Sufficient conditions are obtained for every solution of first order nonlinear neutral delay forced impulsive differential equations with positive and negative coefficients tends to a constant as t ∞. Mathematics Subject Classification [MSC 2010]:34A37

1. IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 2 (Mar. - Apr. 2013), PP 62-69
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www.iosrjournals.org 63 | Page
Asymptotic Behavior of Solutions of Nonlinear Neutral Delay
Forced Impulsive Differential Equations with Positive and
Negative Coefficients
Y.Balachandran, S.Pandian, G.Purushothaman,
1
Department of Mathematics, Presidency College, Chennai-5, Tamilnadu, India
2
College of arts and science, Thiruvalluvar University, Vandavasi, Tamilnadu, India
3
Department of Mathematics, St.Joseph’s College of Engineering, Chennai-119, Tamilnadu, India
Abstract:Sufficient conditions are obtained for every solution of first order nonlinear neutral delay forced
impulsive differential equations with positive and negative coefficients tends to a constant as t ∞.
Mathematics Subject Classification [MSC 2010]:34A37
Keywords: Asymptotic behavior, neutral, non linear, forced, differential equation, impulse.
I. Introduction
In recent years, the theory of impulsive differential equations had obtained much attention and a number of
papers have been published in this field. This is due to wide possibilities for their applications in Physics,
Chemical technology, Biology and Engineering [1,2].The asymptotic behavior of solutions of various impulsive
differential equations are systematically studied (see [4,7-10]). In [10] the authors investigated the asymptotic
behavior of solutions of following nonlinear impulsive delay differential equations with positive and negative
coefficients
 x(t) R(t)x(t τ) P(t) (x(t ρ)) Q(t) (x(t σ)) 0, t t ,t t
0 k
f f        
k k
k k
t t
k k k k
t ρ t σ
x(t ) b x(t ) (1 b ) P(s ρ) (x(s))ds Q(s σ) (x(s))dsf f
 
 
      
 
 
  for k = 1,2,…
In this paper, we adapt the same technique applied in [10] and obtain the asymptotic behavior of solutions
of nonlinear neutral delay forced impulsive differential equations with positive and negative coefficients
II. Preliminaries
Consider the nonlinear neutral delay forced impulsive differential equations with positive and negative
coefficients
 x(t) R(t)x(t τ) P(t) (x(t ρ)) Q(t) (x(t σ)) ( ), t t ,t t
0 k
f f e t         (1)
k
k
k
k k
t
k k k k
t ρ
t
t σ t
(2)
x(t ) b x(t ) (1 b ) P(s ρ) (x(s))ds
Q(s σ) (x(s))ds ( 1) e(s)ds, 1,2,...k
f
f b k




   
    

 
where τ > 0, ρ ≥ σ > 0, 

k
k
k10 tlimwithttt0 , f C(R,R), R(t)  PC([t0,∞),R), P(t),
Q(t),e(t)  C([t0,∞),[0,∞)) and bk are constants k = 1,2,3…, )( 
ktx denotes the left limit of x(t) at t = tk. With
equations (1) and (2), one associates an initial condition of the form
x(t0+s) = υ(s), s[-,0],  = max{τ,ρ} (3)
where υPC([-,0],R)={υ : [-,0]R such that υ is continuous everywhere except at the finite number of
points  and υ(+
) and υ(–
) exist with υ(+
) = υ()}.
A real valued function x(t) is said to be a solution of the initial value problem (1)-(3) if
(i) x(t) = υ(t-t0) for t0- ≤ t ≤ t0, x(t) is continuous for t ≥ t0 and t ≠ tk , k = 1,2,3,…
(ii)  τ)R(t)x(t(t) x is continuously differentiable for t > t0, t ≠ tk t ≠ tk+τ, t ≠ tk+ρ, t ≠ tk+σ,
k = 1,2,3,…and satisfies (1).

