Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349
DOI 10.1186/s13662-015-0687-0
RESEARCH Open Access
Exis ence o solu ions con e ging o ze o
o nonlinea delayed di e en ial sys ems
Jose Rebenda1and Zdenˇ
ek Šma da1,2*
*Co espondence:
sma da@ eec. u b .cz
1CEITEC BUT, B no Uni e si y o
Technology, Technicka 3058/10,
B no, 61600, Czech Republic
2Depa men o Ma hema ics, B no
Uni e si y o Technology, B no,
Czech Republic
Abs ac
We p esen a esul abou an in e es ing asymp o ic p ope y o eal wo-dimensional
delayed diffe en ial sys ems sa is ying ce ain sufficien condi ions. We employ wo
p e ious esul s, which we e ob ained using a Razumikhin- ype modifica ion o he
Wa˙
zewski opological me hod o e a ded diffe en ial equa ions and he me hod o a
Lyapuno -K aso skii unc ional. The esul is illus a ed by a non i ial explana o y
example.
MSC: 34K12; 34K20
Keywo ds: diffe en ial sys em wi h delays; s abili y o solu ions
1 In oduc ion
Va ious p ope ies o solu ions o diffe en ial equa ions wi h delay we e ex ensi ely s ud-
ied ecen ly.Amongo he swemen ion[–]and he e e ences he ein.The esul scon-
ained in his pape a e a gene aliza ion o p e ious esea ch published in [–]and
[].
Ou aim he eis os udy heasymp o icbeha io o solu ions o he ollowingsys em o
diffe en ial equa ions:
x( )=A( )x( )+ m

k=
Bk( )xθk( )+h ,x( ),xθ( ),...,xθm( ),()
whe e –θk( )≥ a e bounded noncons an delays sa is ying lim →∞θk( )=∞,θk( )a e
eal unc ions,
h( ,x,y)=h( ,x,y,...,ym),h( ,x,y,...,ym)
is a eal ec o unc ion, whe e x=(x,x), yk=(yk,yk), and
A( )=aij( ),Bk( )=bijk( ),i,j=,;k=,...,m,
a e eal squa e ma ices.
In hispape ,wein oduceanin e es ing esul ,whichisacombina iono wo heo ems
p esen edin[], one ega ding heins abili yo solu ions, heo he onedealing wi h he
exis ence o bounded solu ions.
©2015 Rebenda and Šma da. This a icle is dis ibu ed unde he e ms o he C ea i e Commons A ibu ion 4.0 In e na-
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medium, p o ided you gi e app op ia e c edi o he o iginal au ho (s) and he sou ce, p o ide a link o he C ea i e Commons
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Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 2 o 10
I is supposed ha he unc ion hsa isfies he Ca a héodo y condi ions on [ ,∞)×
R(m+), he unc ions bijk a e locally Lebesgue in eg able on [ ,∞), and he unc ions θk,
aij a e locally absolu ely con inuous on [ ,∞).
Since we s udy wo-dimensional sys ems, we use a ans o ma ion in o complex a i-
ables o simpli y he sys em () in o one equa ion wi h complex coefficien s.
Thecomplex a iablesa edefinedasz=x+ix,w=y +iy,...,wm=ym+iym.Using
his ans o ma ion we ge
z( )=a( )z( )+b( )¯
z( )+ m

