On a product-type system of difference equations of second order solvable in closed form

Abstract

(ProQuest: ... denotes formulae and/or non-USASCII text omitted; see image)

It is shown that the following system of difference equations ...... where ......, ......, is solvable in closed form.

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 DOI 10.1186/s13660-015-0835-9

R E S E A R C H Open Access

On a product-type system of difference equations of second order solvable in closed form

http://crossmark.crossref.org/dialog/?doi=10.1186/s13660-015-0835-9&domain=pdf

Web End = Stevo Stevi1,2*, Bratislav Irianin3 and Zdenk marda4,5

*Correspondence: [email protected]

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia

2Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi ArabiaFull list of author information is available at the end of the article

Abstract

It is shown that the following system of dierence equations

zn+1 = zan wbn1

, wn+1 = wcn zdn1

, n

N0,

where a, b, c, d

C, is solvable in closed form.

MSC: Primary 39A10; 39A20Keywords: system of dierence equations; second order system; product-type system; solvable in closed form

1 Introduction

Recently there has been a great interest in studying nonlinear dierence equations and systems not stemming from dierential ones (see, e.g., []). The old area of solving dierence equations and systems has re-attracted recent attention (see, e.g., [, , , , ]). Recent Stevis idea of transforming complicated equations and systems into simpler solvable ones, used for the rst time in explaining the solvability of the equation appearing in [] (an extension of the original result can be found in [], see also []), was employed in several other papers (see, e.g., [, , , , , , , , , ] 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 dierence equations (see, e.g., [, , , , , , , , ]).

Stevi also essentially triggered a systematic study of non-rational concrete dierence equations and systems, from one side those obtained by using the translation operator (see, e.g., [] and also []) and from the other side those obtained by using max-type operators (see, e.g., [, , ]), see also the related references cited therein. We would like to point out that for the equations and systems in [, ] only long-term behavior of their positive solutions are studied. For instance, the boundedness of positive solutions to the system

xn+ = max

a, ypnxqn [bracerightbigg], yn+ =

Z, z1, z0, w1, w0

max

a, xpnyqn [bracerightbigg], n

N, ()

2015 Stevi et al. 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.

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 2 of 15

with min{a, p, q} > , was investigated in []. System () is obviously obtained from the

next product-type one

xn+ = ypnxqn , yn+ =

xpn

N, ()

by acting with the max-type operator ma(t) = max{a, t} onto the right-hand sides of both

equations in () (see also [] and [] for related scalar equations). Note that for the case of positive initial values, system () 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 dierence equations with constant coecients, 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 dierence equations and systems are often studied since many real-life models produce such solutions (see, e.g., [, , ]). It is also interesting to note that there are max-type systems of dierence equations which are solvable (see []). Finally, we want to note that the long-term behavior of solutions to product-type 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 product-type dierence equations and systems with real and/or complex initial values. In [], Stevi and his collaborators started studying the problem with an approach dierent from the ones in [, , , ], but which can be regarded as a modication of some of the methods in [, ]. They showed therein that the system

zn+ = wanzbn , wn+ =

zcn

yqn , n

wdn , n

N, ()

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 dierence equations:

zn+ = zanwbn , wn+ =

wcn

where a, b, c, d

Z and z, z, w, w

zdn , n

N, ()

where a, b, c, d

Z and z, z, w, w

C.

Let us mention here that although systems () and () are similar in appearance, the methods used in dealing with them are quite dierent.

It is easy to see that the domain of undenable solutions [] to system () is the set

U = [braceleftbig](z

, z, w, w)

C : z = or z = or w = or w =

.

Hence, from now on we will assume that our initial values belong to the set C \ U.

A solution (zn, wn)n of system () is called periodic (or eventually periodic) with period p

N if there is n such that

(zn+p, wn+p) = (zn, wn) for n n.

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 3 of 15

N, p < p which is a period of the solution. For p = , the

solution is called eventually constant (see, e.g., []). For some results on the topic, see, e.g., [, ] and the related references therein. If it is said that a solution of system () is periodic with period p, it will need not mean that it is prime.

A system of dierence equations of the form

zn = f (zn, . . . , znk, wn, . . . , wnk)

wn = g(zn, . . . , znk, wn, . . . , wnk), n

N,

Period p is prime if there is no p

N, is said to be solvable in closed form if its general solution can be found in terms of initial values zi, wi, i = , k, delay k and index n only.

