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In this paper, we present the asymptotic behavior of the solutions for a general class of difference equations. We introduce general theorems in order to study the stability and periodicity of the solutions. Moreover, we use a new technique to study the existence of periodic solutions of this general equation. By using our general results, we can study many special cases that have not been studied previously and some problems that were raised previously. Some numerical examples are provided to illustrate the new results.

Dynamics of difference equation x n + 1 = f ( x n − l , x n − k ) $x_{n+1}=f( x_{n-l},x_{n-k})$