we investigate the long-term behavior of solutions of the difference equation[ x_{n+1}=x_{n}x_{n-3}-1 ,, n=0 ,, 1 ,, ldots ,, ]noindent 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.

In this article, we show that the Kamae-Xue complexity function for an infinite sequence classifies eventual periodicity completely. We prove that an infinite binary word x1x2 · · · is eventually periodic if and only if Σ(x1x2 · · ·xn)/n3 has a positive limit, where Σ(x1x2 · · ·xn) is the sum of the squares of all the numbers of appearance of finite words in x1x2 · · ·xn, which was introduced b...

The initialand boundary-value problem for the Benjamin-Bona-Mahony (BBM) equation is studied in this paper. The goal is to understand the periodic behavior (termed as eventual periodicity) of its solutions corresponding to periodic boundary condition or periodic forcing. To this aim, we derive a new formula representing solutions of this initialand boundary-value problem by inverting the operat...