2 Main result

The main result in this paper is proved in this section.

Theorem Assume that a, b, c, d

Z and z, z, w, w

where k

C

\ {}. Then system () is solv-

able in closed form.

Proof Case b = . In this case system () becomes

zn+ = zan, wn+ = wcnzdn , n

N. ()

From the rst equation in () we easily obtain

zn = zan, n

N. ()

Employing () into the second equation in (), we get

wn = wcn zdan

()

for n .

Hence, by using (), we have that

wn = zdan [parenleftbigg]

wcn

zdan [parenrightbigg]c

= wcn zdan+dcan

=

zdan+dcan [parenleftbigg]

wcn

zdan [parenrightbigg]c

= wcn zdan+dcan+dcan

for n .

Assume that we have proved

wn = wcknk zdan+dcan+dcan++dckank

()

for n k + .

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 4 of 15

Then, by using () with n n k into (), we get

wn = zdan+dcan+dcan++dckank [parenleftbigg]

wcnk

zdank [parenrightbigg]c

k

= wck+nk zdan+dcan+dcan++dckank+dckank

()

for n k + .

From (), () and the method of induction we see that () holds for every k such that k n .

By taking k = n into () we get

wn = wcn zdan+dcan+dcan++dcn

()

for n .

Now we have two subcases to consider. Subcase a = c. In this case from () we get

wn = wcn

zd

ancn ac

, n . ()

Using the next relation

w = wczd ()

in () we get

wn = wcn

zd

ancn ac

zdcn

N. ()

Subcase a = c. In this case from () we get

wn = wan zd(n)a

n

, n

()

for n .

Using () with a = c into (), we get

wn = wan zd(n)a

n

zdan

N. ()

Case d = . In this case system () becomes

zn+ = zanwbn , wn+ = wcn, n

N. ()

, n

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 5 of 15

From the second equation in () we have that

wn = wcn, n

N. ()

Employing () into the rst equation in (), we get

zn = zan wbcn

()

for n .

Hence, by using (), we have

zn = wbcn [parenleftbigg]

zan

wbcn [parenrightbigg]a

= zan wbcn+bacn

=

wbcn+bacn [parenleftbigg]

zan

wbcn [parenrightbigg]a

= zan wbcn+bacn+bacn

for n .

Assume that we have proved

zn = zaknk wbcn+bacn+bacn++bakcnk

()

for n k + .

Then, by using () with n n k into (), we get

zn = wbcn+bacn+bacn++bakcnk [parenleftbigg]

zank

wbcnk [parenrightbigg]a

k

= zak+nk wbcn+bacn+bacn++bakcnk+bakcnk

()

for n k + .

From (), () and the method of induction we see that () holds for every k such that k n .

By taking k = n into () we get

zn = zan wbcn+bacn+bacn++bcan+ban

()

for n .

Now we have two subcases to consider. Subcase a = c. In this case from () we get

zn = zan

wb

ancn ac

, n . ()

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 6 of 15

Using the next relation

z = zawb ()

in () we get

zn = zan

wb

ancn ac

wban

N. ()

Subcase a = c. In this case from () we get

zn = zan wb(n)a

n

, n

()

for n .

Using () into () we get

zn = zan wb(n)a

n

wban

N. ()

Case bd = . First note that from the rst equation in (), for every well-dened solution,

we have that

wbn = zanzn+ , n

, n

N, ()

while from the second one it follows that

wbn+ = wbcnzbdn , n

N. ()

Using () into () we obtain

zan+

zn+ =

zacn+

zcn+zbdn , n

N,

which can be written as

zn+ = za+cn+zacn+zbdn, n

N, ()

which is a fourth order product-type dierence equation.

Note also that

z = zawb and z =

za

wb =

za

wbwab . ()

Let

a = a + c, b = ac, c = , d = bd. ()

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 7 of 15

Then equation () can be written as

zn+ = zan+zbn+zcnzdn, n

N. ()

From () with n n we get

zn+ = zan+zbnzcnzdn, n

N. ()

Employing () into () we get

zn+ = zan+zbnzcnzdn

a zbn+zcnzdn

= zaa+bn+zab+cnzac+dnzadn

= zan+zbnzcnzdn ()

N, where

a := aa + b, b := ab + c, c := ac + d, d := ad. ()

From () with n n we get

zn+ = zanzbnzcnzdn ()

for n .

