/ / Article

Research Article | Open Access

Volume 2012 |Article ID 541761 | https://doi.org/10.1155/2012/541761

Stevo Stević, Josef Diblík, Bratislav Iričanin, Zdeněk Šmarda, "On Some Solvable Difference Equations and Systems of Difference Equations", Abstract and Applied Analysis, vol. 2012, Article ID 541761, 11 pages, 2012. https://doi.org/10.1155/2012/541761

# On Some Solvable Difference Equations and Systems of Difference Equations

Accepted27 Sep 2012
Published24 Oct 2012

#### 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., ), 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, .

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  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 , that is, we found formulae for all well-defined solutions of system (1.1).

#### 2. Scaling Indices

In the recent paper  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  follow directly from those in .

##### 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.

The following third-order systems of difference equations , have been studied recently (see  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 .

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 , 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 .

#### 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.

1. A. Andruch-Sobiło and M. Migda, “On the rational recursive sequence ${x}_{n+1}=\alpha xn-1/\left(b+c{x}_{n-1}\right)$,” Tatra Mountains Mathematical Publications, vol. 43, pp. 1–9, 2009. View at: Google Scholar | Zentralblatt MATH
2. I. Bajo and E. Liz, “Global behaviour of a second-order nonlinear difference equation,” Journal of Difference Equations and Applications, vol. 17, no. 10, pp. 1471–1486, 2011.
3. L. Berg and S. Stević, “On difference equations with powers as solutions and their connection with invariant curves,” Applied Mathematics and Computation, vol. 217, no. 17, pp. 7191–7196, 2011. View at: Publisher Site | Google Scholar
4. L. Berg and S. Stević, “On some systems of difference equations,” Applied Mathematics and Computation, vol. 218, no. 5, pp. 1713–1718, 2011.
5. B. Iričanin and S. Stević, “On some rational difference equations,” Ars Combinatoria, vol. 92, pp. 67–72, 2009. View at: Google Scholar | Zentralblatt MATH
6. A. S. Kurbanli, “On the behavior of solutions of the system of rational difference equations ${x}_{n+1}={x}_{n-1}/{y}_{n}{x}_{n-1}-1\right),{y}_{n+1}={y}_{n-1}/\left({x}_{n}{y}_{n-1}-1\right)$ and ${z}_{n+1}={z}_{n-1}/\left({y}_{n}{z}_{n-1}-1\right)$,” Advances in Difference Equations, vol. 2011, 40 pages, 2011. View at: Google Scholar | Zentralblatt MATH
7. H. Levy and F. Lessman, Finite Difference Equations, The Macmillan Company, New York, NY, USA, 1961.
8. G. Papaschinopoulos, C. J. Schinas, and G. Stefanidou, “On the nonautonomous difference equation ${x}_{n+1}={A}_{n}+\left({x}_{n-1}^{p}/{x}_{n}^{q}\right)$,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5573–5580, 2011.
9. G. Papaschinopoulos and G. Stefanidou, “Asymptotic behavior of the solutions of a class of rational difference equations,” International Journal of Difference Equations, vol. 5, no. 2, pp. 233–249, 2010. View at: Google Scholar
10. S. Stević, “More on a rational recurrence relation,” Applied Mathematics E-Notes, vol. 4, pp. 80–85, 2004. View at: Google Scholar
11. S. Stević, “A short proof of the Cushing-Henson conjecture,” Discrete Dynamics in Nature and Society, vol. 2006, Article ID 37264, 5 pages, 2006.
12. S. Stević, “On the recursive sequence ${x}_{n+1}=\text{max}\left\{c,{x}_{n}^{p}/{x}_{n-1}^{p}\right\}$,” Applied Mathematics Letters, vol. 21, no. 8, pp. 791–796, 2008.
13. S. Stević, “Global stability of a max-type difference equation,” Applied Mathematics and Computation, vol. 216, no. 1, pp. 354–356, 2010.
14. S. Stević, “On a nonlinear generalized max-type difference equation,” Journal of Mathematical Analysis and Applications, vol. 376, no. 1, pp. 317–328, 2011.
15. S. Stević, “On a system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 7, pp. 3372–3378, 2011.
16. S. Stević, “On the difference equation ${x}_{n}={x}_{n-2}/\left({b}_{n}+{c}_{n}{x}_{n-1}{x}_{n-2}\right)$,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4507–4513, 2011.
17. S. Stević, “Periodicity of a class of nonautonomous max-type difference equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9562–9566, 2011.
18. S. Stević, “On a system of difference equations with period two coefficients,” Applied Mathematics and Computation, vol. 218, no. 8, pp. 4317–4324, 2011. View at: Publisher Site | Google Scholar
19. S. Stević, “On a third-order system of difference equations,” Applied Mathematics and Computation, vol. 218, no. 14, pp. 7649–7654, 2012.
20. S. Stević, “On some solvable systems of difference equations,” Applied Mathematics and Computation, vol. 218, no. 9, pp. 5010–5018, 2012. View at: Publisher Site | Google Scholar
21. S. Stević, “On the difference equation ${x}_{n}={x}_{n-k}/\left(b+c{x}_{n-1}\cdots {x}_{n-k}\right)$,” Applied Mathematics and Computation, vol. 218, no. 11, pp. 6291–6296, 2012. View at: Publisher Site | Google Scholar
22. S. Stević, J. Diblík, B. Iričanin, and Z. Šmarda, “On a third-order system of difference equations with variable coefficients,” Abstract and Applied Analysis, vol. 2012, Article ID 508523, 22 pages, 2012. View at: Publisher Site | Google Scholar
23. S. Stević, J. Diblík, B. Iričanin, and Z. Šmarda, “On the difference equation ${x}_{n}={a}_{n}{x}_{n-k}/\left({b}_{n}+{c}_{n}{x}_{n-1}{x}_{n-k}\right)$,” Abstract and Applied Analysis, vol. 2012, Article ID Article number409237, 20 pages, 2012. View at: Publisher Site | Google Scholar
24. S. Stević, J. Diblík, B. Iričanin, and Z. Šmarda, “On the difference equation ${x}_{n+1}={x}_{n}{x}_{n-k}/\left({x}_{n-k+1}\left(a+b{x}_{n}{x}_{n-k}\right)\right)$,” Abstract and Applied Analysis, vol. 2012, Article ID Article number108047, 9 pages, 2012. View at: Publisher Site | Google Scholar
25. N. Touafek and E. M. Elsayed, “On the solutions of systems of rational difference equations,” Mathematical and Computer Modelling, vol. 55, no. 7-8, pp. 1987–1997, 2012. View at: Publisher Site | Google Scholar

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