On a close to symmetric system of difference equations of second order

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R E S E A R C H

Open Access

On a close to symmetric system of

difference equations of second order

Stevo Stevi´c

1,2*

_{, Bratislav Iriˇcanin}

3

_{and Zden ˇek Šmarda}

4,5

*_{Correspondence: [email protected]} 1_{Mathematical Institute of the}

Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia

2_{Operator 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

Closed form formulas of the solutions to the following system of diﬀerence equations:

xn=

yn–1yn–2

xn–1(an+bnyn–1yn–2)

, yn=

xn–1xn–2

yn–1(

α

n+

β

nxn–1xn–2)

, n∈N0,

wherean,bn,

α

n,

β

n,n∈N0, and initial valuesx–i,y–i,i∈ {1, 2}are real numbers, are

found. The domain of undeﬁnable solutions to the system is described. The long-term behavior of its solutions is studied in detail for the case of constantan,bn,

α

nand

β

n,

n∈N0.

MSC: Primary 39A10; 39A20

Keywords: system of diﬀerence equations; closed form solution; long-term behavior; periodic solutions

1 Introduction

Studying concrete nonlinear difference equations and systems is a topic of a great recent interest (see,e.g., [–] and the references therein). Studying systems of difference equa-tions, especially symmetric and close to symmetric ones, is a topic of considerable interest (see,e.g., [, , , , –, , , , , –, –, , , , ]). Another topic of interest is solvable difference equations and systems and their applications (see,e.g., [–, , , , , –, –, –]). Renewed interest in the area started after the pub-lication of [] where a formula for a solution of a difference equation was theoretically explained. The most interesting thing in [] was a change of variables which reduced the equation to a linear one with constant coefficients. Related ideas were later used,e.g., in [, , , , , –, –, –].

Quite recently in [] the following systems of diﬀerence equations were presented:

xn=

yn–yn– xn–(±±yn–yn–)

,

yn=

xn–xn– yn–(±±xn–xn–)

, n∈N,

()

wherex–i,y–i,i∈ {, }are real numbers, and some formulas for their solutions are given,

some of which are proved by induction.

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The next system of diﬀerence equations

xn=

yn–yn– xn–(an+bnyn–yn–)

,

yn=

xn–xn– yn–(αn+βnxn–xn–)

, n∈N,

()

where an,bn,αn, βn, n∈N, and initial valuesx–i,y–i,i∈ {, }, are real numbers, is a

generalization of the system in (). Our aim is to show that more general system () is solvable by giving a natural method for getting its solutions. The domain of undeﬁnable solutions to the system is also described. For the case whenan, bn,αn,βn,n∈N, are

constant, the long-term behavior of its solutions is investigated in detail.

A solution (xn,yn)n≥–of system () is calledperiodic, oreventually periodic, with period pif there isn≥– such that

xn+p=xn and yn+p=yn forn≥n.

For some results in the area, see,e.g., [, –, , , , ].

2 Solutions to system (2) in closed form

Assume ﬁrst that x–i= , y–i= , i∈ {, }. Then, by the method of induction and the

equations in (), it follows that for every well-deﬁned solution to system (),xn=  and

yn= , for every n∈N. On the other hand, ifxn=  for somen∈N, then the ﬁrst

equation in () implies thatyn–=  oryn–= . Ifyn–= , thenxn–=  orxn–= ,

while ifyn–= , thenxn–=  orxn–= . Repeating this procedure, we get thatx–i= 

ory–i=  for somei∈ {, }. Similarly, ifyn=  for somen∈N, we getx–i=  ory–i= 

for somei∈ {, }. Hence, for a well-deﬁned solution (xn,yn)n≥– of system (), we have

that

xnyn= , n≥– ()

if and only ifx–iy–i= ,i∈ {, }.

