Asymptotic behavior of solutions of mixed type impulsive ... - Core

Tariboon et al. Advances in Difference Equations 2014, 2014:327 http://www.advancesindifferenceequations.com/content/2014/1/327

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Asymptotic behavior of solutions of mixed type impulsive neutral differential equations Jessada Tariboon1* , Sotiris K Ntouyas2,3 and Chatthai Thaiprayoon1 * Correspondence: [email protected] 1 Nonlinear Dynamic Analysis Research Center, Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology North Bangkok, Bangkok, 10800, Thailand Full list of author information is available at the end of the article

Abstract This paper investigates the asymptotic behavior of solutions of the mixed type neutral differential equation with impulsive perturbations x(β t) = 0, 0 < t0 ≤ t, t = tk , [x(t) + C(t)x(t – τ ) – D(t)x(α t)] + P(t)f (x(t – δ )) + Q(t) t tt x(tk ) = bk x(tk– ) + (1 – bk )( tkk–δ P(s + δ )f (x(s)) ds + βktk Q(s/s β ) x(s) ds), k = 1, 2, 3, . . . . Sufficient conditions are obtained to guarantee that every solution tends to a constant as t → ∞. Examples illustrating the abstract results are also presented. MSC: 34K25; 34K45 Keywords: asymptotic behavior; nonlinear neutral delay differential equation; impulse; Lyapunov functional

1 Introduction The main purpose of this paper is to investigate the asymptotic behavior of solutions of the following mixed type neutral differential equation with impulsive perturbations: ⎧ [x(t) + C(t)x(t – τ ) – D(t)x(αt)] + P(t)f (x(t – δ)) + Q(t) x(βt) = , ⎪ t ⎪ ⎪ ⎨  < t ≤ t, t = t ,  k t t – ⎪ x(t ) = b x(t ) + ( – bk )( tkk–δ P(s + δ)f (x(s)) ds + βtkk Q(s/β) x(s) ds), k k ⎪ k s ⎪ ⎩ k = , , , . . . ,

(.)

where τ , δ > ,  < α, β < , C(t), D(t) ∈ PC([t , ∞), R), P(t), Q(t) ∈ PC([t , ∞), R+ ), f ∈ C(R, R),  < tk < tk+ with limk→∞ tk = ∞ and bk , k = , , , . . . , are given constants. For J ⊂ R, PC(J, R) denotes the set of all functions h : J → R such that h is continuous for tk ≤ t < tk+ and limt→tk– h(t) = h(tk– ) exists for all k = , , . . . . The theory of impulsive differential equations appears as a natural description of several real processes subject to certain perturbations whose duration is negligible in comparison with the duration of the process. Differential equations involving impulse effects occurs in many applications: physics, population dynamics, ecology, biological systems, biotechnology, industrial robotic, pharmacokinetics, optimal control, etc. The reader may refer, for instance, to the monographs by Bainov and Simeonov [], Lakshmikantham et al. [], Samoilenko and Perestyuk [], and Benchohra et al. []. In recent years, there has been increasing interest in the oscillation, asymptotic behavior, and stability theory of impulsive delay differential equations and many results have been obtained (see [–] and the references cited therein). ©2014 Tariboon et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Page 2 of 16

Let us mention some papers from which are motivation for our work. By the construction of Lyapunov functionals, the authors in [] studied the asymptotic behavior of solutions of the nonlinear neutral delay differential equation under impulsive perturbations, [x(t) + C(t)x(t – τ )] + P(t)f (x(t – δ)) = ,  < t ≤ t, t = tk , t x(tk ) = bk x(tk– ) + ( – bk ) tkk–δ P(s + δ)f (x(s)) ds, k = , , , . . . .

(.)

A similar method was used in [] by considering an impulsive Euler type neutral delay differential equation with similar impulsive perturbations [x(t) – D(t)x(αt)] + Q(t) x(βt) = ,  < t ≤ t, t = tk , t t x(tk ) = bk x(tk– ) + ( – bk ) βtkk Q(s/β) x(s) ds, k = , , , . . . . s

(.)

