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Asymptotic behavior of solutions of mixed type impulsive neutral differential equations

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Abstract

This paper investigates the asymptotic behavior of solutions of the mixed type neutral differential equation with impulsive perturbations [ x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) ] +P(t)f(x(tδ))+ Q ( t ) t x(βt)=0, 0< t 0 t, t t k , x( t k )= b k x( t k )+(1 b k )( t k δ t k P(s+δ)f(x(s))ds+ β t k t k Q ( s / β ) s 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.

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 ( α t ) ] + P ( t ) f ( x ( t δ ) ) + Q ( t ) t x ( β t ) = 0 , 0 < t 0 t , t t k , x ( t k ) = b k x ( t k ) + ( 1 b k ) ( t k δ t k P ( s + δ ) f ( x ( s ) ) d s + β t k t k Q ( s / β ) s x ( s ) d s ) , k = 1 , 2 , 3 , ,
(1.1)

where τ,δ>0, 0<α,β<1, C(t),D(t)PC([ t 0 ,),R), P(t),Q(t)PC([ t 0 ,), R 0 + ), fC(R,R), 0< t k < t k + 1 with lim k t k = and b k , k=1,2,3, , are given constants. For JR, PC(J,R) denotes the set of all functions h:JR such that h is continuous for t k t< t k + 1 and lim t t k h(t)=h( t k ) exists for all k=1,2, .

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 [1], Lakshmikantham et al. [2], Samoilenko and Perestyuk [3], and Benchohra et al. [4]. 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 [520] and the references cited therein).

Let us mention some papers from which are motivation for our work. By the construction of Lyapunov functionals, the authors in [8] 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 ) ) = 0 , 0 < t 0 t , t t k , x ( t k ) = b k x ( t k ) + ( 1 b k ) t k δ t k P ( s + δ ) f ( x ( s ) ) d s , k = 1 , 2 , 3 , .
(1.2)

A similar method was used in [21] by considering an impulsive Euler type neutral delay differential equation with similar impulsive perturbations

{ [ x ( t ) D ( t ) x ( α t ) ] + Q ( t ) t x ( β t ) = 0 , 0 < t 0 t , t t k , x ( t k ) = b k x ( t k ) + ( 1 b k ) β t k t k Q ( s / β ) s x ( s ) d s , k = 1 , 2 , 3 , .
(1.3)

In this paper we combine the two papers [8, 21] and we study the mixed type impulsive neutral differential equation problem (1.1). By using a similar method with the help of four Lyapunov functionals, sufficient conditions are obtained to guarantee that every solution of (1.1) tends to a constant as t. We note that problems (1.2) and (1.3) can be derived from the problem (1.1) as special cases, i.e., if D(t)0 and Q(t)0, then (1.1) reduces to (1.2) while if C(t)0 and P(t)0, then (1.1) reduces to (1.3). 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 1 =max{τ,δ}, η 2 =min{α,β}, and η=min{ t 0 η 1 , η 2 t 0 }, we define an initial function as

x(t)=φ(t),t[η, t 0 ],
(1.4)

where φPC([η, t 0 ],R) = {φ:[η, t 0 ]R|φ is continuous everywhere except at a finite number of point s, and φ( s ) and φ( s + )= lim s s + φ(s) exist with φ( s + )=φ(s)}.

A function x(t) is said to be a solution of (1.1) satisfying the initial condition (1.4) if

  1. (i)

    x(t)=φ(t) for ηt t 0 , x(t) is continuous for t t 0 and t t k , k=1,2,3, ;

  2. (ii)

    x(t)+C(t)x(tτ)D(t)x(αt) is continuously differentiable for t> t 0 , t t k , k=1,2,3, , and satisfies the first equation of system (1.1);

  3. (iii)

    x( t k + ) and x( t k ) exist with x( t k + )=x( t k ) and satisfy the second equation of system (1.1).

A solution of (1.1) 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 2.1 Assume that:

(H1) There exists a constant M>0 such that

|x| | f ( x ) | M|x|,xR,xf(x)>0, for x0.
(2.1)

(H2) The functions C, D satisfy

lim t | C ( t ) | =μ<1, lim t | D ( t ) | =γ<1with μ+γ<1,
(2.2)

and

C( t k )= b k C ( t k ) ,D( t k )= b k D ( t k ) .
(2.3)

(H3) t k τ and α t k are not impulsive points, 0< b k 1, k=1,2, , and k = 1 (1 b k )<.

(H4) The functions P, Q satisfy

{ lim sup t [ t δ t + δ P ( s + δ ) d s + β t t + δ Q ( s / β ) s d s + μ ( 1 + P ( t + τ + δ ) P ( t + δ ) ) + γ ( 1 + P ( ( t / α ) + δ ) α P ( t + δ ) ) ] < 2 M
(2.4)

and

{ lim sup t [ t δ t / β P ( s + δ ) d s + β t t / β Q ( s / β ) s d s + μ ( 1 + t Q ( ( t + τ ) / β ) ( t + τ ) Q ( t / β ) ) + γ ( 1 + Q ( t / ( α β ) ) Q ( t / β ) ) ] < 2 .
(2.5)

Then every solution of (1.1) tends to a constant as t.

