Asymptotic behavior of solutions of mixed type impulsive neutral differential equations

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1 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 hp:// R E S E A R C H Open Acce Aympoic behavior of oluion of mixed ype impulive neural differenial equaion Jeada Tariboon 1*, Soiri K Nouya 2,3 Chahai Thaiprayoon 1 * Correpondence: jeada@kmunb.ac.h 1 Nonlinear Dynamic Analyi Reearch Cener, Deparmen of Mahemaic, Faculy of Applied Science, King Mongku Univeriy of Technology Norh Bangkok, Bangkok, 10800, Thail Full li of auhor informaion i available a he end of he aricle Abrac Thi paper inveigae he aympoic behavior of oluion of he mixed ype neural differenial equaion wih impulive perurbaion x+cx τ Dxα + Pfx δ + Q x=0,0< 0, k, x k =b k x k +1b k k k δ P + δfx d + k Q/β k x d, k =1,2,3,...Sufficien condiion are obained o guaranee ha every oluion end o a conan a. Example illuraing he abrac reul are alo preened. MSC: 34K25; 34K45 Keyword: aympoic behavior; nonlinear neural delay differenial equaion; impule; Lyapunov funcional 1 Inroducion The main purpoe of hi paper i o inveigae he aympoic behavior of oluion of he following mixed ype neural differenial equaion wih impulive perurbaion: x+cx τdxα + Pf x δ + Q x=0, 0< 0, k, x k =b k x k +1b k k k δ P + δf x d + k Q/β k x d, k =1,2,3,..., 1.1 where τ, δ >0,0<α, β <1,C, D PC 0,, R, P, Q PC 0,, R + 0, f CR, R, 0 < k < k+1 wih lim k k = b k, k =1,2,3,..., are given conan. For J R, PCJ, R denoe he e of all funcion h : J R uch ha h i coninuou for k < k+1 lim k h=h k exi for all k =1,2,... The heory of impulive differenial equaion appear a a naural decripion of everal real procee ubjec o cerain perurbaion whoe duraion i negligible in comparion wih he duraion of he proce. Differenial equaion involving impule effec occur in many applicaion: phyic, populaion dynamic, ecology, biological yem, bioechnology, indurial roboic, pharmacokineic, opimal conrol, ec. The reader may refer, for inance, o he monograph by Bainov Simeonov 1, Lakhmikanham e al. 2, Samoilenko Pereyuk 3, Benchohra e al. 4. In recen year, here ha been increaing inere in he ocillaion, aympoic behavior, abiliy heory of impulive delay differenial equaion many reul have been obained ee 520he reference cied herein Tariboon e al.; licenee Springer. Thi i an Open Acce aricle diribued under he erm of he Creaive Common Aribuion Licene hp://creaivecommon.org/licene/by/2.0, which permi unrericed ue, diribuion, reproducion in any medium, provided he original work i properly cied.

2 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 2 of 16 hp:// Le u menion ome paper from which are moivaion for our work. By he conrucion of Lyapunov funcional, he auhor in 8 udied he aympoic behavior of oluion of he nonlinear neural delay differenial equaion under impulive perurbaion, x+cx τ + Pf x δ = 0, 0 < 0, k, x k =b k x k +1b k k k δ P + δf x d, k =1,2,3, A imilar mehod wa ued in 21 by conidering an impulive Euler ype neural delay differenial equaion wih imilar impulive perurbaion xdxα + Q x=0, 0< 0, k, x k =b k x k +1b k k Q/β k x d, k =1,2,3, In hi paper we combine he wo paper 8, 21 we udy he mixed ype impulive neural differenial equaion problem 1.1. By uing a imilar mehod wih he help of four Lyapunov funcional, ufficien condiion are obained o guaranee ha every oluion of 1.1endoaconana.Wenoehaproblem1.21.3canbederived from he problem 1.1apecialcae,i.e.,ifD 0Q 0, hen 1.1reduceo 1.2whileifC 0P 0, hen 1.1reduceo1.3. Therefore, he mixed ype of nonlinear delay wih an Euler form of impulive neural differenial equaion give more general reul han he previou one. Seing η 1 = maxτ, δ}, η 2 = minα, β},η = min 0 η 1, η 2 0 }, we define an iniial funcion a x=ϕ, η, 0, 1.4 where ϕ PCη, 0, R =ϕ :η, 0 R ϕ i coninuou everywhere excep a a finie number of poin,ϕ ϕ + =lim + ϕexiwihϕ + =ϕ}. Afuncionx i aid o be a oluionof 1.1 aifying he iniial condiion 1.4 if i x=ϕ for η 0, x i coninuou for 0 k, k =1,2,3,...; ii x+cx τdxα i coninuouly differeniable for > 0, k, k =1,2,3,..., aifie he fir equaion of yem 1.1; iii x k + x k exi wih x+ k =x k aifyheecondequaionofyem 1.1. Aoluionof1.1 i aid o be nonocillaory if i i evenually poiive or evenually negaive. Oherwie, i i aid o be ocillaory. 2 Main reul We are now in a poiion o eablih our main reul. Theorem 2.1 Aume ha: H 1 There exi a conan M >0uch ha x f x M x, x R, xf x>0, for x

