R E S E A R C H
Open Access
Two-dimensional product-type system of
difference equations solvable in closed form
Stevo Stevi´c
1,2*, Bratislav Iriˇcanin
3,4and Zden ˇek Šmarda
4*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 Arabia
Full list of author information is available at the end of the article
Abstract
A solvable two-dimensional product-type system of difference equations of interest is presented. Closed form formulas for its general solution are given.
MSC: Primary 39A10; 39A20
Keywords: system of difference equations; product-type system; solvable in closed form
1 Introduction
Concrete nonlinear difference equations and systems have become of some interest re-cently. 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., [–]). Many of the papers study or are motivated by the study of symmetric systems (see,e.g., [–, , , , –]). It turned out that some of the equa-tions and systems can be solved, which motivated some experts to work on the topic (see,
e.g., [, , , –, –]; for some old results see,e.g., [–]). 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, [, , , ] and numer-ous 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 suit-able changes of varisuit-ables (see,e.g., [, , , ] 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 [], which is a kind of perturbation of some product-type and the system in []; see also the related ref-erences 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 val-ues which are solvable. A detailed study of the problem has been started recently by Stević
et al.in [, , , , ] (some subclasses of the class of difference equations studied in [] 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 []. Based on this idea, quite recently in [] it has been shown that the solvability of the system studied in [] is preserved if two coeffi-cients/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 bepracticallysolved in
closed form, due to the well-known fact that roots of the polynomials of degreed≥ can-not 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
zn+=αzanwbn–, wn+=βwcn–zdn–, n∈N, ()
wherea,b,c,d∈Z,α,β∈Candz–,z,w–,w∈C. It is interesting that none of the
sub-classes of the class in () has been previously treated in our papers on product-type sys-tems, 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 [] and [].
A solution to system () need not be defined if its initial values belong to the set
U=(z–,z,w–,w)∈C:z–= orz= orw–= orw=
.
Thus, from now on we will assume thatz–,z,w–,w∈C\ {}. Since the casesα= and
β= are trivial or produce solutions which are not well defined we will also assume that
αβ= .
Let us also note that we will use the conventionli=kai= , whenl<k, throughout the paper.
2 Main results
The main results in this paper are proved in this section.
Theorem Assume that b,c,d∈Z,a= ,α,β∈C\ {}, and z–,z,w–,w∈C\ {}.
Then system()is solvable in closed form.
Proof Sincea= system () is
zn+=αwbn–, wn+=βwcn–zdn–, n∈N. ()
Using the first equation in () in the second one, we obtain
wn+=βαdwcn–wbdn–, n≥, ()
from which it follows that
wn+=βαdwcn–wbdn–, n∈N, ()
and
Case bd= .In this case equations () and () become
wn+=βαdwcn–, n∈N, ()
and
wn+=βαdwcn, n∈N, ()
from which it follows that
wn+=
βαd
n–
j=cjwcn
=
βαd
n–
j=cjβwc
–z–d
cn
=β
n j=cjαd
n–
j=cjwcn+
– zdc
n
–, n∈N, ()
wn=
βαd
n–
j=cjwcn–
=
βαd
n–
j=cjβwc
zd
cn–
=β
n–
j=cjαd
n–
j=cjwcn
zdc
n–
, n≥. ()
Hence
wn+=β
–cn+
–c αd––cncwcn+
– zdc
n
– , n∈N, ()
wn=β
–cn
–cαd–cn
– –c wcn
zdc
n–
, n≥, ()
whenc= , and
wn+=βn+αdnw–zd–, n∈N, ()
wn=βnαd(n–)wzd, n≥, ()
whenc= .
