Hindawi Publishing Co po a ion
Abs ac and Applied Analysis
Volume 2013, A icle ID 931493, 10 pages
h p://dx.doi.o g/10.1155/2013/931493
Resea ch A icle
Rep esen a ion o a Solu ion o he Cauchy P oblem o
an Oscilla ing Sys em wi h Mul iple Delays and Pai wise
Pe mu able Ma ices
Jose Diblík,1Michal FeIkan,2,3 and Michal Pospíšil4
1Depa men o Ma hema ics, Facul y o Elec ical Enginee ing and Communica ion, B no Uni e si y o Technology,
Technick´
a 3058/10, 616 00 B no, Czech Republic
2Depa men o Ma hema ical Analysis and Nume ical Ma hema ics, Comenius Uni e si y, Mlynsk´
adolina,
842 48 B a isla a, Slo akia
3Ma hema ical Ins i u e o Slo ak Academy o Sciences, ˇ
S e ´
aniko a 49, 814 73 B a isla a, Slo akia
4Cen e o Resea ch and U iliza ion o Renewable Ene gy, Facul y o Elec ical Enginee ing and Communica ion,
B no Uni e si y o Technology, Technick´
a 3058/10, 616 00 B no, Czech Republic
Co espondence should be add essed o Michal Posp´
ıˇ
sil; [email p o ec ed] .cz
Recei ed 13 Janua y 2013; Accep ed 19 Ap il 2013
Academic Edi o : Jaan Janno
Copy igh © 2013 Jose Dibl´
ık e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License,
which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
Nonhomogeneous sys em o linea di e en ial equa ions o second o de wi h mul iple di e en delays and pai wise pe mu able
ma ices de ining he linea pa s is conside ed. Solu ion o co esponding ini ial alue p oblem is ep esen ed using ma ix
polynomials.
1. In oduc ion
Mo i a ed by delayed exponen ial ep esen ing a solu ion
o a sys em o di e en ial o di e ence equa ions wi h
oneo mul iple ixedo a iabledelays[1–6], which has
many applica ions in heo y o con ollabili y, asymp o ic
p ope ies, bounda y- alue p oblems, and so o h [3–5,7–
15], we ex ended ep esen a ion o a solu ion o a sys em o
di e en ial equa ions o second o de wi h delay [1]
𝑥(𝑡)=−𝐵2𝑥(𝑡−𝜏)(1)
o he case o wo delays
𝑥(𝑡)=−𝐵2
1𝑥(𝑡−𝜏1)−𝐵2
2𝑥(𝑡−𝜏2), (2)
whe e he linea pa s we e gi en by pe mu able ma ices [16].
Equa ions (1), (2), and he below-s a ed (11)wi h𝑓≡0a e
gene aliza ions o he scala equa ion
𝑥(𝑡)=−𝑏2𝑥(𝑡)(3)
ep esen ing linea oscilla o , o 𝑁-dimensional space wi h
oneo mul iple ixeddelays.Clea ly,eachsolu iono hela e
equa ion is oscilla ing whene e 0 =𝑏∈R. Analogically,
(1)wi h𝑥∈R𝑁can ha e a leas one oscilla ing solu ion
whene e 𝑁is odd. Indeed, i 𝐵is 𝑁×𝑁ma ix, 𝑁≥3is
odd, and 𝐵has a simple eal nonze o eigen alue 𝜆, hen he e
exis s a egula ma ix 𝑆such ha 𝑆−1𝐵𝑆=𝐽=(𝜆0
0
𝐽)whe e
𝐽is (𝑁−1)×(𝑁−1)ma ix. On le ing 𝑥=𝑆𝑦,onege s
𝑦=−𝐽2𝑦(𝑡−𝜏)(4)
o ew i es as he sys em
𝑦1=−𝜆2𝑦1(𝑡−𝜏),
𝑦2=−𝐽2𝑦2(𝑡−𝜏),(5)
whe e 𝑦=(𝑦
1,𝑦2)∈R×R𝑁−1.No e ha he i s
column Vo 𝑆is he eigen ec o o 𝐵co esponding o 𝜆.
Clea ly, i solu ion 𝑦1o (5)isoscilla ing, hensolu ion𝑦o
(4) is oscilla ing in he i s coo dina e whene e i s ini ial
2Abs ac and Applied Analysis
condi ion sa is ies {𝑦(𝑡)|𝑡∈[−𝜏,0]}⊂R×{0}𝑁−1.
Consequen ly, solu ion 𝑥o (1)isoscilla inginspan{V}
whene e {𝑥(𝑡)|𝑡∈[−𝜏,0]}⊂span{V}.Taking𝑦1(𝑡)=𝑒𝜇𝑡,
one ob ains cha ac e is ic equa ion 𝜇2=−𝜆
2𝑒−𝜇𝜏 o (5),
which has solu ions 𝜇1,2 =𝛼±𝚤𝛽∈Cwi h 𝛽 =0.Thus,𝑦1
is oscilla ing.
On he o he hand, he e can exis a nonoscilla ing
solu ion o he sys em (1)whene e 𝑥∈R𝑁and 𝑁is e en.
Fo ins ance, i 𝑁=2and 𝐵=(01
−1 0 ), hen(1)has he o m
𝑥(𝑡)=𝑥(𝑡−𝜏)(6)
wi h 𝑥∈R2,which,ob iously,doesno ha eanoscilla ing
solu ion sa is ying nonoscilla ing ini ial condi ion. Simila ly,
i can be shown ha sys em wi h odd dimension can possess
a nonoscilla ing solu ion sa is ying an app op ia e ini ial
condi ion.
Fo simplici y, we call he gene aliza ions (1), (2), and (11)
wi h 𝑓≡0,o scala equa ion(3), oscilla ing al hough hei
solu ions do no always ha e o be oscilla ing. Ne e heless, a
he end o his pape , in Co olla y 8 we s a e he ep esen a-
ion o a solu ion o mo e gene al sys em (86)wi hou squa es
o ma ices.
We no e ha he delayed ma ix exponen ial om [1–5]
as well as he ep esen a ion o a solu ion o second-o de
di e en ial equa ions de i ed in [1,16]andin hispape can
lead o new esul s in nonlinea bounda y alue p oblems o
impulsi e unc ional di e en ial equa ions conside ed in [17]
o s ochas ic delayed di e en ial equa ions om [18].
So, in he p esen pape , we ex end ou esul om
[16] o h ee and mo e delays by he assump ion o pai -
wise pe mu able ma ices de ining linea pa s. By such an
assump ion, we a e able o cons uc ma ix unc ions sol ing
homogeneous sys em o di e en ial equa ions o second
o de wi hanynumbe o ixeddelays,and,consequen ly,
we use hese unc ions o ep esen a solu ion o he co -
esponding nonhomogeneous ini ial alue p oblem. As will
be shown in he nex sec ions, ex ending om wo o mo e
delays b ings many echnical di icul ies, o example, he
use o mul inomial coe icien s. Na u ally, he esul s o he
p esen pape hold wi h one o wo di e en delays as well.
Howe e , hese cases can by s udied in a simple way, which
was al eady done in [1,16]. Thus, we ocus ou a en ion on
hecaseo h eeandmo edi e en delays.
Fi s , we ecall ou esul om [16].
Theo em 1. Le 𝜏1,𝜏2>0,𝜏:=max{𝜏1,𝜏2},and𝜑∈
𝐶1([−𝜏,0],R𝑁).Le 𝐵1,𝐵2be 𝑁×𝑁pe mu able ma ices; ha
is, 𝐵1𝐵2=𝐵2𝐵1,andle 𝑓:[0,∞)→R𝑁be a gi en unc ion.
