Abstract
A solvable two-dimensional product-type system of difference equations of interest is presented. Closed form formulas for its general solution are given.
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1 Introduction
Concrete nonlinear difference equations and systems have become of some interest recently. Experts have proposed various classes of the equations and systems hoping that their studies will lead to some new general results or will bring about some new methods in the theory (see, e.g., [1–22]). Many of the papers study or are motivated by the study of symmetric systems (see, e.g., [4–7, 9, 10, 14, 17–22]). It turned out that some of the equations and systems can be solved, which motivated some experts to work on the topic (see, e.g., [1, 8, 11, 13–17, 19–22]; for some old results see, e.g., [23–25]). One of the motivations for the renewed interest in the area has been Stević’s method/idea for transforming some nonlinear equations into solvable linear ones (see, for example, [11, 13, 19, 20] and numerous related references therein). It also turned out that many classes of nonlinear difference equations and systems can be transformed to solvable ones by using some tricks and suitable changes of variables (see, e.g., [8, 13, 16, 19] and the related references therein).
Numerous recent equations and systems are closely related to product-type ones, which are solvable for the case of positive initial values (see, e.g., the equation in [12], which is a kind of perturbation of some product-type and the system in [18]; see also the related references therein). If the initial values are not positive, then there appear several problems. Thus, it is of some interest to describe the product-type systems with complex initial values which are solvable. A detailed study of the problem has been started recently by Stević et al. in [14, 15, 17, 21, 22] (some subclasses of the class of difference equations studied in [16] are also product-type ones). During the investigation we realized that the solvability of some product-type systems is preserved if some coefficients/multipliers are added. The first system of this type was studied in [14]. Based on this idea, quite recently in [22] it has been shown that the solvability of the system studied in [17] is preserved if two coefficients/multipliers are added. On the other hand, it can be seen that there are only several classes of product-type systems of difference equations which can be practically solved in closed form, due to the well-known fact that roots of the polynomials of degree \(d\ge5\) cannot be solved by radicals. Hence, it is of interest to find all the classes of practically solvable product-type systems of difference equations and present formulas for their solutions in terms of the initial values and parameters.
Here we present a new class of product-type systems of difference equations which are solvable under some natural assumptions. Namely, we investigate the solvability of the system
where \(a,b,c,d\in {\mathbb {Z}}\), \(\alpha ,\beta \in {\mathbb {C}}\) and \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\). It is interesting that none of the subclasses of the class in (1) has been previously treated in our papers on product-type systems, so that all the formulas presented here should be new. The formulas are obtained by further developing the methods in our previous papers, especially the ones in [14] and [22].
A solution to system (1) need not be defined if its initial values belong to the set
Thus, from now on we will assume that \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\). Since the cases \(\alpha =0\) and \(\beta =0\) are trivial or produce solutions which are not well defined we will also assume that \(\alpha \beta \ne0\).
Let us also note that we will use the convention \(\sum_{i=k}^{l}a_{i}=0\), when \(l< k\), throughout the paper.
2 Main results
The main results in this paper are proved in this section.
Theorem 1
Assume that \(b,c,d\in {\mathbb {Z}}\), \(a=0\), \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\), and \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\). Then system (1) is solvable in closed form.
Proof
Since \(a=0\) system (1) is
Using the first equation in (2) in the second one, we obtain
from which it follows that
and
Case \(bd=0\) . In this case equations (4) and (5) become
and
from which it follows that
Hence
when \(c\ne1\), and
when \(c=1\).
By using (8) and (9) in the first equation in (2) with \(n\to2n\) and \(n\to2n-1\), respectively, we get
Hence, from (14) and (15) we have
when \(c\ne1\), and
when \(c=1\).
Case \(bd\ne0\) . Let \(\gamma :=\beta \alpha ^{d}\)
Then (4) and (5) can be written as
and
By using (21) with \(n\to n-1\) into (21), we get
for \(n\ge2\), where
Assume that
for some \(k\ge2\) and every \(n\ge k\), where
Using (21) with \(n\to n-k\) into (25) we get
for every \(n\ge k+1\), where
Equalities (23), (24), (27), (28), along with the induction show that (25) and (26) hold for all natural numbers k and n such that \(2\le k\le n\). Moreover, because of (21), equality (25) holds for \(1\le k\le n\).
