Noname manusc ip No.
(will be inse ed by he edi o )
New c i e ia o global asymp o ic s abili y o
linea neu al di e en ial equa ions by a ixed poin
app oach
Mimia Benhad i ·Tom´as Ca aballo
Recei ed: da e / Accep ed: da e
Abs ac New c i e ia ensu ing global asymp o ic s abili y o he ze o solu-
ion o a class o linea neu al di e en ial equa ions in C1a e p o ed, by
using wo auxilia y unc ions on a con ac ion condi ion. Necessa y and su -
icien condi ions o he s abili y o ou equa ion which also imp o es ecen
esul s on his ield a e shown. Finally, an example is p o ided o illus a e
he easibili y and ad an age o ou esul s.
Keywo ds Con ac ion mapping p inciple ·Asymp o ic s abili y ·Neu al
di e en ial equa ions
Ma hema ics Subjec Classi ica ion (2010) 34K20 ·34K13 ·92B20
1 In oduc ion
Theo y and applica ions o unc ional di e en ial equa ions wi h delay ha e
been s udied by many au ho s (see, o example, [19,20] and he e e ences
he ein). Mo e ecen ly, esea che s ha e paid special a en ion o he s udy
o equa ions in which he delay a gumen occu s in he de i a i e o he s a e
a iable as well as in he independen a iable, he so-called neu al di e en ial
equa ions, see o ins ance [18,19]. P ac ical examples o neu al delay di e en-
ial sys ems include biological models o single species g ow h [24], dis ibu ed
ne wo ks con aining lossless ansmission lines [11], popula ion ecology [20],
and o he enginee ing sys ems [21]. In pa icula , quali a i e analysis such as
M. Benhad i
LAMAHIS Lab, Facul y o Sciences, Depa men o Ma hema ics, Uni . Skikda,
P.O. Box 26, Skikda 21000, Alge ia.
E-mail: mbenhad [email protected]
T. Ca aballo
Depa amen o de Ecuaciones Di e enciales y An´alisis Num´e ico, Uni e sidad de Se illa,
c/ Ta ia s/n, 41012-Se illa, Spain.
E-mail: [email p o ec ed]
2 Mimia Benhad i, Tom´as Ca aballo
s abili y o solu ions o neu al di e en ial equa ions has ecei ed much a en-
ion o e he las ew decades. We e e o [1–4,13,14,17,22,23,25] o some
ecen wo k on he subjec o s abili y o neu al equa ions.
Lyapuno ’s di ec me hod has been success ully used o in es iga e s a-
bili y p ope ies o a wide a ie y o di e en ial equa ions. Ne e heless, he
applica ion o his me hod o p oblems o s abili y in di e en ial and in eg o-
di e en ial equa ions wi h delays has encoun e ed se ious obs acles when he
delay is unbounded o when he equa ion has unbounded e ms [5]-[7]. In ecen
yea s, se e al in es iga o s ha e analyzed s abili y by using a new echnique.
Pa icula ly, Bu on and o he esea che s s udied s abili y using a ious ixed
poin heo ems which o e came he di icul ies encoun e ed in he s udy o s a-
bili y by means o Lyapuno ’s di ec me hod. We can e e o [5,15,25,26] and
[28]-[32] o mo e de ails. I u ns ou ha he ixed poin me hod is becoming
a powe ul echnique in dealing wi h s abili y p oblems o s ochas ic di e en-
ial equa ions wi h delays [12,16,22,32].
P e iously, almos all schola s, who used ixed poin heo y o s udy he
asymp o ic s abili y o ze o solu ions o nonlinea neu al di e en ial equa-
ions wi h a iable delays, imposed ha cmus be di e en iable and τ wice
di e en iable and τ0( )6= 1 o ≥0.As dis inguished om his line, in [17],
Jin and Luo s a ed a su icien and necessa y condi ion o he asymp o ic
s abili y in he space C0o he ollowing equa ion (1.1), by using a ixed poin
me hod o con inuous unc ions:
x0( ) = −a( )x( ) + c( )x0( −τ( )) −b( )x( −τ( )), ≥0.(1.1)
In [26], Ra oul ob ained su icien condi ions o he asymp o ic s abili y
o he ze o solu ion, unde app op ia e condi ions, o he ollowing equa ion
x0( ) = −a( )x( ) + c( )x0( −τ( )) + Q(x( ), x( −τ( ))) , ≥0.(1.2)
by using he con ac ion mapping heo em.