Proof Let x(t) be any solution of system (1.1). We will prove that the lim t x(t) exists and is finite. Indeed, the system (1.1) can be written as

[ x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) t δ t P ( s + δ ) f ( x ( s ) ) d s β t t Q ( s / β ) s x ( s ) d s ] + P ( t + δ ) f ( x ( t ) ) + Q ( t / β ) t x ( t ) = 0 , t t 0 , t t k ,
(2.6)
x ( t k ) = b k x ( t k ) + ( 1 b k ) ( t k δ t k P ( s + δ ) f ( x ( s ) ) d s + β t k t k Q ( s / β ) s x ( s ) d s ) , k = 1 , 2 , .
(2.7)

From (H2) and (H4), we choose constants ε,λ,υ,ρ>0 sufficiently small such that μ+ε<1 and γ+λ<1 and T> t 0 sufficiently large, for tT,

[ t δ t + δ P ( s + δ ) d s + β t t + δ Q ( s / β ) s d s + ( μ + ε ) ( 1 + P ( t + τ + δ ) P ( t + δ ) ) + ( γ + λ ) ( 1 + P ( ( t / α ) + δ ) α P ( t + δ ) ) ] 2 M υ ,
(2.8)
[ t δ t / β P ( s + δ ) d s + β t t / β Q ( s / β ) s d s + ( μ + ε ) ( 1 + t Q ( ( t + τ ) / β ) ( t + τ ) Q ( t / β ) ) + ( γ + λ ) ( 1 + Q ( t / ( α β ) ) Q ( t / β ) ) ] 2 ρ ,
(2.9)

and, for tT,

| C ( t ) | μ+ε, | D ( t ) | γ+λ.
(2.10)

From (2.1), (2.10), we have

| C ( t ) | μ + ε 1 f 2 ( x ( t τ ) ) x 2 ( t τ ) , | D ( t ) | γ + λ 1 f 2 ( x ( α t ) ) x 2 ( α t ) ,tT,

which lead to

| C ( t ) | x 2 ( t τ ) ( μ + ε ) f 2 ( x ( t τ ) ) , | D ( t ) | x 2 ( α t ) ( γ + λ ) f 2 ( x ( α t ) ) , t T .
(2.11)

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 1 (t)+ V 2 (t)+ V 3 (t)+ V 4 (t), where

V 1 ( t ) = [ x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) t δ t P ( s + δ ) f ( x ( s ) ) d s β t t Q ( s / β ) s x ( s ) d s ] 2 , V 2 ( t ) = t δ t P ( s + 2 δ ) s t P ( u + δ ) f 2 ( x ( u ) ) d u d s + β t t P ( ( s + β δ ) / β ) β s t Q ( u / β ) u x 2 ( u ) d u d s , V 3 ( t ) = t δ t Q ( ( s + δ ) / β ) s + δ s t P ( u + δ ) f 2 ( x ( u ) ) d u d s + β t t Q ( s / β 2 ) s s t Q ( u / β ) u x 2 ( u ) d u d s ,

and

V 4 ( t ) = ( μ + ε ) t τ t P ( s + τ + δ ) f 2 ( x ( s ) ) d s + ( μ + ε ) t τ t Q ( ( s + τ ) / β ) s + τ x 2 ( s ) d s + ( γ + λ ) α t t Q ( s / ( α β ) ) s x 2 ( s ) d s + γ + λ α α t t P ( ( s / α ) + δ ) f 2 ( x ( s ) ) d s .

Computing d V 1 /dt along the solution of (1.1) and using the inequality 2ab a 2 + b 2 , we have

d V 1 d t = 2 [ x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) t δ t P ( s + δ ) f ( x ( s ) ) d s β t t Q ( s / β ) s x ( s ) d s ] × ( P ( t + δ ) f ( x ( t ) ) + Q ( t / β ) t x ( t ) ) P ( t + δ ) [ 2 x ( t ) f ( x ( t ) ) | C ( t ) | x 2 ( t τ ) | C ( t ) | f 2 ( x ( t ) ) | D ( t ) | x 2 ( α t ) | D ( t ) | f 2 ( x ( t ) ) t δ t P ( s + δ ) f 2 ( x ( s ) ) d s f 2 ( x ( t ) ) t δ t P ( s + δ ) d s β t t Q ( s / β ) s x 2 ( s ) d s f 2 ( x ( t ) ) β t t Q ( s / β ) s d s ] Q ( t / β ) t [ 2 x 2 ( t ) | C ( t ) | x 2 ( t ) | C ( t ) | x 2 ( t τ ) | D ( t ) | x 2 ( t ) | D ( t ) | x 2 ( α t ) t δ t P ( s + δ ) f 2 ( x ( s ) ) d s x 2 ( t ) t δ t P ( s + δ ) d s β t t Q ( s / β ) s x 2 ( s ) d s x 2 ( t ) β t t Q ( s / β ) s d s ] .