Assume now that (xn,yn)n≥–is a solution to system () such that () holds. Then, by

multiplying the ﬁrst equation in () by xn–and the second one byyn–, and using the

following changes of variables

un=



xnxn–

, vn=



ynyn–

, ()

n≥–, system () is transformed in the following one:

un=anvn–+bn, vn=αnun–+βn, n∈N. ()

From () it follows that

un=anαn–un–+anβn–+bn, ()

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This means that (un)n∈N, (un–)n∈N, (vn)n∈N, and (vn–)n∈Nare solutions to two linear

ﬁrst-order diﬀerence equations, which are solvable. Solving these equations, we get

un=u

3 Case of constant coefﬁcients

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Assume that (xn,yn)n≥–is a solution to system () such that () holds. Then we have

Now we present formulae for solutions to system ().

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Case aα= . We have

4 Long-term behavior of solutions to system (14)

Before we formulate and prove the main results regarding the long-term behavior of well-deﬁned solutions to system (), we quote the following well-known asymptotic formula which will be used in the proofs of the main results:

( +x)–=  –x+Ox, asx→. ()

We also deﬁne the following quantities:

L:=u–( –aα) –aβ–b

Finally, we give another auxiliary result.

Lemma  If aα= ,aβ+b= =αb+β.Then system()has two-periodic solutions.

Proof The equilibrium solution to system () is

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The next three results are devoted to the long-term behavior of well-deﬁned solutions to system ().

Theorem  Assume that|aα| = and(xn,yn)n≥–is a well-deﬁned solution to system().

Then the following statements are true.

(a) Ifaβ+b= =αb+βand|aα|< ,then(xn,yn)converges to a,not necessarily

prime,two-periodic solution.

(b) Ifu–=u= (aβ+b)/( –aα),then the sequences(xm)m≥–and(xm+)m≥–are

constant.

(c) Ifv–=v= (αb+β)/( –aα),then the sequences(ym)m≥–and(ym+)m≥–are

constant.

(d) If|aα|> andu–= (aβ+b)/( –aα)=u,thenxm→and|xm+| → ∞,as

m→ ∞.

(e) If|aα|> andu–= (aβ+b)/( –aα) =u,thenxm+→and|xm| → ∞,as

m→ ∞.

(f ) If|aα|> andv–= (aβ+b)/( –aα)=v,thenym→and|ym+| → ∞,as

m→ ∞.

(g) If|aα|> andv–= (aβ+b)/( –aα) =v,thenym+→and|ym| → ∞,as

m→ ∞.

(h) If|aα|> ,u–= (aβ+b)/( –aα)=uand|L|< ,thenxm→,asm→ ∞.

(i) If|aα|> ,u–= (aβ+b)/( –aα)= uand|L|> ,then|xm| → ∞,asm→ ∞.

(j) If|aα|> ,u–= (aβ+b)/( –aα)=uandL= ,then(xm)m≥–is constant.

(k) If|aα|> ,u–= (aβ+b)/( –aα)=uandL= –,then(xm)m≥–and(xm+)m≥–

are convergent.

(l) If|aα|> ,u–= (aβ+b)/( –aα)= uand|L|< ,thenxm+→,asm→ ∞.

(m) If|aα|> ,u–= ( aβ+b)/( –aα)=uand|L|> ,then|xm+| → ∞,asm→ ∞.

(n) If|aα|> ,u–= (aβ+b)/( –aα)=uandL= ,then(xm+)m≥–is constant.

(o) If|aα|> ,u–= (aβ+b)/( –aα)=uandL= –,then(xm+)m≥–and

(xm+)m≥–are convergent.

(p) If|aα|> ,v–= (αb+β)/( –aα)=vand|L|< ,thenym→,asm→ ∞.

(q) If|aα|> ,v–= ( αb+β)/( –aα)=vand|L|> ,then|ym| → ∞,asm→ ∞.

(r) If|aα|> ,v–= (αb+β)/( –aα)=vandL= ,then(ym)m≥–is constant.

(s) If|aα|> ,v–= (αb+β)/( –aα)=vandL= –,then(ym)m≥–and(ym+)m≥–

are convergent.

(t) If|aα|> ,v–= ( αb+β)/( –aα)=vand|L|< ,thenym+→,asm→ ∞.

(u) If|aα|> ,v–= (αb+β)/( –aα)= vand|L|> ,then|ym+| → ∞,asm→ ∞.

