# On some classes of solvable systems of difference equations

- 136 Downloads

## Abstract

In a recent paper several periodic systems of difference equations have been presented. We show in an elegant way that all the results therein follow from some known ones. We also show how some extensions of the systems of difference equations can be solved.

## Keywords

System of difference equations Periodicity General solution Difference equation with interlacing indices## MSC

39A05 39A06## 1 Introduction

Here we use the following standard notations: \({\mathbb{N}}\) stands for the set of all natural numbers, \({\mathbb{Z}}\) stands for the set of all integers, for an \(l\in {\mathbb{Z}}\), \({\mathbb{N}}_{l}\) is defined as \(\{n\in {\mathbb{Z}}:n\ge l\}\), \({\mathbb{R}}\) denotes the set of all real numbers, and \({\mathbb{C}}\) denotes the set of all complex numbers. Let \(k,l\in {\mathbb{Z}}\), \(k\le l\), then the notation \(j= \overline{k,l}\) denotes the set of all \(j\in {\mathbb{Z}}\) such that \(k\le j\le l\).

Solvability of difference equations has been studied for a long time. Basic classes of solvable difference equations and systems were found during the eighteenth century, and since that time the books which contain results on the topic have more or less presented the old original methods or their modifications and refinements which were obtained during the nineteenth century (see, e.g., [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11]). For some applications of solvability methods of difference equations, see, e.g., [1, 3, 4, 5, 8, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19].

*a*,

*b*,

*c*,

*d*and the initial value \(x_{0}\) are real or complex numbers (see, e.g., [1, 2, 12, 13, 15, 20, 21]), which is a basic example of solvable nonlinear difference equation. Its extension with nonconstant coefficients is an equation of a huge interest since it is equivalent to the homogeneous linear difference equation of second-order. How equation (1) is solved can be found, e.g., in [2, 5, 10, 15, 21, 22]. For some recent results on representations of solutions to equation (1) in terms of Fibonacci-type sequences, see [23, 24], and the references therein. Some basics on Fibonacci sequences can be found, e.g., in [25] and [26].

We are witnesses that the area of solvability of difference equations and related ones re-attracts some attention. One of the reasons for the recent interest is the use of computer algebra systems, which can help in finding or guessing closed-form formulas for solutions to some difference equations and systems. The computer algebra systems are certainly useful, but in the majority cases the authors obtain results which are known or easily follow from the known ones (see, e.g., some of our comments in [23, 24, 27, 28, 29] related to the issue).

Another area of recent interest is concrete systems of difference equations, with a special interest in symmetric and close-to-symmetric ones. The study of such systems was essentially initiated and popularized by Papaschinopoulos and Schinas (see, e.g., [37, 38, 39, 40, 41, 42, 43, 44]). Many of our papers are also devoted to the area (see, e.g., [17, 23, 24, 27, 29, 32, 33, 36] and the references therein). For other related results, including the ones on invariants of difference equations, see, e.g., [39, 40, 43, 45, 46] and the references therein. Let us also mention that some more complex solvable difference equations and systems can be found, e.g., in [17, 33, 47, 48, 49, 50, 51], but the idea is essentially the same as in the above-mentioned papers, that is, some connections of studied difference equations and systems to some solvable ones are found.

*eventually periodic*with period \(p\in {\mathbb{N}}\) if there is \(n_{0}\in {\mathbb{N}}_{l}\) such that

*periodic*or more precisely

*p-periodic*. Many authors call both, eventually periodic and periodic sequences, simply as periodic.

*a*,

*b*,

*c*,

*d*and initial values \(x_{-j}\), \(y_{-j}\), \(j=\overline{0,2k+1}\), are complex numbers. Namely, system (2) is obtained with \(a=-b=c=-d=1\), system (3) is obtained with \(a=b=c=d=-1\), system (4) is obtained with \(a=b=-c=d=1\), whereas system (5) is obtained with \(-a=b=c=d=1\).

We first study system (6) and then turn to the special cases of the system in (2)–(5). We show that all the results in [46] follow from some known ones. We also show how some extensions of the systems of difference equations can be solved.

## 2 On system (6) and the results in [46]

In this section we first study the structural form of system (6), and then by using the analysis we discuss systems (2)–(5).