By using () and () in the first equation in () withn→nandn→n– , respectively, we get
zn+=αwbn–=αβ
bnj=–cj
wbc–n, n≥, ()
zn=αwbn–=αβ
bnj=–cjwbcn–
, n≥. ()
Hence, from () and () we have
zn+=αβb
–cn
–cwbcn
–, n≥, ()
zn=αβb
–cn–
–c wbcn–
, n≥, ()
whenc= , and
zn+=αβbnwb–, n≥, ()
zn=αβb(n–)wb, n≥, ()
Case bd= .Letγ :=βαd
a:=c, b=bd, x:= . ()
Then () and () can be written as
wn+=γxwan–w
b
n–, n∈N, ()
and
wn+=γxwanw b
n–, n∈N. ()
By using () withn→n– into (), we get
wn+=γx
γwa
n–w
b
n–
a wb
n–
=γx+awaa+b
n– w
ba
n–
=γxwa
(n–)+w
b
(n–)+, ()
forn≥, where
a:=aa+b, b:=ba, x:=x+a. ()
Assume that
wn+=γxkwak(n–k)+wbk(n–k–)+, ()
for somek≥ and everyn≥k, where
ak:=aak–+bk–, bk:=bak–, xk:=xk–+ak–. ()
Using () withn→n–kinto () we get
wn+=γxk
γwa
(n–k–)+w
b
(n–k–)+
ak
wbk(n–k–)+
=γxk+akwaak+bk
(n–k–)+w
bak
(n–k–)+
=γxk+wak+
(n–k–)+w
bk+
(n–k–)+, ()
for everyn≥k+ , where
ak+:=aak+bk, bk+:=bak, xk+:=xk+ak. ()
Forn=k, () becomes
wn+=γxnwan w
bn
–, n∈N. ()
Using the equalitiesw=βwc–zd–,an+=can+bn, andxn+=xn+anin () it follows that
wn+=
βαdxnβw–c z–danwbn–
=αdxnβxn+wcan+bn
– z
dan
–
=αdxnβxn+wan+
– z
dan
– , n∈N. ()
Using () in the first equation in (), we get
zn+=α+bdxn–βbxnwban– z
bdan–
– , n≥. ()
By using the same procedure it is proved that
wn+=γxkw
ak
(n–k+)w
bk
(n–k), ()
for all natural numberskandnsuch that ≤k≤n, where (ak)k∈N, (bk)k∈N, (xk)k∈Nsatisfy () and ().
Forn=k, () becomes
wn+=γxnwanwbn , n∈N. ()
Sincew=βwczd,xn+=xn+an, andan+=can+bn, from () we have
wn+=
βαdxnβwczdanwbn
=αdxnβxn+wan+
z
dan
, n∈N. ()
Using () in the first equation in (), we get
zn+=α+bdxn–βbxnwban z
bdan–
, n≥. ()
From the first two equations in () we have
ak=aak–+bak–, k≥. ()
From () and sincebk=bak–, we see that (bk)k∈Nis also a solution of ().
From () withk= one obtains
a=aa+b, b=ba, x=x+a. ()
From this and sinceb=bd= , from the second equation in () we geta= , which
This and () withk= imply
=a=aa–+b–, =b=ba–, =x=x–+a–, ()
which along withb= and the second equation in () impliesa–= . This along with
the other two relations in () implies that we must haveb–= andx–= .
Hence (ak)k≥–and (bk)k≥–are solutions to () satisfying the (shifted) initial conditions
a–= , a= ; b–= , b= , ()
while (xk)k≥–satisfies the third equation in () and
x–=x= , x= . ()
From the third equation in () along withx= anda= , we have
xk= + k–
j=
aj= k–
j=
aj. ()
The characteristic equation associated to () isλ–cλ–bd= , from which it follows that
λ,=
c±√c+ bd
,
are the corresponding characteristic roots. Ifc+ bd= , then
an=cλn+cλn,
which along witha–= anda= yields
an=
λn+–λn+ λ–λ
. ()
From this and sincebn=ban–, we have
bn=bd
λn–λn λ–λ
. ()
Ifc+bd= , which is equivalent toλ= =λ, from () and (), it follows that
xn= n–
j=
λj+–λj+ λ–λ
=(λ– )λ n+
– (λ– )λn++λ–λ
(λ– )(λ– )(λ–λ)
. ()
Ifc+bd= , that is, if one of the characteristic roots is one, sayλ, thenλ= –bd, so that
xn= n–
j=
λj+–
λ–
= (λ– )
λ
λn–
λ–
–n =(–bd)
n++ (n+ )bd+n
Ifc+ bd= , then
an= (ˆc+cˆn)
c
n
.
