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

For the difference equation

${x}_{n+2}+{b}_{n}{x}_{n+1}+{c}_{n}{x}_{n}=0$

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.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A10 Additive difference equations
##### References:
 [1] Agarwal, R. P.: Difference equations and inequalities, (1992) [2] Elaydi, S. N.: An introduction to difference equations, (2005) [3] Domshlak, Yu.: Sturmian comparison method in the oscillation study for discrete difference equations I, Differential integral equations 7, 571-582 (1994) · Zbl 0790.39003 [4] Baštinec, J.; Diblík, J.: Subdominant positive solutions of the discrete equation ${\Delta }u\left(k+n\right)=-p\left(k\right)u\left(k\right)$, Abstr. appl. Anal., No. 6, 461-470 (2004) · Zbl 1078.39004 · doi:10.1155/S1085337504306056 [5] Berezansky, L.; Braverman, E.: On existence of positive solutions for linear difference equations with several delays, Adv. dyn. Syst. appl. 1, 29-47 (2006) · Zbl 1124.39002 [6] Chen, S.; Wu, C.: Riccati techniques and approximation for a second-order Poincaré difference equation, J. math. Anal. appl. 222, 177-191 (1998) · Zbl 0914.39008 · doi:10.1006/jmaa.1998.5925