## Periodic solutions to a difference equation with maximum.(English)Zbl 1019.39005

Consider the difference equation $x_n=\max \left\{ \frac A{x_{n-1}},\;\frac B{x_{n-3}},\frac C{x_{n-5}}\right\} n=0,1,\dots,\tag{$$*$$}$ where $$A,B,C$$ are any nonnegative real numbers with $$A+B+C>0.$$ The author proves that there exists a positive integer $$T$$ such that every positive solution of $$(*)$$is eventually periodic of period $$T$$. The period $$T$$ $$\in \{2,6,8,10\}$$ and is determined according to the values of $$A$$ , $$B$$ and $$C$$.
Reviewer: Fozi Dannan (Doha)

### MSC:

 39A11 Stability of difference equations (MSC2000) 39B05 General theory of functional equations and inequalities

### Keywords:

periodic solutions; nonlinear difference equation
Full Text:

### References:

 [1] G. Ladas, Open problems and conjectures, J. Diff. Eqns. and Appl., 4(3)(1998), 312. · Zbl 1057.39505 [2] Dean Clark and James T. Lewis, A Collatz-type difference equation, Proceedings of the Twenty-sixth Southeastern International Conference on Combinatorics, Graph Theory and Computing (Boca Raton, FL, 1995), 1995, pp. 129 – 135. · Zbl 0912.11005 [3] G. Ladas, Open problems and conjectures, J. Diff. Eqns. and Appl., 2(1996), 339-341. [4] A. M. Amleh, J. Hoag, and G. Ladas, A difference equation with eventually periodic solutions, Comput. Math. Appl. 36 (1998), no. 10-12, 401 – 404. Advances in difference equations, II. · Zbl 0933.39030 [5] D. Mishev, W.T. Patula, and H.D. Voulov, On a Reciprocal Difference Equation with Maximum, Computers and Mathematics with Applications, 43(2002), 1021-1026. · Zbl 1050.39015 [6] H.D. Voulov, On the Periodic Character of Some Difference Equations, J. Diff. Eqns. and Appl., 8(9)(2002), 799-810. · Zbl 1032.39004
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.