2. Asymptotic behavior of solutions of nonlinear neutral delay forced impulsive differential equations
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(iii) for t = tk , )x(t k

and )x(t k

exist with )x(t k

= )x(t k and satisfies (2).
A solution of (1)-(2) is said to be nonoscillatory if this solution is eventually positive or eventually
negative. Otherwise this solution is said to be oscillatory. A more general form of (1)-(2) was considered in [5-
6] in which the existence and uniqueness of solutions and stability was studied. The main purpose of this paper
is to obtain the sufficient conditions for every solution of (1)-(2) tends to constant as t∞. Our results
generalize the results of [10].
III. Main results
Theorem 1. Assume that the following conditions hold
(A1) there is a constant C > 0 such that ,C)( xxfx  xR, x f (x) > 0 for x≠0; (4)
(A2) tk-τ is not an impulsive point, 0< bk <1 for k = 1,2,3,… and k
k 1
(1 b ) ;


  
(A3) 1R(t)lim
t


, )R(tb)R(t kkk

 and ( ) 0
t
e s ds

 as t∞; (5)
(A4) H(t) = P(t) – Q(t + σ – ρ) > 0 for t ≥t* = t0+ρ-σ (6)
(A5) 0dsσ)Q(slim
σt
ρt
t




(7)
(A6)
t t
t
t ρ t ρ
Q(t σ) H(t τ ρ) 2
lim sup H(s ρ)ds H(s 2 )du μ 1
H(t ρ) H(t ρ) C




 
    
       
    
  (8)
then every solution of (1)-(2) tends to a constant as t∞.
Proof. Let x(t) be any solution of (1)-(2) we shall prove that x(t)lim
t 
exists and is finite. From (1) and (6)
( ) ( ) ( ) ( ) ( ( )) ( ) ( ( )) ( ) ( ) ( ( )) 0
t t
t t t
x t R t x t H s f x s ds Q s f x s ds e s ds H t f x t

 
   
 
 

 
          
  
   (9)
From (2)
( ) ( ) (1 ) ( ) ( ( )) ( ) ( ( )) ( ) ( ( ))
( 1) ( ) ,
k k k
k k k
k
t t t
k k k k
t t t
k
t
x t b x t b P s f x s ds Q s f x s ds Q s f x s ds
b e s ds

  
  


  

  
         
    
 
  

(10)
( ) (1 ) ( ) ( ( )) ( ) ( ( ))
( 1) ( ) , 1,2,3,...,
k k
k k
k
t t
k k k
t t
k
t
b x t b Q s f x s ds H s f x s ds
b e s ds k

 
 


 

 
      
  
  
 

From (5) and (8) we can choose an ε > 0 sufficiently small that + ε < 1 and
CtH
tH
dssH
tH
tQ
sdsH
t
t
t
t
t
2
)(
)(
1)()2(
)(
)(
)(suplim 



















 

 

(11)
also we select T > t0 sufficiently large such that εμR(t)  for t ≥ T (12)
From (4) and (12) we have
))(()()()( 22
  txftxtR , t ≥ T (13)
Let we introduce three functional
2
1( ) ( ) ( ) ( ) ( ) ( ( )) ( ) ( ( )) ( )
t t
t t t
W t x t R t x t H s f x s ds Q s f x s ds e s ds

 
  
 
 
 
        
  
  

3. Asymptotic behavior of solutions of nonlinear neutral delay forced impulsive differential equations
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 

t
t
t
s
dsduuxfuQsHtW ))(()()2()( 2
2
2 2
3 ( ) ( 2 ) ( ) ( ( )) ( ) ( ) ( ( ))
2 ( ) ( ) ( ( )) ,
t t t
t s t
t t
W t H s H u f x s du ds H s f x s ds
e s H u f x u du ds
 
     

 
 
      
 
  
 
Using (9) and the inequality 2ab≤a2
+b2
 
     
2 2 21
2 2 2
( ) 2 ( ) ( ( )) ( ) ( ) ( ) ( ( )) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( )
2 ( ) ( ( )) ( )
t
t
t t t
t t t
t
dW
H t x t f x t R t x t R t f x t H s f x t ds
d t
H s f x s ds Q s f x t ds Q s f x s ds
H t f x t e s ds

 
  
  
  


 
  


       


      

 

  

     

t
t
t
t
dssxfsQtHdssHtxftQ
dt
dW
)()()()2()()( 222
    



t
t
t
t
dssxfsQtHdssHtxftQ )()()()2()()( 22
   
   
2 23
2 2
( ) ( ) ( 2 ) ( ) ( ) ( )
( ) ( ) ( ) ( ) ( ) ( ) 2 ( ) ( ) ( ( )) .
t t
t t
t
dW
H t f x t H s ds H t H s f x s ds
dt
H t f x t H t f x t e s H t f x t ds
 