In this paper we combine the two papers [, ] and we study the mixed type impulsive neutral differential equation problem (.). By using a similar method with the help of four Lyapunov functionals, sufficient conditions are obtained to guarantee that every solution of (.) tends to a constant as t → ∞. We note that problems (.) and (.) can be derived from the problem (.) as special cases, i.e., if D(t) ≡  and Q(t) ≡ , then (.) reduces to (.) while if C(t) ≡  and P(t) ≡ , then (.) reduces to (.). Therefore, the mixed type of nonlinear delay with an Euler form of impulsive neutral differential equations gives more general results than the previous one. Setting η = max{τ , δ}, η = min{α, β}, and η = min{t –η , η t }, we define an initial function as x(t) = ϕ(t),

t ∈ [η, t ],

(.)

where ϕ ∈ PC([η, t ], R) = {ϕ : [η, t ] → R|ϕ is continuous everywhere except at a finite number of point s, and ϕ(s– ) and ϕ(s+ ) = lims→s+ ϕ(s) exist with ϕ(s+ ) = ϕ(s)}. A function x(t) is said to be a solution of (.) satisfying the initial condition (.) if (i) x(t) = ϕ(t) for η ≤ t ≤ t , x(t) is continuous for t ≥ t and t = tk , k = , , , . . . ; (ii) x(t) + C(t)x(t – τ ) – D(t)x(αt) is continuously differentiable for t > t , t = tk , k = , , , . . . , and satisfies the first equation of system (.); (iii) x(tk+ ) and x(tk– ) exist with x(tk+ ) = x(tk ) and satisfy the second equation of system (.). A solution of (.) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, it is said to be oscillatory.

2 Main results We are now in a position to establish our main results. Theorem . Assume that: (H ) There exists a constant M >  such that |x| ≤ f (x) ≤ M|x|,

x ∈ R, xf (x) > , for x = .

(.)


Page 3 of 16

(H ) The functions C, D satisfy lim C(t) = μ < ,

t→∞

lim D(t) = γ <  with μ + γ < ,

(.)

t→∞

and

C(tk ) = bk C tk– ,

D(tk ) = bk D tk– .

(.)

(H ) tk – τ and αtk are not impulsive points,  < bk ≤ , k = , , . . . , and (H ) The functions P, Q satisfy

∞

k= ( – bk ) < ∞.

t+δ t+δ ds lim supt→∞ [ t–δ P(s + δ) ds + βt Q(s/β) s P(t+τ +δ) P((t/α)+δ) + μ( + P(t+δ) ) + γ ( + αP(t+δ) )] < M

(.)

t/β t/β lim supt→∞ [ t–δ P(s + δ) ds + βt Q(s/β) ds s tQ((t+τ )/β) Q(t/(αβ)) + μ( + (t+τ )Q(t/β) ) + γ ( + Q(t/β) )] < .

(.)

and

Then every solution of (.) tends to a constant as t → ∞. Proof Let x(t) be any solution of system (.). We will prove that the limt→∞ x(t) exists and is finite. Indeed, the system (.) can be written as

t

x(t) + C(t)x(t – τ ) – D(t)x(αt) –

P(s + δ)f x(s) ds –

t–δ

t βt

Q(s/β) x(s) ds s

Q(t/β) x(t) = , t ≥ t , t = tk , + P(t + δ)f x(t) + t tk

P(s + δ)f x(s) ds x(tk ) = bk x tk– + ( – bk )

(.)

tk –δ

tk

+ βtk

Q(s/β) x(s) ds , s

k = , , . . . .

(.)

