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

[摘要] We present a closed-form formula for the general solution to the difference equationxn+k−qnxn=fn,n∈N0,$$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$wherek∈N$k\in \mathbb {N}$,(qn)n∈N0$(q_{n})_{n\in \mathbb {N}_{0}}$,(fn)n∈N0⊂C$(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$, in the caseqn=q$q_{n}=q$,n∈N0$n\in \mathbb {N}_{0}$,q∈C∖{0}$q\in \mathbb {C}\setminus\{0\}$. Using the formula, we show the existence of a unique bounded solution to the equation when|q|>1$|q|>1$andsupn∈N0|fn|<∞$\sup_{n\in \mathbb {N}_{0}}|f_{n}|<\infty$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$(q_{n})_{n\in \mathbb {N}_{0}}$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.

[学科分类] 数学（综合）

[关键词] linear difference equation;general solution;existence of bounded solutions;contraction mapping principle;39A14;05A10