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

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.