3 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 3 of 16 hp:// H 2 The funcion C, D aify lim C = μ <1, lim D = γ <1 wih μ + γ < 1, 2.2 C k =b k C k, Dk =b k D k. 2.3 H 3 k τ α k arenoimpulivepoin, 0<b k 1, k =1,2,..., k=1 1 b k<. H 4 The funcion P, Q aify lim up +δ P + δ d + +δ Q/β d + μ1 + P+τ+δ +γ1 + P/α+δ < 2 P+δ αp+δ M 2.4 lim up /β P + δ d + /β Q/β d + μ1 + Q+τ/β +τq/β +γ 1 + Q/αβ Q/β < Then every oluion of 1.1 end o a conan a. Proof Le x be any oluion of yem 1.1. We will prove ha he lim xexi i finie. Indeed, he yem 1.1can be wrien a x +Cx τ Dxα P + δf x d Q/β x d + P + δf x + Q/β x=0, 0, k, 2.6 x k =b k x k k +1bk P + δf x d k δ k Q/β + x d, k =1,2, k From H 2 H 4, we chooe conan ε, λ, υ, ρ >0ufficienlymalluchhaμ + ε <1 γ + λ <1T > 0 ufficienly large, for T, +δ /β +δ P + δ d + Q/β +γ + λ 1+ P/α+δ αp + δ /β Q/β P + δ d + +γ + λ, for T, 1+ Q/αβ Q/β d +μ + ε 1+ P + τ + δ P + δ 2 υ, 2.8 M d +μ + ε 1+ Q + τ/β + τq/β 2ρ, 2.9 C μ + ε, D γ + λ. 2.10

4 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 4 of 16 hp:// From 2.1, 2.10, we have C μ + ε 1 f 2 x τ, x 2 τ D γ + λ 1 f 2 xα, T, x 2 α which lead o C x 2 τ μ + εf 2 x τ, D x 2 α γ + λf 2 xα, T In he following, for convenience, he expreion of funcional equaliie inequaliie will be wrien wihou i domain. Thi mean ha he relaion hold for all ufficienly large. Le V =V 1 +V 2 +V 3 +V 4, where V 1 = x +Cx τ Dxα P + δf x d V 2 = V 3 = + + P +2δ P + βδ/β β Q + δ/β + δ Q/β 2 V 4 =μ + ε +γ + λ τ Pu + δf 2 xu du d Qu/β x 2 u du d, u Pu + δf 2 xu du d Qu/β x 2 u du d, u P + τ + δf 2 x d +μ + ε α Q/αβ x 2 d + γ + λ α α τ Q + τ/β x 2 d + τ P /α+δ f 2 x d. Q/β 2 x d, Compuing dv 1 /d along he oluion of 1.1 uing he inequaliy 2ab a 2 + b 2,we have dv 1 d =2 x +Cx τ Dxα P + δf x Q/β d x d P + δf x + Q/β x P + δ 2xf x C x 2 τ C f 2 x D x 2 α D f 2 x P + δf 2 x d f 2 x P + δ d