Calculating directly for d V i /dt, i=2,3,4, t t k , we have

d V 2 d t = P ( t + δ ) f 2 ( x ( t ) ) t δ t P ( s + 2 δ ) d s P ( t + δ ) t δ t P ( s + δ ) f 2 ( x ( s ) ) d s + Q ( t / β ) β t x 2 ( t ) β t t P ( ( s + β δ ) / β ) d s P ( t + δ ) β t t Q ( s / β ) s x 2 ( s ) d s , d V 3 d t = P ( t + δ ) f 2 ( x ( t ) ) t δ t Q ( ( s + δ ) / β ) s + δ d s Q ( t / β ) t t δ t P ( s + δ ) f 2 ( x ( s ) ) d s + Q ( t / β ) t x 2 ( t ) β t t Q ( s / β 2 ) s d s Q ( t / β ) t β t t Q ( s / β ) s x 2 ( s ) d s ,

and

d V 4 d t = ( μ + ε ) P ( t + τ + δ ) f 2 ( x ( t ) ) ( μ + ε ) P ( t + δ ) f 2 ( x ( t τ ) ) + ( μ + ε ) ( t + τ ) Q ( ( t + τ ) / β ) x 2 ( t ) ( μ + ε ) t Q ( t / β ) x 2 ( t τ ) + ( γ + λ ) Q ( t / ( α β ) ) t x 2 ( t ) ( γ + λ ) Q ( t / β ) t x 2 ( α t ) + ( γ + λ ) α P ( ( t / α ) + δ ) f 2 ( x ( t ) ) ( γ + λ ) P ( t + δ ) f 2 ( x ( α t ) ) .

Summing for d V i /dt, i=1,2,3, we obtain

d V 1 d t + d V 2 d t + d V 3 d t P ( t + δ ) [ 2 x ( t ) f ( x ( t ) ) | C ( t ) | x 2 ( t τ ) | C ( t ) | f 2 ( x ( t ) ) | D ( t ) | x 2 ( α t ) | D ( t ) | f 2 ( x ( t ) ) f 2 ( x ( t ) ) t δ t P ( s + δ ) d s f 2 ( x ( t ) ) β t t Q ( s / β ) s d s f 2 ( x ( t ) ) t δ t P ( s + 2 δ ) d s f 2 ( x ( t ) ) t δ t Q ( ( s + δ ) / β ) s + δ d s ] Q ( t / β ) t [ 2 x 2 ( t ) | C ( t ) | x 2 ( t ) | C ( t ) | x 2 ( t τ ) | D ( t ) | x 2 ( t ) | D ( t ) | x 2 ( α t ) x 2 ( t ) t δ t P ( s + δ ) d s x 2 ( t ) β t t Q ( s / β ) s d s x 2 ( t ) β β t t P ( ( s + β δ ) / β ) d s x 2 ( t ) β t t Q ( s / β 2 ) s d s ] .

Since

t δ t P ( s + 2 δ ) d s = t t + δ P ( s + δ ) d s , β t t Q ( s / β 2 ) s d s = t t / β Q ( s / β ) s d s , t δ t Q ( ( s + δ ) / β ) s + δ d s = t t + δ Q ( s / β ) s d s , 1 β β t t P ( ( s + β δ ) / β ) d s = t t / β P ( s + δ ) d s ,

it follows that

d V 1 d t + d V 2 d t + d V 3 d t P ( t + δ ) [ 2 x ( t ) f ( x ( t ) ) | C ( t ) | x 2 ( t τ ) | C ( t ) | f 2 ( x ( t ) ) | D ( t ) | x 2 ( α t ) | D ( t ) | f 2 ( x ( t ) ) f 2 ( x ( t ) ) t δ t + δ P ( s + δ ) d s f 2 ( x ( t ) ) β t t + δ Q ( s / β ) s d s ] Q ( t / β ) t [ 2 x 2 ( t ) | C ( t ) | x 2 ( t ) | C ( t ) | x 2 ( t τ ) | D ( t ) | x 2 ( t ) | D ( t ) | x 2 ( α t ) x 2 ( t ) t δ t / β P ( s + δ ) d s x 2 ( t ) β t t / β Q ( s / β ) s d s ] .

Adding the above inequality with d V 4 /dt and using condition (2.11), we have

d V 1 d t + d V 2 d t + d V 3 d t + d V 4 d t P ( t + δ ) [ 2 x ( t ) f ( x ( t ) ) | C ( t ) | f 2 ( x ( t ) ) | D ( t ) | f 2 ( x ( t ) ) f 2 ( x ( t ) ) t δ t + δ P ( s + δ ) d s f 2 ( x ( t ) ) β t t + δ Q ( s / β ) s d s ] Q ( t / β ) t [ 2 x 2 ( t ) | C ( t ) | x 2 ( t ) | D ( t ) | x 2 ( t ) x 2 ( t ) t δ t / β P ( s + δ ) d s x 2 ( t ) β t t / β Q ( s / β ) s d s ] + ( μ + ε ) P ( t + τ + δ ) f 2 ( x ( t ) ) + ( μ + ε ) t + τ Q ( ( t + τ ) / β ) x 2 ( t ) + ( γ + λ ) Q ( t / ( α β ) ) t x 2 ( t ) + ( γ + λ ) α P ( ( t / α ) + δ ) f 2 ( x ( t ) ) .

