Families of 6-Cycles of Third Order

The present paper deals with the existence of 6-cycles of a certain type of difference equations of third order. In concrete, it focuses on equations of the form \(x_{n+3} = x_ig(x_j)h(x_k)\) , where \(i,j,k\in \{n,n+1,n+2\}\) are pairwise distinct and \(g,h:(0,\infty )\rightarrow (0,\infty )\) are continuous. The main result of the paper assures that the unique 6-cycle displaying such form is given by the potential one \(x_{n+3} = x_n\left( \frac{x_{n+2}}{x_{n+1}}\right) ^2\) .