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

Abstract

We present some interesting facts connected with the following second-order difference equation: xn+2−qnxn=fn,n∈N0, where (qn)n∈N0 and (fn)n∈N0 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.