Convergence Properties Of Higher Order Difference Equation X_n+1=(X_n-k)/(1+f(X_n))

Ordinary difference equation is made up of a single variable function with discrete values and its difference, it is the scatter of differential equation. There are a lot of phenomena can only use this discrete mathematical model to explain in life sciences, physics, chemistry, economics and other fields. And, as the rapid development of computer technology, we need to calculate the numerical solution for the continuous mathematical model. Therefore, the study of difference equations gets more and more attention from scholars and it becomes hot issue gradually. Recently, the theory of difference equation has new rapid development.Currently, there are many articles about lower-order linear difference equations while the study of higher-order nonlinear difference equations is less. In our study, we discuss the stability and convergence of a class of higher-order nonlinear difference equations through the methods such as iteration and analysis. First of all, we narrate the historical background of the problems and main works of this paper, definitions and properties we need in this paper are also given. Second, we research some lemmas and give out the proof. Last but not the least important, we mainly discuss the convergence of the solutions of the difference equations or more generic in this study and point out that the initial conditions of the convergent solution have some special properties. And we give out some examples of the function we talk about.