k= Ak( )zθk( )+Bk( )¯
zθk( )
+g ,z( ),zθ( ),...,zθm( ),()
whe e we assume (J=[ ,∞)):
•Ak,Bk∈Lloc(J,C)(i.e. locally Lebesgue in eg able complex- alued unc ions on J) o
k=,...,m,
•θk∈ACloc(J,R)(i.e. locally absolu ely con inuous eal- alued unc ions on J) o
k=,...,m,
•a,b∈ACloc(J,C)(i.e. locally absolu ely con inuous complex- alued unc ions on J),
•g∈K(J×Cm+,C)(i.e. a complex- alued unc ion which sa isfies he Ca a héodo y
condi ions on J×Cm+).
Ob iously, he unc ion gis in gene al dependen on ¯
zaswellas one e y ¯
z(θk).Howe e ,
he ac ha he unc iongsa isfies heCa a héodo ycondi ionsenablesus osignifican ly
simpli y he no a ion by using only z, since he alidi y o he Ca a héodo y condi ions is
no iola ed by composing wi h con inuous unc ions ¯
z,¯
wk,andθk.
The ela ions be ween he unc ions a e he ollowing:
a( )=
a( )+a( )+i
a( )–a( ),
b( )=
a( )–a( )+i
a( )+a( ),
Ak( )=
bk( )+bk( )+i
bk( )–bk( ),
Bk( )=
bk( )–bk( )+i
bk( )+bk( ),
g( ,z,w,...,wm)
=h ,
(z+¯
z), 
i(z–¯
z), 
(w+¯
w),..., 
i(wm–¯
wm)
+ih ,
(z+¯
z), 
i(z–¯
z), 
(w+¯
w), 
i(w–¯
w),..., 
i(wm–¯
wm).
Con e sely, pu ing
a( )=Rea( )+b( ),a( )=Imb( )–a( ),
a( )=Ima( )+b( ),a( )=Rea( )–b( ),
bk( )=ReAk( )+Bk( ),bk( )=ImBk( )–Ak( ),
Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 3 o 10
bk( )=ImAk( )+Bk( ),bk( )=ReAk( )–Bk( ),
h( ,x,y,...,ym)=Reg( ,x+ix,y +iy,...,ym+iym),
h( ,x,y,...,ym)=Img( ,x+ix,y +iy,...,ym+iym),
equa ion() can be w i en in he eal o m () as well.
The equi alence o he dynamical in a ian s and asymp o ic p ope ies o he solu ions
o he eal sys em () and he complex equa ion ()isshownin[] o hesimplecase
co e ing o dina y diffe en ial equa ions.
In his pape we conside ()in hecasewhen
limin
→∞ a( )–b( )> ()
and s udy he beha io o he solu ions o () unde his assump ion, which gene ally
means ha de A( )> o sufficien lyla ge.Thissi ua ionco esponds o hecasewhen
he equilib ium poin  o he au onomous homogeneous sys em
x=Ax,()
whe e Ais supposed o be a egula cons an ma ix, is a cen e , a ocus o a node. Such a
si ua ion has some geome ical aspec s, which a e used in an analysis o he ans o med
equa ion(). See [] o mo e de ails.
Fu he , we suppose ha () sa isfies he uniqueness p ope y o solu ions.
2P elimina ies
Th oughou his pape we will assume ha
limin
→∞ a( )–b( )>, ≥θk( )≥ – o ≥ + ,()
whe e >  is a cons an , which means ha he delays θka e bounded. This is he same
caseasconside edin[].Simila esul s o acasediffe en om()we eob ainedin[].
Then he e a e numbe s T≥ + and μ>such ha
a( )>b( )+μ o ≥T.()
Deno e
c( )=¯
a( )b( )
|a( )|,γ( )=a( )+a( )–b( )()
and
ϑ( )=Re(γ( )γ( )–¯
c( )c( ))–|γ( )c( )–γ( )c( )|
γ( )–|c( )|,
α( )=–b( )
a( )sgnRea( ).
()
Mo eo e , we assume he ollowing condi ions o be alid:
Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 4 o 10
(i) The numbe s T≥ + and μ>  sa is y condi ion ().
(ii)The ea e unc ions,κ,κk:[T,∞)→R,whe eiscon inuouson[T,∞),such ha
γ( )g( ,z,w,...,wm)+c( )¯
g( ,z,w,...,wm)
≤κ( )γ( )z+c( )¯
z+m

k=
κk( )γθk( )wk+cθk( )¯
wk+( )
o ≥T,z,wk∈C(k=,...,m).
(iin)The ea enumbe sRn≥ and unc ions κn,κnk :[T,∞)→Rsa is ying he in-
equali y
γ( )g( ,z,w,...,wm)+c( )¯
g( ,z,w,...,wm)
≤κn( )γ( )z+c( )¯
z+m

k=
κnk( )γθk( )wk+cθk( )¯
wk
o ≥τn≥T,|z|+m
k= |wk|>Rn.
(iii) The unc ion β∈ACloc([T,∞),R
–)issuch ha
θ
k( )β( )≤–λk( )a.e.on[T,∞), ()
whe e λkis gi en o ≥Tby
λk( )=κk( )+Ak( )+Bk( )γ( )+|c( )|
γ(θk( ))–|c(θk( ))|.()
(iiin) The unc ion βn∈ACloc([T,∞),R
–)issuch ha
θ
k( )βn( )≤–λnk( )a.e.on[τn,∞), ()
whe e λnk is gi en o ≥Tby
λnk( )=κnk( )+Ak( )+Bk( )γ( )+|c( )|
γ(θk( ))–|c(θk( ))|.()
(i n) The unc ion Λnis eal locally Lebesgue in eg able and he inequali ies β
n( )≥
Λn( )βn( ), Θn( )≥Λn( ) a e sa isfied o almos all ∈[τn,∞), whe e Θnis gi en by
Θn( )=α( )Rea( )+ϑ( )–κn( )+mβn( ).
Fu he mo e, deno e
Θ( )=α( )Rea( )+ϑ( )–κ( ). ()
3 Main esul s
Fi s o all, we ecall he wo esul s om [].
Lemma  Le he assump ions (i), (ii), (iii), (i )be ulfilled o some τ≥T.Suppose
he e exis ≥τand ν∈(–∞,∞)such ha
in
≥ 

