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On the Dynamics of Higher-Order Rational Recursive Sequence

Commun. Math. Anal.
Volume 12, Number 1 (2012), 117 - 133

On the Dynamics of Higher-Order Rational Recursive Sequence

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Abstract

In this paper we investigate the global convergence result, boundedness, and periodicity of solutions of the recursive sequence

\begin{equation*} x_{n+1}=ax_{n}+\dfrac{bx_{n-l}+cx_{n-k}}{dx_{n-l}+ex_{n-k}},\;\;\;n=0,1,..., \end{equation*}

where the parameters 

a,b,c,d\;

and

\;e\;

are positive real numbers and the initial conditions 

x_{-k},x_{-k+1},...,x_{-l},x_{-l+1},...,x_{-1}

 and 

x_{0}\;

are positive real numbers.