For \(n=k\), (25) becomes
Using the equalities \(w_{1}=\beta w_{-1}^{c}z_{-1}^{d}\), \(a_{n+1}=ca_{n}+b_{n}\), and \(x_{n+1}=x_{n}+a_{n}\) in (29) it follows that
Using (30) in the first equation in (2), we get
By using the same procedure it is proved that
for all natural numbers k and n such that \(1\le k\le n\), where \((a_{k})_{k\in {\mathbb {N}}}\), \((b_{k})_{k\in {\mathbb {N}}}\), \((x_{k})_{k\in {\mathbb {N}}}\) satisfy (20) and (26).
For \(n=k\), (32) becomes
Since \(w_{2}=\beta w_{0}^{c}z_{0}^{d}\), \(x_{n+1}=x_{n}+a_{n}\), and \(a_{n+1}=ca_{n}+b_{n}\), from (33) we have
Using (34) in the first equation in (2), we get
From the first two equations in (26) we have
From (36) and since \(b_{k}=b_{1}a_{k-1}\), we see that \((b_{k})_{k\in {\mathbb {N}}}\) is also a solution of (36).
From (26) with \(k=1\) one obtains
From this and since \(b_{1}=bd\ne0\), from the second equation in (37) we get \(a_{0}=1\), which along with the fact \(x_{1}=1\) and the other two relations in (37) implies \(b_{0}=x_{0}=0\).
This and (26) with \(k=0\) imply
which along with \(b_{1}\ne0\) and the second equation in (38) implies \(a_{-1}=0\). This along with the other two relations in (38) implies that we must have \(b_{-1}=1\) and \(x_{-1}=0\).
Hence \((a_{k})_{k\ge-1}\) and \((b_{k})_{k\ge-1}\) are solutions to (36) satisfying the (shifted) initial conditions
while \((x_{k})_{k\ge-1}\) satisfies the third equation in (26) and
From the third equation in (26) along with \(x_{1}=1\) and \(a_{0}=1\), we have
The characteristic equation associated to (36) is \(\lambda ^{2}-c\lambda -bd=0\), from which it follows that
are the corresponding characteristic roots.
If \(c^{2}+4bd\ne0\), then
which along with \(a_{-1}=0\) and \(a_{0}=1\) yields
From this and since \(b_{n}=b_{1}a_{n-1}\), we have
If \(c+bd\ne1\), which is equivalent to \(\lambda _{1}\ne1\ne \lambda _{2}\), from (41) and (42), it follows that
If \(c+bd=1\), that is, if one of the characteristic roots is one, say \(\lambda _{2}\), then \(\lambda _{1}=-bd\), so that
If \(c^{2}+4bd=0\), then
This along with \(a_{-1}=0\) and \(a_{0}=1\) yields
Using the relation \(b_{n}=b_{1}a_{n-1}\) along with the fact \(bd=-c^{2}/4\), we get
if \(c\ne2\). If \(c=2\), we obtain
completing the proof of the result. □
Corollary 1
Consider system (1) with \(b,c,d\in {\mathbb {Z}}\), \(a=0\), and \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\). Assume that \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\). Then the following statements are true.
-
(a)
If \(bd=0\) and \(c\ne1\), then the general solution to system (1) is given by (10), (11), (16), and (17).
-
(b)
If \(bd=0\) and \(c=1\), then the general solution to system (1) is given by (12), (13), (18), and (19).
-
(c)
If \(bd\ne0\), \(c^{2}+4bd\ne0\), and \(c+bd\ne1\), then the general solution to system (1) is given by (30), (31), (34), and (35), where the sequence \((a_{n})_{n\ge-1}\) is given by formula (42), while \((x_{n})_{n\ge-1}\) is given by (44).
