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Nonoscillatory solutions of a second-order difference equation of Poincaré type. (English) Zbl 1169.39004
For the difference equation $x_{n+2}+b_nx_{n+1}+c_nx_n=0$ 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.

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A10 Additive difference equations
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##### References:
 [1] Agarwal, R.P., Difference equations and inequalities, (1992), Marcel Dekker New York · Zbl 0784.33008 [2] Elaydi, S.N., An introduction to difference equations, (2005), Springer New York · Zbl 1071.39001 [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 $$\operatorname{\Delta} u(k + n) = - p(k) u(k)$$, Abstr. appl. anal., 6, 461-470, (2004) · Zbl 1078.39004 [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
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