## On positive solutions of a reciprocal difference equation with minimum.(English)Zbl 1074.39002

The authors consider the behavior of positive solutions of the equation $x_{n+1}=\min\left\{\frac{A}{x_{n}x_{n-1}\cdots x_{n-k}}, \frac{B}{x_{n-(k+2)}\cdots x_{n-(2k+2)}}\right\}, \;\;n\geq0,$ where $$A$$, $$B$$ and the initial values $$x_{-(2k+2)},\ldots,x_0$$ are positive real numbers. It is shown that if $$0<A\leq B$$ then all positive solutions are eventually periodic with period $$k+2$$. A detailed study of the case $$k=0$$ is presented.

### MSC:

 39A11 Stability of difference equations (MSC2000) 39A20 Multiplicative and other generalized difference equations
Full Text:

### References:

 [1] A.M. Amleh, J. Hoag and G. Ladas,A difference equation with eventually periodic solutions, Comput. Math. Appl.36 (10–12) (1998), 401–404. · Zbl 0933.39030 [2] H.M. El-Owaidy, A.M. Ahmed and M.S. Mousa, On asymptotic behaviour of the difference equation $$x_{n + 1} = \alpha + \frac{{x_{n - 1}^p }}{{x_n^p }}$$ , J. Appl. Math. & Computing12 (1–2) (2003), 31–37. · Zbl 1052.39005 [3] G. Ladas,Open problems and conjectures, Differ. Equations Appl.2 (1996), 339–341. [4] G. Ladas,Open problems and conjectures, J. Differ. Equations Appl.4 (3) (1998), 312. · Zbl 1057.39505 [5] D.P. Mishev and W.T. Patula,A reciprocal Difference Equation with Maximum, Comput. Math. Appl.43 (2002), 1021–1026. · Zbl 1050.39015 [6] A.D. Mishkis,On some problems of the theory of differential equations with deviating argument, UMN 32:2 (194) (1977), 173–202. [7] E.P. Popov,Automatic regulation and control, Moscow (1966) (in Russian). [8] S. Stević, On the recursive sequence $$x_{n + 1} = - \frac{1}{{x_n }} + \frac{A}{{x_{n - 1} }}$$ , Int. J. Math. Math. Sci.27 (1) (2001), 1–6. · Zbl 1005.39016 [9] S. Stević, On the recursive sequencex n+1 =g(x n ,x n$$\backslash$$t-1)/(A +x n ), Appl. Math. Lett.15 (2002), 305–308. · Zbl 1029.39007 [10] S. Stević, On the recursive sequencex n+1 =x n$$\backslash$$t-1/g(x n ), Taiwanese J. Math.6 (3) (2002), 405–414. [11] S. Stević, On the recursive sequence $$x_{n + 1} = \frac{{\alpha + \beta x_{n - 1} }}{{1 + g(x_n )}}$$ , Indian J. Pure Appl. Math.33 (12) (2002), 1767–1774. · Zbl 1019.39011 [12] S. Stević, On the recursive sequence $$x_{n + 1} = \alpha _n + \frac{{x_{n - 1} }}{{x_n }}$$ , Dynam. Contin. Discrete Impuls. Systems10a (6) (2003), 911–917. [13] Z. Zhang, B. Ping and W. Dong,Oscillatory of unstable type second-order neutral difference equations, J. Appl. Math. & Computing9 No. 1 (2002), 87–100. · Zbl 0999.39014 [14] Z. Zhou, J. Yu and G. Lei,Oscillations for even-order neutral difference equations, J. Appl. Math. & Computing7 No. 3 (2000), 601–610. · Zbl 0966.39004
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.