Solu ion 𝑥(𝑡)o
𝑥(𝑡)=−𝐵2
1𝑥(𝑡−𝜏1)−𝐵2
2𝑥(𝑡−𝜏2)+𝑓(𝑡)(7)
sa is ying ini ial condi ion
𝑥(𝑡)=𝜑(𝑡),
𝑥(𝑡)=𝜑(𝑡),−𝜏≤𝑡≤0 (8)
has he o m
𝑥(𝑡)={
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
𝜑(𝑡),−𝜏≤𝑡<0,
X(𝑡)𝜑(0)+Y(𝑡)𝜑(0)
−𝐵2
1∫0
−𝜏1
Y(𝑡−𝜏1−𝑠)𝜑(𝑠)𝑑𝑠
−𝐵2
2∫0
−𝜏2
Y(𝑡−𝜏2−𝑠)𝜑(𝑠)𝑑𝑠
+∫𝑡
0Y(𝑡−𝑠)𝑓(𝑠)𝑑𝑠, 0≤𝑡, (9)
whe e
X(𝑡)=X𝐵2
1,𝐵2
2
𝜏1,𝜏2(𝑡)
:= ∑
𝑖,𝑗≥0
𝑖𝜏1+𝑗𝜏2≤𝑡(−1)𝑖+𝑗
×(𝑖+𝑗
𝑖)𝐵2𝑖
1𝐵2𝑗
2(𝑡−𝑖𝜏1−𝑗𝜏2)2(𝑖+𝑗)
(2(𝑖+𝑗))! ,
Y(𝑡)=Y𝐵2
1,𝐵2
2
𝜏1,𝜏2(𝑡)
:= ∑
𝑖,𝑗≥0
𝑖𝜏1+𝑗𝜏2≤𝑡(−1)𝑖+𝑗
×(𝑖+𝑗
𝑖)𝐵2𝑖
1𝐵2𝑗
2(𝑡−𝑖𝜏1−𝑗𝜏2)2(𝑖+𝑗)+1
(2(𝑖+𝑗)+1)! .
(10)
We will deno e Θand 𝐸 he 𝑁×𝑁ze o and iden i y
ma ix, espec i ely.
2. Sys ems wi h Mul iple Delays
In his sec ion, we de i e he ep esen a ion o a solu ion o
𝑥(𝑡)=−𝐵2
1𝑥(𝑡−𝜏1)−⋅⋅⋅−𝐵2
𝑛𝑥(𝑡−𝜏𝑛)+𝑓(𝑡)(11)
sa is ying he ini ial condi ion (8), whe e 𝑛≥3,𝜏1,...,𝜏𝑛>0,
𝜏:=max𝑖=1,...,𝑛𝜏𝑖,𝐵1,...,𝐵𝑛a e 𝑁×𝑁pai wise pe mu able
ma ices; ha is, 𝐵𝑖𝐵𝑗=𝐵𝑗𝐵𝑖 o each 𝑖,𝑗∈{1,...,𝑛},𝜑∈
𝐶1([−𝜏,0],R𝑁),and𝑓:[0,∞)→R𝑁a e gi en unc ions.
The solu ion 𝑥(𝑡)will be ep esen ed using ma ix unc ions
analogical o (10)andwillbes a edinSec ion 3.Weno e ha
hesamep oblemswi h𝑛=1,2we e s udied in [1,16].
F om now on, we assume he p ope y o emp y sum and
emp y p oduc ; ha is,
∑
𝑖∈0𝑓(𝑖)=0, ∑
𝑖∈0𝐹(𝑖)=Θ,
∏
𝑖∈0𝑓(𝑖)=1, ∏
𝑖∈0𝐹(𝑖)=𝐸 (12)
o any unc ion 𝑓and ma ix unc ion 𝐹, whe he hey a e
de ined o no o indica ed a gumen .
Abs ac and Applied Analysis 3
We ecall ha (𝑗1,...,𝑗𝑛)!is a mul inomial coe icien [19]
gi en by
(𝑗1,...,𝑗𝑛)!=(𝑗1+⋅⋅⋅+𝑗𝑛)!
𝑗1!⋅⋅⋅𝑗𝑛!.(13)
No e ha i 𝑛=2, hen(𝑗1,𝑗2)=(𝑗1+𝑗2
𝑗1)and (20) coincides
wi h (10).
We will need a p ope y o mul inomial coe icien s
desc ibed in he nex lemma.
Lemma 2. Le 𝑛≥2be ixed. Then
(𝑖1,𝑖2,...,𝑖𝑛)!=(𝑖1−1,𝑖2,...,𝑖𝑛)!
+(𝑖1,𝑖2−1,𝑖3,...,𝑖𝑛)!+(𝑖1,...,𝑖𝑛−1,𝑖𝑛−1)!
(14)
o any 𝑖1,...,𝑖𝑛≥1.
P oo . I 𝑛=2, hen he s a emen ollows om he p ope y
o binomial coe icien s:
(𝑖1,𝑖2)!=(𝑖1+𝑖2
𝑖1)=(𝑖1+𝑖2−1
𝑖1−1)+(𝑖1+𝑖2−1
𝑖1)
=(𝑖1−1,𝑖2)!+(𝑖1,𝑖2−1)!. (15)
Le he s a emen be ue o 𝑛−1. Nex , we use he p ope y
o mul inomial coe icien
(𝑖1,𝑖2,𝑖3,...,𝑖𝑛)!=(𝑖1+𝑖2,𝑖3,...,𝑖𝑛)!(𝑖1,𝑖2)! (16)
wi h induc i e hypo hesis o de i e
(𝑖1,𝑖2,𝑖3,...,𝑖𝑛)!
=[(𝑖1+𝑖2−1,𝑖3,...,𝑖𝑛)!+(𝑖1+𝑖2,𝑖3−1,...,𝑖𝑛)!
+⋅⋅⋅+(𝑖1+𝑖2,𝑖3,...,𝑖𝑛−1)!](𝑖1,𝑖2)!. (17)
Clea ly, om (16), we ge
(𝑖1+𝑖2,𝑖3−1,...,𝑖𝑛)!(𝑖1,𝑖2)!=(𝑖1,𝑖2,𝑖3−1,...,𝑖𝑛)!,
.
.
.
(𝑖1+𝑖2,𝑖3,...,𝑖𝑛−1)!(𝑖1,𝑖2)!=(𝑖1,𝑖2,𝑖3,...,𝑖𝑛−1)!.(18)
Applying he case 𝑛=2(p ope yo binomialcoe icien )
and (16), we ge
(𝑖1+𝑖2−1,𝑖3,...,𝑖𝑛)!(𝑖1,𝑖2)!
=(𝑖1+𝑖2−1,𝑖3,...,𝑖𝑛)![(𝑖1−1,𝑖2)!+(𝑖1,𝑖2−1)!]
=(𝑖1−1,𝑖2,𝑖3,...,𝑖𝑛)!+(𝑖1,𝑖2−1,𝑖3,...,𝑖𝑛)!. (19)
Pu ing (18)and(19)in(17), we ob ain ha he s a emen
holds o 𝑛and he p oo is comple e.
In u he wo k, we w i e ({𝑗 | 𝑗 ∈ 𝑀})! o he
mul inomial coe icien o elemen s o he ini e se 𝑀,and
(𝑖,{𝑗|𝑗∈𝑀})! o he mul inomial coe icien o 𝑖and
elemen s o he ini e se 𝑀; o example, i 𝑀={1,2}, hen
(𝑎,{𝑗|𝑗∈𝑀})!=(𝑎,1,2)!. Fo he comple eness, we de ine
({𝑗|𝑗∈0})!:=1.
