New class of solvable systems of difference equations

Accepted Manuscript New class of solvable systems of difference equations Stevo Stevi´c PII: DOI: Reference:

S0893-9659(16)30219-1 http://dx.doi.org/10.1016/j.aml.2016.07.025 AML 5060

To appear in:

Applied Mathematics Letters

Received date: 8 June 2016 Revised date: 24 July 2016 Accepted date: 24 July 2016 Please cite this article as: S. Stevi´c, New class of solvable systems of difference equations, Appl. Math. Lett. (2016), http://dx.doi.org/10.1016/j.aml.2016.07.025 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

*Manuscript Click here to view linked References

NEW CLASS OF SOLVABLE SYSTEMS OF DIFFERENCE EQUATIONS ´ STEVO STEVIC Abstract. We present a new class of solvable systems of difference equations of interest by describing a method for finding its general solution.

1. Introduction Concrete difference equations and systems have attracted a great recent interest (see, e.g., [1–3,5,7,10–30]). One of the reasons for this is that many real life problems are modeled by the equations/systems (see, e.g., the equations and systems in [3,16]), so that any new technique, method or formula can give an insight in better understanding of the behavior of their solutions. Considerable interest in studying concrete systems of difference equations started after the publication of papers [10–12] by Papaschinopoulos and Schinas (see, e.g., [2, 7, 14, 17, 19, 20, 22–29]). One of the classical problems in the theory is the solvability of the equations and systems. Some known classes of difference equations and systems, including solvable ones, can be found, e.g., in [4, 6, 8, 9]. The publication of note [15], as well as [16], in which was shown that a biological model is solvable, has renewed an interest in the topic and their applications (see, e.g., [1–3, 13, 17–30]). The transformation method in [15] was later developed in a series of papers (see, e.g., [13, 18, 21, 30]). For some results on the corresponding systems, see, [17, 19, 24, 28], while [2, 20, 29] studies some systems related to the equation in [16]. For solvable max-type or product-type systems see, e.g., [22, 26, 27]. One of interesting nonlinear solvable two-dimensional systems of difference equations, related to known, highly applicable, bilinear one ([4, 9, 25]), is the following β2 β3 β4 β1 + , yn+1 = + , n ∈ N0 , (1) xn+1 = xn yn xn yn where βi , 1 ≤ i ≤ 4, x0 , y0 , are real or complex numbers (see, e.g., [23]). A natural question is to find a three-dimensional relative of (1) solvable in closed form. Here, we present such a system. Namely, we consider the following system b1 c1 a1 + + xn+1 = xn yn yn zn zn xn a2 b2 c2 yn+1 = + + (2) xn yn yn zn zn xn a3 b3 c3 zn+1 = + + , xn yn yn zn zn xn for n ∈ N0 , ai , bi , ci , x0 , y0 , z0 ∈ R (or C), i = 1, 2, 3, and show that it is solvable. 2000 Mathematics Subject Classification. Primary 39A10. Key words and phrases. System of difference equations, solvable system. 1

´ STEVO STEVIC

2

2. Main result In this section we formulate and prove the main result in the paper. Theorem 1. Assume that ai , bi , ci , i = 1, 2, 3, x0 , y0 , z0 are real or complex numbers. Then system of difference equations (2) is solvable in closed form. Proof. If ai = bi = ci = 0, i = 1, 2, 3, then from (2) we have xn = yn = zn = 0, n ∈ N, which are formulas for solutions to system (2) in this case. Hence, from now on we assume that at least one of the numbers ai , bi , ci , i = 1, 2, 3, is different from zero. In this case, a solution (xn , yn , zn )n∈N0 to system (2) is well-defined if and only if xn yn zn 6= 0 for every n ∈ N0 , what will be also assumed from now on, and the rank of the matrix   a1 b 1 c1 A :=  a2 b2 c2  , a3 b 3 c3

i.e., rank (A), can be equal to one, two or three, so we have three cases to consider. Case rank (A) = 1. In this case two rows in matrix A are linearly dependent on the third one. Without loss of generality we may assume that (ai , bi , ci ) = di (a1 , b1 , c1 ),

i = 2, 3,

(3)

for some numbers di , i = 2, 3 (the other two cases are treated similarly). Using (3) in the second and third equation in system (2) we get     a1 b1 c1 b1 c1 a1 and zn+1 = d3 , + + + + yn+1 = d2 xn yn yn zn zn xn xn yn yn zn zn xn for n ∈ N0 , which along with the first equation in (2) implies yn = d2 xn

and

zn = d3 xn ,

(4)

n ∈ N.

