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Bounded and periodic solutions to the linear first-order difference equation on the integer domain
- Stevo Stević^{1, 2}Email author
https://doi.org/10.1186/s13662-017-1350-8
© The Author(s) 2017
Received: 14 July 2017
Accepted: 4 September 2017
Published: 12 September 2017
Abstract
The existence of bounded solutions to the linear first-order difference equation on the set of all integers is studied. Some sufficient conditions for the existence of solutions converging to zero when \(n\to -\infty\), as well as when \(n\to+\infty\), are also given. For the case when the coefficients of the equation are periodic, the long-term behavior of non-periodic solutions is studied.
Keywords
- linear first-order difference equation
- bounded solution
- periodic solution
- difference equation on integer domain
MSC
- 39A06
- 39A22
- 39A23
- 39A45
1 Introduction
Many nonlinear difference equations of interest are closely related to equation (1). For example, some of the nonlinear equations in [17, 19–21] and systems in [19, 22] have been solved by transforming them to some special equations of the form in (1), which shows its importance (some of the equations and systems, such as the one in [22], are transformed into special cases of the delayed version of equation (1), which is obviously also solvable; see also [19] and the comments on scaling indices of difference equations and systems). A deeper analysis can show that even the solvability of some product-type equations and systems is essentially influenced by the solvability of equation (1) (see, e.g., [23] and the references therein).
Usefulness of (1) has also been recently shown in [18], where, among others, a small but nice result on convergence of its solutions was proved by using formula (2), partially extending the result in the following problem in [7] (in the eight edition of the Russian version of the problem book from 1972 it is Problem 637.2).
Problem 1
For some other solvable equations, their applications, as well as invariants for some classes of equations, see, for example, [1–6, 9, 11–16].
Note that the domain of the above defined sequences \(x_{n}\) is \({\mathbb {N}}_{0}\), and it is difficult to find papers which consider equation (1) on \({\mathbb {Z}}\)-the set of all integers, in the literature on difference equations. One of our aims is to fulfill the possible gap in the study of the equation. Motivated also by our recent paper [18], here, among others, we study bounded solutions to equation (1), but also on the whole \({\mathbb {Z}}\). We also present some sufficient conditions for the existence of solutions to (1) converging to zero when \(n\to-\infty\) as well as when \(n\to +\infty\). Some of the results presented in the next section could be folklore, but we could not locate them in the literature.
Periodic solutions to (1) on \({\mathbb {N}}\) were studied in [2], which was another motivation for this paper. Our main result on periodicity is a nice complement to that in [2]. Namely, for the case when the coefficients of equation (1) are periodic, we describe the long-term behavior of its non-periodic solutions when \(n\to-\infty\) as well as when \(n\to+\infty\).
2 Bounded solutions to equation (1) on \({\mathbb {Z}}\)
In this section we study the existence of bounded solutions to equation (1). The cases when the domains are \({\mathbb {N}}_{0}\) and \({\mathbb {Z}}\setminus {\mathbb {N}}\) are treated separately, while the results in the case when the domain is \({\mathbb {Z}}\) are obtained as some consequences of the considerations on the domains \({\mathbb {N}}_{0}\) and \({\mathbb {Z}}\setminus {\mathbb {N}}\).
Our first result is, among others, an extension of the result in Problem 1, so it could be folklore.
Theorem 1
Proof
Remark 1
Now we consider the case \(\liminf_{n\to+\infty} \vert q_{n} \vert >1\). In this case, we may assume that \(q_{n}\ne0\), \(n\in {\mathbb {N}}_{0}\), otherwise we can consider (1) for sufficiently large n for which, due to the condition \(\liminf_{n\to+\infty} \vert q_{n} \vert >1\), will hold \(q_{n}\ne 0\).
Theorem 2
- (a)
There is a unique bounded solution to equation (1).
- (b)
If \(f_{n}\to0\) as \(n\to+\infty\), then the bounded solution also converges to zero as \(n\to+\infty\).
Proof
Remark 2
Closed form formulas (2) and (32) together present the general solution to equation (1) on \({\mathbb {Z}}\), when \(q_{n}\ne0\), for \(n\in {\mathbb {Z}}\setminus {\mathbb {N}}_{0}\).
Now we formulate and prove the corresponding results to Theorems 1 and 2 concerning bounded solutions to equation (31). The results are dual to Theorems 1 and 2 and are essentially obtained from them by using the change of variables \(y_{n}=x_{-n}\). However, there are some different details which are used later in the text. Because of this and for the completeness, we will sketch their proofs.
Theorem 3
Proof
Letting \(n\to+\infty\) in (39) and using the fact that ε is an arbitrary positive number, the result follows. □
Theorem 4
- (a)
There is a unique bounded solution to (31).
- (b)
If \(\lim_{n\to+\infty}f_{-n}=0\), then the bounded solution \(x_{-n}\) also converges to zero as \(n\to+\infty\).
