Document Type : Research Paper

Authors

• C. M. Kent ¹
• W. Kosmala ²

¹ Department of Mathematics and Applied Mathematics, Virginia Commonwealth University, P. O. Box 842014, Richmond, Virginia 23284-2014 USA.

² Department of Mathematical Sciences, Appalachian State University, Boone, North Carolina 28608 USA.

Abstract

We investigate the long-term behavior of solutions of the difference equation
$$x_{n+1}=x_n{x}_{n-3}-1,n=0,1,\dots$$
where the initial conditions $x_{-3},,x_{-2},,x_{-1},,x_0$ are real numbers.  In particular, we look at the periodicity and asymptotic periodicity of solutions, as well as the existence of unbounded solutions.

Keywords

• Difference equations
• boundedness
• periodicity
• Asymptotic periodicity
• Eventual periodicity
• Invariant interval
• Continued fractions