From (H ) and (H ), we choose constants ε, λ, υ, ρ >  sufficiently small such that μ + ε <  and γ + λ <  and T > t sufficiently large, for t ≥ T, P(t + τ + δ) Q(s/β) ds + (μ + ε)  + s P(t + δ) t–δ βt P((t/α) + δ)  + (γ + λ)  + ≤ – υ, αP(t + δ) M t/β

t/β Q(s/β) tQ((t + τ )/β) ds + (μ + ε)  + P(s + δ) ds + s (t + τ )Q(t/β) t–δ βt Q(t/(αβ)) ≤  – ρ, + (γ + λ)  + Q(t/β)

t+δ

t+δ

P(s + δ) ds +

(.)

(.)

and, for t ≥ T, C(t) ≤ μ + ε,

D(t) ≤ γ + λ.

(.)


Page 4 of 16

From (.), (.), we have |D(t)| f  (x(αt)) ≤≤  , γ +λ x (αt)

f  (x(t – τ )) |C(t)| ≤≤  , μ+ε x (t – τ )

t ≥ T,

which lead to

C(t)x (t – τ ) ≤ (μ + ε)f  x(t – τ ) ,

D(t)x (αt) ≤ (γ + λ)f  x(αt) , t ≥ T.

(.)

In the following, for convenience, the expressions of functional equalities and inequalities will be written without its domain. This means that the relations hold for all sufficiently large t. Let V (t) = V (t) + V (t) + V (t) + V (t), where

V (t) = x(t) + C(t)x(t – τ ) – D(t)x(αt) –

V (t) =

t

P(s + δ) t–δ


t–δ

t

t

t βt

 Q(s/β) x(s) ds , s

P(u + δ)f  x(u) du ds

s

P((s + βδ)/β) t Q(u/β)  x (u) du ds, + β u βt s

t

Q((s + δ)/β) t P(u + δ)f  x(u) du ds V (t) = s+δ t–δ s

t  t Q(s/β ) Q(u/β)  x (u) du ds, + s u βt s t

and

P(s + τ + δ)f x(s) ds + (μ + ε)

t

t



V (t) = (μ + ε) t–τ

t

+ (γ + λ) αt

Q(s/(αβ))  γ +λ x (s) ds + s α

t–τ t

Q((s + τ )/β)  x (s) ds s+τ

P (s/α) + δ f  x(s) ds.

αt

Computing dV /dt along the solution of (.) and using the inequality ab ≤ a + b , we have dV = – x(t) + C(t)x(t – τ ) – D(t)x(αt) dt

t

t

Q(s/β) x(s) ds – P(s + δ)f x(s) ds – s t–δ βt

Q(t/β) x(t) × P(t + δ)f x(t) + t

≤ –P(t + δ) x(t)f x(t) – C(t)x (t – τ ) – C(t)f  x(t) – D(t)x (αt)

– D(t)f  x(t) –

t t–δ

P(s + δ)f  x(s) ds – f  x(t)

t

P(s + δ) ds t–δ


Page 5 of 16

t Q(s/β) Q(s/β)   x (s) ds – f x(t) ds – s s βt βt Q(t/β) x (t) – C(t)x (t) – C(t)x (t – τ ) – D(t)x (t) – t

t

t

P(s + δ)f  x(s) ds – x (t) P(s + δ) ds – D(t)x (αt) –

t

t

t–δ

– βt

Q(s/β)  x (s) ds – x (t) s

t βt

Q(s/β) ds . s

t–δ

Calculating directly for dVi /dt, i = , , , t = tk , we have

t

t

dV  = P(t + δ)f x(t) P(s + δ) ds – P(t + δ) P(s + δ)f  x(s) ds dt t–δ t–δ

t

t

Q(s/β)  Q(t/β)  x (t) x (s) ds, P (s + βδ)/β ds – P(t + δ) + βt s βt βt

t Q((s + δ)/β)

