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

General solution to a higher-order linear difference equation and existence of bounded solutions 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Advances in Difference Equations Springer Journals

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

, Volume 2017 (1) – Dec 2, 2017
12 pages

/lp/springer_journal/general-solution-to-a-higher-order-linear-difference-equation-and-0JFPVpwzGa
Publisher
Springer International Publishing
Subject
Mathematics; Difference and Functional Equations; Mathematics, general; Analysis; Functional Analysis; Ordinary Differential Equations; Partial Differential Equations
eISSN
1687-1847
D.O.I.
10.1186/s13662-017-1432-7
Publisher site
See Article on Publisher Site

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.

Journal

Published: Dec 2, 2017

DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month Explore the DeepDyve Library Unlimited reading Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere. Stay up to date Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates. Organize your research It’s easy to organize your research with our built-in tools. Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve Freelancer DeepDyve Pro Price FREE$49/month

\$360/year
Save searches from