-
(d)
If \(bd\ne0\), \(c^{2}+4bd\ne0\), and \(c+bd=1\), then the general solution to system (1) is given by (30), (31), (34), and (35), where the sequence \((a_{n})_{n\ge-1}\) is given by formula (42), while \((x_{n})_{n\ge-1}\) is given by (45).
-
(e)
If \(bd\ne0\), \(c^{2}+4bd=0\), and \(c\ne2\), then the general solution to system (1) is given by (30), (31), (34), and (35), where the sequence \((a_{n})_{n\ge-1}\) is given by formula (46), while \((x_{n})_{n\ge-1}\) is given by (48).
-
(f)
If \(bd\ne0\), \(c^{2}+4bd=0\), and \(c=2\), then the general solution to system (1) is given by (30), (31), (34), and (35), where the sequence \((a_{n})_{n\ge-1}\) is given by formula (46) with \(c=2\), while \((x_{n})_{n\ge-1}\) is given by (49).
Theorem 2
Assume that \(a,c,d\in {\mathbb {Z}}\), \(b=0\), \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\), and \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\). Then system (1) is solvable in closed form.
Proof
Since \(b=0\), we have
From the first equation in (50) we get
Hence, if \(a\ne1\), we have
while if \(a=1\),
Using (51) in the second equation in (50), it follows that
Using (54) twice, we get
for every \(n\ge3\), and
Assume that, for a natural number k, it has been proved that
for \(n\ge k+1\), and
for every \(n\ge k\).
Using (54) with \(n\to2n-2k-1\) and \(n\to2n-2k\), in (57) and (58), we obtain
for \(n\ge k+2\), and
for every \(n\ge k+1\).
From (55), (56), (59), (60), and the induction it follows that (57) holds for all natural numbers k and n such that \(1\le k\le n-1\), while (58) holds for all k and n such that \(1\le k\le n\).
By taking \(k=n-1\) in (57), we get
By using the relation \(w_{2}=\beta w_{0}^{c}z_{0}^{d}\) in (61) we get
By taking \(k=n\) in (58), and using the relation \(w_{1}=\beta w_{-1}^{c}z_{-1}^{d}\), we get
for \(n\in {\mathbb {N}}\). It is also easy to check that (63) holds also for \(n=0\) when \(c\ne0\).
Subcase \(a\ne1\ne c\) , \(c\ne a^{2}\) . In this case we have
for \(n\in {\mathbb {N}}\), and
for every \(n\in {\mathbb {N}}\).
Subcase \(a\ne1\ne c\) , \(c=a^{2}\) . In this case we have
for every \(n\in {\mathbb {N}}\).
Subcase \(a^{2}\ne1=c\) . In this case we have
for every \(n\in {\mathbb {N}}\).
Subcase \(a=-1\) , \(c=1\) . In this case we have
for every \(n\in {\mathbb {N}}_{0}\).
Subcase \(a=1\) , \(c\ne1\) . In this case we have
for every \(n\in {\mathbb {N}}\).
Subcase \(a=c=1\) . In this case we have
for every \(n\in {\mathbb {N}}\). □
Corollary 2
Consider system (1) with \(a,c,d\in {\mathbb {Z}}\), \(b=0\), and \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\). Assume that \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\). Then the following statements are true.
-
(a)
If \(a\ne1\ne c\) and \(c\ne a^{2}\), then the general solution to system (1) is given by (52), (64), and (65).
-
(b)
If \(a\ne1\ne c\) and \(c=a^{2}\ne0\), then the general solution to system (1) is given by (52), (66), and (67).
-
(c)
If \(a^{2}\ne1=c\), then the general solution to system (1) is given by (52), (68), and (69).
-
(d)
If \(a=-1\) and \(c=1\), then the general solution to system (1) is given by (52), (70), and (71).
-
(e)
If \(a=1\) and \(c\ne1\), then the general solution to system (1) is given by (53), (72), and (73).
-
(f)
If \(a=c=1\), then the general solution to system (1) is given by (53), (74), and (75).
Theorem 3
Assume that \(a,b,c\in {\mathbb {Z}}\), \(d=0\), \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\), and \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\). Then system (1) is solvable in closed form.