Employing () into () we get

zn+ =

zanzbnzcnzdn

a zbnzcnzdn

= zaa+bnzba+cnzca+dnzdan

= zanzbnzcnzdn ()

for n , where

a := aa + b, b := ba + c, c := ca + d, d := da. ()

Assume that for some k n, we have proved that

zn+ = zakn+kzbkn+kzckn+kzdknk ()

for n k , and that

ak = aak + bk, bk = bak + ck,

ck = cak + dk, dk = dak.

()

for n

Then, by using the relation

zn+k = zan+kzbn+kzcnkzdnk,

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 8 of 15

for n k, into () we obtain

zn+ = zan+kzbn+kzcnkzdnk[parenrightbig]ak zbkn+kzckn+kzdknk

= zaak+bkn+kzbak+ckn+kzcak+dknkzdaknk

= zak+n+kzbk+n+kzck+nkzdk+nk ()

for n k, where

ak+ := aak + bk, bk+ := bak + ck,

ck+ := cak + dk, dk+ := dak.

za

an++abn++cn+ zdn+wban+waban+bbn+, n

()

Since d = , from the last equation in () we get a = . Using this fact in other three

equalities in (), we get b = , c = d = .

()

This along with (), () and the method of induction shows that () and () hold for every k n + .

Hence, for k = n + , we have

zn+ = zan+zbn+zcn+zdn+

= [parenleftbigg] wbwab [parenrightbigg]an+[parenleftbigg]

za

wb [parenrightbigg]bn+zcn+zdn+

= za

N. ()

From the recurrent relations () we easily obtain that the sequence (ak)k satises the dierence equation

ak = aak + bak + cak + dak. ()

Since bk = ak aak and equation () is linear, we have that the sequence (bk)kN is also a solution to equation (). From this, the linearity of equation () and since ck =

bk bak, we have that the sequence (ck)kN is also a solution to equation (). Finally, since dk = dak, the linearity of equation () shows that (dk)kN 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 ().

From () with k = we get

a = aa + b, b = ba + c, c = ca + d, d = da. ()

Since bd = d = , from the last equation in () we get a = . Using this fact in the rst

three equalities in (), we get b = c = d = .

From this and by () with k = we get

= a = aa + b, = b = ba + c,

= c = ca + d, = d = da.

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 9 of 15

From this and by () with k = we get

= a = aa + b, = b = ba + c,

= c = ca + d, = d = da.

Since d = , from the last equation in () we get a = . Using this fact in other three

equalities in (), we get b = , c = and d = .

Hence, sequences (ak)k, (bk)k, (ck)k and (dk)k are solutions to linear dierence equation () satisfying the following initial conditions:

a = , a = , a = , a = ;

b = , b = , b = , b = ;

c = , c = , c = , c = ;

d = , d = , d = , d = ,

respectively.

Since dierence equation () is solvable, it follows that closed form formulas for (ak)k, (bk)k, (ck)k and (dk)k can be found. From this fact and () we see that equation () is solvable too.

From the second equation in (), for every well-dened solution, we have that

zdn = wcnwn+ , n

N. ()

Using () into () we obtain

wcn+

wn+ =

which can be written as

wn+ = wa+cn+wacn+wbdn, n

()

Since d = , from the last equation in () we get a = . Using this fact in other three

equalities in (), we get b = , c = and d = .From this and by () with k = we get

= a = aa + b, = b = ba + c,

= c = ca + d, = d = da.

()

()

N, ()

while from the rst one it follows that

zdn+ = zadnwbdn , n

wacn+

wan+wbdn , n

N,

N, ()

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 10 of 15

which is nothing but dierence equation (). However, the sequence (wn)n satises the following initial conditions:

w = wczd and w =

N, ()

where (ak)kN, (bk)kN, (ck)kN and (dk)kN satisfy recurrent relations () with initial conditions ().

From () with k = n + and by using () we get

wn+ = wan+wbn+wcn+wdn+

= [parenleftbigg] zdzcd [parenrightbigg]an+[parenleftbigg]

an++cbn++cn+ wdn+zdan+zcdan+dbn+, n

C which uniquely dene solutions to system (), to avoid multi-valued solutions to the system, we posed the condition a, b, c, d

Z.

From the proof of Theorem we obtain the following corollary.