On he o he hand, Liu and Yang [23] we e he i s o es ablish a necessa y
and su icien condi ion o he global asymp o ic s abili y in C1o he ze o
solu ions o he ollowing nonlinea neu al unc ional di e en ial equa ion
x0( ) = −a( )x( ) + c( )x0( −τ1( )) + Q( , x ( ), x( −τ2( ))) ,(1.3)
by he ixed poin heo y, whe e Qis a Lipschi z con inuous unc ion wi h he
espec o x.
Ve y ecen ly, by he same me hod o Jin and Luo [23], A djouni and Djoudi
[1] imp o ed he esul s o Liu e al. [23] o he gene alized nonlinea neu al
di e en ial equa ion wi h a iable delays o he o m
x0( ) = −a( )x( ) + ( , x( −τ1( )), ..., x( −τN( )))
+h( , x0( −τ1( )), ..., x0( −τN( ))) ,(1.4)
Ti le Supp essed Due o Excessi e Leng h 3
whe e ( , 0, ..., 0) = h( , 0, ..., 0) = 0 and he e exis bounded unc ions
bi, ci∈C([0,∞),(0,∞)), a∈C(R+,R), such ha
| ( , x1, ..., xN)− ( , y1, ..., yN)| ≤
N
X
i=1
bi( )|xi−yi|,(1.5)
|h( , x0
1, ..., x0
N)−h( , y0
1, ..., y0
N)| ≤
N
X
i=1
ci( )|x0
i−y0
i|,(1.6)
o all xi, yi∈R, i = 1, ..., N. Mo e p ecisely, he ollowing esul was es ab-
lished.
Theo em A. (A djouni and Djoudi [1]) Suppose ha assump ions (1.5),(1.6)
hold, and he e exis s a cons an η∈(0,1) such ha o ≥ 0,
lim in
→∞ Z
0
a(s)ds > −∞,(1.4)
and
Z
0
e−R
sa(u)du
N
X
i=1
(|bi(s)|+|ci(s)|)ds ≤η, (1.5)
|a( )|Z
0
e−R
sa(u)du
N
X
i=1
(|bi(s)|+|ci(s)|)ds +
N
X
i=1
(|bi( )|+|ci( )|)≤η. (1.6)
Then he ze o solu ion o equa ion (1.4) is globally asymp o ically s able in
C1i and only i
Z
0
a(s)ds → ∞ as → ∞.(1.7)
By using he con ac ion mapping p inciple, he au ho s es ablished some
new condi ions o ensu e ha he ze o solu ion o equa ion (1.4) is globally
asymp o ically s able in C1. Unlike mos esea ch me hods, hese condi ions
do no equi e a quad a ic di e en iabili y o delay τand τ0( )6= 1 o ≥0.
In addi ion, in [1,2,28], hey all s udied he global asymp o ic s abili y in C1.
Inspi ed by he applica ion o he ixed poin me hod men ioned abo e,
in his pape , we will s a e some new condi ions, which make s abili y condi-
ions mo e easible and he esul s in [1,17,23,26] a e imp o ed. By using wo
auxilia y unc ions gand p o cons uc a con ac ion mapping on a comple e
me ic space S, de ined below, which may depend on he ini ial condi ion ϕ, we
ob ain Theo em 1.3 which will be p o ed in Sec ion 2. Namely, a necessa y and
su icien condi ion ensu ing he global asymp o ic s abili y in C1is p o ed.
In addi ion, an example is e en ually analyzed o illus a e he e ec i eness
o he p o ed esul s.