Applying (2.8), (2.9), and (2.10), it follows that

d V d t = d V 1 d t + d V 2 d t + d V 3 d t + d V 4 d t P ( t + δ ) f 2 ( x ( t ) ) [ 2 x ( t ) f ( x ( t ) ) | C ( t ) | | D ( t ) | t δ t + δ P ( s + δ ) d s β t t + δ Q ( s / β ) s d s ( μ + ε ) P ( t + τ + δ ) P ( t + δ ) ( γ + λ ) α P ( ( t / α ) + δ ) P ( t + δ ) ] Q ( t / β ) t x 2 ( t ) [ 2 | C ( t ) | | D ( t ) | t δ t / β P ( s + δ ) d s β t t / β Q ( s / β ) s d s ( μ + ε ) t t + τ Q ( ( t + τ ) / β ) Q ( t / β ) ( γ + λ ) Q ( t / ( α β ) ) Q ( t / β ) ] P ( t + δ ) f 2 ( x ( t ) ) [ 2 M t δ t + δ P ( s + δ ) d s β t t + δ Q ( s / β ) s d s ( μ + ε ) ( 1 + P ( t + τ + δ ) P ( t + δ ) ) ( γ + λ ) ( 1 + P ( ( t / α ) + δ ) α P ( t + δ ) ) ] Q ( t / β ) t x 2 ( t ) [ 2 t δ t / β P ( s + δ ) d s β t t / β Q ( s / β ) s d s ( μ + ε ) ( 1 + t t + τ Q ( ( t + τ ) / β ) Q ( t / β ) ) ( γ + λ ) ( 1 + Q ( t / ( α β ) ) Q ( t / β ) ) ] P ( t + δ ) f 2 ( x ( t ) ) υ Q ( t / β ) t x 2 ( t ) ρ .
(2.12)

For t= t k , we have

V 1 ( t k ) = [ x ( t k ) + C ( t k ) x ( t k τ ) D ( t k ) x ( α t k ) t k δ t k P ( s + δ ) f ( x ( s ) ) d s β t k t k Q ( s / β ) s x ( s ) d s ] 2 = [ b k x ( t k ) + b k C ( t k ) x ( t k τ ) b k D ( t k ) x ( α t k ) b k ( t k δ t k P ( s + δ ) f ( x ( s ) ) d s + β t k t k Q ( s / β ) s x ( s ) d s ) ] 2 = b k 2 V 1 ( t k ) .

It is easy to see that V 2 ( t k )= V 2 ( t k ), V 3 ( t k )= V 3 ( t k ), and V 4 ( t k )= V 4 ( t k ).

Therefore,

V ( t k ) = V 1 ( t k ) + V 2 ( t k ) + V 3 ( t k ) + V 4 ( t k ) = b k 2 V 1 ( t k ) + V 2 ( t k ) + V 3 ( t k ) + V 4 ( t k ) V 1 ( t k ) + V 2 ( t k ) + V 3 ( t k ) + V 4 ( t k ) = V ( t k ) .
(2.13)

From (2.12) and (2.13), we conclude that V(t) is decreasing. In view of the fact that V(t)0, we have lim t V(t)=ψ exist and ψ0.

By using (2.8), (2.9), (2.12), and (2.13), we have

υ T P(t+δ) f 2 ( x ( t ) ) dt+ρ T Q ( t / β ) t x 2 (t)dtV(T),

which yields

P(t+δ) f 2 ( x ( t ) ) , Q ( t / β ) t x 2 (t) L 1 ( t 0 ,).

Hence, for any ϕ>0 and ξ(0,1), we get

lim t t ϕ t P(s+δ) f 2 ( x ( s ) ) ds=0, lim t ξ t t Q ( s / β ) s x 2 (s)ds=0.

Thus, it follows from (2.4) and (2.5) that

t δ t P ( s + 2 δ ) s t P ( u + δ ) f 2 ( x ( u ) ) d u d s + β t t P ( ( s + β δ ) / β ) β s t Q ( u / β ) u x 2 ( u ) d u d s t δ t + δ P ( s + δ ) d s t δ t P ( u + δ ) f 2 ( x ( u ) ) d u + t δ t / β P ( s + δ ) d s β t t Q ( u / β ) u x 2 ( u ) d u 2 M t δ t P ( u + δ ) f 2 ( x ( u ) ) d u d s + 2 M β t t Q ( u / β ) u x 2 ( u ) d u 0 , as  t , t δ t Q ( ( s + δ ) / β ) s + δ s t P ( u + δ ) f 2 ( x ( u ) ) d u d s + β t t Q ( s / β 2 ) s s t Q ( u / β ) u x 2 ( u ) d u d s β t t + δ Q ( s / β ) s d s t δ t P ( u + δ ) f 2 ( x ( u ) ) d u + β t t / β Q ( s / β ) s d s β t t Q ( u / β ) u x 2 ( u ) d u 2 M t δ t P ( u + δ ) f 2 ( x ( u ) ) d u + 2 β t t Q ( u / β ) u x 2 ( u ) d u 0 , as  t ,

and

( μ + ε ) t τ t P ( s + τ + δ ) f 2 ( x ( s ) ) d s + ( μ + ε ) t τ t Q ( ( s + τ ) / β ) s + τ x 2 ( s ) d s + ( γ + λ ) α t t Q ( s / ( α β ) ) s x 2 ( s ) d s + γ + λ α α t t P ( ( s / α ) + δ ) f 2 ( x ( s ) ) d s = ( μ + ε ) t τ t P ( s + τ + δ ) P ( s + δ ) P ( s + δ ) f 2 ( x ( s ) ) d s + ( μ + ε ) t τ t s Q ( ( s + τ ) / β ) Q ( s / β ) ( s + τ ) Q ( s / β ) s x 2 ( s ) d s + ( γ + λ ) α t t Q ( s / ( α β ) ) Q ( s / β ) Q ( s / β ) s x 2 ( s ) d s + γ + λ α α t t P ( ( s / α ) + δ ) P ( s + δ ) P ( s + δ ) f 2 ( x ( s ) ) d s 2 M t τ t P ( s + δ ) f 2 ( x ( s ) ) d s + 2 t τ t Q ( s / β ) s x 2 ( s ) d s + 2 α t t Q ( s / β ) s x 2 ( s ) d s + 2 M α t t P ( s + δ ) f 2 ( x ( s ) ) d s 0 , as  t .