This along witha–= anda= yields
an= (n+ )
c
n
. ()
Using the relationbn=ban–along with the factbd= –c/, we get
bn=bdn
c
n–
= –n
c
n+
. ()
From () and (), we have
xn= n–
j=
(j+ )
c
j
= – (n+ )( c
)n+n(
c
)n+
( –c
)
, ()
ifc= . Ifc= , we obtain
xn= n–
j=
(j+ ) =n(n+ )
, ()
completing the proof of the result.
Corollary Consider system()with b,c,d∈Z,a= ,andα,β∈C\ {}.Assume that z–,z,w–,w∈C\ {}.Then the following statements are true.
(a) Ifbd= andc= ,then the general solution to system()is given by(), (), (),
and().
(b) Ifbd= andc= ,then the general solution to system()is given by(), (), (),
and().
(c) Ifbd= ,c+ bd= ,andc+bd= ,then the general solution to system()is given
by(), (), (),and(),where the sequence(an)n≥–is given by formula(),
while(xn)n≥–is given by().
(d) Ifbd= ,c+ bd= ,andc+bd= ,then the general solution to system()is given
by(), (), (),and(),where the sequence(an)n≥–is given by formula(),
while(xn)n≥–is given by().
(e) Ifbd= ,c+ bd= ,andc= ,then the general solution to system()is given by
(), (), (),and(),where the sequence(an)n≥–is given by formula(),while
(xn)n≥–is given by().
(f ) Ifbd= ,c+ bd= ,andc= ,then the general solution to system()is given by
(), (), (),and(),where the sequence(an)n≥–is given by formula()with
c= ,while(xn)n≥–is given by().
Theorem Assume that a,c,d∈Z,b= ,α,β∈C\ {},and z–,z,w–,w∈C\ {}.
Proof Sinceb= , we have
zn+=αzan, wn+=βwcn–zdn–, n∈N. ()
From the first equation in () we get
zn=α
n–
j=ajzan
, n∈N. ()
Hence, ifa= , we have
zn=α
–an
–azan
, n∈N, ()
while ifa= ,
zn=αnz, n∈N. ()
Using () in the second equation in (), it follows that
wn+=βαd
n–
j=ajzdan–
wcn–, n≥. ()
Using () twice, we get
wn=βαd
n–
j= ajzdan–
wcn–
=βαd
n–
j= ajzdan–
βαd
n–
j= ajzdan–
wcn–
c
=β+cαd
n–
j= aj+dc
n–
j= ajzdan–+dcan–
wc
n–, ()
for everyn≥, and
wn+=βαd
n–
j= ajzdan–
wcn–
=βαd
n–
j= ajzdan–
βαd
n–
j= ajzdan–
wcn–
c
=β+cαd
n–
j= aj+dc
n–
j= ajzdan–+dcan–
wc
n–, n≥. ()
Assume that, for a natural numberk, it has been proved that
wn=β
k–
j=cjαd
k–
i=ci
n–i–
j= ajzd
k–
j=cjan–j–
wc
k
n–k, () forn≥k+ , and
wn+=β
k–
j=cjαd
k–
i=ci
n–i–
j= ajzd
k–
j=cjan–j–
wc
k
n–k+, ()
Using () withn→n– k– andn→n– k, in () and (), we obtain
(a) Ifa= =candc=a,then the general solution to system()is given by(), (),
and().
(b) Ifa= =candc=a= ,then the general solution to system()is given by(), (),
and().
(c) Ifa= =c,then the general solution to system()is given by(), (),and().
(d) Ifa= –andc= ,then the general solution to system()is given by(), (),and
().
(e) Ifa= andc= ,then the general solution to system()is given by(), (),and
().
(f ) Ifa=c= ,then the general solution to system()is given by(), (),and().
Theorem Assume that a,b,c∈Z, d= ,α,β∈C\ {},and z–,z,w–,w∈C\ {}.
Then system()is solvable in closed form.