(v) If|aα|> ,v–= (αb+β)/( –aα)=vandL= ,then(ym+)m≥–is constant.

(w) If|aα|> ,v–= (αb+β)/( –aα)=vandL= –,then(ym+)m≥–and

(ym+)m≥–are convergent.

Proof Let

pm=

aβ+b+ (aα)m₍_u

–( –aα) –aβ–b) aβ+b+ (aα)m₍_u_{( –}_a_α_{) –}_a_β_–_b₎,

ˆ

pm=

aβ+b+ (aα)m(u( –aα) –aβ–b)

aβ+b+ (aα)m+₍_u

–( –aα) –aβ–b)

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qm=

αb+β+ (aα)m(v–( –aα) –αb–β) αb+β+ (aα)m₍_v

( –aα) –αb–β)

,

ˆ

qm=

αb+β+ (aα)m₍_v_{( –}_a_α_{) –}_α_b_–_β₎

αb+β+ (aα)m+₍_v–_{( –}_a_α_{) –}_α_b_–_β₎

form∈N.

(a) By using () we have

pm=

 + (aα)m₍_u

–( –aα) –aβ–b)(aβ+b)–

 + (aα)m₍_u_{( –}_a_α_{) –}_a_β_–_b₎₍_a_β₊_b₎–

=  + (u––u)( –aα)(aβ+b)–(aα)m+o(aα)m, ()

ˆ

pm=

 + (aα)m₍_u_{( –}_a_α_{) –}_a_β_–_b₎₍_a_β₊_b₎–

 + (aα)m+₍_u–_{( –}_a_α_{) –}_a_β_–_b₎₍_a_β₊_b₎–

=  +( –aα)(u–aαu––aβ–b)

aβ+b (aα)

m₊_o₍_a_α₎m_, _()

qm=

 + (aα)m₍_v–_{( –}_a_α_{) –}_α_b_–_β₎₍_α_b₊_β₎–

 + (aα)m₍_v_{( –}_a_α_{) –}_α_b_–_β₎₍_α_b₊_β₎–

=  + (v––v)( –aα)(αb+β)–(aα)m+o(aα)m, ()

ˆ

qm=

 + (aα)m₍_v_{( –}_a_α_{) –}_α_b_–_β₎₍_α_b₊_β₎–

 + (aα)m+₍_v

–( –aα) –αb–β)(αb+β)–

=  +( –aα)(v–aαv––αb–β) αb+β (aα)

m₊_o₍_a_α₎m _()

for suﬃciently largem.

From ()-(), by using the condition|aα|<  and a well-known criterion for the con-vergence of products, the statement easily follows.

(b) By using the conditionu–=u= (aβ+b)/( –aα) in () and (), the statement

immediately follows.

(c) By using the conditionv–=v= (αb+β)/( –aα) in () and (), the statement immediately follows.

(d) By using the conditionu–= (aβ+b)/( –aα)=u, we get

pm=

aβ+b

aβ+b+ (aα)m₍_u_{( –}_a_α_{) –}_a_β_–_b₎, ()

ˆ

pm=

aβ+b+ (aα)m(u( –aα) –aβ–b)

aβ+b . ()

Lettingm→ ∞in () and () and using the condition|aα|> , we havepm→ and

|ˆpm| → ∞, from which along with () and () the statement easily follows.

(e) By using the conditionu–= (aβ+b)/( –aα) =u, we get

pm=

aβ+b+ (aα)m₍_u–_{( –}_a_α_{) –}_a_β_–_b₎

aβ+b , ()

ˆ

pm=

aβ+b

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Lettingm→ ∞in () and () and using the condition|aα|> , we have|pm| → ∞and

ˆ

pm→, from which along with () and () the statement easily follows.

(f ) By using the conditionv–= (aβ+b)/( –aα)=v, we get

qm→, from which along with () and () the statement easily follows.

(h), (i) Note thatlimm→∞pm=L. Hence, from the assumptions|L|< , that is,|L|> 

along with (), the statements easily follow.

