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On a producttype system of difference equations of second order solvable in closed form
Journal of Inequalities and Applications volume 2015, Article number: 327 (2015)
Abstract
It is shown that the following system of difference equations
where \(a,b,c,d\in \mathbb {Z}\), \(z_{1}, z_{0}, w_{1}, w_{0}\in \mathbb {C}\), is solvable in closed form.
1 Introduction
Recently there has been a great interest in studying nonlinear difference equations and systems not stemming from differential ones (see, e.g., [1–30]). The old area of solving difference equations and systems has reattracted recent attention (see, e.g., [1–6, 12, 15, 19–26, 28–30]). Recent Stević’s idea of transforming complicated equations and systems into simpler solvable ones, used for the first time in explaining the solvability of the equation appearing in [6] (an extension of the original result can be found in [20], see also [22]), was employed in several other papers (see, e.g., [1, 2, 4, 12, 15, 19, 21, 24–26, 29, 30] and the related references therein). Another area of some recent interest, essentially initiated by Papaschinopoulos and Schinas, is studying symmetric and close to symmetric systems of difference equations (see, e.g., [5, 8–10, 13, 14, 19, 21, 23, 25, 27–29]).
Stević also essentially triggered a systematic study of nonrational concrete difference equations and systems, from one side those obtained by using the translation operator (see, e.g., [16] and also [11]) and from the other side those obtained by using maxtype operators (see, e.g., [17, 18, 27]), see also the related references cited therein. We would like to point out that for the equations and systems in [16–18, 27] only longterm behavior of their positive solutions are studied. For instance, the boundedness of positive solutions to the system
with \(\min\{a, p, q\}>0\), was investigated in [27]. System (1) is obviously obtained from the next producttype one
by acting with the maxtype operator \(m_{a}(t)=\max\{a,t\}\) onto the righthand sides of both equations in (2) (see also [17] and [18] for related scalar equations). Note that for the case of positive initial values, system (2) can be solved by taking the logarithm to the both sides of both equations therein, since this transforms the system to a linear second order system of difference equations with constant coefficients, which is solvable. Note that the method does not work if initial values are not positive. Let us also mention here that positive solutions to difference equations and systems are often studied since many reallife models produce such solutions (see, e.g., [7, 15, 31]). It is also interesting to note that there are maxtype systems of difference equations which are solvable (see [23]). Finally, we want to note that the longterm behavior of solutions to producttype systems and those obtained from them by acting with some ‘reasonable good’ transformations are frequently closely related, which is another reason for studying these systems.
Hence, a natural problem is to investigate the solvability of producttype difference equations and systems with real and/or complex initial values. In [26], Stević and his collaborators started studying the problem with an approach different from the ones in [5, 21, 24, 25], but which can be regarded as a modification of some of the methods in [16–18, 27]. They showed therein that the system
where \(a,b,c,d\in \mathbb {Z}\) and \(z_{1}, z_{0}, w_{1}, w_{0}\in \mathbb {C}\), is solvable in closed form and presented numerous applications of obtained formulas.
In this paper we continue our investigation by studying the solvability of the following system of difference equations:
where \(a,b,c,d\in \mathbb {Z}\) and \(z_{1}, z_{0}, w_{1}, w_{0}\in \mathbb {C}\).
Let us mention here that although systems (3) and (4) are similar in appearance, the methods used in dealing with them are quite different.
It is easy to see that the domain of undefinable solutions [24] to system (4) is the set
Hence, from now on we will assume that our initial values belong to the set \(\mathbb {C}^{4}\setminus{\mathcal{U}}\).
A solution \((z_{n},w_{n})_{n\ge1}\) of system (4) is called periodic (or eventually periodic) with period \(p\in \mathbb {N}\) if there is \(n_{0}\ge1\) such that
Period p is prime if there is no \(\hat{p}\in \mathbb {N}\), \(\hat{p}< p\) which is a period of the solution. For \(p=1\), the solution is called eventually constant (see, e.g., [32]). For some results on the topic, see, e.g., [7, 15] and the related references therein. If it is said that a solution of system (4) is periodic with period p, it will need not mean that it is prime.
A system of difference equations of the form
where \(k\in \mathbb {N}\), is said to be solvable in closed form if its general solution can be found in terms of initial values \(z_{i}\), \(w_{i}\), \(i=\overline {1,k}\), delay k and index n only.
2 Main result
The main result in this paper is proved in this section.
Theorem 1
Assume that \(a,b,c,d\in \mathbb {Z}\) and \(z_{1}, z_{0}, w_{1}, w_{0}\in \mathbb {C}\setminus\{0\}\). Then system (4) is solvable in closed form.
Proof
Case \(b=0\). In this case system (4) becomes
From the first equation in (5) we easily obtain
Employing (6) into the second equation in (5), we get
for \(n\ge 2\).
Hence, by using (7), we have that
for \(n\ge4\).
Assume that we have proved
for \(n\ge k+1\).
Then, by using (7) with \(n\to nk\) into (8), we get
for \(n\ge k+2\).
From (7), (9) and the method of induction we see that (8) holds for every k such that \(1\le k\le n1\).
By taking \(k=n1\) into (8) we get
for \(n\ge2\).
Now we have two subcases to consider.
Subcase \(a\ne c\). In this case from (10) we get
Using the next relation
in (11) we get
