Opuscula Math. 37, no. 6 (2017), 795-819
http://dx.doi.org/10.7494/OpMath.2017.37.6.795

Opuscula Mathematica

# The second Cushing-Henson conjecture for the Beverton-Holt q-difference equation

Martin Bohner
Sabrina H. Streipert

Abstract. In this paper, we study the second Cushing-Henson conjecture for the Beverton-Holt difference equation with periodic inherent growth rate and periodic carrying capacity in the quantum calculus setting. We give a short summary of recent results regarding the Beverton-Holt difference and $$q$$-difference equation and introduce the theory of quantum calculus briefly. Next, we analyze the second Cushing-Henson conjecture. We extend recent studies in [The Beverton-Holt q-difference equation with periodic growth rate, Difference Equations, Discrete Dynamical Systems, and Applications, Springer-Verlag, Berlin, Heidelberg, New York, 2015, pp. 3-14] and state a modified formulation of the second Cushing-Henson conjecture for the Beverton-Holt $$q$$-difference equation as a generalization of existing formulations.

Keywords: Beverton-Holt equation, Cushing-Henson conjectures, $$q$$-difference equation, periodic solution.

Mathematics Subject Classification: 39A12, 39A13, 92D25.

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• Martin Bohner
• Department of Mathematics and Statistics, Missouri S&T, Rolla, MO 65409-0020, USA
• Sabrina H. Streipert
• Department of Mathematics and Statistics, Missouri S&T, Rolla, MO 65409-0020, USA
• Communicated by Marek Galewski.
• Revised: 2017-02-12.
• Accepted: 2017-02-14.
• Published online: 2017-09-28.