Using the equations in (4) into the first equation in (2) is obtained xn+1 = d/x2n , n ∈ N, where d := ad12 + d2b1d3 + dc13 , from which it follows that xn+1 =

x4n−1 , d

n ≥ 2.

(5)

From (5) it follows that 4n−1 −1  j  1 Pn−2  1  4n−1 2n−1 a1 b1 c1 j=0 4 3 4n−1 x2n = x2 = + + (x0 y0 z0 )2 , (6) d d α1 α2 α2 α3 α1 α3 n ∈ N, where αi = ai z0 + bi x0 + ci y0 , i = 1, 2, 3, and j  1  4n3−1  a z + b x + c y 4n  1 Pn−1 j=0 4 1 0 1 0 1 0 4n x1 = , x2n+1 = d d x0 y0 z0

n ∈ N0 .

(7)

Using (6) and (7) into (4) we get formulas for y2n , y2n+1 , z2n and z2n+1 . Case rank (A) = 2. In this case a row in matrix A linearly depends on the other two ones. Hence, we may assume that (a3 , b3 , c3 ) = d4 (a1 , b1 , c1 ) + d5 (a2 , b2 , c2 ),

(8)

for some numbers di , i = 4, 5 (the other two cases are treated similarly). From (2) and (8) it follows that zn = d4 xn + d5 yn ,

n ∈ N.

(9)

NEW CLASS OF SOLVABLE SYSTEMS OF DIFFERENCE EQUATIONS

3

The fact xn yn zn 6= 0, n ∈ N0 , along with the following consequence of (2) b2 xn + c2 yn + a2 zn b1 xn + c1 yn + a1 zn , yn+1 = , xn+1 = xn yn zn xn yn zn (10) b3 xn + c3 yn + a3 zn zn+1 = , n ∈ N0 , xn yn zn implies bi xn + ci yn + ai zn 6= 0, n ∈ N0 , i = 1, 2, 3. From the first two equations in (10) it follows that b1 xn + c1 yn + a1 zn xn+1 = , n ∈ N0 . (11) yn+1 b2 xn + c2 yn + a2 zn By using (9) into (11) we get xn+1 (b1 + a1 d4 )xn + (c1 + a1 d5 )yn = , yn+1 (b2 + a2 d4 )xn + (c2 + a2 d5 )yn

(12)

n ∈ N.

Let wn = xn /yn ,

(13)

n ∈ N.

Then (12) becomes the next bilinear difference equation with constant coefficients wn+1 =

(b1 + a1 d4 )wn + c1 + a1 d5 , (b2 + a2 d4 )wn + c2 + a2 d5

(14)

n ∈ N,

with the initial condition w1 = x1 /y1 , which can be easily solved (see, e.g., [4,8,9]). After finding formula for wn , we use relation (13) in (9) and get zn = (d4 wn + d5 )yn ,

(15)

n ∈ N.

