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Asymptotic behavior of solutions of dynamic equations. (English. Russian original) 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 ${\Bbb T}$, which include ordinary differential equations (${\Bbb T}={\Bbb R}$), ordinary difference equations (${\Bbb T}={\Bbb Z}$) or $q$-difference equations (${\Bbb T}=\{q^n\}_{n\geq 0}$, $q>1$) as special cases. More detailed, the following results are discussed: (1) If the linear dynamic equation $$ y^\Delta=A(t)y \tag *$$ in ${\Bbb R}^n$ possesses an ordinary dichotomy, then there exists a homeomorphism between the bounded solutions of $(\ast)$ and of the perturbed equation $$ x^\Delta=[A(t)+R(t)]x, $$ provided the perturbation matrix $R(t)$ is of class $L^1$ w.r.t. the ${\Bbb T}$-integral. (2) In order to apply this result to the special case of a diagonal matrix $A(t)= \text{diag}(\lambda_1(t),\ldots,$ $\lambda_n(t))$, sufficient criteria for the existence of an ordinary dichotomy are given in terms of the functions $\lambda_i(t)$. They are illustrated for the classical examples ${\Bbb T}={\Bbb R}$ and ${\Bbb T}={\Bbb Z}$. (3) The authors discuss possible strategies to transform equation $(\ast)$ into $$x^\Delta=[\Lambda(t)+R(t)]x$$ with a diagonal matrix $\Lambda(t)$ using an invertible linear transformation $P(t)$. This yields a representation $Y(t)=P(t)[I+o(1)]D(t)$ for $t\to\infty$ of a fundamental matrix for $(\ast)$, with the Hardy-Littlewood symbol $o(1)$ and a diagonal matrix $D(t)$. Such an approach is illustrated for the special case $A(t)=C+R(t)$ with a diagonalizable constant matrix $C$. Then it is possible to establish a representation of $Y(t)$ 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.

34D05Asymptotic stability of ODE
39A11Stability of difference equations (MSC2000)
39A70Difference operators
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