Proof
In this case system (1) becomes
From the second equation in (76) it easily follows that
which, for the case \(c\ne1\), implies that
and
while, for the case \(c=1\), we have
and
Employing (77) in the first equation in (76) we obtain
Combining (82) and (83) it follows that
for \(n\ge2\), and
Assume that, for some natural number k we have proved that
for \(n\ge k+1\) and
for every \(n\ge k\).
By using (84) with \(n\to n-k\) into (86), and (85) with \(n\to n-k\) into (87), it follows that
for \(n\ge k+2\) and
for every \(n\ge k+1\).
From the equalities in (84), (85), (88), (89), and by induction we see that (86) holds for all natural numbers k and n such that \(1\le k\le n-1\), while (87) holds for \(1\le k\le n\).
If we choose \(k=n-1\) in (86) and \(k=n\) in (87) we get
for every \(n\in {\mathbb {N}}\), and
for every \(n\in {\mathbb {N}}\).
Subcase \(c\ne a^{2}\ne1\ne c\) . In this case we have
for every \(n\in {\mathbb {N}}\).
Subcase \(a^{2}\ne1\ne c\) , \(c=a^{2}\) . In this case we have
for every \(n\in {\mathbb {N}}\).
Subcase \(a^{2}\ne1=c\) . In this case we have
for every \(n\ge-1\).
Subcase \(a=-1\) , \(c=1\) . In this case we have
for every \(n\in {\mathbb {N}}\).
Subcase \(a=1\ne c\) . In this case we have
for \(n\in {\mathbb {N}}\).
Subcase \(a=c=1\) . In this case we have
for every \(n\in {\mathbb {N}}\), completing the proof. □
Corollary 3
Consider system (1) with \(a,b,c\in {\mathbb {Z}}\), \(d=0\), and \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\). Assume that \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\). Then the following statements are true.
-
(a)
If \(c\ne a^{2}\ne1\ne c\), then the general solution to system (1) is given by (78), (79), (92), and (93).
-
(b)
If \(c=a^{2}\ne1\ne c\), then the general solution to system (1) is given by (78), (79), (94), and (95).
-
(c)
If \(a^{2}\ne1=c\), then the general solution to system (1) is given by (80), (81), (96), and (97).
-
(d)
If \(a=-1\) and \(c=1\), then the general solution to system (1) is given by (80), (81), (98), and (99).
-
(e)
If \(a=1\) and \(c\ne1\), then the general solution to system (1) is given by (78), (79), (100), and (101).
-
(f)
If \(a=c=1\), then the general solution to system (1) is given by (80), (81), (102), and (103).
Theorem 4
Assume that \(a,b,c,d\in {\mathbb {Z}}\), \(bd\ne0\), \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\), and \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\). Then system (1) is solvable in closed form.
Proof
First note that the conditions \(\alpha ,\beta \in {\mathbb {C}}\setminus\{ 0\}\) and \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\) along with the equations in (1) imply \(z_{n}w_{n}\ne0\) for \(n\ge-1\). Hence, for every such a solution the first equation in (1) yields
while from the second one it follows that
From (104) and (105) one obtains
which is a fourth order product-type difference equation.
Note also that
Let \(\delta =\alpha ^{1-c}\beta ^{b}\),
Then equation (106) can be written as
Using (109) with \(n\to n-1\) into (109) we get
for \(n\in {\mathbb {N}}\), where
Assume that, for a k such that \(2\le k \le n+1\), we have proved that
for \(n\ge k-1\), and that
Using (109) with \(n\to n-k\) into (110) one obtains
for \(n\ge k\), where
This along with (110), (111), and the method of induction shows that (112), (113), and (114), hold for every k and n such that \(2\le k\le n+1\). In fact (112) holds for \(1\le k\le n+1\) (see (109)).
