A note for “On the rational recursive sequence \(x_{n+1}=\frac{A+\sum^k_{i=0} \alpha_i x_{n-i}}{\sum^k_{i=0} \beta_ix_{n-i}}\)”. (English) Zbl 1320.39015

Li, Xianyi; Zhou, Li

Arab J. Math. Sci. 18, No. 1, 15-24 (2012).

This note is essentially a critique of the paper [E. M. E. Zayed and M. A. El-Moneam, Math. Bohem. 133, No. 3, 225–239 (2008; Zbl 1199.39025)]. In Theorem 5, Zayed and El-Moneam claimed that if all roots of the polynomial \(\lambda^{k+1}-\sum_{j=0}^k b_j\lambda^{k-j}\) lie in the open unit disk, then \(\sum_{j=0}^k | b_j| < 1\). This note gives the well-known example, for the case \(k=1\), namely \(\lambda^2 +\lambda+a\) for \(a\in (0, 1/4]\), where both roots of a quadratic polynomial belong to the unit disk, but \(| b_0| +| b_j| > 1\), which refutes the above claim.
In Theorem 7, Zayed and El-Moneam claimed that positive increasing solutions of the difference equation \[ x_{n+1} = \frac{A+\sum_{i=0}^k \alpha_ix_{n-i}}{\sum_{i=0}^k \beta_ix_{n-i}} \tag{1} \] are bounded and persist. In this note the authors prove that positive solutions of equation (1) are bounded and persist. However, this result is also well known and can be found, for example, in the reviewer’s paper [J. Appl. Math. Comput. 24, No. 1–2, 295–303 (2007; Zbl 1127.39043)].
In Theorem 2.1, Li and Zhou also prove the existence of non-oscillatory solutions of equation (1) by using an inclusion theorem by L. Berg [J. Difference Equ. Appl. 10, 399–408 (2004; Zbl 1056.39003)]. They modify the proofs of similar results presented in some papers by this reviewer, for example, in [J. Math. Anal. Appl. 316, No. 1, 60–68 (2006; Zbl 1090.39009); Appl. Math. Lett. 20, No. 1, 28–31 (2007; Zbl 1131.39009); C. Çinar, S. Stević and İ. Yalçınkaya, Rostocker Math. Kolloq. 59, 41–49 (2005); Zbl 1083.39003)], but do not cite any of these papers.

Reviewer: Stevo Stević (MR2494777)

MSC:

39A22	Growth, boundedness, comparison of solutions to difference equations
39A21	Oscillation theory for difference equations

Keywords:

rational difference equation; existence of non-oscillatory solution; boundedness and persistence; global asymptotic stability; inclusion theorem

References:

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