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

• Main Author: Stevo Stević
• Format: Article
• Language: English
• Published: SpringerOpen 2017-12-01
• Series: Advances in Difference Equations
• Subjects: linear difference equation; general solution; existence of bounded solutions; contraction mapping principle
• Online Access: http://link.springer.com/article/10.1186/s13662-017-1432-7

Abstract We present a closed-form formula for the general solution to the difference equation x n + k − q n x n = f n , n ∈ N 0 , $$x_{n+k}-q_{n}x_{n}=f_{n},\quad n\in \mathbb {N}_{0}, $$ where k ∈ N $k\in \mathbb {N}$ , ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ , ( f n ) n ∈ N 0 ⊂ C $(f_{n})_{n\in \mathbb {N}_{0}}\subset \mathbb {C}$ , in the case q n = q $q_{n}=q$ , n ∈ N 0 $n\in \mathbb {N}_{0}$ , 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 $|q|>1$ and sup n ∈ N 0 | f n | < ∞ $\sup_{n\in \mathbb {N}_{0}}|f_{n}|<\infty$ 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 ( q n ) n ∈ N 0 $(q_{n})_{n\in \mathbb {N}_{0}}$ 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.