De ine he unc ions X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛,Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛:R→𝐿(R𝑁)as
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)
:= ∑
𝑗1,...,𝑗𝑛≥0
𝑗1𝜏1+⋅⋅⋅+𝑗𝑛𝜏𝑛≤𝑡(−1)𝑗1+⋅⋅⋅+𝑗𝑛(𝑗1,...,𝑗𝑛)!
×𝑛
∏
𝑖=1𝐵2𝑗𝑖
𝑖(𝑡−𝑗1𝜏1−⋅⋅⋅−𝑗𝑛𝜏𝑛)2(𝑗1+⋅⋅⋅+𝑗𝑛)
(2(𝑗1+⋅⋅⋅+𝑗𝑛))! ,
Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)
:= ∑
𝑗1,...,𝑗𝑛≥0
𝑗1𝜏1+⋅⋅⋅+𝑗𝑛𝜏𝑛≤𝑡(−1)𝑗1+⋅⋅⋅+𝑗𝑛(𝑗1,...,𝑗𝑛)!
×𝑛
∏
𝑖=1𝐵2𝑗𝑖
𝑖(𝑡−𝑗1𝜏1−⋅⋅⋅−𝑗𝑛𝜏𝑛)2(𝑗1+⋅⋅⋅+𝑗𝑛)+1
(2(𝑗1+⋅⋅⋅+𝑗𝑛)+1)!
(20)
o any 𝑡∈R.
We will need unc ions X𝐵2
𝜏,Y𝐵2
𝜏:R→𝐿(R𝑁) o 𝜏>0
and 𝑁×𝑁complex ma ix 𝐵(c . [16]) de ined as
X𝐵2
𝜏(𝑡):=∑
𝑖≥0
𝑖𝜏≤𝑡(−1)𝑖𝐵2𝑖 (𝑡−𝑖𝜏)2𝑖
(2𝑖)!,
Y𝐵2
𝜏(𝑡):=∑
𝑖≥0
𝑖𝜏≤𝑡(−1)𝑖𝐵2𝑖 (𝑡−𝑖𝜏)2𝑖+1
(2𝑖+1)!(21)
wi h he p ope ies
X𝐵2
𝜏(𝑡)=−𝐵2Y𝐵2
𝜏(𝑡−𝜏),X𝐵2
𝜏(𝑡)=−𝐵2X𝐵2
𝜏(𝑡−𝜏),
Y𝐵2
𝜏(𝑡)=X𝐵2
𝜏(𝑡),Y𝐵2
𝜏(𝑡)=−𝐵2Y𝐵2
𝜏(𝑡−𝜏)(22)
o any 𝑡∈R, conside ing he one-sided de i a i es a −𝜏,0.
Some o p ope ies o unc ions X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛and Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛a e
concluded in Lemma 4,bu op o ei wewillneed henex
lemma.
Lemma 3. Le 𝑛≥1and 𝜏1,...,𝜏𝑛>0.Le 𝐵1,...,𝐵𝑛be
𝑁×𝑁pai wise pe mu able ma ices, ha is, 𝐵𝑖𝐵𝑗=𝐵𝑗𝐵𝑖 o
each 𝑖,𝑗∈{1,...,𝑛}.Then o any𝑡∈R,
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=∑
𝑀⊂{1,...,𝑛}𝑆𝑀(𝑡),
Y𝐵2
1,...𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=∑
𝑀⊂{1,...,𝑛}𝑆𝑀(𝑡),(23)
4Abs ac and Applied Analysis
whe e he sums a e aken o e all subse s o {1,...,𝑛}including
he i ial ones, and
𝑆𝑀(𝑡):= ∑
𝑗𝑖≥1,𝑖∈𝑀
∑𝑖∈𝑀 𝑗𝑖𝜏𝑖≤𝑡(−1)∑𝑖∈𝑀 𝑗𝑖({𝑗𝑖|𝑖∈𝑀})!
×∏
𝑖∈𝑀𝐵2𝑗𝑖
𝑖(𝑡−∑𝑖∈𝑀 𝑗𝑖𝜏𝑖)2∑𝑖∈𝑀 𝑗𝑖
(2∑𝑖∈𝑀 𝑗𝑖)! ,(24)
𝑆𝑀(𝑡):= ∑
𝑗𝑖≥1,𝑖∈𝑀
∑𝑖∈𝑀 𝑗𝑖𝜏𝑖≤𝑡(−1)∑𝑖∈𝑀 𝑗𝑖({𝑗𝑖|𝑖∈𝑀})!
×∏
𝑖∈𝑀𝐵2𝑗𝑖
𝑖(𝑡−∑𝑖∈𝑀 𝑗𝑖𝜏𝑖)2∑𝑖∈𝑀 𝑗𝑖+1
(2∑𝑖∈𝑀 𝑗𝑖+1)! .(25)
P oo . Deno e N0,N he se o all nonnega i e, posi i e
in ege s, espec i ely; ha is, N0={0}∪N.Thus,weha e
he i ial iden i y
N0×⋅⋅⋅×N0
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝑛
=({0}×N0×⋅⋅⋅×N0
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝑛−1 )∪(N×N0×⋅⋅⋅×N0
⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟⏟
𝑛−1 )
=⋅⋅⋅= ⋃
𝑀1,...,𝑀𝑛∈{{0},N}𝑀1×⋅⋅⋅×𝑀𝑛.(26)
Analogically, o any 𝑡∈Reach 𝑛- uple 𝑗1,...,𝑗𝑛≥0
such ha ∑𝑛
𝑖=1 𝑗𝑖𝜏𝑖≤𝑡canbedi idedin wodis inc se so
𝑖-s so ha 𝑗𝑖≥1i 𝑖∈𝑀⊂{1,...,𝑛}and 𝑗𝑖=0i 𝑖∈
{1,...,𝑛} 𝑀.Tha is,𝑀deno es he se o all indices 𝑖such
ha 𝑗𝑖=0.Mo eo e ,∑𝑛
𝑖=1 𝑗𝑖𝜏𝑖=∑𝑖∈𝑀 𝑗𝑖𝜏𝑖. Acco dingly, we
can w i e
{(𝑗1,...,𝑗𝑛)∈N𝑛
0|𝑛
∑
𝑖=1𝑗𝑖𝜏𝑖≤𝑡}
=⋃
𝑀⊂{1,...,𝑛}{(𝑗1,...,𝑗𝑛)∈N𝑛
0|
𝑗𝑖=0∀𝑖∉𝑀,∑
𝑖∈𝑀 𝑗𝑖𝜏𝑖≤𝑡},
(27)
whe e he union is aken o e all subse s o {1,...,𝑛}
including he i ial ones. So, in he iew o de ini ion (20),
he s a emen o X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛 ollows.
S a emen o Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛canbep o edinasimila way.