Now, using (13) and (15) into the second equation in (2) we get Bn yn+1 = 2 , n ∈ N. yn where Bn :=

a2 wn

+

b2 d4 wn +d5

+

c2 wn (d4 wn +d5 ) ,

yn+1 =

and

from which it follows that 4n−j n  Y n−1 B2j−1 y2n = y24 2 B2j−2 j=2 

b2 c2 a2 + + = α1 α2 α2 α3 α1 α3 4 n  Y B2j = 2 B2j−1 j=1

n−j

y2n+1

n y14

from which it follows that

Bn 4 yn−1 , 2 Bn−1

From (17) we have that B2n+1 4 y2n+2 = y2n 2 B2n

=

4n−1 

(16)

n ≥ 2.

y2n+1 =

(17)

B2n 4 y2n−1 , 2 B2n−1

n ∈ N,

4 n  Y B2j−1

n−j

22n−1

(x0 y0 z0 )

j=2

a2 z0 + b2 x0 + c2 y0 x0 y0 z0

2 B2j−2

,

(18)

4n Y 4n−j n  B2j , (19) 2 B2j−1 j=1

for n ∈ N, where αi = ai z0 + bi x0 + ci y0 , i = 1, 2, 3. By using (18) and (19) into (13) and (15) we get formulas for x2n , x2n+1 , z2n and z2n+1 .

´ STEVO STEVIC

4

Case rank(A) = 3. From (10) we have that for every well-defined solution to (2) yn+1 zn+1 xn+1 = = , n ∈ N0 . (20) b1 xn + c1 yn + a1 zn b2 xn + c2 yn + a2 zn b3 xn + c3 yn + a3 zn Hence, for every p, q, r such that (p, q, r) 6= (0, 0, 0), we have that

pxn+1 + qyn+1 + rzn+1 (b1 p + b2 q + b3 r)xn + (c1 p + c2 q + c3 r)yn + (a1 p + a2 q + a3 r)zn xn+1 yn+1 zn+1 = = = , b1 xn + c1 yn + a1 zn b2 xn + c2 yn + a2 zn b3 xn + c3 yn + a3 zn

(21)

for n ∈ N0 . Now we find those λ-s such that the following linear system b1 p + b2 q + b3 r = λp c1 p + c2 q + c3 r = λq

(22)

a1 p + a2 q + a3 r = λr, has a nontrivial solution in variables p, q and r. Note that the matrix appearing on b has also rank equal to three. the left-hand sides of equalities (22), denoted by A, Since system (22) is homogeneous it has a nontrivial solution if and only if the determinant of the system is equal to zero, i.e., when the following polynomial b1 − λ b2 b3 , c2 − λ c3 (23) P3 (λ) = c1 a1 a2 a3 − λ

is equal to zero. Let λi , i = 1, 2, 3, be the zeros of P3 (λi 6= 0, i = 1, 2, 3, since P3 (0) 6= 0 due to b = 3). If for these λi -s system (22) has three linearly independent the fact rank(A) nontrivial solutions (pi , qi , ri ), i = 1, 2, 3, then from (21) we have p2 xn+1 + q2 yn+1 + r2 zn+1 p1 xn+1 + q1 yn+1 + r1 zn+1 = λ1 (p1 xn + q1 yn + r1 zn ) λ2 (p2 xn + q2 yn + r2 zn ) p3 xn+1 + q3 yn+1 + r3 zn+1 , n ∈ N0 . = λ3 (p3 xn + q3 yn + r3 zn )

Hence, from (24) we have pi xn+1 + qi yn+1 + ri zn+1 λi = p3 xn+1 + q3 yn+1 + r3 zn+1 λ3 for i = 1, 2, and consequently pi xn + qi yn + ri zn = p3 xn + q3 yn + r3 zn



λi λ3



 pi xn + qi yn + ri zn , p3 xn + q3 yn + r3 zn

n

pi x0 + qi y0 + ri z0 , p3 x0 + q3 y0 + r3 z0

n ∈ N0 ,

n ∈ N0 .