Q(t/β) t dV = P(t + δ)f  x(t) ds – P(s + δ)f  x(s) ds dt s+δ t t–δ t–δ

t

t  Q(t/β) Q(t/β)  Q(s/β ) Q(s/β)  x (t) ds – x (s) ds, + t s t s βt βt and

dV = (μ + ε)P(t + τ + δ)f  x(t) – (μ + ε)P(t + δ)f  x(t – τ ) dt

(μ + ε) (μ + ε) Q (t + τ )/β x (t) – Q(t/β)x (t – τ ) + (t + τ ) t Q(t/(αβ))  Q(t/β)  x (t) – (γ + λ) x (αt) t t

(γ + λ) + P (t/α) + δ f  x(t) – (γ + λ)P(t + δ)f  x(αt) . α + (γ + λ)

Summing for dVi /dt, i = , , , we obtain dV dV dV + + dt dt dt


– D(t)f  x(t) – f  x(t)

– f x(t)

P(s + δ) ds – f  x(t)

t t–δ

t



P(s + δ) ds – f t–δ



x(t)

t

t–δ

t

Q(s/β) ds s βt Q((s + δ)/β) ds s+δ

Q(t/β) x (t) – C(t)x (t) – C(t)x (t – τ ) – D(t)x (t) – t

t

t Q(s/β) ds P(s + δ) ds – x (t) – D(t)x (αt) – x (t) s t–δ βt

t

x (t) t Q(s/β  ) – ds . P (s + βδ)/β ds – x (t) β βt s βt


Page 6 of 16

Since

t

t

t–δ

t/β Q(s/β  ) Q(s/β) ds = ds, s s t

t/β

 t P (s + βδ)/β ds = P(s + δ) ds, β βt t

t

P(s + δ) ds,

t–δ

t+δ

P(s + δ) ds = t

Q((s + δ)/β) ds = s+δ

βt

t+δ t

Q(s/β) ds, s

it follows that dV dV dV + + dt dt dt


– D(t)f  x(t) – f  x(t)

t+δ

P(s + δ) ds – f



x(t)

t–δ

t+δ

βt

Q(s/β) ds s

Q(t/β) x (t) – C(t)x (t) – C(t)x (t – τ ) – D(t)x (t) – t

t/β

t/β Q(s/β) ds . P(s + δ) ds – x (t) – D(t)x (αt) – x (t) s t–δ βt Adding the above inequality with dV /dt and using condition (.), we have dV dV dV dV + + + dt dt dt dt

≤ –P(t + δ) x(t)f x(t) – C(t)f  x(t) – D(t)f  x(t)

– f  x(t)

t+δ

P(s + δ) ds – f  x(t)

t–δ

t+δ

βt

Q(s/β) ds s

Q(t/β) x (t) – C(t)x (t) – D(t)x (t) – t

t/β

t/β Q(s/β) ds P(s + δ) ds – x (t) – x (t) s t–δ βt

(μ + ε) Q (t + τ )/β x (t) + (μ + ε)P(t + τ + δ)f  x(t) + t+τ

Q(t/(αβ))  (γ + λ) + (γ + λ) x (t) + P (t/α) + δ f  x(t) . t α Applying (.), (.), and (.), it follows that dV dV dV dV dV = + + + dt dt dt dt dt

t+δ

x(t) ≤ –P(t + δ)f  x(t) – C(t) – D(t) – P(s + δ) ds f (x(t)) t–δ

t+δ P(t + τ + δ) (γ + λ) P((t/α) + δ) Q(s/β) ds – (μ + ε) – – s P(t + δ) α P(t + δ) βt

t/β Q(t/β)  – x (t)  – C(t) – D(t) – P(s + δ) ds t t–δ


Page 7 of 16

(μ + ε)t Q((t + τ )/β) Q(t/(αβ)) Q(s/β) ds – – (γ + λ) s t+τ Q(t/β) Q(t/β) βt

t+δ

t+δ  Q(s/β) – ds ≤ –P(t + δ)f  x(t) P(s + δ) ds – M s t–δ βt P((t/α) + δ) P(t + τ + δ) – (γ + λ)  + – (μ + ε)  + P(t + δ) αP(t + δ)

t/β

t/β Q(t/β)  Q(s/β) – x (t)  – ds P(s + δ) ds – t s t–δ βt Q(t/(αβ)) t Q((t + τ )/β) – (γ + λ)  + – (μ + ε)  + t + τ Q(t/β) Q(t/β)

t/β

–

Q(t/β)  ≤ –P(t + δ)f  x(t) υ – x (t)ρ. t

(.)