### 2.1 Analysis of the structural form of system (6)

Here we investigate the structural form of system (6).

Hence, by induction we have proved that in calculation of all terms of the sequences \((x_{m(k+1)+1})_{m\ge -2}\) and \((y_{m(k+1)+1})_{m\ge -2}\) also only terms \(x_{-(2k+1)}\), \(y_{-(2k+1)}\), \(x_{-k}\), and \(y_{-k}\) are used.

The same argument shows that for each fixed \(j\in \{1,2,\ldots ,k+1\}\), in calculation of all terms of the sequences \((x_{m(k+1)+j})_{m\ge -2}\) and \((y_{m(k+1)+j})_{m\ge -2}\), only terms \(x_{j-(2k+2)}\), \(y_{j-(2k+2)}\), \(x_{j-(k+1)}\), and \(y_{j-(k+1)}\) are used.

*l*systems/equations, which are not all systems/equations with interlacing indices, we say that the system/equation is a

*system/equation with interlacing indices of order*

*l*.

It has been recently shown in [29] that system (15) is solvable, where also the long-term behavior of its solutions has been described in detail in many cases. Hence, the long-term behavior of solutions to system (6) practically directly follows from the long-term behavior of solutions to system (15).

### 2.2 On the results on systems (2)–(5) quoted in [46]

Here we discuss in detail the results quoted in [46].

The first result quoted in [46] is the following theorem, which was proved by a long calculatory-inductive argument.

### Theorem 1

*Let*\((x_{n}, y_{n})_{n\ge -(2k+1)}\)

*be a well*-

*defined solution to system*(2).

*Then the following statements hold*.

- (i)
*Sequences*\((x_{n})_{n\ge -(2k+1)}\)*and*\((y_{n})_{n\ge -(2k+1)}\)*are periodic with period*\(6(k+1)\). - (ii)
*We have*$$\begin{aligned} &x_{6(k+1)m+l}= \textstyle\begin{cases} \frac{x_{l-k-1}y_{l-2k-2}}{y_{l-2k-2}-y_{l-k-1}} , & 1\le l\le k+1, \\ \frac{y_{l-3k-3}(-x_{l-3k-3}+x_{l-2k-2})}{y_{l-3k-3}-y_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{y_{l-3k-3}(-x_{l-4k-4}+x_{l-3k-3})}{y_{l-4k-4}-y_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{y_{l-4k-4}x_{l-5k-5}}{-y_{l-5k-5}+y_{l-4k-4}}, & 3k+4\le l \le 4k+4, \\ x_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \\ &y_{6(k+1)m+l}= \textstyle\begin{cases} \frac{y_{l-k-1}x_{l-2k-2}}{x_{l-2k-2}-x_{l-k-1}} , & 1\le l\le k+1, \\ \frac{x_{l-3k-3}(-y_{l-3k-3}+y_{l-2k-2})}{x_{l-3k-3}-x_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{x_{l-3k-3}(-y_{l-4k-4}+y_{l-3k-3})}{x_{l-4k-4}-x_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{x_{l-4k-4}y_{l-5k-5}}{-x_{l-5k-5}+x_{l-4k-4}}, & 3k+4\le l \le 4k+4, \\ y_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \end{aligned}$$

*for*\(m\in {\mathbb{N}}_{0}\).

In [29, Theorem 4] (see also Example 4 therein for details) it was proved that every well-defined solution to system (15) with \(a=-b=c=-d=1\) is six-periodic. Bearing in mind this fact, as well as the above consideration which shows that the sequences defined in (12) and (13) are \(k+1\) independent solutions to the system, it immediately follows that for every well-defined solution to system (2) the sequences \((x_{n})_{n\ge -(2k+1)}\) and \((y_{n})_{n\ge -(2k+1)}\) are periodic with period \(6(k+1)\). So, Theorem 1(i) is a very simple consequence of known results.

The second result in [46] is the following theorem, for which it was only said that it is proved similarly to Theorem 1 (by an inductive argument).

### Theorem 2

*Let*\((x_{n}, y_{n})_{n\ge -(2k+1)}\)

*be a well*-

*defined solution to system*(3).

*Then the following statements hold*.