4 Mimia Benhad i, Tom´as Ca aballo
No ice ha he condi ion (1.6) in Theo em A is mainly dependen o he
cons ain N
X
i=1
|bi( )|+
N
X
i=1
|ci( )|<1.
Howe e , he e a e some in e es ing examples whe e he cons ain is no sa -
is ied. I is ou aim in his pape o emo e his cons ain condi ion and
conside he global s abili y in C1o he special case o (1.4) when
( , x( −τ1( )), ..., x( −τN( ))) =
N
X
i=1
bi( )x( −τi( )),
and
h( , x0( −τ1( )), ..., x0( −τN( ))) =
N
X
i=1
ci( )x0( −τi( )).
In pa icula , we in oduce wo auxilia y con inuous unc ions gand p o de ine
an app op ia e mapping, and p esen new c i e ia o he global asymp o ic
s abili y o equa ion (1.2) which can be applied o he case
N
X
i=1
|bi( )|+
N
X
i=1
|ci( )| ≥ 1,
as well.
2 P elimina ies
Le us conside he ollowing class o neu al di e en ial equa ions wi h a i-
able delays,
x0( ) = −a( )x( ) +
N
X
i=1
bi( )x0( −τi( )) +
N
X
i=1
ci( )x( −τi( )) , ≥ 0,
(2.1)
deno e x( )∈R he solu ion o (2.1) wi h he ini ial condi ion
x( ) = ϕ( ) o ∈[m( 0), 0],
whe e ϕ∈C([m( 0), 0],R).We assume ha a, bi, ci∈C(R+,R), τi∈
C(R+,R+) sa is y
−τi( )→ ∞ as → ∞, i = 1,2, ..., N, (2.2)
and o each 0≥0,mi( 0) = in { −τi( ), ≥ 0}, m( 0) = min{mi( 0), i =
1,2, ..., N}.
Fo each 0∈[0,∞),deno e C1
0=C1([m( 0), 0],R) wi h he no m
de ined by
|x| 0:= max
∈[m( 0), 0]{|x( )|,|x0( )|} ,
Ti le Supp essed Due o Excessi e Leng h 5
o x∈C1
0=C1([m( 0), 0],R). In addi ion, deno e Φ 0,whe e
Φ 0: = (ϕ∈C1
0:ϕ0
−( 0) = −a( )ϕ( 0) +
N
X
i=1
bi( 0)ϕ0( 0−τi( 0))
+
N
X
i=1
ci( 0)ϕ( 0−τi( 0))).
Fo each 0∈[0,∞), we choose ini ial unc ions o equa ion (2.1) o he
ype ϕ∈Φ 0.
Le us ecall he de ini ions o s abili y ha will be used in he nex sec ion.
De ini ion 1.1. Fo each ini ial alue ( 0, ϕ)∈[0,∞)×Φ 0,xis said o be
a solu ion o equa ion (2.1) h ough ( 0, ϕ)i x∈C1([m( 0),∞),R)sa is ies
equa ion (2.1)on [ 0,∞)and x( ) = ϕ( ) o ∈[m( 0), 0]. Such a solu ion
will be deno ed by x( ) = x( , 0, ϕ).
De ini ion 1.2. i) The Ze o solu ion o equa ion (2.1) is said o be s able in
C1i , o any 0∈[0,∞),ε > 0, he e is a δ=δ(ε, 0)such ha ϕ∈Φ 0and
|ϕ| 0< δ implies
max
s∈[m( 0), 0]{|x(s, 0, ϕ)|,|x0(s, 0, ϕ)|} < ε o ≥ 0.
ii) The ze o solu ion o equa ion (2.1) is said o be globally asymp o ically
s able in C1i i is s able in C1,and o any 0∈[0,∞), ϕ ∈Φ 0implies
lim
→∞x( , 0, ϕ) = lim
→∞x0( , 0, ϕ)=0.
A ligh o he p e ious de ini ion o solu ion o equa ion (2.1), i is clea
ha he condi ions imposed on he ini ial unc ions a e sensible.