Lemma 4. Le 𝑛≥3and 𝜏1,...,𝜏𝑛>0.Le 𝐵1,...,𝐵𝑛be
𝑁×𝑁pai wise pe mu able ma ices; ha is, 𝐵𝑖𝐵𝑗=𝐵𝑗𝐵𝑖 o
each 𝑖,𝑗∈{1,...,𝑛}. Then he ollowing holds o any 𝑡∈R:
(1) i 𝐵𝑖=Θ o some 𝑖∈{1,...,𝑛}, hen
X𝐵2
1,...,𝐵2
𝑖−1,𝐵2
𝑖,𝐵2
𝑖+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑖−1,𝜏𝑖,𝜏𝑖+1,...,𝜏𝑛(𝑡)=X𝐵2
1,...,𝐵2
𝑖−1,𝐵2
𝑖+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑖−1,𝜏𝑖+1,...,𝜏𝑛(𝑡),(28)
(2) i 𝜏𝑖=𝜏𝑘 o 𝑖<𝑘,𝑖,𝑘∈{1,...,𝑛}, hen
X𝐵2
1,...,𝐵2
𝑖−1,𝐵2
𝑖,𝐵2
𝑖+1,...,𝐵2
𝑘−1,𝐵2
𝑘,𝐵2
𝑘+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑖−1,𝜏𝑖,𝜏𝑖+1,...,𝜏𝑘−1,𝜏𝑘,𝜏𝑘+1,...,𝜏𝑛(𝑡)
=X𝐵2
1,...,𝐵2
𝑖−1,𝐵2
𝑖+𝐵2
𝑘,𝐵2
𝑖+1,...,𝐵2
𝑘−1,𝐵2
𝑘+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑖−1,𝜏𝑖,𝜏𝑖+1,...,𝜏𝑘−1,𝜏𝑘+1,...,𝜏𝑛(𝑡),(29)
(3) o any bijec i e mapping 𝜎:{1,...,𝑛}→{1,...,𝑛}
we ge
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=X𝐵2
𝜎(1),...,𝐵2
𝜎(𝑛)
𝜏𝜎(1),...,𝜏𝜎(𝑛) (𝑡),(30)
(4) aking he one-sided de i a i es a 0,𝜏1,...,𝜏𝑛, hen
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=−𝐵2
1X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡−𝜏1)
−⋅⋅⋅−𝐵2
𝑛X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡−𝜏𝑛), (31)
(5) conside ing he one-sided de i a i es a 0 ( hey bo h
equal Θ), hen
Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡).(32)
S a emen s (1)–(4) hold wi h Yins ead o X.
P oo . S a emen (1) ollows easily om de ini ion o X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛,
because Θ2𝑖 =𝐸i 𝑖=0and Θ2𝑖 =Θwhene e 𝑖>0. Nex , i
𝜏𝑖=𝜏𝑘, hen
∑
𝑗1,...,𝑗𝑛≥0
𝑗1𝜏1+⋅⋅⋅+𝑗𝑛𝜏𝑛≤𝑡𝐹(𝑗1,...,𝑗𝑛)
=∑
𝑗1,...,𝑗𝑖−1,𝑙,𝑗𝑖+1,...,𝑗𝑘−1,𝑗𝑘+1,...,𝑗𝑛≥0
𝑗1𝜏1+⋅⋅⋅+𝑗𝑖−1𝜏𝑖−1+𝑙𝜏𝑖+𝑗𝑖+1𝜏𝑖+1
+⋅⋅⋅+𝑗𝑘−1𝜏𝑘−1+𝑗𝑘+1𝜏𝑘+1+⋅⋅⋅+𝑗𝑛𝜏𝑛≤𝑡 ∑
𝑗𝑖,𝑗𝑘≥0
𝑗𝑖+𝑗𝑘=𝑙𝐹(𝑗1,...,𝑗𝑛)(33)
o any ma ix unc ion 𝐹.Thus,using hep ope yo
mul inomial coe icien (see (16))
(𝑗1,...,𝑗𝑛)!
=(𝑗1,...,𝑗𝑖−1,𝑗𝑖+𝑗𝑘,𝑗𝑖+1,...,𝑗𝑘−1,𝑗𝑘+1,...,𝑗𝑛)!(𝑗𝑖,𝑗𝑘)!,
(34)
Abs ac and Applied Analysis 5
o (2), we ob ain
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)
=∑
𝑗1,...,𝑗𝑖−1,𝑙,𝑗𝑖+1,...,𝑗𝑘−1,𝑗𝑘+1,...,𝑗𝑛≥0
𝑗1𝜏1+⋅⋅⋅+𝑗𝑖−1𝜏𝑖−1+𝑙𝜏𝑖+𝑗𝑖+1𝜏𝑖+1
+⋅⋅⋅+𝑗𝑘−1𝜏𝑘−1+𝑗𝑘+1𝜏𝑘+1+⋅⋅⋅+𝑗𝑛𝜏𝑛≤𝑡(−1)∑𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 𝑗𝑠+𝑙
×(𝑗1,...,𝑗𝑖−1,𝑙,𝑗𝑖+1,...,𝑗𝑘−1,𝑗𝑘+1,...,𝑗𝑛)!
×(∑
𝑗𝑖,𝑗𝑘≥0
𝑗𝑖+𝑗𝑘=𝑙 (𝑗𝑖,𝑗𝑘)!𝐵2𝑗𝑖
𝑖𝐵2𝑗𝑘
𝑘)
×∏
𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 𝐵2𝑗𝑠
𝑠(𝑡−∑𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 ∑𝑗𝑠𝜏𝑠−𝑙𝜏𝑖)2(∑𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 𝑗𝑠+𝑙)
(2(∑𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 ∑𝑗𝑠+𝑙))!
=∑
𝑗1,...,𝑗𝑖−1,𝑙,𝑗𝑖+1,...,𝑗𝑘−1,𝑗𝑘+1,...,𝑗𝑛≥0
𝑗1𝜏1+⋅⋅⋅+𝑗𝑖−1𝜏𝑖−1+𝑙𝜏𝑖+𝑗𝑖+1𝜏𝑖+1
+⋅⋅⋅+𝑗𝑘−1𝜏𝑘−1+𝑗𝑘+1𝜏𝑘+1+⋅⋅⋅+𝑗𝑛𝜏𝑛≤𝑡(−1)∑𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 𝑗𝑠+𝑙
×(𝑗1,...,𝑗𝑖−1,𝑙,𝑗𝑖+1,...,𝑗𝑘−1,𝑗𝑘+1,...,𝑗𝑛)!(𝐵2
𝑖+𝐵2
𝑘)𝑙
×∏
𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 𝐵2𝑗𝑠
𝑠(𝑡−∑𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 ∑𝑗𝑠𝜏𝑠−𝑙𝜏𝑖)2(∑𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 𝑗𝑠+𝑙)
(2(∑𝑠∈{1,...,𝑛}
𝑠 =𝑖,𝑘 ∑𝑗𝑠+𝑙))!
=X𝐵2
1,...,𝐵2
𝑖−1,𝐵2
𝑖+𝐵2
𝑘,𝐵2
𝑖+1,...,𝐵2
𝑘−1,𝐵2
𝑘+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑖−1,𝜏𝑖,𝜏𝑖+1,...,𝜏𝑘−1,𝜏𝑘+1,...,𝜏𝑛(𝑡).(35)
P ope y (3) is i ial.
Now, we p o e he s a emen (4). I 𝜏:=𝜏1=⋅⋅⋅=𝜏𝑛,
hen X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=X𝐵2
1+⋅⋅⋅+𝐵2
𝑛
𝜏(𝑡)
=−(𝐵2
1+⋅⋅⋅+𝐵2
𝑛)X𝐵2
1+⋅⋅⋅+𝐵2
𝑛
𝜏(𝑡−𝜏)
=−𝐵2
1X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡−𝜏1)
−⋅⋅⋅−𝐵2
𝑛X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡−𝜏𝑛)
(36)
by (2) and om he p ope y o X𝐵2
1+⋅⋅⋅+𝐵2
𝑛
𝜏(𝑡)(see (22)).
Hence, wi hou any loss o gene ali y, we assume ha
𝜏𝑖=𝜏𝑗 o each 𝑖 =𝑗,𝑖,𝑗∈{1,...,𝑛}(in he o he case, we
collec ma ices as s a ed in (2)). No e he case 𝑛=2was
p o ed in [16, Lemma 2.3.] Now, assume ha X𝐵2
1,...,𝐵2
𝑛−1
𝜏1,...,𝜏𝑛−1 (𝑡)
sol es 𝑥(𝑡)=−𝐵2
1𝑥(𝑡−𝜏1)−⋅⋅⋅−𝐵2
𝑛−1𝑥(𝑡−𝜏𝑛−1), (37)
ha is, ha he s a emen is ul illed o 𝑛−1di e en delays.