5 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 5 of 16 hp:// Q/β d Q/β x 2 d f 2 x Q/β 2x 2 C x 2 C x 2 τ D x 2 D x 2 α Q/β x 2 d x 2 P + δf 2 x d x 2 Q/β d. Calculaing direcly for dv i /d, i =2,3,4, k,wehave dv 2 d dv 3 d = P + δf 2 x + Q/β x 2 P +2δ d P + δ P + βδ/β d P + δ = P + δf 2 x Q + δ/β d Q/β + δ + Q/β x 2 Q/β 2 d Q/β P + δ d P + δf 2 x d Q/β x 2 d, P + δf 2 x d Q/β x 2 d, dv 4 d =μ + εp + τ + δf 2 x μ + εp + δf 2 x τ + μ + ε + τ Q + τ/β x 2 +γ + λ Q/αβ + μ + ε Q/βx 2 τ x 2 γ + λ Q/β x 2 α γ + λ P /α+δ f 2 x γ + λp + δf 2 xα. α Summing for dv i /d, i =1,2,3,weobain dv 1 + dv 2 + dv 3 d d d P + δ 2xf x C x 2 τ C f 2 x D x 2 α D f 2 x f 2 x P + δ d f 2 x Q/β d f 2 x P +2δ d f 2 x Q + δ/β d + δ Q/β 2x 2 C x 2 C x 2 τ D x 2 D x 2 αx 2 x2 β P + βδ/β d x 2 P + δ d x 2 Q/β Q/β 2 d. d

6 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 6 of 16 hp:// Since P +2δ d = Q + δ/β + δ +δ d = P + δ d, +δ Q/β d, Q/β 2 1 β /β Q/β d = P + βδ/β d = d, /β P + δ d, i follow ha dv 1 + dv 2 + dv 3 d d d P + δ 2xf x C x 2 τ C f 2 x D x 2 α D f 2 x f 2 x +δ P + δ d f 2 x +δ Q/β 2x 2 C x 2 C x 2 τ D x 2 D x 2 αx 2 /β Q/β /β P + δ d x 2 Q/β d. Adding he above inequaliy wih dv 4 /d uing condiion 2.11, we have d dv 1 + dv 2 + dv 3 + dv 4 d d d d P + δ 2xf x C f 2 x D f 2 x f 2 x +δ P + δ d f 2 x +δ Q/β x 2 /β 2x 2 C x 2 D x 2 Q/β /β P + δ d x 2 Q/β d +μ + εp + τ + δf 2 x + +γ + λ Q/αβ x 2 + Applying 2.8, 2.9, 2.10, i follow ha d μ + ε + τ Q + τ/β x 2 γ + λ P /α+δ f 2 x. α dv d = dv 1 d + dv 2 d + dv 3 d + dv 4 d P + δf 2 x 2x f x C D +δ Q/β P + τ + δ d μ + ε P + δ Q/β x 2 2 C D /β +δ γ + λ α P + δ d P + δ d P/α+δ P + δ

7 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 7 of 16 hp:// /β Q/β μ + ε Q + τ/β d + τ Q/β Q/β μ + ε +δ P + δf 2 x 2 M P + τ + δ μ + ε 1+ P + δ /β x 2 2 For = k,wehave 1+ + τ P + δ d γ + λ /β P + δ d Q + τ/β Q/β γ + λ Q/αβ Q/β +δ Q/β d 1+ P/α+δ αp + δ γ + λ Q/β d 1+ Q/αβ Q/β P + δf 2 x υ Q/β x 2 ρ V 1 k = x k +C k x k τd k xα k k P + δf x k d k δ Q/β k 2 x d = b k x k + bk C k x k τ b k D k x α k b k k k δ = b 2 k V 1 k. P + δf x k d + Q/β k 2 x d I i eay o ee ha V 2 k =V 2 k, V 3 k =V 3 k, V 4 k =V 4 k. Therefore, V k =V 1 k +V 2 k +V 3 k +V 4 k = b 2 k V 1 k + V2 k + V3 k + V4 k V 1 k + V2 k + V3 k + V4 k = V k From , we conclude ha V i decreaing. In view of he fac ha V 0, we have lim V =ψ exi ψ 0. By uing 2.8, 2.9, 2.12, 2.13, we have υ T which yield P + δf 2 x d + ρ T P + δf 2 x, Q/β x 2 L 1 0,. Q/β x 2 d V T,