3 S abili y by con ac ion mapping
As we men ioned p e iously, he esul s o his wo k ex end and imp o e p e-
iously known esul s. Mo e exac ly, we will conside a linea scala neu al
delay di e en ial equa ion wi h a iable delays and gi e new condi ions o en-
su e ha he ze o solu ion is global asymp o ically s able in C1by means o
ixed poin heo y. By weakening he assump ions on he neu al coe icien
ciand delays τi, τ0
i( )6= 1,∀ ≥0,and by ob aining some c i e ia, easie o
check in applica ions, which does no sa is y he cons ain
N
X
i=1
|bi( )|+
N
X
i=1
|ci( )|<1.
Howe e , he ma hema ical analysis used in his esea ch o cons uc he
mapping o employ ixed poin heo em is di e en om ha o [1]. The esul s
o his a icle a e new and hey ex end and imp o e p e iously known esul s.

6 Mimia Benhad i, Tom´as Ca aballo
To he bes o ou knowledge, he e a e ew au ho s who ha e used he
ixed poin heo em o p o e he exis ence and uniqueness o solu ion and he
s abili y o i ial equilib ium o se e al special cases o (2.1) all a once [3,
29,31]. In ou s udy, as we a e mainly conce ned wi h he s abili y analysis o
ou model, we will assume ha he e exis s a unique solu ion o (2.1) globally
de ined in ime.
Theo em 3.1.Conside he neu al delay di e en ial equa ion (2.1)and sup-
pose he ollowing condi ions a e sa is ied:
H1) Suppose he e exis s a bounded unc ion p: [m( 0),∞[→(0,∞)wi h
p( ) = 1 o ∈[m( 0), 0]such ha p0( )exis s o all ∈[m( 0),∞[,
H2) he e exis s an a bi a y bounded con inuous unc ion g∈C([m( 0),∞[,R+)
and
lim in
→∞ Z
0
g(s)ds > −∞,(3.1)
H3) he e exis s a cons an η∈(0,1
2)such ha o ≥ 0,
1( ) := Z
0
e−R
sg(s)ds 
g(s)−a(s) + p0(s)
p(s)
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
+
N
X
i=1 
bi(s)p0(s−τi(s))
p(s)
+
N
X
i=1 
ci(s)p(s−τi(s))
p(s)#ds ≤η, (3.2)
and
2( ) := |g( )|Z
0
e−R
sg(u)du 
g(s)−a(s) + p0(s)
p(s)
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
+
N
X
i=1 
bi(s)p0(s−τi(s))
p(s)
+
N
X
i=1 
ci(s)p(s−τi(s))
p(s))ds
+
g( )−a( ) + p0( )
p( )
+
N
X
i=1 
bi( )p( −τi( ))
p( )
+
N
X
i=1 
bi( )p0( −τi( ))
p( )
+
N
X
i=1 
ci( )p( −τi( ))
p( )
≤η.(3.3)
Then he ze o solu ion o equa ion (2.1) is globally asymp o ically s able in
C1i and only i
Z
0
g(s)ds → ∞ as → ∞.(3.4)
Ti le Supp essed Due o Excessi e Leng h 7
P oo . (⇐:) Fi s , suppose ha R
0g(s)ds → ∞ as → ∞. Fo each
0∈[0,∞),we de ine Sas he ollowing space
S=nz∈C1([m( 0),∞),R) : lim
→∞z( ) = lim
→∞z0( )=0o,
wi h he me ic de ined by
kzk:= max
∈[m( 0),∞){|z( )|,|z0( )|} .
Then Sis a comple e me ic space.
Fo any ini ial unc ion ϕ∈Φ 0, le
Dl
ϕ=z∈S:z( ) = ϕ( ) o ∈[m( 0), 0] and max
≥ 0
{|z( )|,|z0( )|} ≤ l,
which is a nonemp y, closed con ex subse o S.