Λ(s)ds–lnγ( )+c( )≥ν.()
Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 5 o 10
I z( )is any solu ion o () sa is ying
min
θ( )≤s≤ z(s)>R,Δ( )>Re–ν, ()
whe e
θ( )= min
k=,...,mθk( ),
Δ( )=γ( )–c( )z( )+β( )max
θ( )≤s≤ z(s)m

k= 
θk( )γ(s)+c(s)ds,
hen
z( )≥Δ( )
γ( )+|c( )|exp

Λ(s)ds()
o all ≥  o which z( )is defined.
Lemma Le he condi ions (i), (ii), (iii) be ulfilled and Λ,θ
k(k=,...,m)be con inuous
unc ions such ha inequali yΛ( )≤Θ( )holdsa.e.on [T,∞),whe e Θisdefinedby().
Suppose ha ξ:[T– ,∞)→Ris a con inuous unc ion such ha
Λ( )+β( )m

k=
θ
k( )exp–
θk( )ξ(s)ds–ξ( )>( )C– exp–
Tξ(s)ds()
o ∈[T,∞]and some cons an C >.Then he e exis s a >T and a solu ion z( )o
() sa is ying
z( )≤C
γ( )–|c( )|exp
Tξ(s)ds()
o ≥ .
I we combine he p e ious wo esul s, we a e able o p o e he ollowing heo em,
which is he undamen al esul o his pape .
Theo em  Assume ha he hypo heses (i), (ii), (iin), (iii), (iiin), (i n)a e alid o T ≤τn,
whe e <Rn,n∈N,in n∈NRn=.Suppose ha θ
k,Λa e con inuous unc ions such ha
inequali yΘ( )≥Λ( )issa isfiedalmos e e ywhe eon[T,∞),whe eΘ( )=α( )Rea( )+
ϑ( )–κ( ).Le ξ:[T– ,∞)→Rbe a con inuous unc ion sa is ying he inequali y
( )C– exp–
Tξ(s)ds<Λ( )+β( )m

k=
θ
k( )exp–
θk( )ξ(s)ds–ξ( )()
o some cons an C >and ∈[T,∞). Assume
in
τn≤s≤ <∞
sΛn(σ)dσ–lnγ( )+c( )≥ν,()

Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 6 o 10
limsup
→∞ 
TΛn(s)–ξ(s)ds+ln γ( )–|c( )|
γ( )+|c( )|=∞,()
lim
→∞βn( )max
θ( )≤s≤
exp[s
Tξ(σ)dσ]
γ(s)–|c(s)|
m

k= 
θk( )γ(s)+c(s)ds=, ()
o n ∈N,whe e ν∈(–∞,∞)and θ( )=mink=,...,mθk( ). Then he e is a solu ion z( )o
() wi h he p ope y
lim
→∞ min
θ( )≤s≤ z(s)=. ()
P oo Using Lemma we ob ain he exis ence o T≤ and a solu ion z( )o ()sa is-
ying o ≥  he inequali y
z( )≤C
γ( )–|c( )|exp
Tξ(s)ds.()
F om ()wege
in
τ≤ <∞
τ
ΛN(s)ds–lnγ( )+c( )≥ν>–∞.
Lemma yields
Ψ(τ)
γ( )+|c( )|exp
τ
ΛN(s)ds≤z( )()
o τ≤ ,whe eΨis gi en by
Ψ(τ)=γ(τ)–c(τ)z(τ)+βN(τ)max
θ(τ)≤s≤τz(s)m

k= τ
θk(τ)γ(s)+c(s)ds.
Assume ha () does no hold. This implies he exis ence o ε>  sa is ying
limsup →∞minθ( )≤s≤ |z(s)|>ε.We akeN∈Nsuch ha max{RN,
μRNe–ν}<ε.Then
maxRN,
μRNe–ν<min
θ(τ)≤s≤τz(s)()
holds o some τ>max{T,τN, }.Taking() in o accoun we may assume ha
βN(τ)Cmax
θ(τ)≤s≤τ
exp[s
Tξ(σ)dσ]
γ(s)–|c(s)|
m

