Periodicities which preserve and periodicities which destroy boundedness

  • E. CamouzisEmail author
  • E. A. Grove
  • G. Ladas
  • S. W. Schultz
Original Research


It is known that every positive solution of the difference equation with positive parameter β > 0
$$ x_{n + 1} = \beta + \frac{{x_{n - 2} }} {{x_n }}, n = 0,1, \ldots $$
is bounded. In this note we study the difference equation
$$ x_{n + 1} = \beta _n + \frac{{x_{n - 2} }} {{x_n }}, n = 0,1, \ldots $$
. We show that every positive solution of this equation is bounded when {β n } n=0 is a period-2 sequence of positive real numbers, that is, “Period-2 Preserves Boundedness.” We also show that there exist prime period-3m, sequences {β n } n=0 of positive real numbers such that the equation has unbounded solutions. That is,“Period-3m Destroys Boundedness.”


Existence of unbounded solutions Periodic coefficients Rational difference equations 

Mathematics Subject Classification (2000)

39A10 39A11 


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  1. 1.
    Camouzis E., On the boundedness of some rational difference equations, J. Difference Equ. Appl., 12, 69–94, (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Camouzis E. and Ladas G., Dynamics of Third-Order Rational Difference Equations; With Open Problems and Conjectures, Chapman & Hall/CRC Press, November (2007)Google Scholar

Copyright information

© Foundation for Scientific Research and Technological Innovation 2010

Authors and Affiliations

  • E. Camouzis
    • 1
    Email author
  • E. A. Grove
    • 2
  • G. Ladas
    • 2
  • S. W. Schultz
    • 3
  1. 1.Department of MathematicsAmerican College of GreeceAghia Paraskevi, AthensGreece
  2. 2.Department of MathematicsUniversity of Rhode IslandKingstonUSA
  3. 3.Department of Mathematics and Computer ScienceProvidence CollegeProvidenceUSA

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