The echnique o cons uc ing a con ac ion mapping comes om an idea
in [31]. Indeed, le z( ) = ϕ( ) on ∈[m( 0), 0] and o ≥ 0
x( ) = p( )z( ).(3.5)
Replacing (3.5) in o (2.1),we ha e
z0( ) = −a( ) + p0( )
p( )z( )
+
N
X
i=1
bi( )p( −τi( ))
p( )z0( −τi( ))
+
N
X
i=1
ci( )p( −τi( )) + bi( )p0( −τi( ))
p( )z( −τi( )) .(3.6)
I zsa is ies (3.6) hen i can be e i ied ha xsa is ies (2.1).
Since pis a posi i e bounded unc ion, o ob ain global asymp o ic s abili y
o he ze o solu ion o (2.1), i emains o p o e ha he ze o solu ion o (3.6)
is globally asymp o ically s able in C1.
Mul iplying bo h sides o (3.6) by eR
0g(u)du and in eg a ing om 0 o ,
Z
0heRs
0g(u)duz(s)i0
ds
=
Z 0
eRs
0g(u)du g(s)−a(s) + p0(s)
p(s)z(s)ds
+
Z 0
eRs
0g(u)du
N
X
i=1
bi(s)p(s−τi(s))
p(s)z0(s−τi(s)) ds
+
Z 0
eRs
0g(u)du
N
X
i=1
ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s))
p(s)z(s−τi(s)) ds.
8 Mimia Benhad i, Tom´as Ca aballo
As a consequence, we a i e a
z( )eR
0g(u)du
=ϕ( 0) +
Z 0
eRs
0g(u)du g(s)−a(s) + p0(s)
p(s)z(s)ds
+
Z 0
eRs
0g(u)du
N
X
i=1
bi(s)p(s−τi(s))
p(s)z0(s−τi(s)) ds
+
Z 0
eRs
0g(u)du
N
X
i=1
ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s))
p(s)z(s−τi(s)) ds.
Di iding bo h sides o he abo e equa ion by eR
0g(s)ds, we ob ain
z( ) = e−R
0g(s)dsϕ( 0) + Z
0
e−R
sg(u)du g(s)−a(s) + p0(s)
p(s)z(s)ds
+Z
0
e−R
sg(u)du
N
X
i=1
bi(s)p(s−τi(s))
p(s)z0(s−τi(s)) ds
+Z
0
e−R
sg(u)du
N
X
i=1
ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s))
p(s)
×z(s−τi(s)) ds.
Clea ly, Ψ(z) : R→Ris con inuous wi h (Ψz) ( ) = ϕ( ) o ∈[m( 0), 0],
and o ≥ 0,
(Ψz) ( ) = e−R
0g(s)dsϕ( 0) + Z
0
e−R
sg(u)du g(s)−a(s) + p0(s)
p(s)z(s)ds
+Z
0
e−R
sg(u)du
N
X
i=1
bi(s)p(s−τi(s))
p(s)z0(s−τi(s)) ds
+Z
0
e−R
sg(u)du
N
X
i=1
ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s))
p(s)
×z(s−τi(s)) ds. (3.7)
Ti le Supp essed Due o Excessi e Leng h 9
Ini ially, we show ha , Ψ:Dl
ϕ→Dl
ϕ. In iew o (3.7), we can de i e, o
≥ 0,
(Ψz)0( ) = −ϕ( 0)g( )e−R
0g(s)ds
+g( )−a( ) + p0( )
p( )z( )
+
N
X
i=1
bi( )p( −τi( ))
p( )z0( −τi( ))
+
N
X
i=1
ci( )p( −τi( )) + bi( )p0( −τi( ))
p( )z( −τi( ))
−g( )Z
0
e−R
sg(u)du g(s)−a(s) + p0(s)
p(s)z(s)ds
−g( )Z
0
e−R
sg(u)du
N
X
i=1
bi(s)p(s−τi(s))
p(s)z0(s−τi(s)) ds
−g( )Z
0
e−R
sg(u)du
N
X
i=1
ci(s)p(s−τi(s)) + bi(s)p0(s−τi(s))
p(s)
×z(s−τi(s)) ds.
