New criteria for global asymptotic stability of linear neutral differential equations by a fixed point approach

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). 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