Boundedness character of a fourth order system of difference equations

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R E S E A R C H

Open Access

Boundedness character of a fourth-order

system of difference equations

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

The boundedness character of positive solutions of the following system of diﬀerence equations:xn+1=A+ y

p n

xr_n_–3,yn+1=A+ xpn

y_nr_–3,n∈N0, when min{A,r}> 0 andp≥0, is

studied.

MSC: Primary 39A10; 39A20

Keywords: system of diﬀerence equations; bounded solutions; unbounded solutions; positive solutions

1 Introduction

Concrete nonlinear difference equations and systems, especially those which are not closely related to differential ones, have attracted a lot of attention recently (see, for example, [–] and the references therein). Among them, symmetric and close to symmetric systems of difference equations, whose study was essentially initiated by Papaschinopoulos and Schinas in the mid-s, have attracted a considerable interest (see, for example, [, –, , , –]). For example, in [] Papaschinopoulos and Schinas studied the oscillatory behavior, the boundedness character, and the global sta-bility of positive solutions of the following close to symmetric system of difference equa-tions:

xn+=A+

yn xn–p

, yn+=A+

xn yn–q

, n∈N,

whereA>  andp,q∈N. It should be noted that the system is rational. On the other hand, for the casep=qthe system obviously becomes symmetric, that is, it is of the following form:

xn=f(xn–k,yn–l), yn=f(yn–k,xn–l), n∈N,

for somek,l∈N.

On the other hand, a systematic study of positive solutions of nonlinear diﬀerence equa-tions containing non-integer powers of their dependent variables began by Stevićet al., approximately since the publication of [], where the ﬁrst nontrivial results related to the

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following diﬀerence equation were given:

xn+=α+

xp_n_– xpn

, n∈N, ()

wheremin{α,p}> .

A good prototype including () is the following diﬀerence equation:

xn=α+x

p n–k xr

n–l

, n∈N, ()

wherek,l∈N,k=l,min{α,r}> , andp≥, which was proposed for studying by Stević at numerous talks. Some special cases of this, the corresponding max-type diﬀerence equa-tion or related equaequa-tions has been studied considerably (see, for example, [, , , , –, , ] and the references therein).

Motivated by these two lines of investigations Stević has proposed recently studying symmetric and close to symmetric systems of diﬀerence equations which, among others, stem from special cases of ().

Motivated by all above mentioned work, and especially by [], here we investigate the boundedness character of the solutions of the next system of diﬀerence equations

xn+=A+

ypn xr

n–

, yn+=A+

xpn yr

n–

, n∈N, ()

whenmin{A,r}> ,p≥, andx–i,y–i> ,i∈ {, , , }. Our results extend and

comple-ment some results in [].

By using the induction and the equations in () we see that ifx–i,y–i> ,i∈ {, , , },

then

min{xn,yn}> , n≥–,

which means that positive initial values generate positive solutions of system (). More-over, we have

min{xn,yn}>A, n∈N. ()

The casep=  is simple. Namely, in this case by using () into () is obtained

max{xn+,yn+}<A+



Ar, n≥,

which means that all positive solutions of system () in this case are bounded. In fact, since

A<min{xn,yn} ≤max{xn,yn}<A+ 

Ar, n≥,

they are persistent.

For a solution (xn,yn)n≥–of system () it is said that it isunboundedif

sup

n≥–

(xn,yn)_R=sup n≥–

x

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Otherwise, the solution isbounded, that is, if there is a nonnegative constantMsuch that

sup

n≥–

(xn,yn)_R≤M< +∞.

2 Main results

In this section we prove the main results in this paper, all of which are related to the bound-edness character, that is, the boundbound-edness of all positive solutions of system () or the ex-istence of an unbounded solution of the system depending on the values of parametersA,

p, andr.

Theorem  Assume thatmin{A,p,r}> andp< r.Then all positive solutions of

system()are bounded.

Proof Using the equations in (), we have

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Now using the ﬁrst equation in () in () we get

Assume that for somek≥ we have proved that the following equalities hold:

xn+=A+

where the sequencesak,bk, andckare deﬁned by

ak+=

Using again the equations in () and the recurrent relations in (), we have

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=A+

From this and by using recurrent relations (), it follows thatak,bk, andck increase, as far asak<p. On the other hand, () implies

and thatx∗is a solution of the equation

f(x) =x(p–x)–r= .

We havef() =f(p) = –randf (x) = (x–p)₍_p_{– }_x_{). Hence}_max

x∈[,p]f(x) =f(p/). Since

by a condition of the theorem

f(p/) =p



 –r< ,

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This guarantees the existence of the smallestl∈Nsuch thatal–<pandal≥p. This,

of system () the boundedness of yn follows and consequently the boundedness of the

solution. Ifl= k– , then from () it follows that

Due to the symmetry of system () we also have

yn+≤A+

forn≥k+ , which along with the previous inequality implies the boundedness of the

solution.

forn≥, from which the boundedness follows in the case. Due to the symmetry of system () we see that the inequality

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holds forn≥, from which along with the previous inequality the boundedness of the

solution follows.

Remark  Note that ifak=pfor somek∈N, thenak+,bk+, andck+are not deﬁned.

However, if this happens then above mentioned indexlis chosen to be thisk. For such chosenlis obtained an upper bound for positive solutions of system () in the way de-scribed in the proof of Theorem .

Theorem  Assume thatmin{A,p,r}> , p_≥__r_,_{and p}_≥_{/ (}_{where at least one of}

these two inequalities is strict),or r<p–  < /.Then system()has positive unbounded solutions.