Proof
(b) Since \(\lim_{n\to+\infty}f_{-n}=0\), we have that for every \(\varepsilon >0\), there is \(n_{12}\in {\mathbb {N}}\) such that (37) holds for \(n\ge n_{12}\).
From Theorems 1-4 the following four interesting corollaries are obtained.
From Theorems 1 and 3 we obtain the following corollary.
Corollary 1
From Theorems 1 and 4 we obtain the following corollary.
Corollary 2
From Theorems 2 and 3 we obtain the following corollary.
Corollary 3
From Theorems 2 and 4 we obtain the following corollary.
Corollary 4
- (a)There is a unique bounded solution to (1) on \({\mathbb {Z}}\) if and only if$$x_{0}=\sum_{j=1}^{\infty}f_{-j}\prod_{l=1}^{j-1}q_{-l}=- \sum_{i=0}^{\infty}\frac{f_{i}}{\prod_{j=0}^{i}q_{j}}. $$
- (b)
If (47) holds, then, for the bounded solution \((x_{n})_{n\in {\mathbb {Z}}}\), we have \(\lim_{n\to\pm\infty}x_{n}=0\).
3 Periodic solutions to equation (1)
In this section we study equation (1) in the case when the sequences \(q_{n}\) and \(f_{n}\) are periodic with the same period T. Note that if sequence \(q_{n}\) is periodic with period \(T_{1}\) and sequence \(f_{n}\) is periodic with period \(T_{2}\), where \(T_{1}\ne T_{2}\), then \(T:={\mathrm{lcm}}(T_{1}, T_{2})\) (the least common multiple of integers \(T_{1}\) and \(T_{2}\)) is a common period of these two sequences, since \(T=k_{1}T_{1}=k_{2}T_{2}\) for some integers \(k_{1}\) and \(k_{2}\). Hence, the case leads to the investigation of equation (1) whose coefficients are periodic with the same period.
Periodic solutions to equation (1) have been studied in [2]. Among others, the authors of [2] quote, in an equivalent form, the following basic result, which is certainly folklore.
Theorem 5
- (a)Ifthen (1) has a unique T-periodic solution given by the initial condition$$\begin{aligned} \lambda :=\prod_{j=0}^{T-1}q_{j} \ne1, \end{aligned}$$(48)$$\begin{aligned} x_{0}=\frac{\sum_{i=0}^{T-1}f_{i}\prod_{j=i+1}^{T-1}q_{j}}{1-\lambda }. \end{aligned}$$(49)
- (b)Ifthen (1) has a one-parameter family of T-periodic solutions.$$\begin{aligned} \lambda =1\quad\textit{and}\quad\sum_{i=0}^{T-1}f_{i} \prod_{j=i+1}^{T-1}q_{j}=0, \end{aligned}$$(50)
- (c)Ifthen (1) has no T-periodic solutions.$$\begin{aligned} \lambda =1\quad\textit{and}\quad\sum_{i=0}^{T-1}f_{i} \prod_{j=i+1}^{T-1}q_{j}\ne 0, \end{aligned}$$(51)
Theorem 6
- (a)Ifthen (31) has a unique T-periodic solution given by the initial condition$$\begin{aligned} \hat{\lambda }:=\prod_{j=1}^{T}q_{-j} \ne1, \end{aligned}$$(52)$$\begin{aligned} \hat{x}_{0}=\frac{\sum_{j=1}^{T}f_{-j}\prod_{l=1}^{j-1}q_{-l}}{1-\hat{\lambda }}. \end{aligned}$$(53)
- (b)Ifthen all the solutions to (31) are T-periodic.$$\begin{aligned} \hat{\lambda }=1\quad\textit{and}\quad\sum_{j=1}^{T}f_{-j} \prod_{l=1}^{j-1}q_{-l}=0, \end{aligned}$$(54)
- (c)Ifthen (31) has no T-periodic solutions.$$\begin{aligned} \hat{\lambda }=1\quad\textit{and}\quad\sum_{j=1}^{T}f_{-j} \prod_{l=1}^{j-1}q_{-l}\ne 0, \end{aligned}$$(55)
Proof
Now we formulate and prove the main result in this section. The result deals with the asymptotic behavior of solutions to equation (1) in the case when the quantity in (48) is different from 0 and 1.
Theorem 7
- (a)
If \(\vert \lambda \vert <1\), then all the solutions to (1) converge geometrically to the periodic one as \(n\to+\infty\), while they are getting away geometrically from the periodic one as \(n\to-\infty\).
- (b)
If \(\vert \lambda \vert >1\), then all the solutions to (1) converge geometrically to the periodic one as \(n\to-\infty\), while they are getting away geometrically from the periodic one as \(n\to+\infty\).
Proof
Now we are going to consider the case \(n\le0\). From Theorem 6(a) we see that equation (31) has a unique T-periodic solution if (52) holds with \(\hat{x}_{0}\) given in (53).
Declarations
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Authors’ Affiliations
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