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Periodicities which preserve and periodicities which destroy boundedness

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

Abstract

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.”

Keywords

Existence of unbounded solutions Periodic coefficients Rational difference equations 

Mathematics Subject Classification (2000)

39A10 39A11 

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References

  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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