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BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OF SECOND ORDER
INTEGRODIFFERENTIAL EQUATION WITH RETARDED ARGUMENT
D. B. DHAIGUDE
1& J. V. PATIL
2*
1
Department of Mathematics, Dr.Babasaheb Ambedkar Marathwada,University, Aurangabad – 431 004, India
E-mail: [email protected]
2
Department of Mathematics,Vasantrao Naik college, Aurangabad,431 005, India
E-mail: [email protected]
(Received on: 05-11-11; Accepted on: 18-11-11)
________________________________________________________________________________________________
ABSRACT
T
he integral inequality with retarded argument is obtained and the qualitative properties such as boundedness and asymptotic behavior of solutions of certain class of second order integrodifferential equation with retarded argument is studied as an application of this integral inequality. The results are illustrated by suitable examples.KEYWORDS: integral inequality, retarded argument, asymptotic behavior.
SUBJECT CLASSIFICATION: 26D15
________________________________________________________________________________________________
INTRODUCTION:
The purpose of the paper is to study the boundedness and asymptotic behavior of solutions of second order integrodifferential equations with retarded argument of the form,
)]
)
(
(
,
,
(
)),
(
(
,
[
=
)
(
)
)
(
(
0 '
'
ds
s
x
s
t
k
t
x
t
f
x
t
a
x
t
r
tt
α
α
+
(1.1)by comparing with the solutions of the second order linear differential equation,
0
=
)
(
)
)
(
(
r
t
x
' '+
a
t
x
(1.2)
In equation (1.1) and (1.2) we will assume that
f
∈
C
(
I
×
R
×
R
,
R
)
whereI
=
[
t
0,
T
)
andR
denotes the set of real numbers,k
(
t
,
s
,
x
(
α
))
∈
C
(
I
×
I
×
R
,
R
)
,r
(t
)
>
0
,r
(t
)
anda
(t
)
are continuous function defined on)
,
[
=
t
0T
I
.I t
( ) ( ,
∈
I R
+)
be non increasing withα
(
t
)
=
t
−
l
(
t
)
≥
0
onI
.l
'(
t
)
<
1
,l
(
t
0)
=
0
.)
,
(
)
(
t
∈
C
'I
I
α
,α
(
t
)
≤
t
Boundedness and asymptotic behavior of solution of equations (1.1) and (1.2) with1
=
)
(t
r
,f
t
x
t
k
t
s
x
s
ds
b
t
x
t
t
(
,
,
(
(
)))
=
(
)
))),
(
(
,
(
0
α
α
have been previously studied in [2] and [3]. In recent yearsthere have been several extensions of the results given in [2] and [3] see for examples [1],[4],[5],[8],[9], and some of the references given in.Recenty boundedness of solutions of particular case of integrodifferential equation of (1.1) have been studied by Pachpatte in [6].In this paper we study boundedness and asymptotic behaviour of (1.1) by using retarded integral inequality. We organize the paper as follows. Section 2 is devoted for linear integral inequality with retarded argument.This inequality is successfully applied to study the qualitative behavior such as boundedness and asymptotic behavior of solution of integrodifferential equation with retarded argument (1.1) in section 3 .The last section consists of illustrative examples .
D. B. DHAIGUDE & J. V. PATIL*/ BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OF SECOND ORDER INTEGRODIFFERENTIAL EQUATION WITH RETARDED ARGUMENT / IJMA- 2(11), Nov.-2011, Page: 1-8
! ! "
2 NONLINEAR INTEGRAL INEQUALITY WITH RETARDED ARGUMENT:
The integral inequality for p = 1 is establish by Pachpatte in [7,pp-149] Now we prove the following integral inequality with retarded argument for p,
p
∈
(0,1)
.Theorem: 2.1Let
