t_0$}\\ u(t_0)=x\in D(A),& \end{array} \right. \end{equation*} in terms of the stability of the homogeneous equation \begin{equation*} \left\lbrace \begin{array}{l l} u^{\Delta}(t)=Au(t), & \text{ $t\in\mathbb{T},~t>t_0$}\\ u(t_0)=x\in D(A),& \end{array} \right. \end{equation*} where $f$ is rd-continuous in $t\in \mathbb T$ and with values in a Banach space $X$, with $f(t,0)=0$, and $A$ is the generator of a $C_0$-semigroup $\{T(t): t\in\mathbb{T}\}\subset L(X)$, the space of all bounded linear operators from $X$ into itself. Here $D(A)$ is the domain of $A$ and $\mathbb{T}\subseteq \mathbb R^{\geq 0}$ is a time scale which is an additive semigroup with property that $a-b\in\mathbb T$ for any $a,b\in\mathbb T$ such that $a>b$. Finally, we give illustrative examples.
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