Thus
(Ψz)0( ) = −g( ) (Ψz) ( ) + g( )−a( ) + p0( )
p( )z( )
+
N
X
i=1
bi( )p( −τi( ))
p( )z0( −τi( )) (3.8)
+
N
X
i=1
ci( )p( −τi( )) + bi( )p0( −τi( ))
p( )z( −τi( )) .
By he de ini ion o Φ 0,(3.8) yields
(Ψz)0
+( 0) = −g( 0)ϕ( 0)
+g( 0)−a( 0) + p0( 0)
p( 0)z( 0)
+
N
X
i=1
bi( 0)p( 0−τi( 0))
p( 0)z0( 0−τi( 0))
+
N
X
i=1
ci( 0)p( 0−τi( 0)) + bi( 0)p0( 0−τi( 0))
p( 0)
×z( 0−τi( 0))
=ϕ0
−( 0).
16 Mimia Benhad i, Tom´as Ca aballo
implies ha he ze o solu ion o equa ion (3.6) is globally asymp o ically s able
in C1.This shows ha he ze o solu ion o (2.1) is asymp o ically s able i (3.4)
holds.
(:⇒) Assume ha he ze o solu ion o equa ion (2.1) is globally asymp o -
ically s able in C1. Now, we p o e ha (3.4) holds. I no , le us assume ha
(3.1) does no hold. O he wise, se
J= lim in
→∞ Z
0
g(s)ds 0
, K = sup
≥ 0
e−R
0g(s)ds and 0
A= sup
≥ 0
{|g( )|} .
Thus, i ollows om (3.1) ha J∈(−∞,∞), 0
K, 0
A∈[0,∞).
The e o e, he e exis s an inc easing sequence { n} ⊂ [0,∞) such ha lim
n→∞
n=∞and
lim
n→∞ Z n
0
g(s)ds =J, n = 1,2, ... (3.13)
Deno e
In=Z n
0
eRs
0g(u)du 
g(s)−a(s) + p0(s)
p(s)
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
+
N
X
i=1 
ci(s)p(s−τi(s))
p(s)
+
N
X
i=1 
bi(s)p0(s−τi(s))
p(s)!ds,
n= 1,2, ...
F om (3.2), i ollows ha
In=eR n
0g(u)du Z n
0
eRs
0g(u)du 
g(s)−a(s) + p0(s)
p(s)
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
+
N
X
i=1 
ci(s)p(s−τi(s))
p(s)
+
N
X
i=1 
bi(s)p0(s−τi(s))
p(s)!ds
≤ηeR n
0g(u)du < eJ.
This, oge he wi h (3.13), implies ha he sequence {In}is bounded.
Fu he mo e, he e exis s a con e gen subsequence. Fo b e i y o no a ion,
we s ill assume ha {In}is con e gen . The e o e, he e exis s a posi i e

Ti le Supp essed Due o Excessi e Leng h 17
in ege msuch ha o any in ege n > m,
Z n
m
eRs
0g(u)du 
g(s)−a(s) + p0(s)
p(s)
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
+
N
X
i=1 
ci(s)p(s−τi(s))
p(s)
+
N
X
i=1 
bi(s)p0(s−τi(s))
p(s)!ds
<1−η
8B(e−J+ 1),(3.14)
and
e−R n
mg(u)du >1
2, e−R n
0g(u)du < e−J+ 1, eR m
0g(u)du < eJ+ 1,(3.15)
whe e
B= max 0
KeJ+ 1,0
K0
AeJ+ 1,1.