- (i)
*Sequences*\((x_{n})_{n\ge -(2k+1)}\)*and*\((y_{n})_{n\ge -(2k+1)}\)*are periodic with period*\(6(k+1)\). - (ii)
*We have*$$\begin{aligned} &x_{6(k+1)m+l}= \textstyle\begin{cases} -\frac{x_{l-k-1}y_{l-2k-2}}{y_{l-2k-2}+y_{l-k-1}} , & 1\le l\le k+1, \\ \frac{y_{l-3k-3}(x_{l-3k-3}+x_{l-2k-2})}{y_{l-3k-3}+y_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{y_{l-3k-3}(x_{l-4k-4}+x_{l-3k-3})}{y_{l-4k-4}+y_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ -\frac{y_{l-4k-4}x_{l-5k-5}}{y_{l-5k-5}+y_{l-4k-4}}, & 3k+4\le l \le 4k+4, \\ x_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \\ &y_{6(k+1)m+l}= \textstyle\begin{cases} -\frac{y_{l-k-1}x_{l-2k-2}}{x_{l-2k-2}+x_{l-k-1}} , & 1\le l\le k+1, \\ \frac{x_{l-3k-3}(y_{l-3k-3}+y_{l-2k-2})}{x_{l-3k-3}+x_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{x_{l-3k-3}(y_{l-4k-4}+y_{l-3k-3})}{x_{l-4k-4}+x_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ -\frac{x_{l-4k-4}y_{l-5k-5}}{x_{l-5k-5}+x_{l-4k-4}}, & 3k+4\le l \le 4k+4, \\ y_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \end{aligned}$$

*for*\(m\in {\mathbb{N}}_{0}\).

Using again Theorem 4 in [29], we see that every well-defined solution to system (30) is six-periodic, from which along with (29) it follows that every well-defined solution to system (28) is six-periodic. Hence, we have that every well-defined solution to system (3) is periodic with period \(6(k+1)\). So, Theorem 2(i) is also a very simple consequence of known results.

The next two results quoted in [46, Corollary 4.1] are the following theorems, for which it was also only said that they can be proved similarly to Theorem 1.

### Theorem 3

*Let*\((x_{n}, y_{n})_{n\ge -(2k+1)}\)

*be a well*-

*defined solution to system*(4).

*Then the following statements hold*.

- (i)
*Sequences*\((x_{n})_{n\ge -(2k+1)}\)*and*\((y_{n})_{n\ge -(2k+1)}\)*are periodic with period*\(6(k+1)\). - (ii)
*We have*$$\begin{aligned} &x_{6(k+1)m+l}= \textstyle\begin{cases} \frac{x_{l-k-1}y_{l-2k-2}}{y_{l-2k-2}+y_{l-k-1}} , & 1\le l\le k+1, \\ \frac{y_{l-3k-3}(-x_{l-3k-3}+x_{l-2k-2})}{y_{l-3k-3}+y_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{y_{l-3k-3}(x_{l-4k-4}-x_{l-3k-3})}{y_{l-4k-4}+y_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{y_{l-4k-4}x_{l-5k-5}}{y_{l-5k-5}+y_{l-4k-4}}, & 3k+4\le l\le 4k+4, \\ x_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \\ &y_{6(k+1)m+l}= \textstyle\begin{cases} \frac{y_{l-k-1}x_{l-2k-2}}{-x_{l-2k-2}+x_{l-k-1}} , & 1\le l\le k+1, \\ \frac{x_{l-3k-3}(y_{l-3k-3}+y_{l-2k-2})}{x_{l-3k-3}-x_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{x_{l-3k-3}(y_{l-4k-4}+y_{l-3k-3})}{-x_{l-4k-4}+x_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{x_{l-4k-4}y_{l-5k-5}}{x_{l-5k-5}-x_{l-4k-4}}, & 3k+4\le l\le 4k+4, \\ y_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \end{aligned}$$

*for*\(m\in {\mathbb{N}}_{0}\).

Using Theorem 4 in [29] we see that every well-defined solution to system (45) is six-periodic, from which along with (44) it follows that every well-defined solution to system (43) is six-periodic. Hence, every well-defined solution to system (4) is \(6(k+1)\)-periodic. So, Theorem 3(i) is also a simple consequence of known results.