Le 𝜏𝑘:=max𝑖=1,...,𝑛𝜏𝑖.I 𝑡<𝜏𝑘, hen𝑡−𝜏𝑘<0, ha is,
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡−𝜏𝑘)=Θ, (38)
and om de ini ion (20)i holds
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=X𝐵2
1,...,𝐵2
𝑘−1,𝐵2
𝑘+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑘−1,𝜏𝑘+1,...,𝜏𝑛(𝑡)(39)
o such 𝑡.Consequen ly,
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=X𝐵2
1,...,𝐵2
𝑘−1,𝐵2
𝑘+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑘−1,𝜏𝑘+1,...,𝜏𝑛(𝑡)
=−∑
𝑖=1,...,𝑛
𝑖 =𝑘 𝐵2
𝑖X𝐵2
1,...,𝐵2
𝑘−1,𝐵2
𝑘+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑘−1,𝜏𝑘+1,...,𝜏𝑛(𝑡−𝜏𝑖)
=−𝑛
∑
𝑖=1𝐵2
𝑖X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡−𝜏𝑖)
(40)
by he induc i e hypo hesis.
Now, le 𝑡≥max𝑖=1,...,𝑛𝜏𝑖. Applying Lemma 3,wege
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=∑
𝑀⊂{1,...,𝑛}𝑆𝑀(𝑡)(41)
wi h 𝑆𝑀(𝑡)gi en by (24) and he sum aken o e all subse s
o {1,...,𝑛}including he i ial ones. No e ha
𝑆0(𝑡)=∑
𝑗𝑖≥1,𝑖∈0
0≤𝑡 (−1)0({𝑗𝑖|𝑖∈0})!𝐸(𝑡−0)0
0!
=∑
0≤𝑡𝐸=𝐸𝜒[0,∞) (𝑡)(42)
wi h a cha ac e is ic unc ion 𝜒
𝑀o a se 
𝑀gi en by
𝜒
𝑀(𝑡)={1, 𝑡∈
𝑀,
0, 𝑡∉
𝑀. (43)
Since each 𝑀⊂{1,...,𝑛}is a ini e se , Lemma 2 yields
({𝑗𝑖|𝑖∈𝑀})!=∑
𝑖∈𝑀(𝑗𝑖−1,{𝑗𝑘|𝑘∈𝑀 {𝑖}})!. (44)
We apply his iden i y o de i e a o mula o he second
de i a i e o 𝑆𝑀 o any 0 =𝑀⊂{1,...,𝑛}:
𝑆󸀠󸀠
𝑀(𝑡)
=∑
𝑗𝑖≥1,𝑖∈𝑀
∑𝑖∈𝑀 𝑗𝑖𝜏𝑖≤𝑡(−1)∑𝑖∈𝑀 𝑗𝑖({𝑗𝑖|𝑖∈𝑀})!
×∏
𝑖∈𝑀𝐵2𝑗𝑖
𝑖(𝑡−∑𝑖∈𝑀 𝑗𝑖𝜏𝑖)2(∑𝑖∈𝑀 𝑗𝑖−1)
(2(∑𝑖∈𝑀 𝑗𝑖−1))!
=∑
𝑖∈𝑀 ∑
𝑗𝑘≥1,𝑘∈𝑀
∑𝑘∈𝑀 𝑗𝑘𝜏𝑘≤𝑡(−1)∑𝑘∈𝑀 𝑗𝑘(𝑗𝑖−1,{𝑗𝑘|𝑘∈𝑀 {𝑖}})!
×∏
𝑘∈𝑀𝐵2𝑗𝑘
𝑘(𝑡−𝜏𝑖−∑𝑘∈𝑀 {𝑖}𝑗𝑘𝜏𝑘−(𝑗𝑖−1)𝜏𝑖)2(∑𝑘∈𝑀 𝑗𝑘−1)
(2(∑𝑘∈𝑀 𝑗𝑘−1))! .
(45)

6Abs ac and Applied Analysis
Nex , o any ixed 𝑖∈{1,...,𝑛}we spli he second sum o
𝑗𝑖=1and 𝑗𝑖≥2, ha is,
∑
𝑗𝑘≥1,𝑘∈𝑀
∑𝑘∈𝑀 𝑗𝑘𝜏𝑘≤𝑡𝐹(𝑗1,...,𝑗𝑖−1,𝑗𝑖,𝑗𝑖+1,...,𝑗𝑛)
=∑
𝑗𝑘≥1,𝑘∈𝑀 {𝑖}
∑𝑘∈𝑀 {𝑖} 𝑗𝑘𝜏𝑘≤𝑡−𝜏𝑖𝐹(𝑗1,...,𝑗𝑖−1,1,𝑗𝑖+1,...,𝑗𝑛)
+∑
𝑗𝑘≥1,𝑘∈𝑀 {𝑖}
𝑗𝑖≥2
∑𝑘∈𝑀 𝑗𝑘𝜏𝑘≤𝑡 𝐹(𝑗1,...,𝑗𝑖−1,𝑗𝑖,𝑗𝑖+1,...,𝑗𝑛), (46)
anduse heequali y
∑
𝑗𝑘≥1,𝑘∈𝑀 {𝑖}
𝑗𝑖≥2
∑𝑘∈𝑀 𝑗𝑘𝜏𝑘≤𝑡 𝐹(𝑗1,...,𝑗𝑖−1,𝑗𝑖,𝑗𝑖+1,...,𝑗𝑛)
=∑
𝑗𝑘≥1,𝑘∈𝑀
∑𝑘∈𝑀 𝑗𝑘𝜏𝑘≤𝑡−𝜏𝑖𝐹(𝑗1,...,𝑗𝑖−1,𝑗𝑖+1,𝑗𝑖+1,...,𝑗𝑛)(47)
since∑
𝑘∈𝑀𝑗𝑘𝜏𝑘≤𝑡⇐⇒ ∑
𝑘∈𝑀 {𝑖}𝑗𝑘𝜏𝑘+(𝑗𝑖−1)𝜏𝑖≤𝑡−𝜏𝑖.(48)
So we ob ain
𝑆󸀠󸀠
𝑀(𝑡)=−∑
𝑖∈𝑀𝐵2
𝑖(𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)+𝑆𝑀(𝑡−𝜏𝑖)) (49)
o each 0 =𝑀 ⊂ {1,...,𝑛}.Ob iously,𝑆󸀠󸀠
0(𝑡) = Θ.
Consequen ly,
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)
=− ∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∈𝑀𝐵2
𝑖(𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)+𝑆𝑀(𝑡−𝜏𝑖))
=− ∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
−∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀(𝑡−𝜏𝑖). (50)
Now, we add and sub ac
∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀(𝑡−𝜏𝑖)(51)
o he igh -hand side o (50) oge
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)
=− ∑
0 =𝑀⊂{1,...,𝑛}
𝑛
∑
𝑖=1𝐵2
𝑖𝑆𝑀(𝑡−𝜏𝑖)
+∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀(𝑡−𝜏𝑖)
−∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
(52)
and apply 𝑀=𝑀 {𝑖}whene e 𝑖∉𝑀:
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)
=− ∑
0 =𝑀⊂{1,...,𝑛}
𝑛
∑
𝑖=1𝐵2
𝑖𝑆𝑀(𝑡−𝜏𝑖)
+∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
−∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖).