Let un = xn /zn and vn = yn /zn , n ∈ N0 , then (26) can be written as  n λi pi un + qi vn + ri = ai3 (p3 un + q3 vn + r3 ), n ∈ N0 , λ3

(24)

(25)

(26)

(27)

+qi y0 +ri z0 , i = 1, 2, that is, where ai3 := pp3ixx00+q 3 y0 +r3 z0  n    n   n  λi λi λi pi −ai3 p3 un + qi −ai3 q3 vn = ai3 r3 −ri , n ∈ N0 . (28) λ3 λ3 λ3

NEW CLASS OF SOLVABLE SYSTEMS OF DIFFERENCE EQUATIONS

5

For each fixed n ∈ N0 , (28) is a two-dimensional linear system whose solution is a r λ1
n − r q − a q λ1
n 1 1 13 3 λ3 13 3 λ3

n n a23 r3 λλ32 − r2 q2 − a23 q3 λλ23 (29) un :=

n n λ q1 − a13 q3 λλ13 p1 − a13 p3 λ13

n n p2 − a23 p3 λλ32 q2 − a23 q3 λλ23

and

vn :=


λ1 n λ3
λ2 n λ3
n a13 p3 λλ31
n a23 p3 λλ32


λ1 n λ3
λ2 n λ3

p1 − a13 p3 p2 − a23 p3

a13 r3 a23 r3

p1 − p2 −

q1 − a13 q3 q2 − a23 q3

− r1 − r2
λ1 n λ3
λ2 n λ3

.

(30)

From (29) and (30) and since xn = un zn and yn = vn zn , it follows that xn =

α1 λn1 + β1 λn2 + γ1 λn3 zn δ1 λn1 + η1 λn2 + ν1 λn3

and

yn =

α2 λn1 + β2 λn2 + γ2 λn3 zn , δ2 λn1 + η2 λn2 + ν2 λn3

(31)

for some αi , βi , γi , δi , ηi , νi , i = 1, 2, and for every n ∈ N0 . Using (31) into the third equation in (2) we get zn+1 =

An , zn2

(32)

n ∈ N0 ,

where An :=

(δ1 λn1 + η1 λn2 + ν1 λn3 )(δ2 λn1 + η2 λn2 + ν2 λn3 ) b3 c3 a3 + + = a3 (33) un vn vn un (α1 λn1 + β1 λn2 + γ1 λn3 )(α2 λn1 + β2 λn2 + γ2 λn3 ) δ2 λn1 + η2 λn2 + ν2 λn3 δ1 λn1 + η1 λn2 + ν1 λn3 + b3 + c3 , n n n α2 λ1 + β2 λ2 + γ2 λ3 α1 λn1 + β1 λn2 + γ1 λn3

and consequently z2n =

A2n−1 4 z A22n−2 2n−2

and

From (34) it follows that 4n−j n  Y n A2j−1 z2n = z04 , 2 A 2j−2 j=1 4 n  Y A2j A22j−1 j=1

n−j

z2n+1 =

n

z14 =

z2n+1 =

A2n 4 z , A22n−1 2n−1

(34)

n ∈ N.

(35) 

a3 z0 + b3 x0 + c3 y0 x0 y0 z0

Y n  j=1

A2j A22j−1

4n−j

,

(36)

for n ∈ N. Using (35) and (36) into (31), we get the closed form formulas for x2n , x2n+1 , y2n and y2n+1 , in this case. If for these λi -s, (22) has two linearly independent nontrivial solutions (pi , qi , ri ), i = 1, 3, which happens when two of the λi -s are equal and different from the third b corresponding to λ3 one, say, if λ1 6= λ2 = λ3 , and the Jordan block of matrix A has 1 on the superdiagonal, then there are also p2 , q2 and r2 such that

´ STEVO STEVIC

6

p1 xn+1 + q1 yn+1 + r1 zn+1 p2 xn+1 + q2 yn+1 + r2 zn+1 = λ1 (p1 xn + q1 yn + r1 zn ) λ3 (p2 xn + q2 yn + r2 zn ) + p3 xn + q3 yn + r3 zn p3 xn+1 + q3 yn+1 + r3 zn+1 = , n ∈ N0 . (37) λ3 (p3 xn + q3 yn + r3 zn ) From (37) we see that (27) holds for i = 1, and that the second equality implies p2 xn−1 + q2 yn−1 + r2 zn−1 1 p2 xn + q2 yn + r2 zn = + , n ∈ N, (38) p3 xn + q3 yn + r3 zn p3 xn−1 + q3 yn−1 + r3 zn−1 λ3 from which it follows that p2 un + q2 vn + r2 n p2 xn + q2 yn + r2 zn = = a23 + , n ∈ N0 , (39) p3 xn + q3 yn + r3 zn p3 un + q3 vn + r3 λ3 that is,           n n n p2 − a23 + p3 un + q2 − a23 + q3 vn = a23 + r3 −r2 , n ∈ N0 . (40) λ3 λ3 λ3