Subcase \(a=c\). In this case from (10) we get
for \(n\ge2\).
Using (12) with \(a=c\) into (14), we get
Case \(d=0\). In this case system (4) becomes
From the second equation in (16) we have that
Employing (17) into the first equation in (16), we get
for \(n\ge 2\).
Hence, by using (18), we have
for \(n\ge4\).
Assume that we have proved
for \(n\ge k+1\).
Then, by using (18) with \(n\to nk\) into (19), we get
for \(n\ge k+2\).
From (18), (20) and the method of induction we see that (19) holds for every k such that \(1\le k\le n1\).
By taking \(k=n1\) into (19) we get
for \(n\ge2\).
Now we have two subcases to consider.
Subcase \(a\ne c\). In this case from (21) we get
Using the next relation
in (22) we get
Subcase \(a=c\). In this case from (21) we get
for \(n\ge2\).
Case \(bd\ne0\). First note that from the first equation in (4), for every welldefined solution, we have that
while from the second one it follows that
Using (27) into (28) we obtain
which can be written as
which is a fourth order producttype difference equation.
Note also that
Let
Then equation (29) can be written as
From (32) with \(n\to n1\) we get
Employing (33) into (32) we get
for \(n\in \mathbb {N}\), where
From (32) with \(n\to n2\) we get
for \(n\ge2\).
Employing (36) into (34) we get
for \(n\ge2\), where
Assume that for some \(2\le k \le n\), we have proved that
for \(n\ge k1\), and that
Then, by using the relation
for \(n\ge k\), into (39) we obtain
for \(n\ge k\), where
This along with (34), (35) and the method of induction shows that (39) and (40) hold for every \(2\le k\le n+1\).
Hence, for \(k=n+1\), we have
From the recurrent relations (40) we easily obtain that the sequence \((a_{k})_{k\ge5}\) satisfies the difference equation
Since \(b_{k1}=a_{k}a_{1}a_{k1}\) and equation (44) is linear, we have that the sequence \((b_{k})_{k\in \mathbb {N}}\) is also a solution to equation (44). From this, the linearity of equation (44) and since \(c_{k1}=b_{k}b_{1}a_{k1}\), we have that the sequence \((c_{k})_{k\in \mathbb {N}}\) is also a solution to equation (44). Finally, since \(d_{k}=d_{1}a_{k1}\), the linearity of equation (44) shows that \((d_{k})_{k\in \mathbb {N}}\) is also a solution to the equation.
Now, we show that these four sequences can be prolonged for some negative indices of use. This enables easier getting formulas for solutions to system (4).
From (42) with \(k=0\) we get
Since \(bd=d_{1}\ne0\), from the last equation in (45) we get \(a_{0}=1\). Using this fact in the first three equalities in (45), we get \(b_{0}=c_{0}=d_{0}=0\).
From this and by (42) with \(k=1\) we get
Since \(d_{1}\ne0\), from the last equation in (46) we get \(a_{1}=0\). Using this fact in other three equalities in (46), we get \(b_{1}=1\), \(c_{1}=d_{1}=0\).
From this and by (42) with \(k=2\) we get
Since \(d_{1}\ne0\), from the last equation in (47) we get \(a_{2}=0\). Using this fact in other three equalities in (47), we get \(b_{2}=0\), \(c_{2}=1\) and \(d_{2}=0\).
From this and by (42) with \(k=3\) we get
Since \(d_{1}\ne0\), from the last equation in (48) we get \(a_{3}=0\). Using this fact in other three equalities in (48), we get \(b_{3}=0\), \(c_{3}=0\) and \(d_{3}=1\).
Hence, sequences \((a_{k})_{k\ge3}\), \((b_{k})_{k\ge3}\), \((c_{k})_{k\ge3}\) and \((d_{k})_{k\ge3}\) are solutions to linear difference equation (44) satisfying the following initial conditions:
respectively.
Since difference equation (44) is solvable, it follows that closed form formulas for \((a_{k})_{k\ge3}\), \((b_{k})_{k\ge3}\), \((c_{k})_{k\ge3}\) and \((d_{k})_{k\ge3}\) can be found. From this fact and (43) we see that equation (29) is solvable too.
From the second equation in (4), for every welldefined solution, we have that
while from the first one it follows that
Using (50) into (51) we obtain
which can be written as
which is nothing but difference equation (29). However, the sequence \((w_{n})_{n\ge1}\) satisfies the following initial conditions:
Hence, the above presented procedure can be repeated, and it can be obtained that for \(1\le k \le n+1\),
where \((a_{k})_{k\in \mathbb {N}}\), \((b_{k})_{k\in \mathbb {N}}\), \((c_{k})_{k\in \mathbb {N}}\) and \((d_{k})_{k\in \mathbb {N}}\) satisfy recurrent relations (40) with initial conditions (31).
From (54) with \(k=n+1\) and by using (53) we get
Also the sequences \((a_{k})_{k\in \mathbb {N}}\), \((b_{k})_{k\in \mathbb {N}}\), \((c_{k})_{k\in \mathbb {N}}\) and \((d_{k})_{k\in \mathbb {N}}\) satisfy the difference equation (44) with initial conditions in (49), respectively.
As above the solvability of equation (44) shows that closed form formulas for \((a_{k})_{k\ge3}\), \((b_{k})_{k\ge3}\), \((c_{k})_{k\ge3}\) and \((d_{k})_{k\ge3}\) can be found. This fact along with (55) implies that equation (52) is solvable too. A direct calculation shows that the sequences \((z_{n})_{n\ge1}\) in (43) and \((w_{n})_{n\ge1}\) in (55) are solutions to system (4) with initial values \(z_{1}\), \(z_{0}\), that is, \(w_{1}\), \(w_{0}\) respectively. Hence, system (4) is also solvable in this case, finishing the proof of the theorem. □
Remark 1
Note that difference equation (44) is not only theoretically but also practically solvable since the characteristic polynomial
associated to the difference equation is of fourth order, which means that we can explicitly find its roots.
Remark 2
Since we are interested in those initial values \(z_{1}, z_{0}, w_{1}, w_{0}\in \mathbb {C}\) which uniquely define solutions to system (4), to avoid multivalued solutions to the system, we posed the condition \(a,b,c,d\in \mathbb {Z}\).
From the proof of Theorem 1 we obtain the following corollary.
Corollary 1
Consider system (4) with \(a,b,c,d\in \mathbb {Z}\). Assume that \(z_{1}, z_{0}, w_{1}, w_{0}\in \mathbb {C}\setminus\{0\}\). Then the following statements are true.