k= τ
θk(τ)γ(s)+c(s)ds<
RNe–ν.()
Hence, wi h espec o (), (), (), (), (), and he nonposi i eness o βN,weob ain
γ(τ)–c(τ)z(τ)+βN(τ)max
θ(τ)≤s≤τz(s)m

k= τ
θk(τ)γ(s)+c(s)ds
≥γ(τ)–c(τ)z(τ)+βN(τ)Cmax
θ(τ)≤s≤τ
exp[s
Tξ(σ)dσ]
γ(s)–|c(s)|
×m

k= τ
θk(τ)γ(s)+c(s)ds≥μ
μRNe–ν–
RNe–ν>RNe–ν.
Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 7 o 10
The inequali ies ()and() gi e he es ima ion
Ψ(τ)
γ( )+|c( )|exp
τ
ΛN(s)ds≤C
γ( )–|c( )|exp
Tξ(s)ds,
which means ha

TΛN(s)–ξ(s)ds+ln γ( )–|c( )|
γ( )+|c( )|≤τ
TΛN(s)ds–lnC–Ψ(τ)
o τ≤ , which is in con adic ion o (). The p oo is comple e. 
Rema k  Theo em  co e s mo e gene al si ua ions han Theo em  in [], whe e he
diffe en undamen al assump ion limin →∞(|Ima( )|–|b( )|)>issupposed ohold.
Indeed, i we ake o example a( )≡+iand b( )≡i, hencondi ion()in hispape is
sa isfied bu he condi ion limin →∞(|Ima( )|–|b( )|)> is no alid.
The ollowingnon i ialexamplewascons uc ed oillus a eanapplica iono he he-
o e ical esul p esen ed in Theo em .
Example  Conside he wo-dimensional sys em o he nonlinea delayed diffe en ial
equa ions
x( )=x( )+ 
x( )–x( )+ m

k=

me– x –e–k +e– ,()
x( )=x( )+x( )+ 
x( )– m

k=

me– x –e–k .()
This sys em can be w i en in ma ix o m (), whe e
A( )=+ 
–
+

,Bk( )= 
me– 
e
– ,and
h ,x( ),xθ( ),...,xθm( )=e–
.
Following ou app oach, we use a ans o ma ion in o he complex plane and ob ain he
delayed diffe en ial equa ion () wi h complex- alued coefficien s,
z( )=(+i)z( )+i¯
z( )+ 
z( )+ m

k=

me– ¯
z –e–k +e– ,()
whe e a( )≡+i,b( )≡i,Ak( )≡, Bk( )≡, θk( )= –e
–k o k= ,...,m,
g( ,z,w,...,wm)= 
z+m
k= 
me– wk+e– .
Ob iously –≤θk( )≤ and θ
k( )=+ke–k ≥> o ≥.
Suppose =andT≥. Then γ( )=|a( )|+|a( )|–|b( )|≡+
√, c( )=
¯
a( )b( )/|a( )|≡+i
.
Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 8 o 10
Fu he ,
γ( )g( ,z,w,...,wm)+c( )¯
g( ,z,w,...,wm)
≤γ+|c|
γ–|c|
γ( )z+c( )¯
z+γ+|c|
γ–|c|
m

k= 
me– γθk( )wk+cθk( )¯
wk+e–
=√
√

γ( )z+c( )¯
z+√
√
e–
m
m

k= γθk( )wk+cθk( )¯
wk+e– .
Thus condi ions (i) and (ii) a e ulfilled wi h κ( )≡√
√,κk( )=e– √
m√,and( )=e– .To
mee condi ion (iin), we es ima e o Rn=
n,τn=n,and|z|>Rn
γ( )g( ,z,w,...,wm)+c( )¯
g( ,z,w,...,wm)
≤γ+|c|
γ–|c|
+( )
Rnγ( )z+c( )¯
z
+γ+|c|
γ–|c|
m

k= 
me– γθk( )wk+cθk( )¯
wk
=√
√
+ne– γ( )z+c( )¯
z+√
√
e–
m
m

k= γθk( )wk+cθk( )¯
wk,
whe e κn( )=√
√[
+ne– ]andκnk( )=√
√e–
m.
Condi ion (iii) holds wi h
–λk( )θ
k( )– =–e– √
m√

