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A note on periodic character of a difference equation. (English) Zbl 1057.39005
Author’s abstract: We study positive solutions of the difference equation $x_{n+1}=p+{x_{n-(2s-1)}\over x_{n-(2l+1)s+1}}, n=0,1\dots$ where $$p\in[1,\infty)$$ and $$s,l\in N$$. We prove that if $$p>1$$, then every positive solution converges to the positive equilibrium $$x^*=p+1$$ and if $$p=1$$, then every positive solution converges to a $$2s$$-periodic solution. The second result generalizes the main result of W. T. Patula and H. D. Voulov [Proc. Am. Math. Soc. 131, No. 3, 905–909 (electronic) (2003; Zbl 1014.39010)].

##### MSC:
 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations, e.g., of Lyness type
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##### References:
 [1] DOI: 10.1090/S0002-9939-02-06611-X · Zbl 1014.39010 · doi:10.1090/S0002-9939-02-06611-X [2] Karakostas G, Commun. Appl. Nonlinear Anal. (2004) [3] Stević S, Indian J. Math. 43 pp 277– (2001) [4] Stević S, Indian J. Pure Appl. Math. 33 pp 45– (2002)
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