(53)
Deno ing #𝑀 henumbe o elemen so hese 𝑀,wespli
he las wo e ms o he igh -hand side o he la e equali y
wi h espec o ∑
0 =𝑀⊂{1,...,𝑛}=∑
𝑀⊂{1,...,𝑛}
1≤#𝑀≤𝑛−1 +∑
𝑀⊂{1,...,𝑛}
#𝑀=𝑛
=∑
𝑀⊂{1,...,𝑛}
#𝑀=1 +∑
𝑀⊂{1,...,𝑛}
2≤#𝑀≤𝑛 .(54)
Hence, we ha e
∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
−∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
=∑
𝑀⊂{1,...,𝑛}
1≤#𝑀≤𝑛−1 ∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
+∑
𝑀={1,...,𝑛}∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
−∑
𝑀∈{{1},...,{𝑛}} ∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
−∑
𝑀⊂{1,...,𝑛}
2≤#𝑀≤𝑛 ∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖).
(55)
Now, we show ha
∑
𝑀⊂{1,...,𝑛}
1≤#𝑀≤𝑛−1 ∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
=∑
𝑀⊂{1,...,𝑛}
2≤#𝑀≤𝑛 ∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖). (56)
Le 𝑀⊂{1,...,𝑛}, and le 𝑖∉𝑀be a bi a y and ixed such
ha 1≤#𝑀≤𝑛−1. Then, clea ly,
𝐵𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)=𝐵𝑖𝑆(𝑀∪{𝑖}) {𝑖} (𝑡−𝜏𝑖)(57)
and 2≤#(𝑀∪{𝑖})≤𝑛,𝑖∈𝑀∪{𝑖}.Mo eo e ,i 𝑀1,𝑀2⊂
{1,...,𝑛},𝑖∉𝑀1,2 a e such ha 𝑀1=𝑀2,1≤#𝑀1,2 ≤𝑛−1,
hen 𝑀1∪{𝑖}=𝑀2∪{𝑖}.
Abs ac and Applied Analysis 7
On he o he side, i 𝑀⊂{1,...,𝑛},𝑖∈𝑀a e a bi a y
and ixed such ha 2≤#𝑀≤𝑛, hen
𝐵𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)=𝐵𝑖𝑆(𝑀 {𝑖}) {𝑖} (𝑡−𝜏𝑖)(58)
and 1≤#(𝑀 {𝑖})≤𝑛−1,𝑖∉𝑀 {𝑖}.Fu he mo e,i
𝑀1,𝑀2⊂{1,...,𝑛},𝑖∈𝑀1,2 a e such ha 𝑀1=𝑀2,2≤
#𝑀1,2 ≤𝑛, hen,𝑀1 {𝑖}=𝑀2 {𝑖}.Inconclusion, he eis
1−1co espondence be ween he e ms on he le -hand side
o (56) and he e ms on he igh -hand side. So (56)is alid.
Pu ing (56)in(55)weob ain
∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
−∑
0 =𝑀⊂{1,...,𝑛}∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
=∑
𝑀={1,...,𝑛}∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
−∑
𝑀∈{{1},...,{𝑛}} ∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀{𝑖} (𝑡−𝜏𝑖).
(59)
Nex , by he p ope y o emp y sum, we ge
∑
𝑀={1,...,𝑛}∑
𝑖∉𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)= ∑
𝑀={1,...,𝑛}Θ=Θ. (60)
Mo eo e , i holds
∑
𝑀∈{{1},...,{𝑛}} ∑
𝑖∈𝑀𝐵2
𝑖𝑆𝑀 {𝑖} (𝑡−𝜏𝑖)
=𝑛
∑
𝑖=1𝐵2
𝑖𝑆0(𝑡−𝜏𝑖)=∑
𝑀=0
𝑛
∑
𝑖=1𝐵2
𝑖𝑆𝑀(𝑡−𝜏𝑖). (61)
The e o e, pu ing (60)and(61)in(59)and he esul in(53),
we ob ain
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=− ∑
0 =𝑀⊂{1,...,𝑛}
𝑛
∑
𝑖=1𝐵2
𝑖𝑆𝑀(𝑡−𝜏𝑖)
−∑
𝑀=0
𝑛
∑
𝑖=1𝐵2
𝑖𝑆𝑀(𝑡−𝜏𝑖)
=−𝑛
∑
𝑖=1𝐵2
𝑖∑
𝑀⊂{1,...,𝑛}𝑆𝑀(𝑡−𝜏𝑖)
=−𝑛
∑
𝑖=1𝐵2
𝑖X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡−𝜏𝑖).
(62)
Hence, X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)sol es (31) o all𝑡≥0.Clea ly, hesame
is ue o 𝑡<0.
Fo Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡),s a emen s(1)–(3)canbep o edas o
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡). Nex , i 𝜏:=𝜏1=⋅⋅⋅=𝜏𝑛,weapply hepoin (2)
o his lemma and p ope y (22) o Y𝐵2
1+⋅⋅⋅+𝐵2
𝑛
𝜏(𝑡) o see ha
Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=Y𝐵2
1+⋅⋅⋅+𝐵2
𝑛
𝜏(𝑡)
=−(𝐵2
1+⋅⋅⋅+𝐵2
𝑛)Y𝐵2
1+⋅⋅⋅+𝐵2
𝑛
𝜏(𝑡−𝜏)
=−𝐵2
1Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡−𝜏1)
−⋅⋅⋅−𝐵2
𝑛Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡−𝜏𝑛).
(63)
So, Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)is a solu ion o (31) when all delays a e he
same.
Again, he case 𝑛=2wi h di e en delays was p o ed
in [16]; hus, we assume ha he s a emen is ul illed o 𝑛−
1,𝑛≥3and ha 𝜏𝑖=𝜏𝑗 o each 𝑖 =𝑗,𝑖,𝑗∈{1,...,𝑛}.As
be o e, i 𝑡<𝜏𝑘and 𝜏𝑘:=max𝑖=1,...,𝑛𝜏𝑖, hen
Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=Y𝐵2
1,...,𝐵2
𝑘−1,𝐵2
𝑘+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑘−1,𝜏𝑘+1,...,𝜏𝑛(𝑡)(64)
by de ini ion (20), and he s a emen ollows om he
induc i e hypo hesis. Fo 𝑡≥max𝑖=1,...,𝑛𝜏𝑖, we apply Lemma 3
o see ha
Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)=∑
𝑀⊂{1,...,𝑛}𝑆𝑀(𝑡)(65)
wi h 𝑆𝑀(𝑡)gi en by (25). This ime
𝑆0(𝑡)=∑
𝑗𝑖≥1,𝑖∈0
0≤𝑡 (−1)0({𝑗𝑖|𝑖∈0})!𝐸(𝑡−0)1
0!
=∑
0≤𝑡𝐸𝑡=𝑡𝐸𝜒[0,∞) (𝑡)(66)
and 𝑆󸀠󸀠
0(𝑡)=Θ. The es p oceeds analogically o X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡).
The inal s a emen ollows di ec ly om de ini ion (20).
Rema k 5. Ano he p oo o s a emen s (1)–(3) o he p e-
iouslemmacanbemadewi h heaido s a emen (4)
o he same lemma and uses he uniqueness o a solu ion
o he co esponding ini ial alue p oblem. Fo ins ance in
s a emen (1) o he lemma, bo h
X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡),X𝐵2
1,...,𝐵2
𝑖−1,𝐵2
𝑖+1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑖−1,𝜏𝑖+1,...,𝜏𝑛(𝑡)(67)
sol e 𝑥(𝑡)=−𝐵2
1𝑥(𝑡−𝜏1)−⋅⋅⋅−𝐵2
𝑖−1𝑥(𝑡−𝜏𝑖−1)
−𝐵2
𝑖+1𝑥(𝑡−𝜏𝑖+1)−⋅⋅⋅−𝐵2
𝑛𝑥(𝑡−𝜏𝑛)(68)
wi h ini ial condi ion
𝑥(𝑡)={Θ, −𝜏≤𝑡<0,
𝐸, 𝑡=0, 𝑥(𝑡)=Θ,−𝜏≤𝑡≤0 (69)
and 𝜏=max𝑖=1,...,𝑛𝜏𝑖.
