S e i´
ce al.Jou nal o Inequali ies and Applica ions (2022) 2022:25
h ps://doi.o g/10.1186/s13660-022-02761-9
RESEARCH Open Access
No e on di e ence equa ions wi h he
igh -hand side unc ion noninc easing in
each a iable
S e o S e i´
c1,2*, B a isla I iˇ
canin3,4, Wi old Kosmala5and Zdenˇ
ek Šma da6
*Co espondence: [email p o ec ed]
1Ma hema ical Ins i u e o he
Se bian Academy o Sciences, Knez
Mihailo a 36/III, 11000 Beog ad,
Se bia
2Depa men o Medical Resea ch,
China Medical Uni e si y Hospi al,
China Medical Uni e si y, Taichung
40402, Taiwan, Republic o China
Full lis o au ho in o ma ion is
a ailable a he end o he a icle
Abs ac
We p esen an example o a diffe ence equa ion o a bi a y o de , possessing he
igh -hand side unc ion ha is homogeneous o a ce ain deg ee and noninc easing
in each a iable, which has a unique posi i e equilib ium, as well as solu ions ha do
no con e ge o he equilib ium. The example shows ha he main esul in he
pape : O. Moaaz, Dynamics o diffe ence equa ion xn+1 = (xn–l,xn–k)(Ad .Diffe .Equ.
2018:447, 2018), is inco ec .
MSC: 39A10; 39A22
Keywo ds: Diffe ence equa ion; Unbounded solu ions; Nondec easing unc ion;
Homogeneous unc ion
1 In oduc ion
Le , as usual, Nbe he se o posi i e in ege s, Z he se o in ege s, R he se o eals, and
C he se o he complex numbe s. By Nk,whe ek∈Z,wedeno e hese o allj∈Zsuch
ha j≥k.I p,q∈Zand such ha p≤q, hen he no a ion j=p,qmeans ha j akes he
alues o all in ege s be ween pand q(including pand q).
Diffe ence equa ions and sys ems o diffe ence equa ions ha e been s udied analy i-
cally o mo e han h ee hund ed yea s. The fi s impo an esul s we e ob ained by
de Moi e [3,4] and D. Be noulli [2]. Fo some classical esul s, see, o example, [5,7–
13,15].All hese e e encesa e osomeex en de o ed ofindingclosed- o m o mulas o
solu ions o linea o nonlinea diffe ence equa ions and sys ems o diffe ence equa ions.
The e has been some enewed in e es in he sol abili y o diffe ence equa ions and sys-
ems o diffe ence equa ions, hei in a ian s, and applica ions o ob ained closed- o m
o mulas o hei solu ions and/o in a ian s (see, o example, [1,16–28]and ela ed
e e ences he ein). I is a common ac ha many sol able diffe ence equa ions and sys-
ems a e ans o med by some sui able changes o a iables o some sol able linea ones
[1,19,22,23,25–28].
O hemanyclasseso diffe enceequa ionsandsys emso diffe enceequa ionssol able
inclosed o m,he ewemen ionanimpo an class,whichisusedin hisno e.Inaddi ion,
wealsomen ionasimpleme hod o ge ingasequenceo diffe enceequa ionso sys ems
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o diffe ence equa ions om a gi en one, which can some imes be used o ob ain some
coun e examples in he heo y o diffe ence equa ions.
1.1 P oduc - ype di e ence equa ions
Some ecen in es iga ionsonsol abili ya ede o ed op oduc - ypediffe enceequa ions
andsys emso diffe enceequa ions,o osomediffe enceequa ionsandsys ems ha can
be educed o hemusingsomesui able ans o ma ions(see,e.g.,[22–24]and he ela ed
e e ences he ein).
Gene al p oduc - ype diffe ence equa ion is
xn+k=anxα(k–1)
n
n+k–1xα(k–2)
n
n+k–2 ···xα(0)
n
n,n∈N0,(1)
whe e he sequences (an)n∈N0,(α(j)
n)n∈N0,j=0,k–1,aswellas heini ial aluesxj,j=
0,k–1, a e eal o complex.
In some cases, equa ion (1)canbesol edby aking heloga i hm,bu in hecaseo
complex ini ial alues xj,j=0,k–1, coefficien s and exponen s, some o he me hods can
beused(see,e.g.,[22–24] and he ela ed e e ence he ein). Since he equa ion is ela ed
o he gene al linea diffe ence equa ion, i is o g ea impo ance.