   
        
 

     
         
 

Set W(t) = W1(t) +W2(t) +W3(t); and then using (12) and (13), we obtain
   
    










t
t
t
t
t
t
dssQtxfdssHtxf
dssHtxftxftxftxtH
dt
dW
)()()2()(
)()()()())(()(2)(
22
22
 )(
)(
)( 2
txf
tH
tQ


 

t
t
dssH )2(   





)(
)(
)()( 2
tH
tH
txf
 







  


t
t
t
t
dssH
tH
tQ
dssH
txf
tx
txftH






 )2(
)(
)(
)(
))((
)(2
)()( 2












 


)(
)(
1)()(
tH
tH
dssQ
t
t
From (4) and (7), we have
 
t ρ t
2
t ρ t ρ
2 Q(t σ)
( ) ( ) H(s ρ)ds H(s 2 )ds
C H(t ρ)
dW
H t f x t
dt
 

 
  
       
  
 
H(t τ ρ)
(μ ε) 1 0
H(t ρ)
  
    
  
t≠tk (14)
As t =tk we have
   
2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
k k
k k k
t t
k k k k
t t t
W t x t R t x t H s f x s ds Q s f x s ds e s ds

 
  
 
 
 
        
  
  

4. Asymptotic behavior of solutions of nonlinear neutral delay forced impulsive differential equations
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   2 2
( 2 ) ( ) ( ) 2
k k k k
k k
t t t t
t s t ρ s
H s Q u f x u duds H(s ρ) H(u ρ)f x(u) duds

 
 
        
 2
2 ( ) ( ) ( ( ))
k
k k k
t s
t τ t t
(μ ε) H(s τ ρ)f x(s) ds e s H u f x u du ds


       
   
2
2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
k k
k k k
t t
k k k k
t t t
b x t R t x t H s f x s ds Q s f x s ds e s ds
 
  

  
 
 
        
 
  
      

k
k
kk
k
k t
t
t
s
t
t
t
s
dsduuxfuHsHdsduuxfuQsH )()()2()()()2( 22
 2
( ) ( ) ( ) 2 ( ) ( ) ( ( ))
k
k k k
t s
t t t
H s f x s ds e s H u f x u du ds

    


       
= )( 
ktW (15)
Which together with (7), (8) and (14) we get   ),()()( 0
12
 tLtxftH (16)
and hence for any h ≥ 0 we have   0)()(lim 2


dssxfsH
t
ht
t
(17)
On the other hand by (8), (14), (15) we see that W(t) is eventually decreasing.
  dsduuxfuQsHtW
t
t
t
s
 








 )()()2()(0 2
2
 22
( ) ( ) 0 as t
t
t
H s f x u du
C 


    
   2 2
Also, ( 2 ) ( ) ( ) ( ) ( ) ( )
t t t
t s t
H s H u f x u du ds H s f x s ds
 
     
 
       
      
tasdssxfsHduuxfuH
C
t
t
t
t
0)()(2)()(
2 22
and ( ) ( ) ( ( )) ( ) ( ) ( ( )) 0as
s
t t t t
e s H u f x u du ds e s H u f x u du ds t 
  
        
Hence 

)(lim)(lim 1 tWtW
tt
, which is finite. That is
t
lim    
2
( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t t
t t t
x t R t x t H s f x s ds Q s f x s ds e s ds

 
   
 
 
 
        
  
  
(18)
Next we shall prove that
t
lim    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t t
t t t
x t R t x t H s f x s ds Q s f x s ds e s ds

 
  
 
 
 
       
  
  
exists and is finite.
Let    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t t
t t t
y t x t R t x t H s f x s ds Q s f x s ds e s ds

 
  
 
 
         
From (9) we have   0)()()(  txftHty  and from (10) we have
   ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
k k
k k k
t t
k k k k k
t t t
y t b x t R t x t H s f x s ds Q s f x s ds e s ds

 
  
 
 
 
 
        
 
  
)( 
 kk tyb

5. Asymptotic behavior of solutions of nonlinear neutral delay forced impulsive differential equations
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Therefore system (9)-(10) can be rewritten as   ktt,tt,)t(xf)t(H)t(y  00
)()( 
 kkk tybty k=1,2,… (19)
In view of (18) we have 

