IJPAM: Volume 78, No. 4 (2012)

A SUPPLEMENT TO POINCARÉ'S THEOREM
ON DIFFERENCE EQUATIONS

Victor Adukov
Department of Differential Equations and Dynamical Systems
South Ural State University
Chelyabinsk, 454080, RUSSIAN FEDERATION


Abstract. The second-order linear difference equation of Poincaré type

\begin{displaymath} u(n+2)+(a+\alpha(n))u(n+1)+(b+\beta(n))u(n)=0, \ n=0,1, \ldots, \end{displaymath}

with Buslaev's restrictions on coefficients

\begin{displaymath} \limsup\limits_{n\to\infty}\sqrt[n]{\vert\alpha(n)\vert}\le... ...limsup\limits_{n\to\infty}\sqrt[n]{\vert\beta(n)\vert}\le q<1 \end{displaymath}

is considered. It is assumed that the characteristic roots of the equation have the same modulus. The set $\cal A$ of all accumulation points for the sequence $\{\frac{u(n+1)}{u(n)}\}_{n=0}^{\infty}$ for any nontrivial solutions of the equation is described.

Received: June 9, 2012

AMS Subject Classification: 39A10

Key Words and Phrases: second-order difference equation of Poincaré type, Poincaré's theorem, asymptotic behavior, Padé approximants

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Source: International Journal of Pure and Applied Mathematics
ISSN printed version: 1311-8080
ISSN on-line version: 1314-3395
Year: 2012
Volume: 78
Issue: 4