1.2 Di e ence equa ions wi h in e lacing indices
The e is a simple me hod enabling o cons uc a amily o ‘cloned’ diffe ence equa ions
omagi enone.Theequa ionsa ecalled hediffe enceequa ionswi hin e lacingindices
[27]. The equa ions appea om ime o ime in he li e a u e, and i seems ha some
au ho sa eno awa e ha heya eob ainedbycloningsomesimple diffe enceequa ions
(see he examples and analyses conduc ed in [25–28]). Howe e , he equa ions can be
use ul in p o iding some coun e examples in he heo y o diffe ence equa ions (see, e.g.,
[6]). Sys ems o diffe ence equa ions wi h in e lacing indices can also be cons uc ed by
he cloning me hod. Now we b iefly desc ibe he me hod.
The gene al o m o he diffe ence equa ion wi h in e lacing indices is he ollowing
yn=h(yn–k,yn–2k,...,yn–lk), n∈N0,(2)
whe e l∈N,andk≥2.
I we define kse s o indices by
Il
j:=(s–l)k+j:s∈N0,j=0,k–1,
we ob iously ha e
Il
i∩Il
j=∅,i=j,
and
k–1

j=0 Il
j=N–lk.
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Le
y(j)
s:=y(s–l)k+j,s∈N0,j=0,k–1,
hen om (2), we ha e
y(j)
s=hy(j)
s–1,y(j)
s–2,...,y(j)
s–l,s∈N0,
o j=0,k–1.
This means ha he sequences (y(j)
s)s∈N0,j=0,k–1a eksolu ions o he equa ion
zn=h(zn–1,zn–2,...,zn–l), n∈N0,(3)
wi h un ela ed ini ial alues.
Using hisp ocedu ein he e e sedi ec ion, omequa ion(3),onecanob aina amily
o cloned equa ions in (2), i.e., a amily o diffe ence equa ions wi h in e lacing indices.
1.3 Global con e gence esul s and a claim
One o he main p oblems in he heo y o diffe ence equa ions is he con e gence o he
solu ions o he equa ions. The e a e many esul s on con e gence in he li e a u e, some
o which can be ound in he li e a u e men ioned abo e.
The ollowing diffe ence equa ion
xn+1 = (xn–k,xn–m), n∈N0,(4)
whe e k,m∈N0, has been ecen ly s udied in [14].
The ollowing claim is hemain esul conce ning hediffe enceequa ionsappea ingin
[14] (Theo em 3.3 he ein).
Theo emA Assume ha has non-posi i e pa ial de i a i es and is homogeneous wi h
deg ee s.Then equa ion (4)has a unique posi i e equilib ium x∗,and e e y solu ion o he
equa ion con e ges o x∗.
In his no e, we show ha he claim in Theo em Ais no ue by gi ing an example o
equa ion (4) such ha he unc ion is homogeneous (wi h a deg ee o be chosen app o-
p ia ely) and has non-posi i e pa ial de i a i es, and equa ion (4) has a unique posi i e
equilib ium x∗and solu ions ha do no con e ge o he equilib ium.
2 A coun e example o Theo em A
In hissec ion,wegi eacoun e example oTheo emA.Tocons uc hecoun e example,
we a e looking o a diffe ence equa ion belonging o he abo e classes o equa ions; ha
is, we a e looking o a p oduc - ype diffe ence equa ion wi h in e lacing indices ha is
sol able in he closed o m.
Example1 Conside he diffe ence equa ion
xn+k=1
xα
n,n∈N0,(5)
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whe ek∈N,andα> 1,whichisaspecialcaseo hegene aldiffe enceequa iono highe
o de
xn+k= (xn+k–1,...,xn), n∈N0,(6)
whe e k∈N,and is an a bi a y unc ion.
He e we a e in e es ed in posi i e solu ions o equa ion (5). Hence, we assume ha
xj∈(0,+∞), j=0,k–1.
Fo he case o equa ion (5), we ha e ha
( 1,..., k)= 1
α
k(7)
is he igh -hand side unc ion, which gene a es hei solu ions along wi h he ini ial al-
ues.
Since
∂
∂ j( 1,..., k)=0, j=1,k–1
and
∂
∂ k( 1,..., k)=– α
α+1
k<0,
when k> 0, we ha e ha he unc ion (7) has non-posi i e pa ial de i a i es on he se
(0,+∞)k, om which we choose ou ini ial alues.
On he o he hand, since
(λ 1,...,λ k)= 1
(λ k)α=1
λα
1
α
k=λ–α ( 1,..., k)
holds o e e y λ> 0, we see ha he unc ion defined in (7) is homogeneous wi h deg ee
–α.
Fu he , i
xn≡x∗,n∈N0
is an equilib ium solu ion o equa ion (5), hen i mus be
x∗=1
(x∗)α,
omwhichi immedia ely ollows ha x∗=1,whichmeans ha equa ion(5)hasaunique
posi i e equilib ium.
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Nowno e ha equa ion(5) is a diffe ence equa ion wi h in e lacing indices, which is
ob ained by cloning he ollowing p oduc - ype diffe ence equa ion o fi s o de
yn+1 =1
yα
n,n∈N0,(8)
ha is, equa ion (5)consis so kcopies o equa ion (8) wi h ini ial alues no connec ed
o each o he .