)(lim 2
ty
t
. (20)
If  = 0, then 0)(lim 2


ty
t
.
If  > 0 , then there exists a sufficiently large T1 such that y(t) ≠ 0 for any t>T1.Otherwise there is a sequence
,..,..,, 321 k with 

k
k
lim such that y(τk) = 0 so y2
(τk)=0 as k∞ .This is contradiction to  >
0.Therefore for tk >T1 , t[tk,tk+1) we have y(t) > 0 or y(t) < 0 because y(t) is continuous on [tk,tk+1). Without loss
of generality we assume that y(t) > 0 on t[tk,tk+1), it follows that y(tk+1) )( 11

 kk tyb >0 , thus y(t) > 0 on
[tk+1,tk+2). By induction, we can conclude that y(t) >0 on [tk,∞).
From (20) we have


)(lim ty
t t
lim    ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )
t t
t t t
x t R t x t H s f x s ds Q s f x s ds e s ds

 
   
 
 
 
        
  
   (21)
where λ  and is finite.
In view of (19) we have     
 


t
t ttt
kk
k
tytytytydssxfsH
 
 )()()()()()(
 


ttt
kkk
k
tytybtyty

 )()()()(
 


ttt
kk
k
tybtyty

 )(1)()(
Let t  ∞   

1
1
k
kb
, we have
t
lim   0)()( 
t
t
dssxfsH

 (22)
By condition
t
lim ( ) 0
t
e s ds

 and from (21), we have
t
lim   











 


t
t
dssxfsQtxtRtx )()()()()( (23)
Next we shall prove that
t
lim  )()()(  txtRtx exists and is finite. To prove this, first prove that )(tx is
bounded. If )(tx is unbounded then there is a sequence {sn} such that sn  ∞, )( _
nsx  ∞ as n  ∞ and
x(t)Sup)x(s
n0 stt
_
n

 where if sn is not an impulsive point then )( _
nsx = )( nsx . Thus from (7) we have
 




n
n
s
s
nnn dssxfsQsxsRsx )()()()()(









 



nasdssQMsx
n
n
s
s
n )()(1)(
which contradicts (23). So )(tx is bounded. Also by (7) we obtain
     






tasdssxsQMdssxfsQdssxfsQ
t
t
t
t
t
t
0)()()()()()(0
which together with (23) gives
t
lim     )()()( txtRtx (24)
Next we prove that )(lim tx
t 
exists and finite .

6. Asymptotic behavior of solutions of nonlinear neutral delay forced impulsive differential equations
www.iosrjournals.org 68 | Page
If  = 0 then 

)(lim tx
t
which is finite.
If 0<  < 1 then R(t) is eventually positive or negative and also we can find a sufficiently large T2 such that
1)( tR for t > T2.. Set Ltx
t


)(suplim , ltx
t


)(inflim , then we can choose two sequences {an}
and {bn} such that an ∞ , bn∞ as n∞ and .)(lim,)(lim lbxLax n
n
n
n


Since 1)( tR for t > T2,
we have the following two possible cases.
Case 1. If 0<R(t) <1 for t>T2 then
n
lim   lLaxaRax nnn   )()()( and
n
lim   LlbxbRbx nnn   )()()(
Therefore L+l ≤ l+L. i.e.). L ≤ l . But L ≥ l , it follows that L = l. Hence L = l = λ/(1+).
which shows that x(t)lim
t 
exists and finite.
Case 2. If -1<R(t) < 0 for t>T2 then  )()()()()(lim)(lim  

nnnnn
n
n
n
axaRaxaRaxax
therefore l = λ+l . i.e. l = λ/ (1- ).
Similarly  )()()()()(lim)(lim 

nnnnn
n
n
n
bxbRbxbRbxbx
Therefore L = λ+L i.e.) L = λ/(1-).Finally we get, L= l = λ/(1-). This shows that
)(lim tx
t 
exists and finite.
Theorem 2. Assume that the conditions in theorem (1) hold, then every oscillatory solution of (1)-(2) tends to
zero as t  ∞.
Theorem 3. The conditions in theorem (1) together with
(i) for any  > 0 there exists  > 0 such that )(xf for x and (25)
(ii) 


