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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=), ordinary difference equations (𝕋=) or q-difference equations (𝕋={q n } n0 , q>1) as special cases. More detailed, the following results are discussed:

(1) If the linear dynamic equation

y Δ =A(t)y(*)

in n possesses an ordinary dichotomy, then there exists a homeomorphism between the bounded solutions of (*) and of the perturbed equation

x Δ =[A(t)+R(t)]x,

provided the perturbation matrix R(t) 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(t)=diag(λ 1 (t),..., λ n (t)), sufficient criteria for the existence of an ordinary dichotomy are given in terms of the functions λ i (t). They are illustrated for the classical examples 𝕋= and 𝕋=.

(3) The authors discuss possible strategies to transform equation (*) into

x Δ =[Λ(t)+R(t)]x

with a diagonal matrix Λ(t) using an invertible linear transformation P(t). This yields a representation Y(t)=P(t)[I+o(1)]D(t) for t of a fundamental matrix for (*), 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.

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