On a Third-Order System of Difference Equations with Variable Coefficients

Stević, Stevo; Diblík, Josef; Iricanin, Bratislav; Šmarda, Zdenek

doi:https://doi.org/10.1155/2012/508523

Abstract and Applied Analysis

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Research Article | Open Access

Volume 2012 | Article ID 508523 | https://doi.org/10.1155/2012/508523

On a Third-Order System of Difference Equations with Variable Coefficients

Stevo Stević,¹Josef Diblík,^2,3Bratislav Iricanin,⁴and Zdenek Šmarda³

Academic Editor: Agacik Zafer

Received03 Mar 2012

Revised20 Mar 2012

Accepted22 Mar 2012

Published29 May 2012

Abstract

We show that the system of three difference equations , , and , , where all elements of the sequences , , , , , and initial values , , , , are real numbers, can be solved. Explicit formulae for solutions of the system are derived, and some consequences on asymptotic behavior of solutions for the case when coefficients are periodic with period three are deduced.

1. Introduction

Studying nonlinear difference equations and systems is an area of a great interest nowadays (see, e.g., [1–39] and the references therein).

This paper studies the system of three difference equations where all elements of the sequences , , , , , and initial values , , , are real numbers. The cases when both and are equal to zero for some fixed and an , are not interesting so they are excluded. In [37] we have shown that system (1.1) for the case when the sequences , , , , , are constant can be explicitly solved (if solutions are well defined). Some recent results on solving difference equations can be found, for example, in [6, 7, 12, 24, 25, 34, 35, 38, 39]. For some old results see, for example, classic book [14].

Note that the solutions of (1.1) such that all sequences , , , , , and initial values in system (1.1) are positive, are also positive, that is, so that there are a lot of well defined solutions of the system. In fact, for “majority” initial values of system (1.1), solutions are well defined, but we will not discuss the problem here. Instead of that we assume, throughout the paper, that solutions of (1.1) are well defined. We also adopt the customary notation and .

We show that in the main case, system (1.1) is transformed to a third-order system of nonhomogeneous linear first-order difference equations, which can be explicitly solved.

This idea appeared for the first time in [24] for the case of the scalar equation with constant coefficients corresponding to system (1.1) and was also used later in [1, 4]. Some related transformations are used also in [25, 30]. For a different approach in dealing with the scalar difference equation see [2, 3]. For some related scalar difference equations see, for example, [8, 13, 26] and the related references therein. Some related results on systems of difference equations can be found in [11, 15–22] (see also references therein).

Here we give explicit formulae for solutions of system (1.1) and present some consequences on asymptotic behavior of the solutions for the case when coefficients are periodic with period three.

2. Case for Some and All

If , then the first equation in (1.1) becomes so that from the second and the third equations and since the solution is well defined we get, , from which it follows that

If , , then the second equation in (1.1) becomes so that from the first and the third equations we get, from which it follows that

Finally, if , , then the third equation in (1.1) becomes so that from the first and the second equation we get, from which it follows that

3. Explicit Formulae for the Case for All and

Here we consider system (1.1) in the case when for all and . Noticing that in this case, system (1.1) can be written in the form where , , , we see that we may assume that , for every and for each .

Hence we consider, without loss of generality, the system using the same notation for coefficients as in (1.1) except for the coefficients , assuming that for all and .

First we consider the case when some of initial values of solutions of system (3.2) is equal to zero.

If for some then from (3.2) it follows that , for each such that . Hence, we have that or or .

If , then , , which implies and consequently

If , then , which implies and consequently

If or for some then similar results are obtained analogously.

3.1. Main Case

Here we study well-defined solutions of system (1.1) when neither of the sequences , , or initial conditions , , , , is equal to zero. Recall that we may assume that , for every and for each .

Following the idea in [37], we use a transformation which reduces nonlinear systems (1.1) and (3.2) to third-order systems of nonhomogeneous linear difference equations.

If we multiply the first equation in system (3.2) by , the second by and the third by , and then using in such obtained system the change of variables the system is, for , transformed into System (3.10) implies that for where values for , , are computed by (3.9) with .

Equation (3.11) implies that the sequences , , are solutions of the first-order linear difference equation

Applying the well-known formula for solutions of first-order difference equation we have that the general solution of equation (3.14) is for every and each .

From (3.12) we get that the sequences , , are solutions of the first-order linear difference equation from which it follows that for every and each .

From (3.13) we get that the sequences , , are solutions of the first-order linear difference equation

Hence, for every and each , we have that

Now note that from (3.9) we have

Hence

Applying (3.15), (3.17) and (3.19) in (3.21)–(3.23), we get explicit solutions of system (3.2) in terms of sequences , , , .

The results in this section can be summed up in Table 1.

Remark 3.1. Formulae for the solutions of system (3.2) when some of the numbers , , , are zero follow from the formulae given in Table 1.

4. Some Consequences

4.1. Case for and

First we use the formulae in Section 3 to get solutions of system (1.1), when for and .

For this we replace sequences in formulas of Section 3 with sequences We arrange these formulae in Table 2.

4.2. Case , , , Are Period-Three Sequences

Now we get formulae for solutions of system (3.2) when the sequences , , , are periodic with period three.

If this holds then from (3.15) we have that for every , when , for some , while when , for some . Here we regard that , for some , and , when .

Similarly, we get for every , if , for some , and for every , when , for some .

Finally for every , when , for some , and for every , when , for some .

If we assume that for all and , that these three sequences are also periodic with period three, and replace the sequences in (4.1) with the corresponding in (4.2) we get formulae for solutions of system (1.1) when the sequences , , and , , are periodic with period three.

