# Bounded Solutions to Nonhomogeneous Linear Second-Order Difference Equations

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## Abstract

**:**

## 1. Introduction

## 2. Bounded Solutions to Equation (5) on ${\mathbb{N}}_{0}$

**Lemma**

**1.**

- (a)
- If ${p}^{2}\ne 4q$, then the general solution to Equation (8) is given by the following formula$$\begin{array}{c}\hfill {x}_{n}={\lambda}_{1}^{n}\left({c}_{0}-{\displaystyle \sum _{k=0}^{n-1}}\frac{{f}_{k}}{{\lambda}_{1}^{k+1}({\lambda}_{2}-{\lambda}_{1})}\right)+{\lambda}_{2}^{n}\left({d}_{0}+{\displaystyle \sum _{k=0}^{n-1}}\frac{{f}_{k}}{{\lambda}_{2}^{k+1}({\lambda}_{2}-{\lambda}_{1})}\right),\end{array}$$$$\begin{array}{c}\hfill {\lambda}_{1,2}=\frac{-p\pm \sqrt{{p}^{2}-4q}}{2}.\end{array}$$
- (b)
- If ${p}^{2}=4q$, then the general solution to Equation (8) is given by the following formula$$\begin{array}{c}\hfill {x}_{n}={\lambda}^{n}\left({c}_{0}-{\displaystyle \sum _{k=0}^{n-1}}\frac{(k+1){f}_{k}}{{\lambda}^{k+2}}\right)+n{\lambda}^{n}\left({d}_{0}+{\displaystyle \sum _{k=0}^{n-1}}\frac{{f}_{k}}{{\lambda}^{k+2}}\right),\end{array}$$

**Proof.**

**Remark**

**1.**

**Corollary**

**1.**

- (a)
- If ${p}^{2}\ne 4q$, then the solution to Equation (8) with the initial values ${x}_{0}$ and ${x}_{1}$ is given by the following formula$$\begin{array}{c}\hfill {x}_{n}=\frac{1}{{\lambda}_{2}-{\lambda}_{1}}\left({\lambda}_{1}^{n}\left({\lambda}_{2}{x}_{0}-{x}_{1}-{\displaystyle \sum _{k=0}^{n-1}}\frac{{f}_{k}}{{\lambda}_{1}^{k+1}}\right)+{\lambda}_{2}^{n}\left({x}_{1}-{\lambda}_{1}{x}_{0}+{\displaystyle \sum _{k=0}^{n-1}}\frac{{f}_{k}}{{\lambda}_{2}^{k+1}}\right)\right),\end{array}$$
- (b)
- If ${p}^{2}=4q$, then the solution to Equation (8) with the initial values ${x}_{0}$ and ${x}_{1}$ is given by the following formula$$\begin{array}{c}\hfill {x}_{n}={\lambda}^{n}\left({x}_{0}-{\displaystyle \sum _{k=0}^{n-1}}\frac{(k+1){f}_{k}}{{\lambda}^{k+2}}\right)+n{\lambda}^{n-1}\left({x}_{1}-\lambda {x}_{0}+{\displaystyle \sum _{k=0}^{n-1}}\frac{{f}_{k}}{{\lambda}^{k+1}}\right),\end{array}$$

**Proof.**

**Remark**

**2.**

**Corollary**

**2.**

**Proof.**

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

**Theorem**

**3.**

**Proof.**

**Theorem**

**4.**

- (a)
- If $|{\lambda}_{1}|<1<|{\lambda}_{2}|,$ then a solution to Equation (8) is bounded if and only if$$\begin{array}{c}\hfill {\lambda}_{1}{x}_{0}-{x}_{1}={\displaystyle \sum _{j=0}^{\infty}}\frac{{f}_{j}}{{\lambda}_{2}^{j+1}}.\end{array}$$
- (b)
- If $|{\lambda}_{2}|<1<|{\lambda}_{1}|,$ then a solution to Equation (8) is bounded if and only if$$\begin{array}{c}\hfill {\lambda}_{2}{x}_{0}-{x}_{1}={\displaystyle \sum _{j=0}^{\infty}}\frac{{f}_{j}}{{\lambda}_{1}^{j+1}}.\end{array}$$

**Proof.**

**Theorem**

**5.**

**Proof.**

**Remark**

**3.**

**Theorem**

**6.**

**Proof.**

**Theorem**

**7.**

**Proof.**

## 3. Bounded Solutions to Equation (5) on the Domain $\mathbb{Z}\backslash {\mathbb{N}}_{2}$

**Lemma**

**2.**

- (a)
- If ${p}^{2}\ne 4q$, then the solution to Equation (78) with initial/end values ${x}_{0}$ and ${x}_{1}$ is given by$$\begin{array}{c}\hfill {x}_{-n}=\frac{{\lambda}_{1}^{-n}({\lambda}_{2}{x}_{0}-{x}_{1}+{\sum}_{j=1}^{n}{f}_{-j}{\lambda}_{1}^{j-1})-{\lambda}_{2}^{-n}({\lambda}_{1}{x}_{0}-{x}_{1}+{\sum}_{j=1}^{n}{f}_{-j}{\lambda}_{2}^{j-1})}{{\lambda}_{2}-{\lambda}_{1}},\end{array}$$
- (b)
- If ${p}^{2}=4q$, then the solution to Equation (78) with initial/end values ${x}_{0}$ and ${x}_{1}$ is given by$$\begin{array}{c}\hfill {x}_{-n}={\lambda}^{-(n+1)}\left(\lambda {x}_{0}+(\lambda {x}_{0}-{x}_{1})n+{\displaystyle \sum _{j=1}^{n}}{f}_{-j}(n-j+1){\lambda}^{j-1}\right),\end{array}$$

**Remark**

**4.**

**Corollary**

**3.**

**Theorem**

**8.**

**Proof.**

**Theorem**

**9.**

**Proof.**