General solution to a higher-order linear difference equation and existence of bounded solutions

Abstract

We present a closed-form formula for the general solution to the difference equation xn+k – qnxn = fn, n ∈ N0, where k∈ N, (qn)n∈N0, (fn)n∈N0⊂C, in the case qn= q, n∈ N0, q∈ C∖ { 0 }. Using the formula, we show the existence of a unique bounded solution to the equation when | q| > 1 and supn∈N0|fn|<∞ by finding a solution in closed form. By using the formula for the bounded solution we introduce an operator that, together with the contraction mapping principle, helps in showing the existence of a unique bounded solution to the equation in the case where the sequence (qn)n∈N0 is real and nonconstant, which shows that, in this case, there is an elegant method of proving the result in a unified way. We also obtain some interesting formulas.