Equa ion(8) ob iously has he same equilib ium. Hence, o show ha he claim o The-
o em Ais no ue, i is enough o show ha equa ion (8) has solu ions, which do no
con e ge o he equilib ium.
Now no e ha by i e a ing equa ion (8), we ge
y2n+2 =yα2
2n(9)
and
y2n+3 =yα2
2n+1, (10)
o n∈N0.
Byasimpleinduc i ea gumen om ela ions(9),(10),andequa ion(8)wi hn=1,one
can easily ob ain
y2n=yα2n
0(11)
and
y2n+1 =yα2n
1=1
yα2n+1
0, (12)
o n∈N0.
Taking α>1,weha e
lim
n→+∞α2n=lim
n→+∞α2n+1 =+∞. (13)
Hence, i y0∈(0,1), hen le ing n→+∞in (11)and(12)andusing(13), i ollows ha
lim
n→+∞y2n=0 (14)
and
lim
n→+∞y2n+1 =+∞. (15)
Besides his, i y0> 1, hen le ing n→+∞in (11)and(12)andusing(13), we ob ain
lim
n→+∞y2n=+∞(16)

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and
lim
n→+∞y2n+1 =0. (17)
F om he ela ions (14)–(17), we see ha i any ini ial alue o equa ion (8)belongs o
he se (0,+∞) {1}, he solu ion is unbounded. This ac implies ha i any ini ial alue
o equa ion (5) belongs o he se (0,+∞) {1}, he solu ion is unbounded.
The abo e analysis shows ha he claim o Theo em Ais no ue.
Rema k 1 No e ha he only bounded posi i e solu ion o equa ion (5) is he one which
is gene a ed by he ini ial alues
x0=x1=···=xk–1 =1. (18)
The solu ion is, in ac , he equilib ium solu ion xn≡1, n∈N0,whichiseasilyp o ed
using he ini ial alues (18)inequa ion(5), oge he wi h a simple induc i e a gumen .
Rema k2 Theabo econside a ionalsoshows ha heequilib iumsolu ionis heunique
posi i e solu ion o equa ion (5) ha con e ges, which indica es o wha ex en he claim
in Theo em A ails.
Rema k 3 No e also ha he linea ized equa ion associa ed wi h equa ion (8)abou he
equilib iumpoin is
zn+1 =–αzn,n∈N0, (19)
om which, along wi h he assump ion α>1, i ollows ha he equilib ium is a epelle .
No e ha om (19), we ha e |zn|=αn|z0|,n∈N0, om which oge he wi h (13), we see
ha o each z0=0, hesolu ion o(19) goes o infini y as n→+∞.
Gi en ha equa ion(8) is sol ableand hasaclosed- o m o mula o i s solu ions, om
which hei long- e mbeha io iseasilydesc ibed,linea iza ionisno equi ed.Howe e ,
he linea iza ion a gumen also sugges s in which di ec ion coun e examples should be
sough .
Acknowledgemen s
The wo k o Zdenˇ
ek Šma da was suppo ed by he p ojec FEKT-S-20-6225 o B no Uni e si y o Technology.
Funding
B no Uni e si y o Technology, p ojec FEKT-S-20-6225.
A ailabili y o da a and ma e ials
No applicable.
Decla a ions
Compe ing in e es s
The au ho s decla e ha hey ha e no compe ing in e es s.
Au ho s’ con ibu ions
SS in es iga ed possibili ies o finding a coun e example o Theo em A. BI, WK and ZŠ analyzed some o iginal ideas in
o de o find as much as possible simple coun e example o he heo em. All au ho s ead and app o ed he final
manusc ip .
S e i´
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Au ho de ails
1Ma hema ical Ins i u e o he Se bian Academy o Sciences, Knez Mihailo a 36/III, 11000 Beog ad, Se bia. 2Depa men
o Medical Resea ch, China Medical Uni e si y Hospi al, China Medical Uni e si y, Taichung 40402, Taiwan, Republic o
China. 3Facul y o Elec ical Enginee ing, Uni e si y o Belg ade, Bule a K alja Aleksand a 73, 11000 Beog ad, Se bia.
4Facul y o Mechanical and Ci il Enginee ing in K alje o, Uni e si y o K aguje ac, K alje o, Se bia. 5Depa men o
Ma hema ical Sciences, Appalachian S a e Uni e si y, Boone, NC 28608, USA. 6Depa men o Ma hema ics, Facul y o
Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology, Technicka 3058/10, CZ-616 00 B no, Czech
Republic.
Publishe ’s No e
Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and ins i u ional affilia ions.
Recei ed: 11 Oc obe 2021 Accep ed: 7 Feb ua y 2022
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