These formulae follows from above obtained ones and are summarized in Table 3.

5. Some Applications

Using above listed formulae the behavior of solutions of system (1.1) or (3.2) can be described. We will present here some results which can be obtained from these formulae, to demonstrate how they can be used. Before we formulate the results note that if , , are periodic with period three, then

Theorem 5.1. Consider system (3.2). Let the sequences , , , be periodic with period three, when , and for some the following condition holds: Then if for every , the following statements hold: (a)if , then , as ;(b)if , then , as ;(c)if and , then , as ;(d)if and , then , as ;(e)if and , then is convergent;(f)if and , then , as ;(g)if and , then , as , so that and , or and as ;(h)If and , then and are convergent, as .If , then the following statements hold:(i)if , then , as ;(j)if , then

Proof. (a), (b) From (3.21), (5.4) and (5.1), we have that for each for every .
Since the results in (a) and (b) follow from (5.6) easily.
(c), (d), (e) Since , we have that From (5.8) and by using well-known asymptotic relation where we assume and ; the results in (c), (d), and (e) easily follow.
(f), (g), (h) Since , we have that From (5.6), (5.10) and asymptotic relation (5.9) the results in (f), (g), and (h) follow.
(i), (j) These two statements follow easily from (5.6).

Remark 5.2. If for an , then by (3.2) we get , for , which is the reason why we posed the condition , for every , in Theorem 5.1. Similar conditions will be posed in the theorems which follow.

Theorem 5.3. Consider system (3.2). Let the sequences , , , be periodic with period three, when , and that for some the following condition holds Then if for every , the following statements hold: (a)if , then , as ;(b)if , then , as ;(c)if and , then , as ;(d)if and , then , as ;(e)if and , then is convergent;(f)if and , then , as ;(g)if and , then , as , so that and , or and as ;(h)if and , then and are convergent, as .If , then the following statements hold:(i)if , then , as ;(j)if , then

Proof. From (3.22), by using condition (5.12) and the second equality in (5.1), we have that for each and every , from which the results in this theorem follows similar to Theorem 5.1.

Theorem 5.4. Consider system (3.2). Let the sequences , , , be periodic with period three, when , and for some the following condition holds Then if for every , the following statements hold: (a)if , then , as ;(b)if , then , as ;(c)if and , then , as ;(d)if and , then , as ;(e)if and , then is convergent;(f)if and , then , as ;(g)if and , then , as , so that and , or and as ;(h)if and , then and are convergent, as .If , then the following statements hold:(i)if , then , as ;(j)if , then

Proof. From (3.23), by using conditions (5.16) and (5.2), we have that for each and every , from which the results in this theorem follows similar to Theorem 5.1.

Theorem 5.5. Consider system (3.2). Let the sequences , , , be periodic with period three and for some the following condition holds: Then if for every , the following statements hold: (a)if , then , as ;(b)if , , and , then , as ;(c)if , , then (d)if and (5.20) holds, then (e)if and , then as ;(f)if and , then as ;(g)if and , then converges, as ;(h)if and , then and converge, as ;(i)if and , where then as ;(j)if and , then as ;(k)if and , then converges, as ;(l)if and , then and converge, as .

Proof. From (3.21) and by using conditions (5.19) and (5.2), we have that for every and for each . Using (5.24) the statements in (a)–(d) easily follows.
(e)–(h) Let . Then where , are defined by (5.23). We have From this and since , the results in these four cases easily follow.
(i)–(l) We have From this and since , the results in these four cases easily follow.

Theorem 5.6. Consider system (3.2). Let the sequences , , , be periodic with period three and for some the following condition holds: Then if for every , the following statements hold: (a)if , then , as ;(b)if , , and , then , as ;(c)if , , then (d)if and (5.29) holds, then (e)if and , then as ;(f)if and , then as ;(g)if and , then converges, as ;(h)if and , then and converge, as ;(i)if and , where then as ;(j)if and , then as ;(k)if and , then converges, as ;(l)if and , then and converge, as .

Proof. From (3.22) and by using condition (5.28), we have that for every and for each , from which the statements in (a)–(l) follows similar as the corresponding ones in Theorem 5.5.

Theorem 5.7. Consider system (3.2). Let the sequences , , , be periodic with period three and for some the following condition holds Then if for every , the following statements hold: (a)if , then , as ;(b)if , ,, then , as ;(c)if , , then (d)if and (5.35) holds, then (e)if and , then as ;(f)if and , then as ;(g)if and , then converges, as ;(h)if and , then and converge, as ;(i)if and , where then as ;(j)if and , then as ;(k)if and , then converges, as ;(l)if and , then and converge, as .

Proof. From (3.23) and by using condition (5.34), we have that for every and for each , from which the statements in (a)–(l) follow similar as the corresponding ones in Theorem 5.5.

Acknowledgments

The first author is supported by Development Programme of The Ministry of Education, Youth and Sports of Czech Republic no. 25/16—PoZaP—Support for Foreign Teachers (Stays at BUT) in programme “5 support of international collaboration in university education’’ and subprogramme “5b Support to improve conditions for the functioning of major foreign academics in universities.” The second author is supported by Grant 201/10/1032 of the Czech Grant Agency (Prague) and by the Council of Czech Government Grant MSM 00 216 30519. The fourth author is supported by Grant FEKT-S-11-2-921 of Faculty of Electrical Engineering and Communication, Brno University of Technology. This paper is partially also supported by the Serbian Ministry of Science Projects III 41025, III 44006, and OI 171007.

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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.

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