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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 β<0, c n β 2 /4 as n, it is shown that every non-oscillatory solution has the Poincaré property x n+1 x n β. Note that β is a double zero of the corresponding characteristic equation.

39A11Stability of difference equations (MSC2000)
39A10Additive difference equations
[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 Δu(k+n)=-p(k)u(k), 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