For each fixed n ∈ N0 , (27) for i = 1 along with (40) makes a two-dimensional linear system from which un and vn can be found. By using the relations xn = un zn and yn = vn zn into the third equation in (2) we get that (32) and (33) hold with different un and vn , from which it follows that (35) and (36) hold with the corresponding An -s, and consequently are obtained formulas for x2n , x2n+1 , y2n and y2n+1 . If for these λi -s, (22) has only one linearly independent nontrivial solution (p1 , q1 , r1 ), which happens when all λi -s are equal and the Jordan form of matrix b has units on the superdiagonal, then there are pi , qi , ri , i = 1, 2, such that A p1 xn+1 + q1 yn+1 + r1 zn+1 = λ3 (p1 xn + q1 yn + r1 zn ) + p2 xn + q2 yn + r2 zn p3 xn+1 + q3 yn+1 + r3 zn+1 p2 xn+1 + q2 yn+1 + r2 zn+1 = , (41) λ3 (p2 xn + q2 yn + r2 zn ) + p3 xn + q3 yn + r3 zn λ3 (p3 xn + q3 yn + r3 zn )

n ∈ N0 . Then (39) and (40) hold. Using (39) in the first equation in (41) is obtained   a23 n (n − 1)n p1 xn + q1 yn + r1 zn = a13 + (p3 xn + q3 yn + r3 zn ), n ∈ N0 . (42) + λ3 2λ23 + (n−1)n , n ∈ N. Then from (42) we easily get 2λ23

p1 − tn p3 un + q1 − tn q3 vn = tn r3 − r1 , n ∈ N0 .

Let tn := a13 +

a23 n λ3

(43)

For each fixed n ∈ N0 , (40) along with (43) makes a two-dimensional linear system from which un and vn can be found. By using the relations xn = un zn and yn = vn zn into the third equation in (2) we get that (32) and (33) hold with different un and vn , from which it follows that (35) and (36) hold with the corresponding An -s, and consequently are obtained formulas for x2n , x2n+1 , y2n and y2n+1 .  References [1] A. Andruch-Sobilo and M. Migda, Further properties of the rational recursive sequence xn+1 = axn−1 /(b + cxn xn−1 ), Opuscula Math. 26 (3) (2006), 387-394. [2] L. Berg and S. Stevi´ c, On some systems of difference equations, Appl. Math. Comput. 218 (2011), 1713-1718. [3] L. Berezansky and E. Braverman, On impulsive Beverton-Holt difference equations and their applications, J. Differ. Equations Appl. 10 (9) (2004), 851-868.