We a e eady o s a e and p o e ou main esul .
8Abs ac and Applied Analysis
3. Main Resul
He ewe indasolu iono heini ial aluep oblem(11), (8)
in he sense o he nex de ini ion.
De ini ion 6. Le 𝜏1,...,𝜏𝑛>0,𝜏:=max𝑖=1,...,𝑛𝜏𝑖,and𝜑∈
𝐶1([−𝜏,0],R𝑁),andle 𝐵1,...,𝐵𝑛be 𝑁×𝑁ma ices, and
le 𝑓:[0,∞)→R𝑁be a gi en unc ion. Func ion 𝑥:
[−𝜏,∞) → R𝑁is a solu ion o (11) and ini ial condi ion
(8), i 𝑥∈𝐶
1([−𝜏,∞),R𝑁)∩𝐶2([0,∞),R𝑁)( aken he
second igh -hand de i a i e a 0) sa is ies (11)on[0,∞)and
condi ion (8)on[−𝜏,0].
Theo em 7. Le 𝑛≥3,𝜏1,...,𝜏𝑛>0,𝜏:=max𝑖=1,...,𝑛𝜏𝑖,and
𝜑∈𝐶1([−𝜏,0],R𝑁),andle 𝐵1,...,𝐵𝑛be 𝑁×𝑁pai wise
pe mu able ma ices; ha is, 𝐵𝑖𝐵𝑗=𝐵
𝑗𝐵𝑖 o each 𝑖,𝑗 ∈
{1,...,𝑛},andle 𝑓:[0,∞)→R𝑁be a gi en unc ion.
Solu ion 𝑥(𝑡)o (11)sa is ying ini ial condi ion (8)has he o m
𝑥(𝑡)={
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
𝜑(𝑡),−𝜏≤𝑡<0,
X(𝑡)𝜑(0)+Y(𝑡)𝜑(0)
−𝑛
∑
𝑖=1𝐵2
𝑖∫0
−𝜏𝑖
Y(𝑡−𝜏𝑖−𝑠)𝜑(𝑠)𝑑𝑠
+∫𝑡
0Y(𝑡−𝑠)𝑓(𝑠)𝑑𝑠, 0≤𝑡, (70)
whe e X(𝑡)=X𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡)and Y(𝑡)=Y𝐵2
1,...,𝐵2
𝑛
𝜏1,...,𝜏𝑛(𝑡).
P oo . Ob iously, 𝑥(𝑡)sa is ies he ini ial condi ion on [−𝜏,0),
and, om de ini ion (20), 𝑥(0)=𝜑(0). Fo he de i a i e, i
holds lim𝑡→0−𝑥(𝑡)= 𝜑(0).Mo eo e ,i 0≤𝑡<min𝑖=1,...,𝑛𝜏𝑖,
hen 𝑥(𝑡)=𝜑(0)+𝑡𝜑(0)
−𝑛
∑
𝑖=1𝐵2
𝑖∫𝑡−𝜏𝑖
−𝜏𝑖(𝑡−𝜏𝑖−𝑠)𝜑(𝑠)𝑑𝑠
+∫𝑡
0(𝑡−𝑠)𝑓(𝑠)𝑑𝑠
(71)
since
Y(𝑡−𝜏𝑖−𝑠)={(𝑡−𝜏𝑖−𝑠)𝐸, 𝑠∈[−𝜏𝑖,𝑡−𝜏𝑖],
Θ, 𝑠∈(𝑡−𝜏𝑖,0] (72)
o each 𝑖=1,...,𝑛.Thus
𝑥(𝑡)=𝜑(0)−𝑛
∑
𝑖=1𝐵2
𝑖∫𝑡−𝜏𝑖
−𝜏𝑖𝜑(𝑠)𝑑𝑠+∫𝑡
0𝑓(𝑠)𝑑𝑠 (73)
and lim𝑡→0+𝑥(𝑡)= 𝜑(0).Clea ly,
𝑥∈𝐶1((−𝜏,∞),R𝑁)∩𝐶2((0,∞) {𝜏1,...,𝜏𝑛},R𝑁).
(74)
We show ha , al hough X(𝑡)is no 𝐶2a 𝜏1,...,𝜏𝑛,
unc ion 𝑥(𝑡)is 𝐶2a hese poin s and, he e o e, in (0,∞).
A once, we p o e ha 𝑥(𝑡)is a solu ion o (11).
Assume ha 0≤𝑡<min𝑖=1,...,𝑛𝜏𝑖. Then iden i ies (71)and
(73) a e alid, and by di e en ia ing (73) o such𝑡we ge
𝑥(𝑡)=−𝑛
∑
𝑖=1𝐵2
𝑖𝜑(𝑡−𝜏𝑖)+𝑓(𝑡)=−𝑛
∑
𝑖=1𝐵2
𝑖𝑥(𝑡−𝜏𝑖)+𝑓(𝑡)(75)
since 𝑥(𝑡−𝜏𝑖)=𝜑(𝑡−𝜏𝑖) o each 𝑖=1,...,𝑛.
Now, le 0 =𝑀1,2 ⊂{1,...,𝑛}be such ha 𝜏𝑖≤𝑡<𝜏𝑗 o
each 𝑖∈𝑀1,𝑗∈𝑀2.Then
Y(𝑡−𝜏𝑗−𝑠)={Y(𝑡−𝜏𝑗−𝑠), 𝑠∈[−𝜏𝑗,𝑡−𝜏𝑗],
Θ, 𝑠∈(𝑡−𝜏𝑗,0] (76)
whene e 𝑗∈𝑀2,and(70)becomes
𝑥(𝑡)=X(𝑡)𝜑(0)+Y(𝑡)𝜑(0)
−∑
𝑖∈𝑀1𝐵2
𝑖∫0
−𝜏𝑖
Y(𝑡−𝜏𝑖−𝑠)𝜑(𝑠)𝑑𝑠
−∑
𝑗∈𝑀2𝐵2
𝑗∫𝑡−𝜏𝑗
−𝜏𝑗
Y(𝑡−𝜏𝑗−𝑠)𝜑(𝑠)𝑑𝑠
+∫𝑡
0Y(𝑡−𝑠)𝑓(𝑠)𝑑𝑠.