+ke–k ≥–e– √
m√=β( ).
Condi ion (iiin)issa isfied o
βn( )=–e–( –)
m≤–e– √
m√=–λnk( )θ
k( )–.
We ge condi ion (i n) by se ing
n( )=Θn( )=
–√
√
+ne– –e–( –) >.
Fu he , we pu ( )=Θ( )=
–√
√.
Then condi ion ()holds o
ξ( )=
–√
 √–√
√
e–
m–e–
and ξ( )> o ≥T=.
Nowi isno difficul o e i ycondi ions()and(),sincen( )–ξ( )>andn( )>
 o n∈N.In es iga ing he ac o so hep oduc in pa en hesesin (),we come o he
Rebenda and Šma da Ad ances in Diffe ence Equa ions (2015) 2015:349 Page 9 o 10
conclusion ha
βn=Oe– ,exps
Tξ(σ)dσ=Oe. 
and
m

k= 
θk( )γ(s)+c(s)ds=Om

k=
e–k =Oe– .
Consequen ly, he p oduc o hese ac o s is asymp o ically equal o O(e–δ ), whe e δ>
and huscondi ion ()is sa isfied. All assump ionso Theo em a e ulfilled and we can
conclude ha he e exis >  and a solu ion z( )o ()sa is ying() o ≥ .
Rema k  This esul is sligh ly su p ising. Howe e , i is in good ag eemen wi h he
well-known ac ha in oducing delay in o an uns able sys em wi hou delay can cause a
changeo beha io o hesys em.Suchsi ua ionsa edesc ibedandco esponding esul s
a e o mula ed e.g.in [].
4Conclusion
We p o ed an in e es ing esul abou he s abili y o wo-dimensional sys ems wi h
boundeddelays.Sufficien condi ions o hes abili yo ano iginallyuns ablesys emwe e
p esen ed. The esul is in pe ec ag eemen wi h he esul s s a ed and p o ed in he es-
ablished li e a u e. An example showed how his esul can be used in p ac ice.
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
The au ho s ha e made con ibu ions o he same significance. All au ho s ead and app o ed he final manusc ip .
Acknowledgemen s
The wo k o he au ho s was ealized in CEITEC - Cen al Eu opean Ins i u e o Technology wi h esea ch in as uc u e
suppo ed by he p ojec CZ.1.05/1.1.00/02.0068 financed om Eu opean Regional De elopmen Fund and he second
au ho was suppo ed by he p ojec FEKT-S-11-2-921 o Facul y o Elec ical Enginee ing and Communica ion, B no
Uni e si y o Technology. This suppo is g a e ully acknowledged.
Recei ed: 15 Sep embe 2015 Accep ed: 5 No embe 2015
Re e ences
1. Baculíko á, B, Džu ina, J, Rogo chenko, YV: Oscilla ion o hi d o de inomial delay diffe en ial equa ions. Appl. Ma h.
Compu . 218(13), 7023-7033 (2012)
2. Be ezansky, L, B a e man, E: On nonoscilla ion and s abili y o sys ems o diffe en ial equa ions wi h a dis ibu ed
delay. Au oma ica 48(4), 612-618 (2012)
3. Be ezansky, L, Diblík, J, S oboda, Z, Šma da, Z: Simple uni o m exponen ial s abili y condi ions o a sys em o linea
delayed diffe en ial equa ions. Appl. Ma h. Compu . 250(1), 605-614 (2012)
4. B a e man, E, Be ezansky, L: New s abili y condi ions o linea diffe en ial equa ions wi h se e al delays. Abs . Appl.
Anal. 2011, A icle ID 178568 (2011)
5. Bu on, TA: S abili y by Fixed Poin Theo y o Func ional Diffe en ial Equa ions, 348 pp. Do e , New Yo k (2006)
6. Olach, R: Posi i e pe iodic solu ions o delay diffe en ial equa ions. Appl. Ma h. Le . 26, 1141-1145 (2013)
7. Šma da, Z, Rebenda, J: Asymp o ic beha iou o a wo-dimensional diffe en ial sys em wi h a fini e numbe o
noncons an delays unde he condi ions o ins abili y. Abs . Appl. Anal. 2012, A icle ID 952601 (2012)
8. S e i´
c, S: On q-diffe ence asymp o ic solu ions o a sys em o nonlinea unc ional diffe en ial equa ions. Appl. Ma h.
Compu . 219(15), 8295-8301 (2013)
9. S e i´
c, S: Solu ions con e ging o ze o o some sys ems o nonlinea unc ional diffe en ial equa ions wi h i e a ed
de ia ing a gumen . Appl. Ma h. Compu . 219(8), 4031-4035 (2012)
10. Kalas, J: Asymp o ic beha iou o a wo-dimensional diffe en ial sys em wi h delay unde he condi ions o ins abili y.
Nonlinea Anal. 62, 207-224 (2005)