(a)
If \(b=0\) and \(a\ne c\), then the general solution to system (4) is given by (6) and (13).

(b)
If \(b=0\) and \(a=c\), then the general solution to system (4) is given by (6) and (15).

(c)
If \(d=0\) and \(a\ne c\), then the general solution to system (4) is given by (17) and (24).

(d)
If \(d=0\) and \(a=c\), then the general solution to system (4) is given by (17) and (26).

(e)
If \(bd\ne0\), then the general solution to system (4) is given by (43) and (55).
Let \(\lambda _{i}\), \(i=\overline {1,4}\), be the roots of the characteristic polynomial (56) of difference equation (44). If they satisfy the condition
then it is known that a general solution to equation (44) has the following form:
where \(\alpha _{i}\), \(i=\overline {1,4}\), are arbitrary constants. Since for the case \(d_{1}\ne0\) the solution can be prolonged for nonpositive indices, then we may assume that formula (57) holds also for \(n\ge3\) (or \(n\ge4\) if necessary).
In order to find, in this case, a general solution to system (4) in closed form, we will need the following known lemma. We give a proof of it for the completeness and benefit of the reader.
Lemma 1
Assume that \(\lambda _{j}\), \(j=\overline {1,k}\), are pairwise different zeros of the polynomial
Then
for \(l=\overline {0,k2}\), and
Proof
The functions
are meromorphic on the Riemann sphere. Hence, by the residue theorem, we have that
for every \(l\in \mathbb {N}\).
Now note that the Laurent expansion of \(f_{l}\) at zero is
for some complex numbers \(b_{s}\), \(s\in \mathbb {N}\).
On the other hand, since \(\lambda _{j}\), \(j=\overline {1,k}\), are simple poles of \(f_{l}\), we have that
From this and since \(\operatorname{Res}_{z=\infty}f_{l}(z)\) is equal to the negative value of the coefficient at \(1/z\) in the Laurent expansion, it follows that \(\operatorname{Res}_{z=\infty}f_{l}(z)=0\) when \(l=\overline {0,k2}\) and \(\operatorname{Res}_{z=\infty}f_{k1}(z)=1/\alpha _{k}\). Using these facts in (58) the lemma follows. □
If we apply Lemma 1 to polynomial \(p_{4}\) in (56), and since \(p_{4}(t)=\prod_{l=1}^{4}(t\lambda _{j})\) (note that \(\alpha _{4}=1\)), we have
for \(l=\overline {0,2}\), and
From this, since from (49) we have \(a_{3}=a_{2}=a_{1}=0\) and \(a_{0}=1\), and a general solution of (4) has the form in (57), we obtain
for \(n\ge3\).
On the other hand, from (40) we get
for \(n\ge3\).
By using (59) into (60) we get
for \(n\ge3\).
By using (59) and (63) into (61) we get
for \(n\ge3\).
By using (59) into (62) we get
for \(n\ge3\), where we have used the fact that (59) also holds for \(n=4\) (in fact, we may assume that equality (59) holds for every \(n\ges\), for any fixed \(s\in \mathbb {N}\), since due to the assumption \(d_{1}\ne0\), any solution of equation (44) can be prolonged for any nonpositive value of index n).
By using (59), (63), (64) and (65) into (43) and (55), we get formulas for general solutions to system (4) in closed form.
Formulas obtained in this section can be used in describing the longterm behavior of solutions to system (4) in many cases. We will formulate and prove here only one result, just as an example. The formulations and proofs of other results, which are similar and whose proofs use standard techniques, we leave to the reader as some exercises.
Theorem 2
Assume that \(b=c=0\) and \(a,d\in \mathbb {Z}\). Then the following statements hold:

