Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation

摘要

We present some interesting facts connected with the following second-order difference equation: x n + 2 − q n x n = f n , n ∈ N 0 , $$x_{n+2}-q_{n}x_{n}=f_{n},quad nin mathbb{N}_{0}, $$ where ( q n ) n ∈ N 0 $(q_{n})_{ninmathbb{N}_{0}}$ and ( f n ) n ∈ N 0 $(f_{n})_{ninmathbb {N}_{0}}$ are given sequences of numbers. We give some sufficient conditions for the existence of a unique bounded solution to the difference equation and present an elegant proof based on a combination of theory of linear difference equations and the Banach fixed point theorem. We also deal with the equation by using theory of solvability of difference equations. A global convergence result of solutions to a linear first-order difference equation is given. Some comments on an abstract version of the linear first-order difference equation are also given.