Corollary Consider system () with a, b, c, d

Z. Assume that z, z, w, w

Then the following statements are true.(a) If b = and a = c, then the general solution to system () is given by () and ().

wc

zd =

wc

zdzcd . ()

Hence, the above presented procedure can be repeated, and it can be obtained that for k n + ,

wn+ = wakn+kwbkn+kwckn+kwdknk, n

wc

zd [parenrightbigg]bn+wcn+wdn+

= wc

wc

N. ()

Also the sequences (ak)kN, (bk)kN, (ck)kN and (dk)kN satisfy the dierence equation () with initial conditions in (), respectively.

As above the solvability of equation () shows that closed form formulas for (ak)k, (bk)k, (ck)k and (dk)k can be found. This fact along with () implies that equation () is solvable too. A direct calculation shows that the sequences (zn)n in () and (wn)n in () are solutions to system () with initial values z, z, that is, w, w respectively. Hence, system () is also solvable in this case, nishing the proof of the theorem.

Remark Note that dierence equation () is not only theoretically but also practically solvable since the characteristic polynomial

p() = a b c d ()

associated to the dierence equation is of fourth order, which means that we can explicitly nd its roots.

Remark Since we are interested in those initial values z, z, w, w

C

\ {}.

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 11 of 15

(b) If b = and a = c, then the general solution to system () is given by () and ().(c) If d = and a = c, then the general solution to system () is given by () and ().

(d) If d = and a = c, then the general solution to system () is given by () and ().(e) If bd = , then the general solution to system () is given by () and ().

Let i, i = , , be the roots of the characteristic polynomial () of dierence equation (). If they satisfy the condition

i = j for i = j,

then it is known that a general solution to equation () has the following form:

un = n + n + n + n, n

N, ()

where i, i = , , are arbitrary constants. Since for the case d = the solution can be

prolonged for nonpositive indices, then we may assume that formula () holds also for n (or n if necessary).

In order to nd, in this case, a general solution to system () in closed form, we will need the following known lemma. We give a proof of it for the completeness and benet of the reader.

Lemma Assume that j, j = , k, are pairwise dierent zeros of the polynomial

P(z) = kzk + kzk + + z + .

Then

k

[summationdisplay]

j=

lj

P (j) =

for l = , k , and

k

[summationdisplay]

j=

k .

Proof The functions

fl(z) = zl

P(z), l

N,

kj

P (j) =

are meromorphic on the Riemann sphere. Hence, by the residue theorem, we have that

k

[summationdisplay]

j=

Resz=j fl(z) + Resz= fl(z) = ()

for every l

N.

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 12 of 15

Now note that the Laurent expansion of fl at zero is

fl(z) = zl

k

kj=(z j)

= zlk

k

bszlks

kj=( j/z)

=

k zlk +

N.

On the other hand, since j, j = , k, are simple poles of fl, we have that

Resz=j fl(z) =

lj

for some complex numbers bs, s

P (j), j = , k.

From this and since Resz= fl(z) is equal to the negative value of the coecient at /z in the Laurent expansion, it follows that Resz= fl(z) = when l = , k and Resz= fk(z) =

/k. Using these facts in () the lemma follows.

If we apply Lemma to polynomial p in (), and since p(t) =

l=(t j) (note that

= ), we have

lj

p (j) =

for l = , , and

j

p (j) = .

From this, since from () we have a = a = a = and a = , and a general solution of () has the form in (), we obtain

an =

[summationdisplay]

j=

n+j

p (j)

n+

= ( )( )( ) +

n+

( )( )( )

n+

+ ( )( )( ) +

n+

( )( )( ) ()

for n .

On the other hand, from () we get

bn = an+ aan, ()

cn = bn+ ban, ()

dn = dan ()

for n .

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 13 of 15

By using () into () we get

bn =

j a

p (j)

n+j ()

for n .

By using () and () into () we get

cn =

(j a)j b

p (j)

n+j ()

for n .

By using () into () we get

dn =

[summationdisplay]

j=

n+j ()

for n , where we have used the fact that () also holds for n = (in fact, we may

assume that equality () holds for every n s, for any xed s

d p (j)

N, since due to the assumption d = , any solution of equation () can be prolonged for any nonpositive

value of index n).

By using (), (), () and () into () and (), we get formulas for general solutions to system () in closed form.

Formulas obtained in this section can be used in describing the long-term behavior of solutions to system () 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 Assume that b = c = and a, d

Z. Then the following statements hold:(a) If a = , then every solution to system () is eventually constant.(b) If a = , then zn = wn = , n .

