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Asymptotic behavior of solutions of dynamic equations. (English) Zbl 1083.34035

This paper reviews some earlier results of the authors given in the papers by the second and third author [J. Difference Equ. Appl. 7, No. 1, 21–50 (2001; Zbl 0972.39004)], and the first and the third author [Dyn. Syst. Appl. 12, No. 1–2, 23–43 (2003; Zbl 1053.39029)]. They address linear dynamic equations on time scales $𝕋$, which include ordinary differential equations ($𝕋=ℝ$), ordinary difference equations ($𝕋=ℤ$) or $q$-difference equations ($𝕋={\left\{{q}^{n}\right\}}_{n\ge 0}$, $q>1$) as special cases. More detailed, the following results are discussed:

(1) If the linear dynamic equation

${y}^{{\Delta }}=A\left(t\right)y\phantom{\rule{2.em}{0ex}}\left(*\right)$

in ${ℝ}^{n}$ possesses an ordinary dichotomy, then there exists a homeomorphism between the bounded solutions of $\left(*\right)$ and of the perturbed equation

${x}^{{\Delta }}=\left[A\left(t\right)+R\left(t\right)\right]x,$

provided the perturbation matrix $R\left(t\right)$ is of class ${L}^{1}$ w.r.t. the $𝕋$-integral.

(2) In order to apply this result to the special case of a diagonal matrix $A\left(t\right)=\text{diag}\left({\lambda }_{1}\left(t\right),...,$ ${\lambda }_{n}\left(t\right)\right)$, sufficient criteria for the existence of an ordinary dichotomy are given in terms of the functions ${\lambda }_{i}\left(t\right)$. They are illustrated for the classical examples $𝕋=ℝ$ and $𝕋=ℤ$.

(3) The authors discuss possible strategies to transform equation $\left(*\right)$ into

${x}^{{\Delta }}=\left[{\Lambda }\left(t\right)+R\left(t\right)\right]x$

with a diagonal matrix ${\Lambda }\left(t\right)$ using an invertible linear transformation $P\left(t\right)$. This yields a representation $Y\left(t\right)=P\left(t\right)\left[I+o\left(1\right)\right]D\left(t\right)$ for $t\to \infty$ of a fundamental matrix for $\left(*\right)$, with the Hardy-Littlewood symbol $o\left(1\right)$ and a diagonal matrix $D\left(t\right)$. Such an approach is illustrated for the special case $A\left(t\right)=C+R\left(t\right)$ with a diagonalizable constant matrix $C$. Then it is possible to establish a representation of $Y\left(t\right)$ as above.

Finally, results on the asymptotic behavior of the scalar exponential function on time scales from Bodine and Lutz [loc cit] conclude the paper.

##### MSC:
 34D05 Asymptotic stability of ODE 39A11 Stability of difference equations (MSC2000) 39A70 Difference operators
##### Keywords:
dynamic equation; time scale; ordinary dichotomy