Therefore, from the above estimations, we have lim t V 2 (t)=0, lim t V 3 (t)=0, and lim t V 4 (t)=0, respectively.

Thus, lim t V 1 (t)= lim t V(t)=ψ, that is,

lim t [ x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) t δ t P ( s + δ ) f ( x ( s ) ) d s β t t Q ( s / β ) s x ( s ) d s ] 2 = ψ .
(2.14)

Now, we will prove that the limit

lim t [ x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) t δ t P ( s + δ ) f ( x ( s ) ) d s β t t Q ( s / β ) s x ( s ) d s ]
(2.15)

exists and is finite. Setting

y ( t ) = x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) t δ t P ( s + δ ) f ( x ( s ) ) d s β t t Q ( s / β ) s x ( s ) d s ,
(2.16)

and using (1.1) and condition (H3), we have

y ( t k ) = x ( t k ) + C ( t k ) x ( t k τ ) D ( t k ) x ( α t k ) t k δ t k P ( s + δ ) f ( x ( s ) ) d s β t k t k Q ( s / β ) s x ( s ) d s = b k [ x ( t k ) + C ( t k ) x ( t k τ ) D ( t k ) x ( α t k ) t k δ t k P ( s + δ ) f ( x ( s ) ) d s β t k t k Q ( s / β ) s x ( s ) d s ] = b k y ( t k ) .
(2.17)

In view of (2.14), it follows that

lim t y 2 (t)=ψ.

In addition, from (2.16) and (2.17), system (2.6)-(2.7) can be written as

{ y ( t ) + P ( t + δ ) f ( x ( t ) ) + Q ( t / β ) t x ( t ) = 0 , 0 < t 0 t , t t k , y ( t k ) = b k y ( t k ) , k = 1 , 2 , 3 , .
(2.18)

If ψ=0, then lim t y(t)=0. If ψ>0, then there exists a sufficiently large T such that y(t)0 for any t> T . Otherwise, there is a sequence { a k } with lim k a k = such that y( a k )=0, and so y 2 ( a k )0 as k. This contradicts ψ>0. Therefore, for any t k > T , and t[ t k , t k + 1 ), we have y(t)>0 or y(t)<0 from the continuity of y on [ t k , t k + 1 ). Without loss of generality, we assume that y(t)>0 on [ t k , t k + 1 ). It follows from (H3) that y( t k + 1 )= b k y( t k + 1 )>0, and thus y(t)>0 on [ t k + 1 , t k + 2 ). By using mathematical induction, we deduce that y(t)>0 on [ t k ,). Therefore, from (2.14), we have

lim t y ( t ) = lim t [ x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) t δ t P ( s + δ ) f ( x ( s ) ) d s β t t Q ( s / β ) s x ( s ) d s ] = κ ,
(2.19)

where κ= ψ and is finite. In view of (2.18), for sufficient large t, we have

β t δ t P ( s + δ ) f ( x ( s ) ) d s + β t δ t Q ( s / β ) s x ( s ) d s = y ( β t δ ) y ( t ) β t δ < t k < t [ y ( t k ) y ( t k ) ] = y ( β t δ ) y ( t ) β t δ < t k < t ( 1 b k ) y ( t k ) .

Taking t and using (H3), we have

lim t [ β t δ t P ( s + δ ) f ( x ( s ) ) d s + β t δ t Q ( s / β ) s x ( s ) d s ] =0,

which leads to

lim t t δ t P(s+δ)f ( x ( s ) ) ds=0and lim t β t t Q ( s / β ) s x(s)ds=0.

This implies that

lim t [ x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) ] =κ.
(2.20)

Next, we shall prove that

lim t x(t) exists and is finite.
(2.21)

Further, we first show that |x(t)| is bounded. Actually, if |x(t)| is unbounded, then there exists a sequence { z n } such that z n , |x( z n )|, as n and

| x ( z n ) | = sup t 0 t z n | x ( t ) | ,
(2.22)

where, if z n is not an impulsive point, then x( z n )=x( z n ). Thus, we have

| x ( z n ) + C ( z n ) x ( z n τ ) D ( z n ) x ( α z n ) | | x ( z n ) | | C ( z n ) | | x ( z n τ ) | | D ( z n ) | | x ( α z n ) | | x ( z n ) | [ 1 μ ε γ λ ] ,

as n, which contradicts (2.20). Therefore, |x(t)| is bounded.

If μ=0 and γ=0, then lim t x(t)=κ, which implies that (2.21) holds. If 0<μ<1 and 0<γ<1, then we deduce that C(t) and D(t) are eventually positive or eventually negative. Otherwise, there are two sequences { w k } and { w j } with lim k w k = and lim j w j = such that C( w k )=0 and D( w j )=0. Therefore, C( w k )0 and D( w j )0 as k,j. It is a contradiction to μ>0 and γ>0.

Now, we will show that (2.21) holds. By condition (H2), we can find a sufficiently large T 1 such that for t> T 1 , |C(t)|+|D(t)|<1. Set

ω= lim inf t x(t),θ= lim sup t x(t).

Then we can choose two sequences { u n } and { v n } such that u n , v n as n, and

lim n x( u n )=ω, lim n x( v n )=θ.

For t> T 1 , we consider the following eight possible cases.

