On a general system of difference equations defined by homogeneous functions
Nouressadat Touafek
Abstract
Abstract The aim of this paper is to study the following second order system of difference equations $$\begin{array}{} x_{n+1} = f(y_{n},y_{n-1}),\quad y_{n+1} = g(x_{n},x_{n-1}) \end{array}$$ where n ∈ ℕ 0 , the initial values x −1 , x 0 , y −1 and y 0 are positive real numbers, the functions f , g : (0, +∞) 2 → (0, +∞) are continuous and homogeneous of degree zero. In this study, we establish results on local stability of the unique equilibrium point and to deal with the global attractivity, and so the global stability, some general convergence theorems are provided. Necessary and sufficient conditions on existence of prime period two solutions of our system are given. Also, a result on oscillatory solutions is proved. As applications of the obtained results, concrete models of systems of difference equations defined by homogeneous functions of degree zero are investigated. Our system generalize some existing works in the literature and our results can be applied to study new models of systems of difference equations. For interested readers, we left in the conclusion as open problems two more general systems of higher order defined by homogenous functions of degree zero.