## 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
Full Text:

### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.