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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 541761, 11 pages
http://dx.doi.org/10.1155/2012/541761
Research Article

On Some Solvable Difference Equations and Systems of Difference Equations

1Mathematical Institute of the Serbian Academy of Sciences and Arts, Knez Mihailova 36/III, 11000 Beograd, Serbia
2Department of Mathematics and Descriptive Geometry, Faculty of Civil Engineering, Brno University of Technology, 60200 Brno, Czech Republic
3Department of Mathematics, Faculty of Electrical Engineering and Communication, Brno University of Technology, 61600 Brno, Czech Republic
4Faculty of Electrical Engineering, University of Belgrade, Bulevar Kralja Aleksandra 73, 11000 Beograd, Serbia

Received 8 August 2012; Accepted 27 September 2012

Academic Editor: Svatoslav Staněk

Copyright © 2012 Stevo Stević et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Here, we give explicit formulae for solutions of some systems of difference equations, which extend some very particular recent results in the literature and give natural explanations for them, which were omitted in the previous literature.

1. Introduction

Recently, there has been a great interest in difference equations and systems (see, e.g., [125]), and among them in those ones which can be solved explicitly (see, e.g., [15, 911, 15, 16, 1824] and the related references therein). For some classical results in the topic see, for example, [7].

Beside the above-mentioned papers, there are some papers which give formulae of some very particular equations and systems which are proved by induction, but without any explanation how these formulae are obtained and how these authors came across the equations and systems. Our explanation of such a formula that we gave in [10] has re-attracted attention to solvable difference equations.

Our aim here is to give theoretical explanations for some of the formulae recently appearing in the literature, as well as to give some extensions of their equations.

Before we formulate our results, we would like to say that the system of difference equations where and are real numbers, was completely solved in [15], that is, we found formulae for all well-defined solutions of system (1.1).

2. Scaling Indices

In the recent paper [25] were given some formulae for the solutions of the following systems of difference equations:

Now we show that the results regarding system (2.1) easily follow from known ones. Indeed, if we use the change of variables then the systems in (2.1) are reduced to the next systems This means that , , are two (independent) solutions of the systems of difference equations However, all the systems of difference equations in (2.4) are particular cases of system (1.1). Hence, formulae for the solutions of systems (2.1) given in [25] follow directly from those in [15].

2.1. An Extension of Systems (2.1)

Systems (2.1) can be extended as follows: where is a fixed natural number.

If we use the change of variables then system (2.5) is reduced to the following systems of difference equations: . This means that , , are (independent) solutions of system (1.1), and solutions of system (2.5) are obtained by interlacing solutions of systems (2.7), .

For example, a natural extension of the systems in (2.1) is obtained for taking , ,  ,   , and in (2.5), that is, the system becomes

In this way it can be obtained countable many, at first sight different, systems of difference equations. Systems of difference equations in (2.1) are artificially obtained in this way. This method can be applied to any equation or system of difference equations, and one can get papers with putative “new” results.

3. Some Third-Order Systems of Difference Equations Related to (1.1)

The following third-order systems of difference equations , have been studied recently (see [6] and the references therein).

As is directly seen, the first two equations in systems (3.1)–(3.3) are the same, and they form a particular case of system (1.1) which is solved in [15].

Since we know solutions for and , it is only needed to find explicit solutions for in the third equations in systems (3.1)–(3.3), that is, in all three equations, the only unknown sequence is . The joint feature for all three cases is that can be solved in closed form.

Now we discuss systems of difference equations given in (3.1)–(3.3).

Case of System (3.1)
From the third equation in (3.1), we get from which it follows that

Case of System (3.3)
From the third equation in (3.3), we get from which it follows that

Remark 3.1. The third equations in systems (3.1) and (3.3) are particular cases of the following difference equation (up to the shifting indices): where . From (3.8), it follows that and consequently,

Case of System (3.2)
If we use the change of variables , the third equation in (3.2) becomes Hence, from which it follows that so that

Remark 3.2. The third equation in system (3.2) is a particular case of the following difference equation: which, by the change of variables , is transformed into from which it follows that so by a well-known formula, we have that for and , and consequently, for and .

Remark 3.3. Note that (3.16) suggests that the third equation in (3.2) can be also of the form where is fixed, that is, to be an equation which consists of (independent) linear first-order difference equation, which is solvable. In fact, the third equation in systems (3.1)–(3.3) can be any difference equation which can be solved in , and in this way we can obtain numerous putative “new” results.

4. A Generalization of System (1.1)

Consider the following system of difference equations: where are increasing functions such that

Now we will find formulae for all well-defined solutions of system (4.1), that is, for the solutions , , such that for every .

If , then from (4.1), (4.2), and (4.3), and by the method of induction, we get . Also, if , then from (4.1), (4.2), and (4.3), and by the method of induction, we get . Similarly, if , then we get , while if , then we get .

If for some , then from (4.1)–(4.3) it follows that , for each such that . Hence, in this case we have that or . Similarly, if for some , then from (4.1)–(4.3) it follows that , for each such that . Hence, in this case we have that or . Thus, in both cases we arrive at a situation explained in the previous paragraph.

Hence, from now on, we assume that none of the initial values , , , and is equal to zero. Then, for every well-defined solution of system (4.1), we have that and , for every , and consequently and , for every .

Let then by taking function to the first equation in system (4.1) and function to the second one, then multiplying the first equation in such obtained system by and the second by , system (4.1) is transformed into: from which it follows that

Hence, if , we have that while if , we have that

We have also that while if , we have that

From (4.5), we have that

Using the relations (4.12), we get

Example 4.1. If we choose and for some , then conditions (4.2) and (4.3) are obviously satisfied, and system (4.1) can be written in the form and from (4.13), we have that its solutions are given by

Remark 4.2. System (4.1) can be generalized by using the method of scaling indices from Section 2.1, that is, the following system is also solvable: , where and are increasing functions satisfying conditions (4.2) and (4.3).
It is easy to see that the change of variables in (2.6) leads to the following systems of difference equations: for .

Remark 4.3. Well-defined solutions of the following system of difference equations: where are increasing functions satisfying conditions (4.2) and (4.3) and , , , and , , are real sequences, can be found similarly. We omit the details.

5. Solutions of a Generalization of a Recent Equation

Explaining some recent formulae appearing in the literature, in our recent paper [24], we have found formulae for well-defined solutions of the following difference equation: where and the parameters ,   as well as initial values , are real numbers.

Equation (5.1) can be extended naturally in the following way: where and the sequences ,  , as well as initial values , , are real numbers.

Employing the change of variables Equation (5.2) is transformed into the linear first-order difference equation whose general solution is

From (5.3), we have that for , which yields for every and .

Using (5.5) in (5.7), we get for every and .

Formula (5.8) generalizes the main formulae obtained in our paper [24].

Acknowledgments

The second author is supported by the Grant P201/10/1032 of the Czech Grant Agency (Prague) and the “Operational Programme Research and Development for Innovations,” no. CZ.1.05/2.1.00/03.0097, as an activity of the regional Centre AdMaS. The fourth author is supported by the Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology and by the Grant P201/11/0768 of the Czech Grant Agency (Prague). This paper is also supported by the Serbian Ministry of Science projects III 41025, III 44006, and OI 171007.

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