Nonoscillatory solutions of a second-order difference equation of Poincaré type. (English) Zbl 1169.39004

Medina, Rigoberto; Pituk, Mihály

Appl. Math. Lett. 22, No. 5, 679-683 (2009).

For the difference equation \[ x_{n+2}+b_nx_{n+1}+c_nx_n=0 \] with real coefficients satisfying \(b_n\to\beta<0\), \(c_n\to\beta^2/4\) as \(n \to\infty\), it is shown that every non-oscillatory solution has the Poincaré property \(\frac{x_{n+1}}{x_n}\to\beta\). Note that \(\beta\) is a double zero of the corresponding characteristic equation.

Reviewer: Lothar Berg (Rostock)

MSC:

39A11	Stability of difference equations (MSC2000)
39A10	Additive difference equations

Keywords:

second-order difference equation; Poincaré’s theorem; non-oscillatory solution; asymptotic behavior

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