8 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 8 of 16 hp:// Hence, for any φ >0ξ 0, 1, we ge lim P + δf 2 x d =0, φ lim ξ Q/β x 2 d =0. Thu, i follow from ha P +2δ P + βδ/β + β +δ + 2 M /β + 2 M Pu + δf 2 xu du d Qu/β x 2 u du d u P + δ d Pu + δf 2 xu du P + δ d Q + δ/β + δ + Pu + δf 2 xu du d Qu/β x 2 u du u Qu/β x 2 u du 0, u Q/β 2 +δ Q/β + 2 M +2 /β d Q/β Pu + δf 2 xu du d d Qu/β x 2 u du d u Pu + δf 2 xu du Qu/β x 2 u du 0, u a, Pu + δf 2 xu du Qu/β x 2 u du u a, μ + ε P + τ + δf 2 x d +μ + ε τ τ +γ + λ =μ + ε +μ + ε +γ + λ α τ τ α Q/αβ x 2 d + γ + λ α P + τ + δ P + δf 2 x d P + δ α Q + τ/β Q/β + τ Q/β x 2 d Q/αβ Q/β Q/β x 2 d Q + τ/β x 2 d + τ P /α+δ f 2 x d

9 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 9 of 16 hp:// + γ + λ α 2 M +2 τ α α P/α+δ P + δf 2 x d P + δ P + δf 2 x d +2 Q/β x 2 d + 2 M α τ Q/β x 2 d P + δf 2 x d 0, a. Therefore, from he above eimaion, we have lim V 2 =0,lim V 3 =0, lim V 4 =0,repecively. Thu, lim V 1 =lim V =ψ,hai, lim x+cx τ Dxα P + δf x d Q/β 2 x d = ψ Now, we will prove ha he limi lim x+cx τ Dxα P + δf x d Q/β x d 2.15 exi i finie. Seing y =x+cx τ Dxα P + δf x d uing 1.1 condiion H 3, we have y k =x k +C k x k τd k xα k k k δ P + δf x d k Q/β x d, 2.16 Q/β k = b k x k + C k x k τ D k x α k k P + δf x k d k δ Q/β k x d x d = b k y k In view of 2.14, i follow ha lim y2 =ψ. In addiion, from , yem can be wrien a y +P + δf x + Q/β x=0, 0< 0, k, y k =b k y k, k =1,2,3,

10 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 10 of 16 hp:// If ψ =0,henlim y =0.Ifψ > 0, hen here exi a ufficienly large T uch ha y 0forany > T.Oherwie,hereiaequencea k } wih lim k a k = uch ha ya k =0,oy 2 a k 0ak. Thi conradic ψ > 0. Therefore, for any k > T, k, k+1, we have y>0ory<0fromheconinuiyofy on k, k+1. Wihou lo of generaliy, we aume ha y>0on k, k+1. I follow from H 3 ha y k+1 =b k y k+1 >0,huy>0on k+1, k+2. By uing mahemaical inducion, we deduce ha y>0on k,. Therefore, from 2.14, we have lim y= lim x+cx τ Dxα P + δf x d Q/β x d = κ, 2.19 where κ = ψ i finie. In view of 2.18, for ufficien large,wehave P + δf x Q/β d + x d β = y δy yk y k β β< k < = y δy β< k < 1 b k y k. Taking uing H 3, we have lim P + δf x d + β β Q/β x d =0, which lead o lim P + δf x d =0 lim Q/β x d =0. Thi implie ha lim x+cx τdxα = κ Nex, we hall prove ha lim xexiifinie Furher, we fir how ha x i bounded. Acually, if x i unbounded, hen here exi a equence z n } uch ha z n, xz n,an x z n = up x, z n