(j) The statement immediately follows by using the conditionL=  in (). (k) SinceL= – and by using (), we have that

From (), by using the condition|aα|>  and a well-known criterion for the convergence of products, the statement easily follows.

(l), (m) Note thatlim_m_→∞pˆm=L. Hence, from the assumptions|L|< , that is,|L|> 

along with (), the statements easily follow.

(n) The statement immediately follows by using the conditionL=  in ().

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From (), by using the condition|aα|>  and a well-known criterion for the convergence of products, the statement easily follows.

(p), (q) Note thatlimm→∞qm=L. Hence, from the assumptions|L|< , that is,|L|> 

along with (), the statements easily follow.

(r) The statement immediately follows by using the conditionL=  in (). (s) SinceL= – and by using (), we have that

qm=

αb+β+ (aα)m₍_v–_{( –}_a_α_{) –}_α_b_–_β₎

αb+β– (aα)m₍_v

–( –aα) –αb–β)

= – +

αb+β (aα)m₍_v

–(–aα)–αb–β)

 –₍_a_α₎m₍_v αb+β

–(–aα)–αb–β)

= –

 + (αb+β)

(aα)m₍_v–_{( –}_a_α_{) –}_α_b_–_β₎+o

 (aα)m

. ()

From (), by using the condition|aα|>  and a well-known criterion for the convergence of products, the statement easily follows.

(t), (u) Note thatlimm→∞qˆm=L. Hence, from the assumptions|L|< , that is,|L|> 

along with (), the statements easily follow.

(v) The statement immediately follows by using the conditionL=  in (). (w) SinceL= – and by using (), we have that

ˆ

qm=

αb+β+ (aα)m₍_v_{( –}_a_α_{) –}_α_b_–_β₎

αb+β– (aα)m₍_v

( –aα) –αb–β)

= – +

αb+β (aα)m₍_v

(–aα)–αb–β)

 –₍_a_α₎m₍_vαb+β

(–aα)–αb–β)

= –

 + (αb+β)

(aα)m₍_v_{( –}_a_α_{) –}_α_b_–_β₎+o

 (aα)m

. ()

From (), by using the condition|aα|>  and a well-known criterion for the convergence

of products, the statement easily follows.

Let

M:=

u–(u––b–aβ)

u(u–b–aβ) , M:=

v–(v––β–αb)

v(v–β–αb) .

Theorem  Assume that aα= –and(xn,yn)n≥–is a well-deﬁned solution to system().

Then the following statements are true.

(a) If|M|< ,thenxm→and|xm+| → ∞,asm→ ∞.

(b) If|M|> ,thenxm+→and|xm| → ∞,asm→ ∞.

(c) IfM= ,then(xn)n≥–is four-periodic.

(d) IfM= –,then(xn)n≥–is eight-periodic.

(e) If|M|< ,thenym→and|ym+| → ∞,asm→ ∞.

(f ) If|M|> ,thenym+→and|ym| → ∞,asm→ ∞.

(g) IfM= ,then(yn)n≥–is four-periodic.

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Proof First, note that sinceaα= –, from ()-() we have

xm=xMm, xm+=x–Mm+, xm+= x Mm

, xm+= x– Mm+



, ()

ym=yMm, ym+=y–Mm+, ym+= y

Mm_ , ym+= y– Mm+



, ()

form∈N. From () and () all the statements easily follow.

Let

N:=u–

u, N:=

v– v.

Theorem  Assume that aα= and(xn,yn)n≥–is a well-deﬁned solution to system().

Then the following statements hold true.

(a) Ifaβ+b= and|N|< ,thenxm→and|xm+| → ∞,asm→ ∞;

(b) Ifaβ+b= and|N|> ,then|xm| → ∞andxm+→,asm→ ∞;

(c) Ifaβ+b= andN= ,then(xm)m≥–and(xm+)m≥–are constant;

(d) Ifaβ+b= andN= –,then(xm+i)m≥–,i= , ,are constant.