### Theorem 4

*Let*\((x_{n}, y_{n})_{n\ge -(2k+1)}\)

*be a well*-

*defined solution to system*(5).

*Then the following statements hold*.

- (i)
*Sequences*\((x_{n})_{n\ge -(2k+1)}\)*and*\((y_{n})_{n\ge -(2k+1)}\)*are periodic with period*\(6(k+1)\). - (ii)
*We have*$$\begin{aligned} &x_{6(k+1)m+l}= \textstyle\begin{cases} \frac{x_{l-k-1}y_{l-2k-2}}{-y_{l-2k-2}+y_{l-k-1}} , & 1\le l\le k+1, \\ \frac{y_{l-3k-3}(x_{l-3k-3}+x_{l-2k-2})}{y_{l-3k-3}-y_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{y_{l-3k-3}(x_{l-4k-4}+x_{l-3k-3})}{-y_{l-4k-4}+y_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{y_{l-4k-4}x_{l-5k-5}}{y_{l-5k-5}-y_{l-4k-4}}, & 3k+4\le l\le 4k+4, \\ x_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \\ &y_{6(k+1)m+l}= \textstyle\begin{cases} \frac{y_{l-k-1}x_{l-2k-2}}{x_{l-2k-2}+x_{l-k-1}} , & 1\le l\le k+1, \\ \frac{x_{l-3k-3}(-y_{l-3k-3}+y_{l-2k-2})}{x_{l-3k-3}+x_{l-2k-2}}, & k+2 \le l\le 2k+2, \\ \frac{x_{l-3k-3}(y_{l-4k-4}-y_{l-3k-3})}{x_{l-4k-4}+x_{l-3k-3}}, & 2k+3 \le l\le 3k+3, \\ \frac{x_{l-4k-4}y_{l-5k-5}}{x_{l-5k-5}+x_{l-4k-4}}, & 3k+4\le l\le 4k+4, \\ y_{l-6k-6}, & 4k+5\le l\le 6k+6, \end{cases}\displaystyle \end{aligned}$$

*for*\(m\in {\mathbb{N}}_{0}\).

*x*and

*y*. Thus, Theorem 4 directly follows from Theorem 3, which as we have already mentioned follows from known results, so it is also the case with Theorem 4.

### Remark 1

### Remark 2

Note that the changes of variables (29) in Theorem 2, (44) in Theorem 3, (59) in Remark 1, along with the interlacing argument explained above show that systems (3)–(5) are equivalent to system (2), so essentially only one result was proved in [46], which as we have explained follows from the known ones.

## 3 Solvability of some extensions of system (6)

Here we show how some extensions of system (6) can be solved.

### 3.1 A three-dimensional extension to system (6)

*a*,

*b*,

*c*,

*d*,

*e*,

*f*and initial values \(x_{-j}\), \(y_{-j}\), \(z_{-j}\), \(j=\overline{0,2k+1}\), are complex numbers, and \(k\in {\mathbb{N}}_{0}\).

As in the previous section, it is proved that system (61) is a system of difference equations with interlacing indices of order \(k+1\).

Using the procedure in Sect. 2, it is seen that equations (68)–(70) are three equations with interlacing indices of order three.

Since bilinear difference equations are solvable, it follows that systems (77)–(79) are also solvable. From this and (71)–(76) it follows that equations (68)–(70) are solvable, from which along with (66) it easily follows that system (61) is also solvable.

### 3.2 A four-dimensional extension to system (6)

*a*,

*b*,

*c*,

*d*,

*e*,

*f*,

*g*,

*h*and initial values \(x_{-j}\), \(y_{-j}\), \(z_{-j}\), \(u_{-j}\), \(j= \overline{0,2k+1}\), are complex numbers, and \(k\in {\mathbb{N}}_{0}\).

As in Sect. 2 it is proved that system (80) is a system of difference equations with interlacing indices of order \(k+1\).

Using the procedure from Sect. 2, it is seen that equations (88)–(91) are four equations with interlacing indices of order four.

Since bilinear difference equations are solvable in closed form, it follows that systems (100)–(103) are also solvable. From this along with (92)–(99) it follows that equations (88)–(91) are solvable, from which and (86) it follows that system (80) is also solvable.