Proof In this case system () becomes
zn+=αzanwbn–, wn+=βwcn–, n∈N. ()
From the second equation in () it easily follows that
wn=β
n–
j=cjwcn
, n∈N and wn+=β
n j=cjwcn+
– , n∈N, ()
which, for the casec= , implies that
wn=β
–cn
–cwcn
, n∈N, ()
and
wn+=β
–cn+
–c wcn+
– , n∈N, ()
while, for the casec= , we have
wn=βnw, n∈N, ()
and
wn+=βn+w–, n∈N. ()
Employing () in the first equation in () we obtain
zn=αβb
n–
j=cjwbcn–
zan–, n≥, ()
zn+=αβb
n–
j=cjwbcn
–zan, n∈N. () Combining () and () it follows that
zn=αβb
n–
j=cjwbcn–
αβb
n–
j=cjwbcn–
– zan–
a
=α+aβb(+a)
n–
j=cjwb
wab–
cn–
forn≥, and
Assume that, for some natural numberkwe have proved that
Corollary Consider system()with a,b,c∈Z,d= ,andα,β∈C\ {}.Assume that z–,z,w–,w∈C\ {}.Then the following statements are true.
(a) Ifc=a= =c,then the general solution to system()is given by(), (), (),and
().
(b) Ifc=a= =c,then the general solution to system()is given by(), (), (),and
().
(c) Ifa= =c,then the general solution to system()is given by(), (), (),and
().
(d) Ifa= –andc= ,then the general solution to system()is given by(), (), (),
and().
(e) Ifa= andc= ,then the general solution to system()is given by(), (), (),
and().
(f ) Ifa=c= ,then the general solution to system()is given by(), (), (),and
().
Theorem Assume that a,b,c,d∈Z,bd= ,α,β∈C\ {},and z–,z,w–,w∈C\ {}.
Then system()is solvable in closed form.
Proof First note that the conditionsα,β∈C\ {}andz–,z,w–,w∈C\ {}along with
the equations in () implyznwn= forn≥–. Hence, for every such a solution the first equation in () yields
wbn–=zn+
αza n
, n∈N, ()
while from the second one it follows that
wbn+=βbwbcn–zbdn–, n∈N. ()
From () and () one obtains
zn+=α–cβbzan+zcn+z–naczbdn–, n∈N, ()
which is a fourth order product-type difference equation. Note also that
z=αzawb–, z=α
αza
wb–
a
wb
=α+aza
wab–wb. ()
Letδ=α–cβb,
a=a, b=c, c= –ac, d=bd, y= . ()
Then equation () can be written as
zn+=δyzan+z
b
n+zcnz d
Using () withn→n– into () we get
zn+=δy
δza
n+zbnz c
n–z
d
n–
a zb
n+zcnz d
n–,
=δy+azaa+b
n+ zbna+cz ca+d
n– z
da
n–
=δyza
n+zbnz c
n–z
d
n–, ()
forn∈N, where
a:=aa+b, b:=ba+c, c:=ca+d,
d:=da, y:=y+a.
()
Assume that, for aksuch that ≤k≤n+ , we have proved that
zn+=δykzakn+–kz bk n+–kz
ck n+–kz
dk
n–k, ()
forn≥k– , and that
ak=aak–+bk–, bk=bak–+ck–,
ck=cak–+dk–, dk=dak–,
()
yk:=yk–+ak–. ()
Using () withn→n–kinto () one obtains
zn+=δyk
δza
n+–kz b
n+–kz c
n–kz d
n–k–
ak
zbkn+–kznck+–kzdkn–k
=δyk+akzaak+bk n+–k z
bak+ck n+–k z
cak+dk n–k z
dak n–k–
=δyk+zak+
n+–kz bk+
n+–kz ck+
n–kz dk+
n–k–, ()
forn≥k, where
ak+:=aak+bk, bk+:=bak+ck,
ck+:=cak+dk, dk+:=dak,
()
yk+:=yk+ak. ()
This along with (), (), and the method of induction shows that (), (), and (), hold for everykandnsuch that ≤k≤n+ . In fact () holds for ≤k≤n+ (see ()).
