Boundedness and uniqueness of solutions to dynamic equations on time scales.

*(English)*Zbl 1072.39017The authors investigate the boundedness and uniqueness of solutions to systems of dynamical equations in the more general time scale setting. They give some basic definitions for the dynamical equations

$${x}^{{\Delta}}=f(t,x),\phantom{\rule{1.em}{0ex}}t\ge 0\phantom{\rule{2.em}{0ex}}\left(1\right)$$

$$x\left({t}_{0}\right)={x}_{0},\phantom{\rule{1.em}{0ex}}{t}_{0}\ge {x}_{0}\in \mathbb{R}\xb7\phantom{\rule{2.em}{0ex}}\left(2\right)$$

They define suitable Lyapunov-type functions on time scales and then formulate appropriate inequalities on these functions that guarantee solutions to (1) and (2) are uniformly bounded and unique. These results are generalizations of known results for the case $T=\mathbb{R}$. Some results on the boundedness of solutions of equations (1) and (2) are deduced with examples. The uniqueness of the solutions is also represented.

Reviewer: Ahmed Hegazi (Mansoura)

##### MSC:

39A12 | Discrete version of topics in analysis |

39A11 | Stability of difference equations (MSC2000) |

34A34 | Nonlinear ODE and systems, general |