11 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 11 of 16 hp:// where, if z n i no an impulive poin, hen xz n =xz n. Thu, we have x z n + C z n x z n τ D z n x αz n x z n C z n x z n τ D z n x αz n x z n 1 μ ε γ λ, a n, which conradic Therefore, x i bounded. If μ =0γ =0,henlim x=κ, which implie ha 2.21hold.If0<μ <1 0<γ <1,henwededucehaCD are evenually poiive or evenually negaive. Oherwie, here are wo equence w k } w j } wih lim k w k = lim j w j = uch ha Cw k =0Dw j = 0. Therefore, Cw k 0Dw j 0ak, j. I i a conradicion o μ >0γ >0. Now, we will how ha 2.21 hold. By condiion H 2, we can find a ufficienly large T 1 uch ha for > T 1, C + D <1.Se ω = lim inf x, θ = lim up x. Then we can chooe wo equence u n } v n } uch ha u n, v n a n, lim n xu n=ω, lim xv n=θ. n For > T 1, we conider he following eigh poible cae. Cae 1. When lim C=01<D<0for > T 1,wehave κ = lim xun Du n xαu n ω + γθ, n κ = lim xvn Dv n xαv n θ + γω. n Thu, we obain ω + γθ θ + γω, ha i, ω1 γ θ1 γ. Since 0 < γ <1θ ω, i follow ha θ = ω.by2.20, we obain θ = ω = κ 1γ, which how ha 2.21hold.

12 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 12 of 16 hp:// Cae 2. When lim D=01<C<0for > T 1,wege κ = lim xun +Cu n xu n τ ω μω n κ = lim xvn +Cv n xv n τ θ μθ, n which lead o ω1 μ θ1 μ. Since 0 < μ <1θ ω,weconcludeha θ = ω = κ 1μ, which implie ha 2.21hold. Cae 3. lim C =0,0<D <1for > T 1. The mehod of proof i imilar o he above wo cae. Therefore, we omi i. Cae 4. lim D =0,0<C <1for > T 1. The mehod of proof i imilar o he above wo fir cae. Therefore, we omi i. Cae 5. When 1 < D<00<C<1for > T 1,wehave κ = lim xun +Cu n xu n τdu n xαu n ω + μθ + γθ n κ = lim xvn +Cv n xv n τdv n xαv n θ + μω + γω, n which yield ω1 μ γ θ1 μ γ. Since 0 < μ + γ <1θ ω,wehaveθ = ω.thu θ = ω = κ 1μ γ, o 2.21hold. Uing imilar argumen, we can prove ha 2.21 alo hold for he following cae: Cae 6. 1 < C<0,0<D<1. Cae 7. 1 < C<0,1<D<0. Cae 8. 0 < C<1,0<D<1. Summarizing he above inveigaion, we conclude ha 2.21 hold o he proof i compleed. Theorem 2.2 Le condiion H 1 -H 4 of Theorem 2.1 hold. Then every ocillaory oluion of 1.1 end o zero a.

13 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 13 of 16 hp:// Corollary 2.1 Aume ha H 3 hold +δ +δ Q/β lim up P + δ d + d < /β /β Q/β lim up P + δ d + d < Then every oluion of he equaion x +Px δ+ Q x=0, 0< 0, k, x k =b k x k +1b k k k δ P + δx d + k Q/β k x d, k =1,2,3,..., 2.25 end o a conan a. Corollary 2.2 The condiion imply ha every oluion of he equaion x +Px δ+ Q x=0, 0< 0, 2.26 end o a conan a. Theorem 2.3 The condiion H 1 -H 4 of Theorem 2.1 ogeher wih 0 P + δ d =, 0 Q/β d =, 2.27 imply ha every oluion of 1.1 end o zero a. Proof From Theorem 2.2, we only have oprove haevery nonocillaoryoluion of 1.1 end o zero a. Wihou lo of generaliy, we aume ha x i an evenually poiive oluion of 1.1. A in he proof of Theorem 2.1,1.1canbewrienainheform Inegraing from 0 o boh ide of he fir equaion of 2.18, one ha 0 P + δf x d + 0 Q/β x d = y 0 y 1 b k y k. 0 < k < Applying 2.19H 3, we have 0 P + δf x d < 0 Q/β x d <. Thi, ogeher wih 2.27, implie ha lim inf f x = 0 lim inf x =0.By Theorem 2.1, lim x=0.thicompleeheproof.

