Existence of a unique bounded solution to a linear second-order difference equation and the linear first-order difference equation
- Stevo Stević^{1, 2}Email author
https://doi.org/10.1186/s13662-017-1227-x
© The Author(s) 2017
Received: 13 March 2017
Accepted: 1 June 2017
Published: 15 June 2017
Abstract
Keywords
MSC
1 Introduction
Difference equations and systems of difference equations have been of a great interest in the last several decades. For some classical results in the research area, see, for example, [1–8], whereas some recent results can be found, for example, in [9–22] (see also the references therein). Book [23] contains a lot of results obtained up to 2,000.
- (a)
If \(q\in(0,1)\), then all the solutions to equation (3) converge to zero.
- (b)
If \(q=1\) then all the solutions to equation (3) are bounded.
- (c)If \(q>1\), then all the solutions to equation (3) are unbounded, except the trivial solutionwhich is obtained for the initial conditions \(x_{0}=x_{1}=0\).$$x_{n}=0,\quad n\in\mathbb{N}_{0}, $$
In terms of the boundedness these statements claim that all the solutions to equation (3) are bounded if and only if \(q\in(0,1]\), while for the case \(q>1\) there is only one bounded solution to the equation.
It is a classical problem to see how bounded perturbations of the right-hand side of equation (3), as well as of coefficient q influence on the boundedness character of the solutions of such obtained equations. We will present two interesting ways how the problem of the existence of a unique bounded solution can be solved for the case of the difference equation (1), where \((q_{n})_{n\in\mathbb{N}_{0}}\) and \((f_{n})_{n\in\mathbb{N}_{0}}\) are two given bounded sequences.
The paper is partially based on several comments and ideas presented by the author at several international conferences and invited talks during the last several years, which has not been published so far. Some of the results could be known, but we could not find specific references for them.
The paper is organized as follows. First, we consider the case when \(q_{n}=q\ne0\) for every \(n\in\mathbb{N}_{0}\) and \((f_{n})_{n\in \mathbb{N}_{0}}\) is a bounded sequence. Then we consider the case when the sequences \((q_{n})_{n\in\mathbb{N}_{0}}\) and \((f_{n})_{n\in\mathbb{N}_{0}}\) are bounded. By a nice combination of the theory of linear difference equations and the Banach fixed point theorem we present some sufficient conditions for the existence of a unique bounded solution to equation (1). In the third section we will present another way for dealing with the unique existence problem by using only the theory of linear difference equations. We finish the paper by giving some comments on the form of general solution to an abstract version of the linear first-order difference equation.
2 Fixed point approach in dealing with equation (1)
In this section we deal with the problem of the existence of a unique bounded solution to the difference equation (1). To do this we use fixed point theory. First we deal with the case when \(q_{n}=q \ne0\) for every \(n\in\mathbb{N}_{0}\) and \((f_{n})_{n\in\mathbb {N}_{0}}\) is a bounded sequence.
Before we formulate and prove the main result in the section we need an auxiliary result which is incorporated into the following lemma. The lemma is essentially folklore, but we will give a proof of it for completeness, for the benefit of the reader and also as a good motivation and better understanding of some ideas appearing in the proof of the main result.
Lemma 1
Proof
For each fixed \(n\in\mathbb{N}_{0}\), equalities (10) and (11) can be regarded as a two-dimensional linear system in the variables \(c_{n+1}-c_{n}\) and \(d_{n+1}-d_{n}\).
The following result solves the problem of existence of a unique bounded solution to equation (1) for the case \(q_{n}=q\), \(n\in\mathbb{N} _{0}\), when \(\vert q\vert >1\).
Theorem 1
Consider the difference equation (6), where q is a complex number such that \(\vert q\vert >1\) and \(f:=(f_{n})_{n\in\mathbb{N} _{0}}\) is a given bounded sequence of complex numbers. Then there is a unique bounded solution to the difference equation.
Proof
Now we are in a position to formulate and prove the main result in the section in an elegant way.
Theorem 2
Proof
We will prove the theorem under condition (21). The proof when (22) holds is similar/dual so is omitted.
A direct calculation shows that this bounded sequence satisfies the difference equation (1) for every \(n\in\mathbb{N}_{0}\), from which the theorem follows. □
Remark 1
Beside the choice of operator (25), which is naturally imposed, a crucial point in the proof of Theorem 2 is the choice of constant q in (23) to get the contractivity of the operator. It is also expected that a modification of the above arguments can be applied to some other related difference equations.
3 Equation (1) versus the linear first-order difference equation
It is important to note that many nonlinear difference equations and systems can be reduced to some special cases of equation (29), which means that they are solvable too. Some interest in the area has been renewed recently. For some recent classes of solvable difference equations see, for example, [11, 13, 14, 16, 17, 19, 25], while some related systems of difference equations can be found in [10, 12, 16, 18]. For some recent results on solvable product-type systems of difference equations, see [15, 20–22] and the references therein. For some classical equations and systems which can be reduced to (29) or solved by some other methods; see, for example, [3, 5, 6, 8, 23].
How useful and powerful equation (31) is shows the following small but nice result, which is an old result by the author which has never been published so far but has been presented at several talks.
Theorem 3
Proof
Remark 2
By applying the arguments from (33) to (35), it is easy to see that the following result holds.
Theorem 4
Remark 3
Theorem 4 also shows that equation (1) has a unique bounded solution if (33) holds and \((f_{n})_{n\in\mathbb{N} _{0}}\) is a bounded sequence, and gives it explicitly, although in not so nice way. Moreover, the conditions in Theorem 4 are somewhat weaker so that its result is somewhat stronger than the one in Theorem 2, which again shows the importance and usefulness of equation (31). Nevertheless, both approaches in dealing with equation (1) are interesting, each of them in its own way.
3.1 An abstract form of equation (29)
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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