(e) Ifaβ+b= and(u––u)/(aβ+b) > ,then|xm| → ∞,asm→ ∞;

(f ) Ifaβ+b= and(u––u)/(aβ+b) < ,thenxm→,asm→ ∞;

(g) Ifaβ+b= andu–=u,then(xm)m≥–is constant;

(h) Ifaβ+b= and(u–u–)/(aβ+b) > ,then|xm+| → ∞,asm→ ∞;

(i) Ifaβ+b= and(u–u–)/(aβ+b) < ,thenxm+→,asm→ ∞;

(j) Ifaβ+b= andu––u=aβ+b,then(xm+)m≥–is constant;

(k) Ifαb+β= and|N|< ,thenym→and|ym+| → ∞,asm→ ∞;

(l) Ifαb+β= and|N|> ,then|ym| → ∞andym+→,asm→ ∞;

(m) Ifαb+β= andN= ,then(ym)m≥–and(ym+)m≥–are constant;

(n) Ifαb+β= andN= –,then(ym+i)m≥–,i= , ,are constant.

(o) Ifαb+β= and(v––v)/(αb+β) > ,then|ym| → ∞,asm→ ∞;

(p) Ifαb+β= and(v––v)/(αb+β) < ,thenym→,asm→ ∞;

(q) Ifαb+β= andv–=v,then(ym)m≥–is constant.

(r) Ifαb+β= and(v–v–)/(αb+β) < ,thenym+→,asm→ ∞;

(s) Ifαb+β= and(v–v–)/(αb+β) > ,then|ym+| → ∞,asm→ ∞;

(t) Ifαb+β= andv––v=αb+β,then(ym+)m≥–is constant.

Proof Let

rm=

u–+ (aβ+b)m

u+ (aβ+b)m, rˆm=

u+ (aβ+b)m u–+ (aβ+b)(m+ ),

sm=

v–+ (αb+β)m

v+ (αb+β)m, sˆm=

v+ (αb+β)m

v–+ (αb+β)(m+ ), m∈N.

(a)-(d) Since in this case we have

xm=x–

u– u

m+

, xm+=x–

u u–

m+

, m∈N,

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(e), (f ) By using () we have

From (), by using the fact that for everyk∈N

and a known criterion for convergence of products, the statements easily follow. (g) Using the conditionu–=uin (), the statement immediately follows. (h), (i) By using () we have

From (), (), () and a known criterion for convergence of products, the statements easily follow.

(j) Using the conditionu=u–+aβ+bin (), the statement immediately follows. (k)-(n) Since in this case we have

ym=y–

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From (), (), () and a known criterion for convergence of products, the statements easily follow.

(q) Using the conditionv=v–in (), the statement immediately follows. (r), (s) By using () we have

ˆ

sm=

v+ (αb+β)m v–+ (αb+β)(m+ )

=

 + v (αb+β)m

 +v–+αb+β (αb+β)m

–

=

 + v (αb+β)m

 –v–+αb+β (αb+β)m +O



m

=  +v–v––αb–β (αb+β)m +O



m

()

From (), (), () and a known criterion for convergence of products, the statements easily follow.

(t) Using the conditionv=v–+αb+βin (), the statement immediately follows.

5 Domain of undeﬁnable solutions to system (2)

In Section  we proved that solutions to system (), for whichx–j=  ory–j=  for some

j∈ {, }, are not deﬁned. The set of all such initial values is characterized here.

Deﬁnition  Consider the system of diﬀerence equations

xn=f(xn–, . . . ,xn–s,yn–, . . . ,yn–s,n),

yn=g(xn–, . . . ,xn–s,yn–, . . . ,yn–s,n), n∈N,

()

wheres∈N, andx–i,y–i∈R,i= ,s. The string of vectors

(x–s,y–s), . . . , (x–,y–), (x,y), . . . , (xn,yn),

wheren≥–, is called anundeﬁned solutionof system () if

xj=f(xj–, . . . ,xj–s,yj–, . . . ,yj–s,j)

and

yj=g(xj–, . . . ,xj–s,yj–, . . . ,yj–s,j)

for ≤j<n+ , andxn+oryn+is not a deﬁned number, that is, the quantity

f(xn, . . . ,xn–s+,yn, . . . ,yn–s+,n+ )

or

g(xn, . . . ,xn–s+,yn, . . . ,yn–s+,n+ )

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The set of all initial values (x–s,y–s), . . . , (x–,y–) which generate undeﬁned solutions to

system () is calleddomain of undeﬁnable solutionsof the system.