14 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 14 of 16 hp:// Corollary 2.3 Aume ha 2.1, 2.2, 2.4, 2.5, 2.27 hold. Then every oluion of he equaion Q x+cx τdxα + Pf x δ + x=0, < 0, end o zero a. 3 Example In hi ecion, we preen wo example o illurae our reul. Example 3.1 Conider he following mixed ype neural differenial equaion wih impulive perurbaion: x+ k+3 x 1 3k+9 3k 2 +3k6 2 x 8k 2 +8k16 e co2 x πx π xk= k2 +6k+8 k+3 2 xk +1 k2 +6k+8 k+3 2 k k π co2 xx d + k k e 2 1 ln +2 x 2++π 5+π 2 1 lne 2 +2 e 2 =0, 1, x d, k =2,3,4, Here C =k +3 /3k 2 +3k 6,D = 3k +9 /8k 2 +8k 16,P =2+ /5 2, Q = 1/ln +2, k 1,k, b k =k 2 +6k + 8/k +3 2, 0 =1,k =2,3,4,..., f x=x1 + 1/4co 2 x, τ = 1/2, δ = π, α =1/e,β =1/e 2. We can find ha i x co2 xx 5 4 x, x R, co2 xx 2 >0for x 0; ii lim C = 1 3 = μ <1, lim D = 3 17 = γ <1wih μ + γ = 8 24 <1, Ck= k2 +6k+8 Ck, Dk= k2 +6k+8 Dk ; k+3 2 k+3 2 iii k 1/2 1/e k are no impulive poin, 0<k 2 +6k + 8/k for k =1,2,..., k=1 1 k2 +6k +8 k +3 2 = k=1 1 k +3 2 < ; iv lim up +δ P + δ d + +δ Q/β d + μ1 + P+τ+δ +γ1 + P/α+δ = 17 P+δ αp+δ 12 < 8 5 lim up /β P + δ d + /β Q/β d + μ1 + Q+τ/β +τq/β +γ 1 + Q/αβ Q/β = <2. Hence, by i-iv all aumpion of Theorem 2.1 are aified. Therefore, we conclude ha every oluion of 3.1endoaconana.

15 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 15 of 16 hp:// Example 3.2 Conider he following mixed ype neural differenial equaion wih impulive perurbaion x+ 12k+16 12k+16 42k 2 +21k21 x 2e 3 x 2 54k 2 +27k in2 x π 2 x π ln +3 x 3e xk= 6k2 +17k+7 6k 2 +17k+12 xk +1 6k2 +17k+7 6k 2 +17k+12 k in2 xx d + k k 3e k π ln3e π 4+3+2π 2 =0, 1, x d, k =2,3,4, Here C = 12k +16 /54k 2 +27k 27,D = 12k +16 /42k 2 +21k 21, P =2 + 1/4 +3 2, Q =4/2ln +3, k 1,k, b k =6k 2 +17k + 7/6k k +12, 0 =1,k =2,3,4,...,f x=x1 + 2/5 in 2 x, τ = 2/3, δ = π/2, α = 1/2e 3, β = 1/3e. We can how ha i x in2 xx 7 5 x, x R, in2 xx 2 >0for x 0; ii lim C = 2 9 = μ <1, lim D = 2 32 = γ <1wih μ + γ = 7 63 <1, Ck= 6k2 +17k+7 6k 2 +17k+12 Ck, Dk= 6k2 +17k+7 6k 2 +17k+12 Dk ; iii k 2/3 1/2e 3 k are no impulive poin, iv v 0<6k 2 +17k + 7/6k 2 +17k for k =1,2,..., k=1 1 6k2 +17k +7 6k 2 +17k +12 = k=1 5 6k 2 +17k +12 < ; lim up +δ P + δ d + +δ Q/β d + μ1 + P+τ+δ +γ1 + P/α+δ = 64 P+δ αp+δ 63 < 10 7 lim up /β P + δ d + /β Q/β d μ1 + Q+τ/β +τq/β +γ 1 + Q/αβ Q/β = < 2; 2 +1+π P + δ d = π d = 2 Q/β 4 d = d =. 1 2 ln3e+3 Hence, all aumpion of Theorem 2.3 are aified herefore every oluion of 3.2 end o zero a. Compeing inere The auhor declare ha hey have no compeing inere.