NEW CLASS OF SOLVABLE SYSTEMS OF DIFFERENCE EQUATIONS

7

[4] L. Brand, A sequence defined by a difference equation, Amer. Math. Monthly 62 (7) (1955), 489-492. [5] B. Iriˇ canin and S. Stevi´ c, Eventually constant solutions of a rational difference equation, Appl. Math. Comput. 215 (2009), 854-856. [6] C. Jordan, Calculus of Finite Differences, Chelsea Publishing Company, New York, 1956. [7] G. Karakostas, The dynamics of a cooperative difference system with coefficient a Metzler matrix, J. Differ. Equations Appl. 20 (5-6) (2014), 685-693. [8] H. Levy and F. Lessman, Finite Difference Equations, Dover Pub. Inc., New York, 1992. [9] D. S. Mitrinovi´ c and J. D. Keˇ cki´ c, Methods for Calculating Finite Sums, Nauˇ cna Knjiga, Beograd, 1984 (in Serbian). [10] G. Papaschinopoulos and C. J. Schinas, On a system of two nonlinear difference equations, J. Math. Anal. Appl. 219 (2) (1998), 415-426. [11] G. Papaschinopoulos and C. J. Schinas, On the behavior of the solutions of a system of two nonlinear difference equations, Comm. Appl. Nonlinear Anal. 5 (2) (1998), 47-59. [12] G. Papaschinopoulos and C. J. Schinas, Invariants for systems of two nonlinear difference equations, Differential Equations Dynam. Systems 7 (2) (1999), 181-196. [13] G. Papaschinopoulos and G. Stefanidou, Asymptotic behavior of the solutions of a class of rational difference equations, Inter. J. Difference Equations 5 (2) (2010), 233-249. [14] G. Stefanidou, G. Papaschinopoulos, and C. J. Schinas, On a system of two exponential type difference equations, Commun. Appl. Nonlinear Anal. 17 (2) (2010), 1-13. [15] S. Stevi´ c, More on a rational recurrence relation, Appl. Math. E-Notes 4 (2004), 80-85. [16] S. Stevi´ c, A short proof of the Cushing-Henson conjecture, Discrete Dyn. Nat. Soc. Vol. 2006, Article ID 37264, (2006), 5 pages. [17] S. Stevi´ c, On a system of difference equations, Appl. Math. Comput. 218 (2011), 3372-3378. [18] S. Stevi´ c, On the difference equation xn = xn−2 /(bn + cn xn−1 xn−2 ), Appl. Math. Comput. 218 (2011), 4507-4513. [19] S. Stevi´ c, On a third-order system of difference equations, Appl. Math. Comput. 218 (2012), 7649-7654. [20] S. Stevi´ c, On some solvable systems of difference equations, Appl. Math. Comput. 218 (2012), 5010-5018. [21] S. Stevi´ c, On the difference equation xn = xn−k /(b+cxn−1 · · · xn−k ), Appl. Math. Comput. 218 (2012), 6291-6296. [22] S. Stevi´ c, Solutions of a max-type system of difference equations, Appl. Math. Comput. 218 (2012), 9825-9830. [23] S. Stevi´ c, On a system of difference equations which can be solved in closed form, Appl. Math. Comput. 219 (2013), 9223-9228. [24] S. Stevi´ c, On the system of difference equations xn = cn yn−3 /(an + bn yn−1 xn−2 yn−3 ), yn = γn xn−3 /(αn + βn xn−1 yn−2 xn−3 ), Appl. Math. Comput. 219 (2013), 4755-4764. [25] S. Stevi´ c, Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences, Electron. J. Qual. Theory Differ. Equ. 2014, Art. no. 67, (2014), 15pp. [26] S. Stevi´ c, Product-type system of difference equations of second-order solvable in closed form, Electron. J. Qual. Theory Differ. Equ. Vol. 2015, Article No. 56, (2015), 16 pages. [27] S. Stevi´ c, M. A. Alghamdi, A. Alotaibi and E. M. Elsayed, Solvable product-type system of difference equations of second order, Electron. J. Differential Equations Vol. 2015, Article No. 169, (2015), 20 pages. ˇ [28] S. Stevi´ c, J. Diblik, B. Iriˇ canin and Z. Smarda, On a third-order system of difference equations with variable coefficients, Abstr. Appl. Anal. Vol. 2012, Article ID 508523, (2012), 22pp. ˇ [29] S. Stevi´ c, J. Diblik, B. Iriˇ canin and Z. Smarda, On a solvable system of rational difference equations, J. Difference Equ. Appl. 20 (5-6) (2014), 811-825. ˇ [30] S. Stevi´ c, J. Diblik, B. Iriˇ canin and Z. Smarda, Solvability of nonlinear difference equations of fourth order, Electron. J. Differential Equations Vol. 2014, Article No. 264, (2014), 14pp. ´, Mathematical Institute of the Serbian Academy of Sciences, Knez Stevo Stevic Mihailova 36/III, 11000 Beograd, Serbia Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia E-mail address: [email protected]