For t = tk , we have V (tk ) = x(tk ) + C(tk )x(tk – τ ) – D(tk )x(αtk )

tk


–

tk tk –δ

βtk

Q(s/β) x(s) ds s



= bk x tk– + bk C tk– x tk– – τ – bk D tk– x αtk–

tk

– bk

P(s + δ)f x(s) ds +

tk –δ

=

bk V

tk βtk

–

tk .

Q(s/β) x(s) ds s



It is easy to see that V (tk ) = V (tk– ), V (tk ) = V (tk– ), and V (tk ) = V (tk– ). Therefore, V (tk ) = V (tk ) + V (tk ) + V (tk ) + V (tk )

= bk V tk– + V tk– + V tk– + V tk–

≤ V tk– + V tk– + V tk– + V tk–

= V tk– .

(.)

From (.) and (.), we conclude that V (t) is decreasing. In view of the fact that V (t) ≥ , we have limt→∞ V (t) = ψ exist and ψ ≥ . By using (.), (.), (.), and (.), we have

∞

υ

P(t + δ)f  x(t) dt + ρ

T

∞ T

Q(t/β)  x (t) dt ≤ V (T), t

which yields

Q(t/β)  P(t + δ)f  x(t) , x (t) ∈ L (t , ∞). t


Page 8 of 16

Hence, for any φ >  and ξ ∈ (, ), we get

P(s + δ)f  x(s) ds = ,

t

lim

t→∞ t–φ

t

Q(s/β)  x (s) ds = . s

lim

t→∞ ξ t

Thus, it follows from (.) and (.) that

t

t

P(s + δ) t–δ


s

P((s + βδ)/β) t Q(u/β)  x (u) du ds + β u βt s

t

t+δ

P(s + δ) ds P(u + δ)f  x(u) du ≤

t

t–δ

t–δ

t/β

+ t–δ

≤

t

P(s + δ) ds

 M

βt

Q(u/β)  x (u) du u


t t–δ

 + M

t

Q(u/β)  x (u) du → , as t → ∞, u βt

t

Q((s + δ)/β) t P(u + δ)f  x(u) du ds s+δ t–δ s

t  t Q(s/β ) Q(u/β)  x (u) du ds + s u βt s

t

t+δ

Q(s/β) ds P(u + δ)f  x(u) du ≤ s βt t–δ

t

t/β Q(s/β) Q(u/β)  ds x (u) du + s u βt βt

 t P(u + δ)f  x(u) du ≤ M t–δ

t Q(u/β)  x (u) du → , as t → ∞, + u βt and

t

(μ + ε)

P(s + τ + δ)f  x(s) ds + (μ + ε)

t–τ

t

Q((s + τ )/β)  x (s) ds s+τ

t–τ t

+ (γ + λ) αt

γ +λ Q(s/(αβ))  x (s) ds + s α

t

P (s/α) + δ f  x(s) ds

αt

P(s + τ + δ) P(s + δ)f  x(s) ds P(s + δ)

t

= (μ + ε) t–τ

t

+ (μ + ε) t–τ

t

+ (γ + λ) αt

sQ((s + τ )/β) Q(s/β)  · x (s) ds Q(s/β)(s + τ ) s Q(s/(αβ)) Q(s/β)  · x (s) ds Q(s/β) s


Page 9 of 16

γ + λ t P((s/α) + δ) P(s + δ)f  x(s) ds α P(s + δ) αt

t

t

Q(s/β)   x (s) ds P(s + δ)f  x(s) ds +  ≤ M t–τ s t–τ

t

Q(s/β)   t x (s) ds + P(s + δ)f  x(s) ds → , + s M αt αt +

as t → ∞.