The next result characterizes the domain of undeﬁnable solutions to system () when

anbnαnβn= ,n∈N.

Theorem  Assume that anbnαnβn= ,n∈N.Then the domain of undeﬁnable solutions to system()is the following set:

U=

m∈N

(x–,x–,y–,y–)∈R:



x–x–

=g_–◦f_–◦ · · · ◦g–__m_–◦f_–_m_–◦g_–_m◦f_–_m_+()

or 

x–x– =g –

 ◦f–◦ · · · ◦g–m–◦f–m–◦g–m()

or 

y–y– =f –

 ◦g–◦ · · · ◦f–m–◦g–m–◦f–m()

or 

y–y–

=f_–◦g_–◦ · · · ◦g_–_m_–◦f_–_m◦g_–_m_+()

∪(x–,x–,y–,y–)∈R:

x–= or x–= or y–= or y–= , ()

where

fn(t) =ant+bn, gn(t) =αnt+βn, n∈N.

Proof We have already proved that the set

(x–,x–,y–,y–)∈R:x–=  orx–=  ory–=  ory–= 

belongs to the domain of undeﬁnable solutions to system ().

Ifx–j= =y–j,j= ,  (i.e.,xn= =ynfor everyn≥–), then such a solution (xn,yn)n≥–

is not deﬁned if and only if

an+bnyn–yn–=  or αn+βnxn–xn–=  ()

for somen∈N, which is equivalent to

vn–= –bn/an or un–= –βn/αn ()

for somen∈N.

Note that

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We have

vm–= (gm–◦fm–◦ · · · ◦f◦g◦f)(v–), ()

vm= (gm◦fm–◦ · · · ◦g◦f◦g)(u–), ()

um–= (fm–◦gm–◦ · · · ◦g◦f◦g)(u–), ()

um= (fm◦gm–◦ · · · ◦f◦g◦f)(v–) ()

form∈N.

From () and () we have that

–bm

am

=vm–= (gm–◦fm–◦ · · · ◦f◦g◦f)(v–)

for somem∈Nif and only if



y–y– =f –

 ◦g–◦ · · · ◦f–m–◦g–m–◦f–m(). ()

From () and () we have that

–bm+

am+

=vm= (gm◦fm–◦ · · · ◦g◦f◦g)(u–)



x–x–

=g_–◦f_–◦ · · · ◦g_–_m_–◦f_–_m_–◦g_–_m◦f_–_m_+(). ()

From () and () we have that

–βm αm

=um–= (fm–◦gm–◦ · · · ◦g◦f◦g)(u–)



x–x– =g –

 ◦f–◦ · · · ◦g–m–◦f–m–◦g–m(). ()

From () and () we have that

–βm+ αm+

=um= (fm◦gm–◦ · · · ◦f◦g◦f)(v–)



y–y– =f –

 ◦g–◦ · · · ◦g–m–◦f–m◦g–m+(). ()

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Remark  Quantities

On the other hand, if

hj(t) =cjt+dj, j∈N,

From ()-() explicit formulas for the quantities in ()-() are easily obtained.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

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Author details

1_{Mathematical Institute of the Serbian Academy of Sciences, Knez Mihailova 36/III, Beograd, 11000, Serbia.}2_Operator

Theory and Applications Research Group, Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. 3_{Faculty of Electrical Engineering, Belgrade University, Bulevar Kralja Aleksandra 73, Beograd, 11000,}

Serbia.4_{CEITEC - Central European Institute of Technology, Brno University of Technology, Technická 3058/10, Brno,}

CZ-616 00, Czech Republic. 5_{FEEC - Faculty of Electrical Engineering and Communication, Department of Mathematics,}

Brno University of Technology, Technická 3058/10, Brno, CZ-616 00, Czech Republic.