Fo any δ0>0, conside he solu ion z( ) = z( , m, ϕ) o equa ion (3.6)
wi h |ϕ| m< δ0and |ϕ( m)|>δ0
2.I ollows om (3.7) ,(3.8) ,(3.15) and
(3.1) −(3.3) , ha o ∈[ m,∞),
|z( )| ≤ δ0e−R
mg(s)ds +Z
m
e−R
sg(u)du 
g(s)−a(s) + p0(s)
p(s)
|z(s)|
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
|z0(s−τi(s))|
+"N
X
i=1 
ci(s)p(s−τi(s))
p(s)
+
N
X
i=1 
bi(s)p0(s−τi(s))
p(s)#|z(s−τi(s))|!ds
≤0
KeJ+ 1δ0+kzk mZ
m
e−R
sg(u)du 
g(s)−a(s) + p0(s)
p(s)
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
+"N
X
i=1 
ci(s)p(s−τi(s))
p(s)
+
N
X
i=1 
bi(s)p0(s−τi(s))
p(s)#!ds
≤Bδ0+ηkzk m,
18 Mimia Benhad i, Tom´as Ca aballo
and
|z0( )|≤|z( m)| |g( )|e−R
mg(s)ds +
g( )−a( ) + p0( )
p( )
|z( )|
+
N
X
i=1 
bi( )p( −τi( ))
p( )
|z0( −τi( ))|
+"N
X
i=1 
ci( )p( −τi( ))
p( )
+
N
X
i=1 
bi( )p0( −τi( ))
p( )#|z( −τi( ))|
+|g( )|Z
m
e−R
sg(u)du 
g(s)−a(s) + p0(s)
p(s)
|z(s)|
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
|z0(s−τi(s))|
+
N
X
i=1 
ci(s)p(s−τi(s))
p(s)
+
bi(s)p0(s−τi(s))
p(s)|z(s−τi(s))|ds
≤0
K0
AeJ+ 1δ0
+kzk m|g( )|Z
m
e−R
sg(u)du 
g(s)−a(s) + p0(s)
p(s)
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
+
N
X
i=1 
ci(s)p(s−τi(s))
p(s)
+
N
X
i=1 
bi(s)p0(s−τi(s))
p(s)#
+
g( )−a( ) + p0( )
p( )
+
N
X
i=1 
ci( )p( −τi( ))
p( )
+
N
X
i=1 
bi( )p0( −τi( ))
p( )
≤Bδ0+ηkzk m.
Hence, kzk m≤Bδ0+ηkzk m, hus we ha e
kzk m≤B
1−ηδ0, o all ≥ m.(3.16)
Ti le Supp essed Due o Excessi e Leng h 19
I ollows om (3.7) ,(3.14) −(3.16) ha , o any n > m,
|z( n)|
≥ |ϕ( m)|e−R n
mg(s)ds −Z n
m
e−R n
sg(u)du g(s)−a(s) + p0(s)
p(s)z(s)
+
N
X
i=1
bi(s)p(s−τi(s))
p(s)z0(s−τi(s))
+
N
X
i=1
ci(s)p(s−τi(s))
p(s)z(s−τi(s))#ds
≥δ0e−R n
mg(u)du −Z n
m
e−R n
sg(u)du g(s)−a(s) + p0(s)
p(s)z(s)
+
N
X
i=1
bi(s)p(s−τi(s))
p(s)z0(s−τi(s))
+
N
X
i=1
ci(s)p(s−τi(s))
p(s)z(s−τi(s))#ds
≥δ0e−R n
mg(u)du
− kzk me−R n
0g(u)du Z n
m
eRs
0g(u)du 
g(s)−a(s) + p0(s)
p(s)
+
N
X
i=1 
bi(s)p(s−τi(s))
p(s)
+
N
X
i=1 
ci(s)p(s−τi(s))
p(s)#ds
≥1
4δ0−δ0B
1−ηe−J+ 11−η
8B(e−J+ 1) =1
4δ0.
The ac s ha lim
n→∞ n=∞and he ze o solu ion o equa ion (3.6) is
globally asymp o ically s able in C1imply lim
n→∞z( , n, ϕ) = lim
n→∞z0( , n, ϕ) =
0,which is in con adic ion wi h (3.17) .Hence condi ion (3.4) is necessa y in
o de ha (3.6) has a solu ion asymp o ically s able in C1. Thus, he ze o
solu ion o (3.6) is asymp o ically s able, and hence he ze o solu ion o (2.1)
is asymp o ically s able in C1. The p oo is comple e.