### Remark 3

We will not present closed-form formulas for solutions to systems (61) and (80) since it is done in a standard way by using a method for solving bilinear difference equations along with the changes of variables given in this section. By using such obtained closed-form formulas for solutions to the systems, the long-term behavior of their solutions can be described, which can be done as in our papers [28, 29, 34]. We leave the standard problem to the reader as an exercise.

### Remark 4

*y*is replaced by letter

*x*), it is known that system (104) must be periodic with period \(18(k+1)\), whereas (105) must be periodic with period \(24(k+1)\).

## Notes

### Acknowledgements

The work of Bratislav Iričanin was supported by the Serbian Ministry of Education and Science projects III 41025 and OI 171007, work of Stevo Stević by projects III 41025 and III 44006, while the work of Zdeněk Šmarda was supported by the project FEKT-S-17-4225 of Brno University of Technology.

### Availability of data and materials

Not applicable.

### Authors’ contributions

All authors contributed equally to the writing of this paper. All authors read and approved the manuscript.

### Funding

FEKT-S-17-4225 of Brno University of Technology.

### Competing interests

The authors declare that they have no competing interests.

## References

- 1.Agarwal, R.P.: Difference Equations and Inequalities: Theory, Methods, and Applications, 2nd edn. Dekker, New York (2000) zbMATHGoogle Scholar
- 2.Boole, G.: A Treatsie on the Calculus of Finite Differences, 3rd edn. Macmillan & Co., London (1880) Google Scholar
- 3.Fort, T.: Finite Differences and Difference Equations in the Real Domain. Clarendion Press, Oxford (1948) zbMATHGoogle Scholar
- 4.Jordan, C.: Calculus of Finite Differences. Chelsea, New York (1956) zbMATHGoogle Scholar
- 5.Krechmar, V.A.: A Problem Book in Algebra. Mir, Moscow (1974) (Russian first edition 1937) Google Scholar
- 6.Markoff, A.A.: Differenzenrechnung, Leipzig (1896) (in German) Google Scholar
- 7.Milne-Thomson, L.M.: The Calculus of Finite Differences. MacMillan & Co., London (1933) zbMATHGoogle Scholar
- 8.Mitrinović, D.S., Kečkić, J.D.: Methods for Calculating Finite Sums. Naučna Knjiga, Beograd (1984) (in Serbian) Google Scholar
- 9.Nörlund, N.E.: Vorlesungen über Differenzenrechnung. Springer, Berlin (1924) (in German) CrossRefGoogle Scholar
- 10.Richardson, C.H.: An Introduction to the Calculus of Finite Differences. Van Nostrand, Toronto (1954) zbMATHGoogle Scholar
- 11.Stević, S.: Bounded solutions to nonhomogeneous linear second-order difference equations. Symmetry
**9**, Article ID 227 (2017) CrossRefGoogle Scholar - 12.Agarwal, R.P., Popenda, J.: Periodic solutions of first order linear difference equations. Math. Comput. Model.
**22**(1), 11–19 (1995) MathSciNetCrossRefGoogle Scholar - 13.Berezansky, L., Braverman, E.: On impulsive Beverton–Holt difference equations and their applications. J. Differ. Equ. Appl.
**10**(9), 851–868 (2004) MathSciNetCrossRefGoogle Scholar - 14.Mitrinović, D.S.: Matrices and Determinants. Naučna Knjiga, Beograd (1989) (in Serbian) Google Scholar
- 15.Mitrinović, D.S., Adamović, D.D.: Sequences and Series. Naučna Knjiga, Beograd (1980) (in Serbian) Google Scholar
- 16.Riordan, J.: Combinatorial Identities. Wiley, New York (1968) zbMATHGoogle Scholar
- 17.Stević, S.: First-order product-type systems of difference equations solvable in closed form. Electron. J. Differ. Equ.