0t
dt)t(H
(26) imply that every solution of (1) –(2) tends to zero as t∞.
Proof. By theorem (2) we only have to prove that every nonoscillatory solution of (1) tends to zero as t∞.Let
x(t) be an eventually positive solution of (1). We shall prove 0)(lim 

tx
t
. By theorem (1) we rewrite (1)-(2)
in the form   0)()()(  txftHty  .
Integrating from t0 to t on both sides we get
   


ttt
kk
t
t k
tybtytydssxftH
00
)()1()()()()( 0
Using 



1
)1(
k
kb and (21) we get   

0
)()(
t
dssxftH
which together with (26) we get 0)((inflim 

txf
t
.We shall prove that 0)(inflim 

tx
t
.Let ms be such
that sm  ∞ as m∞ and 0))((inflim 

m
m
sxf we must have 0)(inflim 

Msx m
m
. In fact, if M > 0,
then there is a subsequence }{ kms such that 2/)( Msx km  for sufficiently large k. By (25) we have
)(( kmsxf for some  > 0 and sufficiently large k, which yields a contradiction because
0))((inflim 
 km
k
sxf .Therefore by Theorem 1, 0)(inflim 

tx
t
holds and hence 0)(lim 

tx
t
.
Remark 1. When e(t) = 0 the results of this paper reduce to the results of [10].
Remark 2. When R(t) = 0 ,Q(t) = 0 and e(t) = 0 the results of the present paper reduce to the results of [9].

7. Asymptotic behavior of solutions of nonlinear neutral delay forced impulsive differential equations
www.iosrjournals.org 69 | Page
Remark 3. When all bk = 1 and f(x) = x, equation (1)-(2) reduced to the differential equation without impulses
whose asymptotic behavior of solutions discussed in [3]
IV. Example
Consider the following impulsive differential equation
 
2 2
2 2
t 1 2 1
x(t) x t [1 sin x(t 2)]x(t 2) [1 sin x(t 1)]x(t 1) , t 2,t k
8k 2 tt 1
t
e
                                
k k2
2 2
k2 2 2 2
k 2 k 1
2
k 1 1 2 1
x(k) x(t ) [1 sin x(s)]x(s)ds [1 sin x(s)]x(s)ds
k k (s 1) (s 1)
1
( ) for k 1,2,3,...
k
e s ds
k

 

    
        
     
 
 

Here P(t) =
 2
1t
2

, Q(t) = 2
t
1
, R(t) =
8k
t
, f (x) = xx]sin[1 2
 , τ = ½ , ρ = 2 ,σ = 1,bk =
2
2
k 1
k

The above equation satisfies all the conditions of Theorem 1. Therefore, every solution of this equation tends to
constant as t∞.
References
[1] V. Lakshmikantham, D.D. Bainov, P.S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
[2] D.D. Bainov and P.S. Simeonov, Impulsive Differential Equations: Periodic Solutions and Applications, Longman, Harlow, 1993.
[3] J.H. Shen and J.S. Yu, Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients, J.
Math. Anal. Appl., 195(1995), 517-526.
[4] Aimin Zhao and Jurang Yan, Asymptotic behavior of solutions of impulsive delay differential equations, J. Math. Anal. Appl.,
201(1996), 943-954.
[5] Jianhua Shen, Zhiguo Luo, Xinzhi Liu, Impulsive stabilization for functional differential equations via Liapunov functional, J. Math.
Anal. Appl. 240 (1999) 1–15.
[6] Xinzhi Liu, G. Ballinger, Existence and continuability of solutions for differential equations with delays and state dependent
impulses, Nonlinear Anal. 51 (2002) 633–647.
[7] Gengping Wei and Jianhua Shen, Asymptotic behavior of solutions of nonlinear impulsive delay differential equations positive and
negative coefficients, Math. & Comp. Modeling, 44(2006), 1089-1096.
[8] Jiowan Luo and Lokenath Debonath, Asymptotic behavior of solutions of forced nonlinear neutral delay differential equations with
impulses J. Appl. Math. & Computing, 12(2003),39-47.
[9] Jianhua Shen and Y.Liu, Asymptotic behavior of solutions of nonlinear delay differential equations with impulses, Appl. Math. and
Comput., 213(2009), 449-454.
[10] S. Pandian, G. Purushothaman and Y. Balachandran, Asymptotic behavior of solutions nonlinear neutral delay impulsive differential
equations with positive and negative coefficients, Far East Journal of Mathematical Sciences, 51(2) (2011), 165--178.