Acknowledgements

The work of the ﬁrst and the second authors was supported by the Serbian Ministry of Education and Science, project III 41025. The work of the ﬁrst author was also supported by the Serbian Ministry of Education and Science, project III 44006. The work of the second author was also supported by the Serbian Ministry of Education and Science, project OI 171007. The work of the third author was realized in CEITEC - Central European Institute of Technology with research

infrastructure supported by project CZ.1.05/1.1.00/02.0068 ﬁnanced from the European Regional Development Fund. The third author was also supported by the project FEKT-S-14-2200 of Brno University of Technology.

Received: 11 June 2015 Accepted: 3 August 2015 References

1. Aloqeili, M: Dynamics of akth order rational diﬀerence equation. Appl. Math. Comput.181, 1328-1335 (2006) 2. Alzahrani, EO, El-Dessoky, MM, Elsayed, EM, Kuang, Y: Solutions and properties of some degenerate systems of

diﬀerence equations. J. Comput. Anal. Appl.18(2), 321-333 (2015)

3. Andruch-Sobilo, A, Migda, M: On the rational recursive sequencexn+1=axn–1/(b+cxnxn–1). Tatra Mt. Math. Publ.43,

1-9 (2009)

4. Bajo, I, Liz, E: Global behaviour of a second-order nonlinear diﬀerence equation. J. Diﬀer. Equ. Appl.17(10), 1471-1486 (2011)

5. Berezansky, L, Braverman, E: On impulsive Beverton-Holt diﬀerence equations and their applications. J. Diﬀer. Equ. Appl.10(9), 851-868 (2004)

6. Berg, L, Stević, S: Periodicity of some classes of holomorphic difference equations. J. Differ. Equ. Appl.12(8), 827-835 (2006)

7. Berg, L, Stevi´c, S: On some systems of diﬀerence equations. Appl. Math. Comput.218, 1713-1718 (2011)

8. Berg, L, Stević, S: On the asymptotics of the difference equationyn(1 +yn–1· · ·yn–k+1) =yn–k. J. Differ. Equ. Appl.17(4),

577-586 (2011)

9. Grove, EA, Ladas, G: Periodicities in Nonlinear Difference Equations. Chapman & Hall/CRC Press, Boca Raton (2005) 10. Iriˇcanin, B, Stević, S: Some systems of nonlinear difference equations of higher order with periodic solutions. Dyn.

Contin. Discrete Impuls. Syst.13a(3-4), 499-508 (2006)

11. Iriˇcanin, B, Stevi´c, S: Eventually constant solutions of a rational diﬀerence equation. Appl. Math. Comput.215, 854-856 (2009)

12. Papaschinopoulos, G, Schinas, CJ: On a system of two nonlinear diﬀerence equations. J. Math. Anal. Appl.219(2), 415-426 (1998)

13. Papaschinopoulos, G, Schinas, CJ: On the behavior of the solutions of a system of two nonlinear diﬀerence equations. Commun. Appl. Nonlinear Anal.5(2), 47-59 (1998)

14. Papaschinopoulos, G, Schinas, CJ: Invariants for systems of two nonlinear diﬀerence equations. Diﬀer. Equ. Dyn. Syst. 7(2), 181-196 (1999)

15. Papaschinopoulos, G, Schinas, CJ: Invariants and oscillation for systems of two nonlinear diﬀerence equations. Nonlinear Anal. TMA46(7), 967-978 (2001)

16. Papaschinopoulos, G, Schinas, CJ: On the dynamics of two exponential type systems of diﬀerence equations. Comput. Math. Appl.64(7), 2326-2334 (2012)

17. Papaschinopoulos, G, Stefanidou, G: Asymptotic behavior of the solutions of a class of rational diﬀerence equations. Int. J. Diﬀerence Equ.5(2), 233-249 (2010)

18. Stefanidou, G, Papaschinopoulos, G, Schinas, CJ: On a system of two exponential type diﬀerence equations. Commun. Appl. Nonlinear Anal.17(2), 1-13 (2010)