Rema k 3.1. When
( , x( −τ1( ))), ..., x( −τN( )))) =
N
X
i=1
bi( )x( −τi( ))
20 Mimia Benhad i, Tom´as Ca aballo
and
h( , x( −τ1( ))), ..., x( −τN( )))) =
N
X
i=1
ci( )x0( −τi( )),
wi h g( )≡a( ) and p( ) = 1, Theo em 3.1 educes o Theo em A.
Rema k 3.2. I ollows om he i s pa o he p oo o Theo em 3.1 ha he
ze o solu ion o (2.1) is globally asymp o ically s able in C1unde (3.1),(3.2),
and (3.3). Mo eo e , Theo em 3.1 s ill holds ue i (3.2),(3.3) a e sa is ied o
≥ σ o some σ∈R+.
4 An Example
In his sec ion, we analyse an example o illus a e wo ac s. On he one
hand, we will show how o apply ou main esul in his pape , Theo em 3.1.
On he o he hand and mos impo an ly, we will highligh he eal in e es and
impo ance o ou esul because he p e ious heo y de eloped by A djouni
and Djoudi [1] canno be applied o his example.
Example 4.1. Conside he ollowing linea neu al delay di e en ial equa ion
x0( ) = −a( )x( ) + b1( )x0( −τ1( )) + c1( )x( −τ1( )),(4.1)
o ≥0,co esponding o equa ion (2.1) when N= 1, τ1( ) = π
2and
a( )=1−6 sin cos
1 + sin2 , c1( ) = sin6 , b1( ) = 1
10 sin6 . By choosing g( )=1
and p( ) = 1 + sin2 3in Theo em 3.1, we ob ain ha

g( )−a( ) + p0( )
p( )
= 0.
By s aigh o wa d compu a ions, we ha e

p( −τ1( ))
p( )c1( )
=2−sin2 3
1 + sin2 3sin6 
≤0.1539,
and

p( −τ1( ))
p( )b1( )
=2−sin2 3
1 + sin2 3
sin6
10 
≤0.1539
10
≤0.0153.
Since |sin cos |<1 o ∈R, hen we deduce

p0( −τ1( ))
p( )b1( )
=6
10 2−sin2 2
1 + sin2 3sin6 
|sin cos |
≤6
10 ×0.1286 × |sin cos |
≤0.0771.
Ti le Supp essed Due o Excessi e Leng h 21
Acco ding o he de ini ion o 1, 2unde condi ions (3.2) and (3.3) o
Theo em 3.1, o ∈[0,∞),we can calcula e and ob ain
1( ) := 0.1539 + 0.0153 + 0.0771 = 0.2463,
and
2( ) := 2 (0.1539 + 0.0153 + 0.0771) = 0.4926,
hen, we ha e 1( )<1/2 and 2( )<1/2. Hence, since −π
2→ ∞ as
→ ∞,and i is easy o e i y ha R
0g(s)ds → ∞ as → ∞,|p( )| ≤ 2,
hen assump ions o Theo em 3.1 a e ul illed. The e o e, he ze o solu ion o
(4.1) is globally asymp o ically s able in C1 hanks o Theo em 3.1.
No e ha
|b1( )|+|c1( )|=1
10 sin6 +sin6 = 1.1 when =kπ+π
2 o k= 0,1,2, ....
Thus, Theo em A canno be applied o equa ion (4.1), when ( , x( −τ1( )))) =
b1( )x( −τ1( )) and h( , x( −τ1( ))) = c1( )x0( −τ1( )) in (1.4).
Rema k 3.3: The me hod in his pape can be applied o mo e gene al
neu al di e en ial sys ems han Eq. (2.1).
Con lic o in e es
The au ho s decla e ha hey ha e no con lic o in e es .
Acknowledgemen s This wo k has been pa ially suppo ed by FEDER and Minis e io
de Ciencia, Inno aci´on y Uni e sidades o Spain (G an PGC2018-096540-B-I00), and Jun a
de Andaluc´ıa, Spain (G an US-1254251)
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