MR2427543
Toshiki NAITO, Pham Huu Anh NGOC and Jong Son SHIN
Toshiki NAITO, Pham Huu Anh NGOC and Jong Son SHIN
Representations and Asymptotic Behavior of Solutions to Periodic Linear Difference Equations
Funkcialaj Ekvacioj. Serio Internacia
51
2008
55--80
http://fe.math.kobe-u.ac.jp/FE/FullPapers/51-1/51_55.pdf
http://www.ams.org/mathscinet-getitem?mr=MR2427543
We give a new representation of solutions of the periodic linear difference equation of the form $x(n+1)=Bx(n)+b(n)$, where $B$ is a complex $p\times p$ matrix and $b(n)\in{\mathbb C}^p$ satisfies the condition $b(n)=b(n+\rho)$, $\rho\in{\mathbb N}$, $\rho\geq 2$. If $B=e^{\tau A}$, $\tau>0$, then the equation has two representations of solutions based on $A$ and $B$. In particular, the representation of solutions based on $A$ is deduced from the one based on $B$ by using the translation formulae from $B$ to $A$. Using these representations, we can obtain the complete classification of the set of initial values according to the behavior of solutions. As applications of these results, by the initial values we characterize necessary and sufficient conditions on the existence of a bounded solution and a $\rho$-periodic solution.
Periodic linear difference equation, Representation of solution, Bounded solution, Periodic solution, Asymptotic behavior of solution, Index of growth order.
39A10, 39A11.
51-55
2008
Representations and Asymptotic Behavior of Solutions to Periodic Linear Difference Equations
Toshiki NAITO, Pham Huu Anh NGOC and Jong Son SHIN
Toshiki NAITO, Pham Huu Anh NGOC and Jong Son SHIN
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