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Exponential stability of dynamic equations on time scales. (English) Zbl 1100.39013
The authors consider the exponential stability of the zero solution of the first-order vector dynamic equation on a time scale $\mathbb{T}$ $$x^\Delta=f(t,x),\quad t\geq 0,$$ where $f:[0, \infty)\times \mathbb{R}^n\to \mathbb{R}^n$ is a continuous function with $f(t,0)=0$ for all $t$ in the time scale interval $[0,\infty):=\{t\in \mathbb{T}: 0\leq t< \infty\}$. By using appropriate Lyapunov-type functions on time scales and then formulating certain inequalities on these functions, they show that the trivial solution of the above time scale dynamic equation is exponentially or uniformly exponentially stable on $[0, \infty)$. The results obtained are illustrated with some examples.

39A11Stability of difference equations (MSC2000)
39A12Discrete version of topics in analysis
Full Text: DOI EuDML