19. Stevi´c, S: A global convergence results with applications to periodic solutions. Indian J. Pure Appl. Math.33(1), 45-53 (2002)

20. Stevi´c, S: More on a rational recurrence relation. Appl. Math. E-Notes4, 80-85 (2004)

21. Stević, S: A short proof of the Cushing-Henson conjecture. Discrete Dyn. Nat. Soc.2006, Article ID 37264 (2006) 22. Stević, S: Periodicity of max difference equations. Util. Math.83, 69-71 (2010)

23. Stevi´c, S: On a system of diﬀerence equations. Appl. Math. Comput.218, 3372-3378 (2011)

24. Stević, S: On a system of difference equations with period two coefficients. Appl. Math. Comput.218, 4317-4324 (2011)

25. Stevi´c, S: On the diﬀerence equationxn=xn–2/(bn+cnxn–1xn–2). Appl. Math. Comput.218, 4507-4513 (2011)

26. Stević, S: On a solvable rational system of difference equations. Appl. Math. Comput.219, 2896-2908 (2012) 27. Stević, S: On a third-order system of difference equations. Appl. Math. Comput.218, 7649-7654 (2012)

28. Stević, S: On some periodic systems of max-type difference equations. Appl. Math. Comput.218, 11483-11487 (2012) 29. Stević, S: On some solvable systems of difference equations. Appl. Math. Comput.218, 5010-5018 (2012)

30. Stevi´c, S: On the diﬀerence equationxn=xn–k/(b+cxn–1· · ·xn–k). Appl. Math. Comput.218, 6291-6296 (2012)

31. Stević, S: Solutions of a max-type system of difference equations. Appl. Math. Comput.218, 9825-9830 (2012) 32. Stević, S: Domains of undefinable solutions of some equations and systems of difference equations. Appl. Math.

Comput.219, 11206-11213 (2013)

33. Stevi´c, S: On a system of diﬀerence equations which can be solved in closed form. Appl. Math. Comput.219, 9223-9228 (2013)

(17)

35. Stevi´c, S: On the systemxn+1=ynxn–k/(yn–k+1(an+bnynxn–k)),yn+1=xnyn–k/(xn–k+1(cn+dnxnyn–k)). Appl. Math. Comput.

219, 4526-4534 (2013)

36. Stević, S: Representation of solutions of bilinear difference equations in terms of generalized Fibonacci sequences. Electron. J. Qual. Theory Differ. Equ.2014, Article ID 67 (2014)

37. Stević, S, Alghamdi, MA, Alotaibi, A, Shahzad, N: On a higher-order system of difference equations. Electron. J. Qual. Theory Differ. Equ.2013, Article ID 47 (2013)

38. Stević, S, Alghamdi, MA, Alotaibi, A, Shahzad, N: Boundedness character of a max-type system of difference equations of second order. Electron. J. Qual. Theory Differ. Equ.2014, Article ID 45 (2014)

39. Stevi´c, S, Alghamdi, MA, Maturi, DA, Shahzad, N: On a class of solvable diﬀerence equations. Abstr. Appl. Anal.2013, Article ID 157943 (2013)

40. Stević, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: On a third-order system of difference equations with variable coefficients. Abstr. Appl. Anal.2012, Article ID 508523 (2012)

41. Stević, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: On some solvable difference equations and systems of difference equations. Abstr. Appl. Anal.2012, Article ID 541761 (2012)

42. Stevi´c, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: On the diﬀerence equationxn+1=xnxn–k/(xn–k+1(a+bxnxn–k)). Abstr. Appl. Anal.

2012, Article ID 108047 (2012)

43. Stevi´c, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: On the diﬀerence equationxn=anxn–k/(bn+cnxn–1· · ·xn–k). Abstr. Appl. Anal.

2012, Article ID 409237 (2012)

44. Stević, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: On a solvable system of rational difference equations. J. Differ. Equ. Appl. 20(5-6), 811-825 (2014)

45. Stević, S, Diblik, J, Iriˇcanin, B, Šmarda, Z: Solvability of nonlinear difference equations of fourth order. Electron. J. Differ. Equ.2014, Article ID 264 (2014)