On quasi-periodic solutions of forced higher order nonlinear difference equations

Consider the following higher order difference equation \begin{equation*} x(n+1)= f(n,x(n))+g(n,x(n-k))+b(n), \qquad n=0, 1, \dots \end{equation*} where $f(n,x), g(n,x): \{0, 1, \dots \}\times [0, \infty) \rightarrow [0,\infty)$ are continuous functions in $x$ and periodic functions with period $\omega$ in $n$, $\{b(n)\}$ is a real sequence, and $k$ is a nonnegative integer. We show that under proper conditions, every nonnegative solution of the equation is quasi-periodic with period $\omega$. Applications to some other difference equations derived from mathematical biology are also given.