Necessary and Sufficient Conditions for Existence of an Equilibrium and a Periodic of Prime Period 2 Solution of a Certain Rational Difference Equation

Ratinan Boonklurb; Julaluk Neammai; Vasana Sukkrasanti; Theeruth Tantasuparuk

Authors

Ratinan Boonklurb
Julaluk Neammai
Vasana Sukkrasanti
Theeruth Tantasuparuk

Keywords:

Equilibrium, Periodic, Prime Period, Rational difference equation

Abstract

This article studies the necessary and sufficient conditions for the existence of positive equilibrium solutions and positive periodic solutions of prime period $gif.latex?2$ of the following rational difference equation.

$gif.latex?x_{n+1}=\dfrac{\alpha+\beta&space;x_{n-k}}{A+B_{0}x_{n}+B_{1}x_{n-1}+\cdots+B_{k}x_{n-k}},&space;\text{&space;for&space;}n&space;\in&space;\{0,1,2,\ldots\}$

where the parameters $gif.latex?\alpha$ > $gif.latex?0$ and $gif.latex?\beta,\,A,\,B_{0},\,B_{1},\,\ldots,\,B_{k}$ and the initial conditions $gif.latex?x_{-k},\,x_{-k+1}\,x_{-k+2},\,\ldots,\,x_{-1},\,x_{0}$ are nonnegative real numbers such that the denominator is always positive.

Necessary and Sufficient Conditions for Existence of an Equilibrium and a Periodic of Prime Period 2 Solution of a Certain Rational Difference Equation

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