Case 1. When lim t C(t)=0 and 1<D(t)<0 for t> T 1 , we have

κ= lim n [ x ( u n ) D ( u n ) x ( α u n ) ] ω+γθ,

and

κ= lim n [ x ( v n ) D ( v n ) x ( α v n ) ] θ+γω.

Thus, we obtain

ω+γθθ+γω,

that is,

ω(1γ)θ(1γ).

Since 0<γ<1 and θω, it follows that θ=ω. By (2.20), we obtain

θ=ω= κ 1 γ ,

which shows that (2.21) holds.

Case 2. When lim t D(t)=0 and 1<C(t)<0 for t> T 1 , we get

κ= lim n [ x ( u n ) + C ( u n ) x ( u n τ ) ] ωμω

and

κ= lim n [ x ( v n ) + C ( v n ) x ( v n τ ) ] θμθ,

which leads to

ω(1μ)θ(1μ).

Since 0<μ<1 and θω, we conclude that

θ=ω= κ 1 μ ,

which implies that (2.21) holds.

Case 3. lim t C(t)=0, 0<D(t)<1 for t> T 1 . The method of proof is similar to the above two cases. Therefore, we omit it.

Case 4. lim t D(t)=0, 0<C(t)<1 for t> T 1 . The method of proof is similar to the above two first cases. Therefore, we omit it.

Case 5. When 1<D(t)<0 and 0<C(t)<1 for t> T 1 , we have

κ= lim n [ x ( u n ) + C ( u n ) x ( u n τ ) D ( u n ) x ( α u n ) ] ω+μθ+γθ

and

κ= lim n [ x ( v n ) + C ( v n ) x ( v n τ ) D ( v n ) x ( α v n ) ] θ+μω+γω,

which yields

ω(1μγ)θ(1μγ).

Since 0<μ+γ<1 and θω, we have θ=ω. Thus

θ=ω= κ 1 μ γ ,

and so (2.21) holds.

Using similar arguments, we can prove that (2.21) also holds for the following cases:

Case 6. 1<C(t)<0, 0<D(t)<1.

Case 7. 1<C(t)<0, 1<D(t)<0.

Case 8. 0<C(t)<1, 0<D(t)<1.

Summarizing the above investigation, we conclude that (2.21) holds and so the proof is completed. □

Theorem 2.2 Let conditions (H1)-(H4) of Theorem 2.1 hold. Then every oscillatory solution of (1.1) tends to zero as t.

Corollary 2.1 Assume that (H3) holds and

lim sup t [ t δ t + δ P ( s + δ ) d s + β t t + δ Q ( s / β ) s d s ] <2
(2.23)

and

lim sup t [ t δ t / β P ( s + δ ) d s + β t t / β Q ( s / β ) s d s ] <2.
(2.24)

Then every solution of the equation

{ x ( t ) + P ( t ) x ( t δ ) + Q ( t ) t x ( β t ) = 0 , 0 < t 0 t , t t k , x ( t k ) = b k x ( t k ) + ( 1 b k ) ( t k δ t k P ( s + δ ) x ( s ) d s x ( t k ) = + β t k t k Q ( s / β ) s x ( s ) d s ) , k = 1 , 2 , 3 , ,
(2.25)

tends to a constant as t.

Corollary 2.2 The conditions (2.23) and (2.24) imply that every solution of the equation

x (t)+P(t)x(tδ)+ Q ( t ) t x(βt)=0,0< t 0 t,
(2.26)

tends to a constant as t.

Theorem 2.3 The conditions (H1)-(H4) of Theorem 2.1 together with

t 0 P(s+δ)ds=, t 0 Q ( s / β ) s ds=,
(2.27)

imply that every solution of (1.1) tends to zero as t.

Proof From Theorem 2.2, we only have to prove that every nonoscillatory solution of (1.1) tends to zero as t. Without loss of generality, we assume that x(t) is an eventually positive solution of (1.1). As in the proof of Theorem 2.1, (1.1) can be written as in the form (2.18). Integrating from t 0 to t both sides of the first equation of (2.18), one has

t 0 t P(s+δ)f ( x ( s ) ) ds+ t 0 t Q ( s / β ) s x(s)ds=y( t 0 )y(t) t 0 < t k < t (1 b k )y ( t k ) .

Applying (2.19) and (H3), we have

t 0 P(s+δ)f ( x ( s ) ) ds<and t 0 Q ( s / β ) s x(s)ds<.

This, together with (2.27), implies that lim inf t f(x(t))=0 and lim inf t x(t)=0. By Theorem 2.1, lim t x(t)=0. This completes the proof. □

Corollary 2.3 Assume that (2.1), (2.2), (2.4), (2.5), and (2.27) hold. Then every solution of the equation

[ x ( t ) + C ( t ) x ( t τ ) D ( t ) x ( α t ) ] +P(t)f ( x ( t δ ) ) + Q ( t ) t x(βt)=0,
(2.28)

0< t 0 t, tends to zero as t.

3 Examples

In this section, we present two examples to illustrate our results.