(77)
By he poin (5) o Lemma 4,wege
𝑥(𝑡)=X(𝑡)𝜑(0)+Y(𝑡)𝜑(0)
−∑
𝑖∈𝑀1𝐵2
𝑖∫0
−𝜏𝑖Y(𝑡−𝜏𝑖−𝑠)𝜑(𝑠)𝑑𝑠
−∑
𝑗∈𝑀2𝐵2
𝑗∫𝑡−𝜏𝑗
−𝜏𝑗
X(𝑡−𝜏𝑗−𝑠)𝜑(𝑠)𝑑𝑠
+∫𝑡
0X(𝑡−𝑠)𝑓(𝑠)𝑑𝑠,
(78)
and o hesecondde i a i ei holds
𝑥(𝑡)=X(𝑡)𝜑(0)+Y(𝑡)𝜑(0)
−∑
𝑖∈𝑀1𝐵2
𝑖∫0
−𝜏𝑖Y(𝑡−𝜏𝑖−𝑠)𝜑(𝑠)𝑑𝑠
−∑
𝑗∈𝑀2𝐵2
𝑗(𝜑(𝑡−𝜏𝑗)+∫𝑡−𝜏𝑗
−𝜏𝑗Y(𝑡−𝜏𝑗−𝑠)𝜑(𝑠)𝑑𝑠)
+𝑓(𝑡)+∫𝑡
0Y(𝑡−𝑠)𝑓(𝑠)𝑑𝑠 (79)
since X(0)=𝐸. Now, we apply he p ope y (4) o Lemma 4
oge he wi h
X(𝑡−𝜏𝑗)=Y(𝑡−𝜏𝑗)=Θ, ∀𝑗∈𝑀2(80)
Abs ac and Applied Analysis 9
o see ha bo h Xand Ya e solu ions o
𝑦(𝑡)=−∑
𝑖∈𝑀1𝐵2
𝑖𝑦(𝑡−𝜏𝑖). (81)
The e o e,
𝑥(𝑡)=−∑
𝑘∈𝑀1𝐵2
𝑘(X(𝑡−𝜏𝑘)𝜑(0)+Y(𝑡−𝜏𝑘)𝜑(0)
−∑
𝑖∈𝑀1𝐵2
𝑖∫0
−𝜏𝑖
Y(𝑡−𝜏𝑖−𝜏𝑘−𝑠)𝜑(𝑠)𝑑𝑠
−∑
𝑗∈𝑀2𝐵2
𝑗∫𝑡−𝜏𝑗
−𝜏𝑗
Y(𝑡−𝜏𝑗−𝜏𝑘−𝑠)𝜑(𝑠)𝑑𝑠
+∫𝑡
0Y(𝑡−𝜏𝑘−𝑠)𝑓(𝑠)𝑑𝑠)
−∑
𝑗∈𝑀2𝐵2
𝑗𝜑(𝑡−𝜏𝑗)+𝑓(𝑡)
=−∑
𝑖∈𝑀1𝐵2
𝑖𝑥(𝑡−𝜏𝑖)−∑
𝑗∈𝑀2𝐵2
𝑗𝜑(𝑡−𝜏𝑗)+𝑓(𝑡).(82)
In ac , his is exac ly o mula (11)since𝑥(𝑡−𝜏𝑗)=𝜑(𝑡−𝜏𝑗)
o each 𝑗∈𝑀2.
Finally, i max𝑖=1,...,𝑛𝜏𝑖≤𝑡,weha e
𝑥(𝑡)=X(𝑡)𝜑(0)+Y(𝑡)𝜑(0)
−𝑛
∑
𝑖=1𝐵2
𝑖∫0
−𝜏𝑖
Y(𝑡−𝜏𝑖−𝑠)𝜑(𝑠)𝑑𝑠
+∫𝑡
0Y(𝑡−𝑠)𝑓(𝑠)𝑑𝑠.
(83)
So, di e en ia ing his o mula wice and applying (4) o
Lemma 4 esul in (11). Hence, one can see ha unc ion 𝑥(𝑡)
gi en by (70) eallysol es(11) and sa is ies ini ial condi ion
(8)and,mo eo e , ha 𝑥∈𝐶2((0,∞),R𝑁).Tosee helas
one, one has o pu 𝜏1,...,𝜏𝑛in o he compu ed de i a i es,
o example, i 𝜏𝑘:=min𝑖=1,...,𝑛𝜏𝑖<𝜏𝑖 o each 𝑖=1,...,𝑘−
1,𝑘+1,...,𝑛, henby(75)and(82)wege
lim
𝑡→𝜏−
𝑘𝑥(𝑡)=−𝑛
∑
𝑖=1
𝑖 =𝑘𝐵2
𝑖𝜑(𝜏𝑘−𝜏𝑖)−𝐵2
𝑘𝜑(0)+𝑓(𝜏𝑘)
=−∑
𝑗∈𝑀2𝐵2
𝑗𝜑(𝜏𝑘−𝜏𝑗)
−𝐵2
𝑘𝑥(0)+𝑓(𝜏𝑘)=lim
𝑡→𝜏+
𝑘𝑥(𝑡),
(84)
whe e 𝑀2={1,...,𝑛} {𝑘}.
I is easy o see ha de ining unc ions

X𝐵1,...,𝐵𝑛
𝜏1,...,𝜏𝑛(𝑡):=X−𝐵1,...,−𝐵𝑛
𝜏1,...,𝜏𝑛(𝑡),

Y𝐵1,...,𝐵𝑛
𝜏1,...,𝜏𝑛(𝑡):=Y−𝐵1,...,−𝐵𝑛
𝜏1,...,𝜏𝑛(𝑡)(85)
leads o he solu ion o
𝑥(𝑡)=𝐵1𝑥(𝑡−𝜏1)+⋅⋅⋅+𝐵𝑛𝑥(𝑡−𝜏𝑛)+𝑓(𝑡)(86)
wi h pai wise pe mu able ma ices 𝐵1,...,𝐵𝑛and ini ial
condi ion (8). Mo e p ecisely, we ha e he ollowing co olla y
o Theo em 7.
Co olla y 8. Le 𝑛≥3,𝜏1,...,𝜏𝑛>0,𝜏:=max𝑖=1,...,𝑛𝜏𝑖,
𝜑∈𝐶1([−𝜏,0],R𝑁),andle 𝐵1,...,𝐵𝑛be 𝑁×𝑁pai wise
pe mu able ma ices; ha is, 𝐵𝑖𝐵𝑗=𝐵
𝑗𝐵𝑖 o each 𝑖,𝑗 ∈
{1,...,𝑛},andle 𝑓:[0,∞)→R𝑁be a gi en unc ion.
Solu ion 𝑥(𝑡)o (86)sa is ying ini ial condi ion (8)has he
o m
𝑥(𝑡)={
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
{
𝜑(𝑡),−𝜏≤𝑡<0,
X(𝑡)𝜑(0)+Y(𝑡)𝜑(0)
+𝑛
∑
𝑖=1𝐵𝑖∫0
−𝜏𝑖
Y(𝑡−𝜏𝑖−𝑠)𝜑(𝑠)𝑑𝑠
+∫𝑡
0Y(𝑡−𝑠)𝑓(𝑠)𝑑𝑠, 0≤𝑡, (87)
whe e X(𝑡)=
X𝐵1,...,𝐵𝑛
𝜏1,...,𝜏𝑛(𝑡)and Y(𝑡)=
Y𝐵1,...,𝐵𝑛
𝜏1,...,𝜏𝑛(𝑡).
P oo . The co olla y can be p o ed exac ly in he same way as
Theo em 7.
Acknowledgmen s
J. Dibl´
ık was suppo ed by he G an GAˇ
CR P201/11/0768.
M. Feˇ
ckan was suppo ed in pa by he G an s VEGA-
MS 1/0507/11, VEGA-SAV 2/0029/13, and APVV-0134-10. M.
Posp´
ıˇ
sil was suppo ed by he P ojec no. CZ.1.07/2.3.00/
30.0005 unded by Eu opean Regional De elopmen Fund.
Re e ences
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ık, M. R˚
uˇ
ziˇ
cko ´
a, and J. Luk´
aˇ
co ´
a,
“Rep esen a ion o a solu ion o he Cauchy p oblem o an
oscilla ing sys em wi h pu e delay,” Nonlinea Oscilla ions, ol.
11, no. 2, pp. 276–285, 2008.
[2] D. Y. Khusaino and G. V. Shuklin, “Linea au onomous ime-
delay sys em wi h pe mu a ion ma ices sol ing,” S udies o he
Uni e si y o ˇ
Zilina, ol.17,no.1,pp.101–108,2003.
[3]M.Med ed’andM.Posp
´
ıˇ
sil, “Su icien condi ions o he
asymp o ic s abili y o nonlinea mul idelay di e en ial equa-
ions wi h linea pa s de ined by pai wise pe mu able ma i-
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[4]M.Med ed’andM.Posp
´
ıˇ
sil, “Rep esen a ion and s abili y o
solu ions o sys ems o di e ence equa ions wi h mul iple delays
and linea pa s de ined by pai wise pe mu able ma ices,”