**2015**, Article ID 308 (2015) MathSciNetCrossRefGoogle Scholar - 18.Stević, S.: Existence of a unique bounded solution to a linear second order difference equation and the linear first order difference equation. Adv. Differ. Equ.
**2017**, Article ID 169 (2017) MathSciNetCrossRefGoogle Scholar - 19.Stević, S., Alghamdi, M.A., Alotaibi, A., Elsayed, E.M.: Solvable product-type system of difference equations of second order. Electron. J. Differ. Equ.
**2015**, Article ID 169 (2015) MathSciNetCrossRefGoogle Scholar - 20.Brand, L.: Differential and Difference Equations. Wiley, New York (1966) zbMATHGoogle Scholar
- 21.Adamović, D.: Solution to problem 194. Mat. Vesn.
**23**, 236–242 (1971) Google Scholar - 22.Brand, L.: A sequence defined by a difference equation. Am. Math. Mon.
**62**(7), 489–492 (1955) MathSciNetCrossRefGoogle Scholar - 23.Stević, S.: Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences. Electron. J. Qual. Theory Differ. Equ.
**2014**, Article ID 67 (2014) MathSciNetCrossRefGoogle Scholar - 24.Stević, S., Iričanin, B., Šmarda, Z.: On a symmetric bilinear system of difference equations. Appl. Math. Lett.
**89**, 15–21 (2019) MathSciNetCrossRefGoogle Scholar - 25.Alfred, B.U.: An Introduction to Fibonacci Discovery, the Fibonacci Association (1965) Google Scholar
- 26.Vorobiev, N.N.: Fibonacci Numbers. Birkhäuser, Basel (2002) (Russian original 1950) CrossRefGoogle Scholar
- 27.Stević, S., Diblik, J., Iričanin, B., Šmarda, Z.: On some solvable difference equations and systems of difference equations. Abstr. Appl. Anal.
**2012**, Article ID 541761 (2012) MathSciNetzbMATHGoogle Scholar - 28.Stević, S., Diblik, J., Iričanin, B., Šmarda, Z.: Solvability of nonlinear difference equations of fourth order. Electron. J. Differ. Equ.
**2014**, Article ID 264 (2014) MathSciNetCrossRefGoogle Scholar - 29.Stević, S., Diblik, J., Iričanin, B., Šmarda, Z.: On the system of difference equations \(x_{n}=x_{n-1}y_{n-2}/(ay_{n-2}+by_{n-1})\), \(y_{n}=y_{n-1}x_{n-2}/(cx _{n-2}+dx_{n-1})\). Appl. Math. Comput.
**270**, 688–704 (2015) MathSciNetGoogle Scholar - 30.Papaschinopoulos, G., Stefanidou, G.: Asymptotic behavior of the solutions of a class of rational difference equations. Int. J. Difference Equ.
**5**(2), 233–249 (2010) MathSciNetGoogle Scholar - 31.Stević, S.: On the difference equation \(x_{n}=x_{n-2}/(b_{n}+c_{n}x _{n-1}x_{n-2})\). Appl. Math. Comput.
**218**, 4507–4513 (2011) MathSciNetzbMATHGoogle Scholar - 32.Stević, S.: On the system of difference equations \(x_{n}=c_{n}y_{n-3}/(a_{n}+b_{n}y_{n-1}x_{n-2}y_{n-3})\), \(y_{n}=\gamma_{n} x_{n-3}/(\alpha_{n}+\beta_{n} x_{n-1}y_{n-2}x_{n-3})\). Appl. Math. Comput.
**219**, 4755–4764 (2013) MathSciNetzbMATHGoogle Scholar - 33.Stević, S., Iričanin, B., Šmarda, Z.: Solvability of a close to symmetric system of difference equations. Electron. J. Differ. Equ.
**2016**, Article ID 159 (2016) MathSciNetCrossRefGoogle Scholar - 34.Stević, S., Diblik, J., Iričanin, B., Šmarda, Z.: On a third-order system of difference equations with variable coefficients. Abstr. Appl. Anal.
**2012**, Article ID 508523 (2012) MathSciNetzbMATHGoogle Scholar - 35.Stević, S., Diblik, J., Iričanin, B., Šmarda, Z.: On a solvable system of rational difference equations. J. Differ. Equ. Appl.
**20**(5–6), 811–825 (2014) MathSciNetCrossRefGoogle Scholar - 36.Berg, L., Stević, S.: On some systems of difference equations. Appl. Math. Comput.