(a)
If \(a=1\), then every solution to system (4) is eventually constant.

(b)
If \(a=0\), then \(z_{n}=w_{n}=1\), \(n\ge3\).

(c)
If \(a=1\), then every solution to system (4) is twoperiodic.

(d)
If \(a>1\) and \(z_{0}<1\), then \(z_{n}\to0\) as \(n\to\infty\).

(e)
If \(a>1\) and \(z_{0}>1\), then \(z_{n}\to\infty\) as \(n\to\infty\).

(f)
If \(a>1\) and \(z_{0}^{d}<1\), then \(w_{n}\to\infty\) as \(n\to\infty\).

(g)
If \(a>1\) and \(z_{0}^{d}>1\), then \(w_{n}\to0\) as \(n\to\infty\).

(h)
If \(a<1\) and \(z_{0}<1\), then \(z_{2n}\to0\) as \(n\to\infty\) and \(z_{2n+1}\to\infty\) as \(n\to\infty\).

(i)
If \(a<1\) and \(z_{0}>1\), then \(z_{2n+1}\to0\) as \(n\to\infty\) and \(z_{2n}\to\infty\) as \(n\to\infty\).

(j)
If \(a<1\) and \(z_{0}^{d}>1\), then \(w_{2n}\to0\) as \(n\to\infty\) and \(w_{2n+1}\to\infty\) as \(n\to\infty\).

(k)
If \(a<1\) and \(z_{0}^{d}<1\), then \(w_{2n+1}\to0\) as \(n\to\infty\) and \(w_{2n}\to\infty\) as \(n\to\infty\).
Proof
(a) If we replace \(a=1\) and \(c=0\) in (6) and (7), we obtain \(z_{n}=z_{0}\) and \(w_{n+2}=1/z_{0}^{d}\), \(n\in \mathbb {N}_{0}\), from which the statement follows.
(b) By replacing \(a=0\) and \(c=0\) in (6) and (7), we get \(z_{n}=1\), \(n\in \mathbb {N}\) and \(w_{n}=1\), \(n\ge3\), from which the statement follows.
(c) By replacing \(a=1\) and \(c=0\) in (6) and (7), we get \(z_{2n}=z_{0}\), \(z_{2n+1}=\frac{1}{z_{0}}\), \(w_{2n}=1/{z_{0}^{d}}\) and \(w_{2n+1}=z_{0}^{d}\), \(n\in \mathbb {N}\), from which the statement follows.
(d)(k) From (6) and (7) with \(c=0\) we get
Using the formulas in (66) all these statements easily follow. □
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Acknowledgements
The work of Stevo Stević is supported by the Serbian Ministry of Education and Science projects III 41025 and III 44006. The work of Bratislav Iričanin is supported by the Serbian Ministry of Education and Science projects III 41025 and OI 171007. The work of Zdeňek Šmarda was realized in CEITEC  Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 financed from European Regional Development Fund. He was also supported by the project FEKTS142200 of Brno University of Technology.
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Stević, S., Iričanin, B. & Šmarda, Z. On a producttype system of difference equations of second order solvable in closed form. J Inequal Appl 2015, 327 (2015). https://doi.org/10.1186/s1366001508359
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DOI: https://doi.org/10.1186/s1366001508359