Hence, choosingk=n+ in (), and using () we have
zn+=δyn+zan+z
bn+
z
cn+
z
dn+
–
=α–cβbyn+α+aza
wab–wb
an+α
zawb–bn+ zcn+
z
dn+
–
=α(–c)yn++(+a)an++bn+βbyn+zaan++abn++cn+
×waban++bbn+
– w
ban+
z
dn+
From () we easily see that (ak)k≥satisfies the difference equation
ak=aak–+bak–+cak–+dak–. ()
Sincebk=ak+–aak,ck=bk+–bak,dk=dak–, and from the linearity of equation
() we see that (bk)k∈N, (ck)k∈N, and (dk)k∈Nare also solutions to the equation.
System () withk= yields
a=aa+b, b=ba+c, c=ca+d,
d=da, y=y+a.
()
The conditiond=bd= along with the fourth equation in () impliesa= . Using
this andy= in the other equalities in () we getb=c=d=y= . Repeating the
procedure fork= , –, –, is easily obtained
a–= , a–= , a–= , a= ;
b–= , b–= , b–= , b= ;
c–= , c–= , c–= , c= ;
d–= , d–= , d–= , d= .
()
Hence, (ak)k≥–, (bk)k≥–, (ck)k≥–, and (dk)k≥– are solutions to () satisfying initial
conditions (), while (yk)k≥–satisfies the following conditions:
y–=y–=y–=y= , y= , ()
and (), from which it follows that
yk= k–
j=
aj. ()
Since equation () is solvable, it follows that closed form formulas for (ak)k≥–,
(bk)k≥–, (ck)k≥–, and (dk)k≥–, can be found. From (), the form of the solution ak, and by using some known summation formulas it follows that the formula for (yk)k≥–can
also be found. From these facts and () we see that equation () is solvable too. From the second equation in (), we have that for every well-defined solution
znd–= wn+
βwc n–
, n∈N, ()
while from the first one it follows that
znd+=αdzadn wbdn–, n∈N. ()
From () into () one obtains
wn+=αdβ–awan+wcn+w–nacwbdn–, n∈N, ()
We have
w=βwc–zd– and w=βwczd. ()
As above one obtains, for all natural numberskandnsuch that ≤k≤n+ ,
wn+=ηykˆ wakn+–kw bk n+–kw
ck n+–kw
dk
n–k, n≥k– , ()
whereη=αdβ–a, (ak)k∈N, (bk)k∈N, (ck)k∈N, and (dk)k∈Nsatisfy () with initial conditions (), while (ˆyk)k∈Nsatisfies () and (), so that () holds whereykis replaced byyˆk.
From () withk=n+ and by using () we get
wn+=ηynˆ +wan+w
bn+
w
cn+
w
dn+
–
=αdβ–aˆyn+β
wczdan+β
wc–zd–bn+ wcn+
w
dn+
–
=αdynˆ +β(–a)ynˆ++an++bn+wcan++cn+
z
dan+
w
cbn++dn+
– z
dbn+
– , ()
forn∈N.
As above the solvability of () shows that formulas for (ak)k≥–, (bk)k≥–, (ck)k≥–, and
(dk)k≥–can be found, and consequently a formula for (yˆk)k≥–. This fact along with ()
implies that equation () is solvable too. Hence, system () is also solvable in this case,
as desired.
Corollary Consider system()with a,b,c,d∈Z, bd= ,α,β∈C\ {}.Assume that z–,z,w–,w∈C\ {}.Then the general solution to system()is given by()and(),
where the sequences (ak)k∈N, (bk)k∈N, (ck)k∈N,and(dk)k∈N satisfy the difference equation ()with initial conditions in(),while(yk)k∈Nand(yˆk)k∈Nare given by()and satisfy
conditions().
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.4Department of Mathematics, FEEC - Faculty of Electrical Engineering and Communication, Brno University of
Technology, Technická 3058/10, Brno, 616 00, Czech Republic.
Acknowledgements
The work of Stevo Stevi´c is supported by the Serbian Ministry of Education and Science projects III 41025 and III 44006. The work of Bratislav Iriˇcanin is supported by the Serbian Ministry of Education and Science projects III 41025 and OI 171007. The work of Zden ˇek 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 Iriˇcanin’s visit of Faculty of Electrical Engineering and Communication at the Brno University of Technology.
Received: 27 August 2016 Accepted: 21 September 2016
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