**218**, 1713–1718 (2011) MathSciNetzbMATHGoogle Scholar - 37.Papaschinopoulos, G., Schinas, C.J.: On a system of two nonlinear difference equations. J. Math. Anal. Appl.
**219**(2), 415–426 (1998) MathSciNetCrossRefGoogle Scholar - 38.Papaschinopoulos, G., Schinas, C.J.: On the behavior of the solutions of a system of two nonlinear difference equations. Commun. Appl. Nonlinear Anal.
**5**(2), 47–59 (1998) MathSciNetzbMATHGoogle Scholar - 39.Papaschinopoulos, G., Schinas, C.J.: Invariants for systems of two nonlinear difference equations. Differ. Equ. Dyn. Syst.
**7**, 181–196 (1999) MathSciNetzbMATHGoogle Scholar - 40.Papaschinopoulos, G., Schinas, C.J.: Invariants and oscillation for systems of two nonlinear difference equations. Nonlinear Anal., Theory Methods Appl.
**46**, 967–978 (2001) MathSciNetCrossRefGoogle Scholar - 41.Papaschinopoulos, G., Schinas, C.J.: Oscillation and asymptotic stability of two systems of difference equations of rational form. J. Differ. Equ. Appl.
**7**, 601–617 (2001) MathSciNetCrossRefGoogle Scholar - 42.Papaschinopoulos, G., Schinas, C.J.: On the dynamics of two exponential type systems of difference equations. Comput. Math. Appl.
**64**(7), 2326–2334 (2012) MathSciNetCrossRefGoogle Scholar - 43.Papaschinopoulos, G., Schinas, C.J., Stefanidou, G.: On a
*k*-order system of Lyness-type difference equations. Adv. Differ. Equ.**2007**, Article ID 31272 (2007) MathSciNetCrossRefGoogle Scholar - 44.Stefanidou, G., Papaschinopoulos, G., Schinas, C.: On a system of max difference equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A
**14**(6), 885–903 (2007) MathSciNetzbMATHGoogle Scholar - 45.Andruch-Sobilo, A., Migda, M.: Further properties of the rational recursive sequence \(x_{n+1}= ax_{n-1}/(b + cx_{n}x_{n-1})\). Opusc. Math.
**26**(3), 387–394 (2006) zbMATHGoogle Scholar - 46.Gocen, M., Cebeci, A.: On the periodic solutions of some systems of higher order difference equations. Rocky Mt. J. Math.
**48**(3), 845–858 (2018) MathSciNetCrossRefGoogle Scholar - 47.Stević, S.: Bounded and periodic solutions to the linear first-order difference equation on the integer domain. Adv. Differ. Equ.
**2017**, Article ID 283 (2017) MathSciNetCrossRefGoogle Scholar - 48.Stević, S.: New class of solvable systems of difference equations. Appl. Math. Lett.
**63**, 137–144 (2017) MathSciNetCrossRefGoogle Scholar - 49.Stević, S.: Solvable product-type system of difference equations whose associated polynomial is of the fourth order. Electron. J. Qual. Theory Differ. Equ.
**2017**, Article ID 13 (2017) MathSciNetCrossRefGoogle Scholar - 50.Stević, S., Iričanin, B., Šmarda, Z.: On a product-type system of difference equations of second order solvable in closed form. J. Inequal. Appl.
**2015**, Article ID 327 (2015) MathSciNetCrossRefGoogle Scholar - 51.Stević, S., Iričanin, B., Šmarda, Z.: Two-dimensional product-type system of difference equations solvable in closed form. Adv. Differ. Equ.
**2016**, Article ID 253 (2016) MathSciNetCrossRefGoogle Scholar - 52.Stević, S.: Periodicity of max difference equations. Util. Math.
**83**, 69–71 (2010) MathSciNetzbMATHGoogle Scholar - 53.Stević, S.: On some periodic systems of max-type difference equations. Appl. Math. Comput.
**218**, 11483–11487 (2012) MathSciNetzbMATHGoogle Scholar - 54.Iričanin, B., Stević, S.: Eventually constant solutions of a rational difference equation. Appl. Math. Comput.
**215**, 854–856 (2009) MathSciNetzbMATHGoogle Scholar

## Copyright information

**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.