Proof Assume that (xn,yn)n≥–is a positive solution of (). Then we have

xn+≥

ypn xr

n–

, ()

yn+≥

xpn

yr_n_–, ()

forn∈N. Let

zn=ln(xnyn), n≥–.

Taking the logarithm of the both sides in (), (), then summing such obtained inequal-ities, it follows that

zn+–pzn+rzn–≥, n∈N. ()

Let

P(λ) =λ–pλ+r. ()

ThenP() =rand

P(λ) =λ(λ– p),

from which it follows that the polynomialP(λ) has a local minimum atλ= p/, and ac-cording to the conditions of the theorem

P(p/) = –p/ +r≤. ()

Ifp> /, then p/ > . From this, () and since

lim

λ→+∞P(λ) = +∞, ()

it follows that there isλ>  such thatP(λ) = . Ifp= /, inequality () is strict, p/ = ,

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Now assume thatr<p–  < /. ThenP() =  –p+r<  and since () holds, we again see that there isλ>  such thatP(λ) = .

Let

P(λ) =P(λ)/(λ–λ) =λ+aλ+bλ+c

and

un=zn+azn–+bzn–+czn–.

Then inequality () can be written in the following form:

un+–λun≥. ()

Choosex–i,y–i,i∈ {, , , }, such thatu> . For example, to getu> , it is enough to

choosex–i,y–i,i∈ {, , , }, such that

xy>|a||x–y–|+|b||x–y–|+|c||x–y–|.

From this and () it follows that

un≥λn_u. ()

Sinceu>  andλ> , by lettingn→ ∞in () we obtainun→+∞asn→ ∞. Ifzn

were bounded thenunwould be also bounded, which would be a contradiction. Hencezn

is unbounded. From this and since

x

n+yn≥xnyn= ezn, n∈N,

we have

sup

n≥–

x

n+yn= +∞,

that is, the solution of system () is unbounded, completing the proof of the theorem.

Theorem  Assume thatmin{A,p,r}> and p=r+ .Then system()has positive

un-bounded solutions.

Proof Let

xy>x–y–>x–y–>x–y–> . ()

Sincep=r+ , system () is

xn+=A+

yr_n+ xr

n–

, yn+=A+

xr_n+ yr

n–

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Multiplying these two equations we easily obtain

xn+yn+

xnyn >

xnyn xn–yn–

r

, ()

from which withn= , it follows that

xy

xy

>

xy

x–y– r

> .

Assume that we have proved

xkyk>xk–yk–, for –≤k≤n. ()

Then from () and () we have

xn+yn+

xnyn

>

xnyn xn–yn–

r

> , n∈N. ()

Hence

xn+yn+>xnyn,

for every n ≥ –. If xnyn was bounded, then there would be a ﬁnite positive limn→∞xnyn=c. Lettingn→ ∞in the product of equations in () we would obtain

c≥A₊_c_{, which would be a contradiction. Hence, all the solutions of () satisfying ()}

are unbounded.

Theorem  Assume thatmin{A,r}> and p∈(, ).Then every positive solution of system

()is bounded.

Proof Sincexn>A,n∈N, we have

xn+≤A+

ypn

Ar, yn+≤A+

xpn Ar,

forn≥, where (xn,yn)n≥–is an arbitrary positive solution of system ().

Hence

xn++yn+≤A+

(xn+yn)p

Ar , ()

forn≥.

Let (zn)n≥be the solution of the equation

zn+= A+

zpn

Ar, n≥, ()

such thatz=x+y.

Since

f(x) = A+x

p

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is increasing onR+, a simple inductive argument shows that

xn+yn≤zn, forn≥. ()

Sincep∈(, ) functionf is concave, which implies that there is a unique ﬁxed pointx∗

off and that the next condition

f(x) –x x–x∗ < , x∈(,∞)\x∗, ()

holds.

Ifz∈(,x∗] condition () implies that (zn)n≥is nondecreasing and bounded above

by x∗, and ifz≥x∗ that it is nonincreasing and bounded below byx∗. Hence (zn)n≥

is bounded, which along with () implies the boundedness of (xn)n≥and (yn)n≥, from

which the result easily follows.

In the next theorem we use the fact that the comparison equation is a linear ﬁrst order diﬀerence equation, which is solvable in closed form. For recent application of this and related equations see, for example, [, , , –, ].

Theorem  Assume that p= ,r> ,and A>√r.Then every positive solution of system

()is bounded.

Proof From the proof of Theorem  we see that any positive solution (xn,yn)n≥–of system

() satisﬁes () withp= .

Let (zn)n≥be the solution of the equation

zn+= A+

zn

Ar, n≥, ()

such thatz=x+y. Then clearly () also holds.

It is well known that () is solvable. Using its solution in closed form is easily proved that

lim

n→∞zn=

Ar+

Ar_{– },

from which the boundedness of (zn)n≥ follows. This fact along with () implies the

boundedness of (xn)n≥and (yn)n≥, from which the result easily follows.

Remark  The boundedness character of positive solutions of system () in the following two cases:

(a) r≤p_/,_{ <}_p_<_r_{+ ,}_r_{< /;}

(b) r≤p_/,_p_{= ,}_A_∈_(,√r ],

is not known to us. Hence, we leave the cases to the interested reader.

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,} 616 00, Czech Republic.5_{Department 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 was realized in CEITEC - Central European Institute of Technology with research infrastructure supported by the project CZ.1.05/1.1.00/02.0068 ﬁnanced from European Regional Development Fund. He was also supported by the project FEKT-S-14-2200 of Brno University of Technology.

Received: 27 July 2015 Accepted: 23 September 2015 References

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