(c) If a = , then every solution to system () is two-periodic.(d) If a > and |z| < , then zn as n .

(e) If a > and |z| > , then |zn| as n .

(f) If a > and |zd| < , then |wn| as n .

(g) If a > and |zd| > , then wn as n .

(h) If a < and |z| < , then zn as n and |zn+| as n .

(i) If a < and |z| > , then zn+ as n and |zn| as n .

(j) If a < and |zd| > , then wn as n and |wn+| as n .

(k) If a < and |zd| < , then wn+ as n and |wn| as n .

Proof (a) If we replace a = and c = in () and (), we obtain zn = z and wn+ = /zd, n

N, from which the statement follows.(b) By replacing a = and c = in () and (), we get zn = , n

N and wn = , n ,

from which the statement follows.(c) By replacing a = and c = in () and (), we get zn = z, zn+ =

z , wn = /zd and

wn+ = zd, n

N, from which the statement follows.

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 14 of 15

(d)-(k) From () and () with c = we get

zn = zan, wn = zdan

Competing interests

The authors declare that they have no competing interests.

Authors contributions

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

Author details

1Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia. 2Operator Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 3Faculty of Electrical Engineering, Belgrade University, Bulevar Kralja Aleksandra 73, Beograd, 11000, Serbia. 4CEITEC - Central European Institute of Technology, Brno University of Technology, Technick 3058/10, Brno,616 00, Czech Republic. 5Department of Mathematics, FEEC - Faculty of Electrical Engineering and Comunication, Brno University of Technology, Technick 3058/10, Brno, 616 00, Czech Republic.

Acknowledgements

The work of Stevo Stevi is supported by the Serbian Ministry of Education and Science projects III 41025 and III 44006. The work of Bratislav Irianin is supported by the Serbian Ministry of Education and Science projects III 41025 and OI 171007. The work of Zdeek marda was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 nanced from European Regional Development Fund. He was also supported by the project FEKT-S-14-2200 of Brno University of Technology.

Received: 14 August 2015 Accepted: 23 September 2015

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18. Stevi, S: On a nonlinear generalized max-type dierence equation. J. Math. Anal. Appl. 376, 317-328 (2011)19. Stevi, S: On a system of dierence equations. Appl. Math. Comput. 218, 3372-3378 (2011)20. Stevi, S: On the dierence equation xn = xn2/(bn + cnxn1xn2). Appl. Math. Comput. 218, 4507-4513 (2011)21. Stevi, S: On a third-order system of dierence equations. Appl. Math. Comput. 218, 7649-7654 (2012)22. Stevi, S: On the dierence equation xn = xnk/(b + cxn1 xnk). Appl. Math. Comput. 218, 6291-6296 (2012)

23. Stevi, S: Solutions of a max-type system of dierence equations. Appl. Math. Comput. 218, 9825-9830 (2012)24. Stevi, S: Domains of undenable solutions of some equations and systems of dierence equations. Appl. Math. Comput. 219, 11206-11213 (2013)

25. Stevi, S: Representation of solutions of bilinear dierence equations in terms of generalized Fibonacci sequences. Electron. J. Qual. Theory Dier. Equ. 2014, Article No. 67 (2014)

, n . ()

Using the formulas in () all these statements easily follow.

Stevi et al. Journal of Inequalities and Applications (2015) 2015:327 Page 15 of 15

26. Stevi, S, Alghamdi, MA, Alotaibi, A, Elsayed, EM: Solvable product-type system of dierence equations of second order. Electron. J. Dier. Equ. 2015, Article No. 169 (2015)

27. Stevi, S, Alghamdi, MA, Alotaibi, A, Shahzad, N: Boundedness character of a max-type system of dierence equations of second order. Electron. J. Qual. Theory Dier. Equ. 2014, Article No. 45 (2014)

28. Tollu, DT, Yazlik, Y, Taskara, N: On fourteen solvable systems of dierence equations. Appl. Math. Comput. 233,

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29. Stevi, S, Diblik, J, Irianin, B, marda, Z: On a solvable system of rational dierence equations. J. Dier. Equ. Appl. 20(5-6), 811-825 (2014)

30. Stevi, S, Diblik, J, Irianin, B, marda, Z: Solvability of nonlinear dierence equations of fourth order. Electron. J. Dier. Equ. 2014, Article No. 264 (2014)

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