Therefore, from the above estimations, we have limt→∞ V (t) = , limt→∞ V (t) = , and limt→∞ V (t) = , respectively. Thus, limt→∞ V (t) = limt→∞ V (t) = ψ , that is, lim x(t) + C(t)x(t – τ ) – D(t)x(αt)

t→∞

t

–


t–δ

t βt

Q(s/β) x(s) ds s

 = ψ.

(.)

Now, we will prove that the limit lim x(t) + C(t)x(t – τ ) – D(t)x(αt)

t→∞

t

–


t–δ

t βt

Q(s/β) x(s) ds s

(.)

exists and is finite. Setting y(t) = x(t) + C(t)x(t – τ ) – D(t)x(αt)

t

t

Q(s/β) x(s) ds, – P(s + δ)f x(s) ds – s t–δ βt

(.)

and using (.) and condition (H ), we have y(tk ) = x(tk ) + C(tk )x(tk – τ ) – D(tk )x(αtk )

tk

tk

Q(s/β) x(s) ds P(s + δ)f x(s) ds – – s tk –δ βtk

= bk x tk– + C tk– x tk– – τ – D tk– x αtk–

tk

–


tk –δ

= bk y tk– .

tk βtk

Q(s/β) x(s) ds s

(.)

In view of (.), it follows that lim y (t) = ψ.

t→∞

In addition, from (.) and (.), system (.)-(.) can be written as x(t) = , y (t) + P(t + δ)f (x(t)) + Q(t/β) t y(tk ) = bk y(tk– ), k = , , , . . . .

 < t ≤ t, t = tk ,

(.)


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If ψ = , then limt→∞ y(t) = . If ψ > , then there exists a sufficiently large T ∗ such that y(t) =  for any t > T ∗ . Otherwise, there is a sequence {ak } with limk→∞ ak = ∞ such that y(ak ) = , and so y (ak ) →  as k → ∞. This contradicts ψ > . Therefore, for any tk > T ∗ , and t ∈ [tk , tk+ ), we have y(t) >  or y(t) <  from the continuity of y on [tk , tk+ ). Without loss of generality, we assume that y(t) >  on [tk , tk+ ). It follows from (H ) that – y(tk+ ) = bk y(tk+ ) > , and thus y(t) >  on [tk+ , tk+ ). By using mathematical induction, we deduce that y(t) >  on [tk , ∞). Therefore, from (.), we have lim y(t) = lim x(t) + C(t)x(t – τ ) – D(t)x(αt)

t→∞

t→∞

t


– t–δ

where κ =

√

t βt

Q(s/β) x(s) ds = κ, s

(.)

ψ and is finite. In view of (.), for sufficient large t, we have

t

t

Q(s/β) x(s) ds s βt–δ y(tk ) – y tk–

P(s + δ)f x(s) ds + βt–δ

= y(βt – δ) – y(t) –

βt–δ T , we consider the following eight possible cases. Case . When limt→∞ C(t) =  and – < D(t) <  for t > T , we have κ = lim x(un ) – D(un )x(αun ) ≤ ω + γ θ , n→∞

and κ = lim x(vn ) – D(vn )x(αvn ) ≥ θ + γ ω. n→∞

Thus, we obtain ω + γ θ ≥ θ + γ ω, that is, ω( – γ ) ≥ θ ( – γ ). Since  < γ <  and θ ≥ ω, it follows that θ = ω. By (.), we obtain θ =ω=

κ , –γ

which shows that (.) holds.