16 Tariboon e al. Advance in Difference Equaion 2014, 2014:327 Page 16 of 16 hp:// Auhor conribuion All auhor conribued equally in hi aricle. They read approved he final manucrip. Auhor deail 1 Nonlinear Dynamic Analyi Reearch Cener, Deparmen of Mahemaic, Faculy of Applied Science, King Mongku Univeriy of Technology Norh Bangkok, Bangkok, 10800, Thail. 2 Deparmen of Mahemaic, Univeriy of Ioannina, Ioannina, , Greece. 3 Nonlinear Analyi Applied Mahemaic NAAM-Reearch Group, Deparmen of Mahemaic, Faculy of Science, King Abdulaziz Univeriy, P.O. Box 80203, Jeddah, 21589, Saudi Arabia. Acknowledgemen We would like o hank he reviewer for heir valuable commen uggeion on he manucrip. Thi reearch wa funded by King Mongku Univeriy of Technology Norh Bangkok. Conrac no. KMUTNB-GOV Received: 29 Augu 2014 Acceped: 9 December 2014 Publihed: 22 Dec 2014 Reference 1. Bainov, DD, Simeonov, PS: Syem wih Impule Effec. Elli Horwood, Chicheer Lakhmikanham, V, Bainov, DD, Simeonov, PS: Theory of Impulive Differenial Equaion. World Scienific, Singapore Samoilenko, AM, Pereyuk, NA: Impulive Differenial Equaion. World Scienific, Singapore Benchohra, M, Henderon, J, Nouya, SK: Impulive Differenial Equaion Incluion, vol. 2. Hindawi Publihing Corporaion, New York Bainov, DD, Dinirova, MB, Dihliev, AB: Ocillaion of he oluion of impulive differenial equaion inequaliie wih a rearded argumen. Rocky M. J. Mah. 28, Luo, Z, Shen, J: Sabiliy boundedne for impulive differenial equaion wih infinie delay. Nonlinear Anal. 46, Liu, X, Shen, J: Aympoic behavior of oluion of impulive neural differenial equaion. Appl. Mah. Le. 12, Shen, J, Liu, Y, Li, J: Aympoic behavior of oluion of nonlinear neural differenial equaion wih impule. J. Mah. Anal. Appl. 332, Shen, J, Liu, Y: Aympoic behavior of oluion for nonlinear delay differenial equaion wih impule. J. Appl. Mah. Compu. 213, Wei, G, Shen, J: Aympoic behavior of oluion of nonlinear impulive delay differenial equaion wih poiive negaive coefficien. Mah. Compu. Model. 44, Luo, J, Debnah, L: Aympoic behavior of oluion of forced nonlinear neural delay differenial equaion wih impule. J. Appl. Mah. Compu. 12, Jiang, F, Sun, J: Aympoic behavior of neural delay differenial equaion of Euler form wih conan impulive jump. Appl. Mah. Compu. 219, Pian, S, Balachran, Y: Aympoic behavior reul for nonlinear impulive neural differenial equaion wih poiive negaive coefficien. Bonfring In. J. Daa Min. 2, Wang, QR: Ocillaion crieria for fir-order neural differenial equaion. Appl. Mah. Le. 8, Tariboon, J, Thiramanu, P: Ocillaion of a cla of econd-order linear impulive differenial equaion. Adv. Differ. Equ. 2012, Jiang, F, Shen, J: Aympoic behavior of oluion for a nonlinear differenial equaion wih conan impulive jump. Aca Mah. Hung. 138, Jiang, F, Shen, J: Aympoic behavior of nonlinear neural impulive delay differenial equaion wih forced erm. Kodai Mah. J. 35, Gunaekar, T, Samuel, FP, Arjunan, MM: Exience reul for impulive neural funcional inegrodifferenial equaion wih infinie delay. J. Nonlinear Sci. Appl. 6, Kumar, P, Pey, DN, Bahuguna, D: On a new cla of abrac impulive funcional differenial equaion of fracional order. J. Nonlinear Sci. Appl. 7, Samuel, FP, Balachran, K: Exience of oluion for quai-linear impulive funcional inegrodifferenial equaion in Banach pace. J. Nonlinear Sci. Appl. 7, Guan, K, Shen, J: Aympoic behavior of oluion of a fir-order impulive neural differenial equaion in Euler form. Appl. Mah. Le. 24, / Cie hi aricle a: Tariboon e al.: Aympoic behavior of oluion of mixed ype impulive neural differenial equaion. Advance in Difference Equaion 2014, 2014:327