Example 3.1 Consider the following mixed type neutral differential equation with impulsive perturbations:

{ [ x ( t ) + ( k + 3 ) t 3 k 2 + 3 k 6 x ( t 1 2 ) ( 3 k + 9 ) t 8 k 2 + 8 k 16 x ( t e ) ] + ( 2 + t 5 t 2 ) ( 1 + 1 4 cos 2 x ( t π ) ) x ( t π ) + 1 t ( ln t + 2 ) x ( t e 2 ) = 0 , t 1 , x ( k ) = k 2 + 6 k + 8 ( k + 3 ) 2 x ( k ) + ( 1 k 2 + 6 k + 8 ( k + 3 ) 2 ) ( t k π t k 2 + s + π 5 ( s + π ) 2 x ( k ) = × ( 1 + 1 4 cos 2 x ( s ) ) x ( s ) d s + t k e 2 t k 1 s ( ln ( e 2 s ) + 2 ) x ( s ) d s ) , k = 2 , 3 , 4 , .
(3.1)

Here C(t)=((k+3)t)/(3 k 2 +3k6), D(t)=((3k+9)t)/(8 k 2 +8k16), P(t)=(2+t)/(5 t 2 ), Q(t)=1/(lnt+2), t[k1,k), b k =( k 2 +6k+8)/( ( k + 3 ) 2 ), t 0 =1, k=2,3,4, , f(x)=x(1+((1/4)( cos 2 x))), τ=1/2, δ=π, α=1/e, and β=1/ e 2 . We can find that

  1. (i)

    |x||(1+ 1 4 cos 2 x)x| 5 4 |x|, xR, (1+ 1 4 cos 2 x) x 2 >0 for x0;

  2. (ii)

    lim t |C(t)|= 1 3 =μ<1, lim t |D(t)|= 3 8 =γ<1 with μ+γ= 17 24 <1, and C(k)= k 2 + 6 k + 8 ( k + 3 ) 2 C( k ), D(k)= k 2 + 6 k + 8 ( k + 3 ) 2 D( k );

  3. (iii)

    t k (1/2) and (1/e) t k are not impulsive points, 0<( k 2 +6k+8)/( ( k + 3 ) 2 )1 for k=1,2, , and

    k = 1 ( 1 k 2 + 6 k + 8 ( k + 3 ) 2 ) = k = 1 1 ( k + 3 ) 2 <;
  4. (iv)
    { lim sup t [ t δ t + δ P ( s + δ ) d s + β t t + δ Q ( s / β ) s d s + μ ( 1 + P ( t + τ + δ ) P ( t + δ ) ) + γ ( 1 + P ( ( t / α ) + δ ) α P ( t + δ ) ) ] = 17 12 < 8 5

and

{ lim sup t [ t δ t / β P ( s + δ ) d s + β t t / β Q ( s / β ) s d s + μ ( 1 + t Q ( ( t + τ ) / β ) ( t + τ ) Q ( t / β ) ) + γ ( 1 + Q ( t / ( α β ) ) Q ( t / β ) ) ] = 109 60 < 2 .

Hence, by (i)-(iv) all assumptions of Theorem 2.1 are satisfied. Therefore, we conclude that every solution of (3.1) tends to a constant as t.

Example 3.2 Consider the following mixed type neutral differential equation with impulsive perturbations

{ [ x ( t ) + ( 12 k + 16 ) t 54 k 2 + 27 k 27 x ( t 2 3 ) ( 12 k + 16 ) t 42 k 2 + 21 k 21 x ( t 2 e 3 ) ] + ( 2 t + 1 ( 4 t + 3 ) 2 ) ( 1 + 2 5 sin 2 x ( t π 2 ) ) x ( t π 2 ) + 4 t ( 2 ln t + 3 ) x ( t 3 e ) = 0 , t 1 , x ( k ) = 6 k 2 + 17 k + 7 6 k 2 + 17 k + 12 x ( k ) + ( 1 6 k 2 + 17 k + 7 6 k 2 + 17 k + 12 ) ( t k π 2 t k 2 s + 1 + π ( 4 s + 3 + 2 π ) 2 x ( k ) = × ( 1 + 2 5 sin 2 x ( s ) ) x ( s ) d s + t k 3 e t k 4 s ( 2 ln ( 3 e s ) + 3 ) x ( s ) d s ) , k = 2 , 3 , 4 , .
(3.2)

Here C(t)=((12k+16)t)/(54 k 2 +27k27), D(t)=((12k+16)t)/(42 k 2 +21k21), P(t)=(2t+1)/( ( 4 t + 3 ) 2 ), Q(t)=4/(2lnt+3), t[k1,k), b k =(6 k 2 +17k+7)/(6 k 2 +17k+12), t 0 =1, k=2,3,4, , f(x)=x(1+((2/5) sin 2 x)), τ=2/3, δ=π/2, α=1/(2 e 3 ), and β=1/(3e). We can show that

  1. (i)

    |x||(1+ 2 5 sin 2 x)x| 7 5 |x|, xR, (1+ 2 5 sin 2 x) x 2 >0 for x0;

  2. (ii)

    lim t |C(t)|= 2 9 =μ<1, lim t |D(t)|= 2 7 =γ<1 with μ+γ= 32 63 <1, and C(k)= 6 k 2 + 17 k + 7 6 k 2 + 17 k + 12 C( k ), D(k)= 6 k 2 + 17 k + 7 6 k 2 + 17 k + 12 D( k );