Hence, choosing \(k=n+1\) in (112), and using (107) we have
From (113) we easily see that \((a_{k})_{k\ge4}\) satisfies the difference equation
Since \(b_{k}=a_{k+1}-a_{1}a_{k}\), \(c_{k}=b_{k+1}-b_{1}a_{k}\), \(d_{k}=d_{1}a_{k-1}\), and from the linearity of equation (119) we see that \((b_{k})_{k\in {\mathbb {N}}}\), \((c_{k})_{k\in {\mathbb {N}}}\), and \((d_{k})_{k\in {\mathbb {N}}}\) are also solutions to the equation.
System (113) with \(k=1\) yields
The condition \(d_{1}=bd\ne0\) along with the fourth equation in (120) implies \(a_{0}=1\). Using this and \(y_{1}=1\) in the other equalities in (120) we get \(b_{0}=c_{0}=d_{0}=y_{0}=0\). Repeating the procedure for \(k=0,-1, -2\), is easily obtained
Hence, \((a_{k})_{k\ge-3}\), \((b_{k})_{k\ge-3}\), \((c_{k})_{k\ge-3}\), and \((d_{k})_{k\ge-3}\) are solutions to (119) satisfying initial conditions (121), while \((y_{k})_{k\ge-3}\) satisfies the following conditions:
and (114), from which it follows that
Since equation (119) is solvable, it follows that closed form formulas for \((a_{k})_{k\ge-3}\), \((b_{k})_{k\ge-3}\), \((c_{k})_{k\ge-3}\), and \((d_{k})_{k\ge-3}\), can be found. From (123), the form of the solution \(a_{k}\), and by using some known summation formulas it follows that the formula for \((y_{k})_{k\ge-3}\) can also be found. From these facts and (118) we see that equation (106) is solvable too.
From the second equation in (1), we have that for every well-defined solution
while from the first one it follows that
From (124) into (125) one obtains
which differs from (106) only by the constant multiplier.
We have
As above one obtains, for all natural numbers k and n such that \(1\le k \le n+1\),
where \(\eta=\alpha ^{d}\beta ^{1-a}\), \((a_{k})_{k\in {\mathbb {N}}}\), \((b_{k})_{k\in {\mathbb {N}}}\), \((c_{k})_{k\in {\mathbb {N}}}\), and \((d_{k})_{k\in {\mathbb {N}}}\) satisfy (113) with initial conditions (108), while \((\hat{y}_{k})_{k\in {\mathbb {N}}}\) satisfies (114) and (122), so that (123) holds where \(y_{k}\) is replaced by \(\hat{y}_{k}\).
From (128) with \(k=n+1\) and by using (127) we get
for \(n\in {\mathbb {N}}_{0}\).
As above the solvability of (119) shows that formulas for \((a_{k})_{k\ge-3}\), \((b_{k})_{k\ge-3}\), \((c_{k})_{k\ge-3}\), and \((d_{k})_{k\ge-3}\) can be found, and consequently a formula for \((\hat{y}_{k})_{k\ge-3}\). This fact along with (129) implies that equation (126) is solvable too. Hence, system (1) is also solvable in this case, as desired. □
Corollary 4
Consider system (1) with \(a,b,c,d\in {\mathbb {Z}}\), \(bd\ne0\), \(\alpha ,\beta \in {\mathbb {C}}\setminus\{0\}\). Assume that \(z_{-1}, z_{0}, w_{-1}, w_{0}\in {\mathbb {C}}\setminus\{0\}\). Then the general solution to system (1) is given by (118) and (129), where 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 (119) with initial conditions in (121), while \((y_{k})_{k\in {\mathbb {N}}}\) and \((\hat{y}_{k})_{k\in {\mathbb {N}}}\) are given by (123) and satisfy conditions (122).
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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 Zdeněk Šmarda is supported by the project FEKT-S-14-2200 of the Brno University of Technology. Some results in the paper are obtained during Bratislav Iričanin’s visit of Faculty of Electrical Engineering and Communication at the Brno University of Technology.
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Stević, S., Iričanin, B. & Šmarda, Z. Two-dimensional product-type system of difference equations solvable in closed form. Adv Differ Equ 2016, 253 (2016). https://doi.org/10.1186/s13662-016-0980-6
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DOI: https://doi.org/10.1186/s13662-016-0980-6