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Case . When limt→∞ D(t) =  and – < C(t) <  for t > T , we get κ = lim x(un ) + C(un )x(un – τ ) ≤ ω – μω n→∞

and κ = lim x(vn ) + C(vn )x(vn – τ ) ≥ θ – μθ , n→∞

which leads to ω( – μ) ≥ θ ( – μ). Since  < μ <  and θ ≥ ω, we conclude that θ =ω=

κ , –μ

which implies that (.) holds. Case . limt→∞ C(t) = ,  < D(t) <  for t > T . The method of proof is similar to the above two cases. Therefore, we omit it. Case . limt→∞ D(t) = ,  < C(t) <  for t > T . The method of proof is similar to the above two first cases. Therefore, we omit it. Case . When – < D(t) <  and  < C(t) <  for t > T , we have κ = lim x(un ) + C(un )x(un – τ ) – D(un )x(αun ) ≤ ω + μθ + γ θ n→∞

and κ = lim x(vn ) + C(vn )x(vn – τ ) – D(vn )x(αvn ) ≥ θ + μω + γ ω, n→∞

which yields ω( – μ – γ ) ≥ θ ( – μ – γ ). Since  < μ + γ <  and θ ≥ ω, we have θ = ω. Thus θ =ω=

κ , –μ–γ

and so (.) holds. Using similar arguments, we can prove that (.) also holds for the following cases: Case . – < C(t) < ,  < D(t) < . Case . – < C(t) < , – < D(t) < . Case .  < C(t) < ,  < D(t) < . Summarizing the above investigation, we conclude that (.) holds and so the proof is completed. Theorem . Let conditions (H )-(H ) of Theorem . hold. Then every oscillatory solution of (.) tends to zero as t → ∞.

Page 12 of 16


Page 13 of 16

Corollary . Assume that (H ) holds and lim sup t→∞

t+δ

t+δ

P(s + δ) ds +

t–δ

βt

Q(s/β) ds <  s

(.)

Q(s/β) ds < . s

(.)

and

t/β

P(s + δ) ds +

lim sup t→∞

t/β

t–δ

βt

Then every solution of the equation ⎧ Q(t) = ,  < t ≤ t, t = tk , ⎪ ⎨x (t) + P(t)x(t – δ) + t x(βt) t – x(tk ) = bk x(tk ) + ( – bk )( tkk–δ P(s + δ)x(s) ds ⎪ t ⎩ + βtkk Q(s/β) x(s) ds), k = , , , . . . , s

(.)

tends to a constant as t → ∞. Corollary . The conditions (.) and (.) imply that every solution of the equation x (t) + P(t)x(t – δ) +

Q(t) x(βt) = , t

 < t ≤ t,

(.)

tends to a constant as t → ∞. Theorem . The conditions (H )-(H ) of Theorem . together with

∞

P(s + δ) ds = ∞,

t

∞

t

Q(s/β) ds = ∞, s

(.)

imply that every solution of (.) tends to zero as t → ∞. Proof From Theorem ., we only have to prove that every nonoscillatory solution of (.) tends to zero as t → ∞. Without loss of generality, we assume that x(t) is an eventually positive solution of (.). As in the proof of Theorem ., (.) can be written as in the form (.). Integrating from t to t both sides of the first equation of (.), one has

t

P(s + δ)f x(s) ds +

t

t

t

Q(s/β) x(s) ds = y(t ) – y(t) – ( – bk )y tk– . s t  for x = ;  (ii) limt→∞ |C(t)| =  = μ < , limt→∞ |D(t)| =  = γ <  with μ + γ =  < , and k  +k+ k  +k+ – – C(k) = k  +k+ C(k ), D(k) = k  +k+ D(k ); (iii) tk – (/) and (/(e ))tk are not impulsive points,  < (k  + k + )/(k  + k + ) ≤  for k = , , . . . , and ∞ k=

–

k  + k +  k  + k + 

=

∞ k=

k 

 < ∞; + k + 

(iv) t+δ t+δ lim supt→∞ [ t–δ P(s + δ) ds + βt Q(s/β) ds s P(t+τ +δ) P((t/α)+δ)  + μ( + P(t+δ) ) + γ ( + αP(t+δ) )] = 