  3. (iii)

    t k (2/3) and (1/(2 e 3 )) t k are not impulsive points, 0<(6 k 2 +17k+7)/(6 k 2 +17k+12)1 for k=1,2, , and

    k = 1 ( 1 6 k 2 + 17 k + 7 6 k 2 + 17 k + 12 ) = k = 1 5 6 k 2 + 17 k + 12 <;
  4. (iv)
    { lim sup t [ t δ t + δ P ( s + δ ) d s + β t t + δ Q ( s / β ) s d s + μ ( 1 + P ( t + τ + δ ) P ( t + δ ) ) + γ ( 1 + P ( ( t / α ) + δ ) α P ( t + δ ) ) ] = 64 63 < 10 7

and

{ lim sup t [ t δ t / β P ( s + δ ) d s + β t t / β Q ( s / β ) s d s + μ ( 1 + t Q ( ( t + τ ) / β ) ( t + τ ) Q ( t / β ) ) + γ ( 1 + Q ( t / ( α β ) ) Q ( t / β ) ) ] = 1.2781996 < 2 ;
  1. (v)
    1 P(s+δ)ds= 1 2 s + 1 + π ( 4 s + 3 + 2 π ) 2 ds=

and

1 Q ( s / β ) s ds= 1 4 s ( 2 ln ( 3 e s ) + 3 ) ds=.

Hence, all assumptions of Theorem 2.3 are satisfied and therefore every solution of (3.2) tends to zero as t.

References

  1. 1.

    Bainov DD, Simeonov PS: Systems with Impulse Effect. Ellis Horwood, Chichester; 1989.

  2. 2.

    Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.

  3. 3.

    Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.

  4. 4.

    Benchohra M, Henderson J, Ntouyas SK 2. In Impulsive Differential Equations and Inclusions. Hindawi Publishing Corporation, New York; 2006.

  5. 5.

    Bainov DD, Dinitrova MB, Dishliev AB: Oscillation of the solutions of impulsive differential equations and inequalities with a retarded argument. Rocky Mt. J. Math. 1998, 28: 25-40. 10.1216/rmjm/1181071821

  6. 6.

    Luo Z, Shen J: Stability and boundedness for impulsive differential equations with infinite delays. Nonlinear Anal. 2001, 46: 475-493. 10.1016/S0362-546X(00)00123-1

  7. 7.

    Liu X, Shen J: Asymptotic behavior of solutions of impulsive neutral differential equations. Appl. Math. Lett. 1999, 12: 51-58.

  8. 8.

    Shen J, Liu Y, Li J: Asymptotic behavior of solutions of nonlinear neutral differential equations with impulses. J. Math. Anal. Appl. 2007, 332: 179-189. 10.1016/j.jmaa.2006.09.078

  9. 9.

    Shen J, Liu Y: Asymptotic behavior of solutions for nonlinear delay differential equation with impulses. J. Appl. Math. Comput. 2009, 213: 449-454. 10.1016/j.amc.2009.03.043

  10. 10.

    Wei G, Shen J: Asymptotic behavior of solutions of nonlinear impulsive delay differential equations with positive and negative coefficients. Math. Comput. Model. 2006, 44: 1089-1096. 10.1016/j.mcm.2006.03.011

  11. 11.

    Luo J, Debnath L: Asymptotic behavior of solutions of forced nonlinear neutral delay differential equations with impulses. J. Appl. Math. Comput. 2003, 12: 39-47. 10.1007/BF02936180

  12. 12.

    Jiang F, Sun J: Asymptotic behavior of neutral delay differential equation of Euler form with constant impulsive jumps. Appl. Math. Comput. 2013, 219: 9906-9913. 10.1016/j.amc.2013.04.022

  13. 13.

    Pandian S, Balachandran Y: Asymptotic behavior results for nonlinear impulsive neutral differential equations with positive and negative coefficients. Bonfring Int. J. Data Min. 2012, 2: 13-21.

  14. 14.

    Wang QR: Oscillation criteria for first-order neutral differential equations. Appl. Math. Lett. 2002, 8: 1025-1033.

  15. 15.

    Tariboon J, Thiramanus P: Oscillation of a class of second-order linear impulsive differential equations. Adv. Differ. Equ. 2012., 2012: Article ID 205

  16. 16.

    Jiang F, Shen J: Asymptotic behavior of solutions for a nonlinear differential equation with constant impulsive jumps. Acta Math. Hung. 2013, 138: 1-14. 10.1007/s10474-012-0282-8

  17. 17.

    Jiang F, Shen J: Asymptotic behaviors of nonlinear neutral impulsive delay differential equations with forced term. Kodai Math. J. 2012, 35: 126-137. 10.2996/kmj/1333027258

  18. 18.

    Gunasekar T, Samuel FP, Arjunan MM: Existence results for impulsive neutral functional integrodifferential equation with infinite delay. J. Nonlinear Sci. Appl. 2013, 6: 234-243.

  19. 19.

    Kumar P, Pandey DN, Bahuguna D: On a new class of abstract impulsive functional differential equations of fractional order. J. Nonlinear Sci. Appl. 2014, 7: 102-114.

  20. 20.

    Samuel FP, Balachandran K: Existence of solutions for quasi-linear impulsive functional integrodifferential equations in Banach spaces. J. Nonlinear Sci. Appl. 2014, 7: 115-125.

  21. 21.

    Guan K, Shen J: Asymptotic behavior of solutions of a first-order impulsive neutral differential equation in Euler form. Appl. Math. Lett. 2011, 24: 1218-1224. 10.1016/j.aml.2011.02.012

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Acknowledgements

We would like to thank the reviewers for their valuable comments and suggestions on the manuscript. This research was funded by King Mongkut’s University of Technology North Bangkok. Contract no. KMUTNB-GOV-57-08.

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Correspondence to Jessada Tariboon.

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Keywords

  • asymptotic behavior
  • nonlinear neutral delay differential equation
  • impulse
  • Lyapunov functional