u
(t
)
,a
(t
)
andb
(t
)
be nonnegative continuous functions defined onI
, andα
(
t
)
∈
C
1(
I
,
I
)
be nondecreasing withα
(
t
)
≤
t
onI
=
[
t
0,
T
)
andK
be a real constant. suppose that the inequality,I
t
for
ds
d
u
b
s
u
s
a
K
t
u
s pt t
t
+
∈
+
(
)[
(
)
(
)
]
,
=
)
(
) 0 ( )
(
) 0
( α
σ
σ
α
α (2.1)
where
p
∈
(0,1)
is a constant . Then]
)
(
))
(
)(
(
[1
)
(
) 0 ( 1
1 0 )
( ) 0
(
a
s
E
s
exp
a
s
ds
K
t
u
st p t
t
−
+
≤
α α
α (2.2)
where,
ds
d
a
p
exp
s
b
p
K
t
E
st t
t p
)
)
(
)
(1
(
)
(
)
(1
1
=
)
(
( )) 0 ( )
(
) 0 ( 1
0
σ
σ
α
α α
α
−
−
−
+
−(2.3)
proof: Let
K
>
0
let us assume thatz
(t
)
is right side of(2.1)
,u
(
t
)
≤
z
(
t
)
)
(
]
)
(
)
(
))
(
(
))[
(
(
=
)
(
() ') 0 (
'
t
a
t
u
t
b
u
d
t
z
t pt
σ
σ
σ
α
α
α
αα
+
(2.4))
(
]
)
(
)
(
))
(
(
))[
(
(
)
(
() ') 0 ( '
t
d
z
b
t
z
t
a
t
z
t pt
σ
σ
σ
α
α
α
αα
+
≤
)
(
]
)
(
)
(
)
(
))[
(
(
)
(
() ') 0 (
'
t
a
t
z
t
b
z
d
t
z
t pt
σ
σ
σ
α
α
αα
+
≤
(2.5)Let
,
)
(
)
(
)
(
=
)
(
()) 0
(
σ
σ
σ
α
α
b
z
d
t
z
t
v
t pt
+
(2.6)such that
z
(
t
)
≤
v
(
t
)
,
v
(
t
0)
=
z
(
t
0)
=
K
,Differentiating (2.6) with respective t,we have,
)
(
))
(
(
))
(
(
)
(
=
)
(
' ''
t
z
t
b
t
z
t
t
v
+
α
pα
α
)
(
)
(
))
(
(
)
(
)
(
)
(
(
)
(
' ''
t
v
t
t
b
t
t
v
t
a
t
v
≤
α
α
+
α
α
psolving above inequality by Bernoulli's equation , we have ,
)
(
))
(
(
)
(
)
(
))
(
(
)
(
' 1 ''
t
t
b
t
v
t
t
a
t
v
v
−p−
α
α
−p≤
α
α
(2.7)solving above inequality by Bernoulli's equation , we have , Put,
p
t
v
t
m
(
)
=
(
(
))
1− (2.8)
)
(
'
)
(1
=
)
(
'
t
v
v
p
t
m
−
−pp
v
t
v
p
t
m
(
)
=
(1
)
(
)
'
'
−
And
(
(
))
(
)
(
)
(
(
))
(
)
)
(1
)
(
' ''
t
t
b
t
m
t
t
a
p
t
m
α
α
α
α
≤
! ! " # p p
K
t
v
t
m
(
0)
=
(
(
0))
1−=
1−As
)
(
))
(
(
)
(
)
(
))
(
(
)
(1
)
(
' ' 't
t
b
t
m
t
t
a
p
t
m
α
α
α
α
≤
−
−
)
(
))
(
(
)
(1
)
(
)
(
))
(
(
)
(1
)
(
' ''
t
p
a
t
t
m
t
p
b
t
t
m
−
−
α
α
≤
−
α
α
multiply both sides by
)
)
(
))
(
(
)
(1
(
' 0ds
s
s
a
p
exp
tt
α
α
−
−
we have)
)
(
))
(
(
)
(1
(
)
(
)
(
))
(
(
)
(1
)
)
(
))
(
(
)
(1
(
)
(
' 0 ' ' 0 'ds
s
s
a
p
exp
t
m
t
t
a
p
ds
s
s
a
p
exp
t
m
t t tt
α
α
−
−
α
α
−
−
α
α
−
−
)
)
(
))
(
(
)
(1
(
)
(
))
(
(
)
(1
' 0 'ds
s
s
a
p
exp
t
t
b
p
tt
α
α
α
α
−
−
−
≤
ds
s
s
a
p
exp
t
t
b
p
ds
s
s
a
p
exp
t
m
dt
d
t t tt
(
(
))
(
)
)]
(1
)
(
(
))
(
)
(
(1
)
(
(
))
(
))
)
(1
(
)
(
[
' 0 ' ' 0α
α
α
α
α
α
≤
−
−
−
−
−
Setting t=s and integrating above inequality from
t
0 to t we have,)
(
)
)
(
))
(
(
)
(1
(
)
(
0 ' 0t
m
ds
s
s
a
p
exp
t
m
t t−
−
−
α
α
p
b
s
s
exp
p
sa
s
d
ds
t t
t
(1
)
(
(
))
(
)
(
(1
)
(
(
))
(
)
)
' 0 ' 0
σ
σ
σ
α
α
α
−
−
−
≤
×
−
−
−
+
≤
{
(
)
(1
)
(
(
))
(
)
(
(1
)
(
(
))
(
)
)
}
)
(
'0 '
0
0
p
b
s
s
exp
p
a
d
ds
t
m
t
m
s t tt
α
α
α
σ
α
σ
σ
exp
p
a
s
s
ds
st
(
(
))
(
))
)
(1
(
' 0α
α
−
−
×
−
−
−
+
≤
− +}
})
)
(
))
(
(
)
(1
(
)
(
))
(
(
)
(1
{
)
(
' 0 ' 0 1ds
d
a
p
exp
s
s
b
p
K
t
m
s t t tp
α
α
α
σ
α
σ
σ
ds
s
s
a
p
exp
st
(
(
))
(
))
)
(1
(
' 0α
α
−
−
By taking change of variable on right hand side of the above inequality,
1 ) ( ) 0 ( ) 0 ( ) ( ) 0 (
1
(
)
(
(1
)
(
)
)
}
((1
)
(
)
)
)
(1
{1
)
(
− −−
−
×
−
−
+
≤
p t t s t t t pK
ds
s
a
p
exp
ds
d
a
p
exp
s
b
K
p
t
m
α α α αα
σ
σ
)
)
(
)
((1
)
(
)
(
() ) 0 ( 0 1ds
s
a
p
exp
t
E
K
t
m
t t p−
≤
− αα
ds
d
a
p
exp
s
b
p
K
t
E
s t t t p)
)
(
)
(1
(
)
(
)
(1
1
=
)
(
) 0 ( ) ( ) 0 ( 10 α
σ
σ
α
α
−
−
−
+
− pt
v
t
m
As
,
(
)
=
(
(
))
1−ds
s
a
p
exp
t
E
K
t
v
t t p p)
(
)
(1
))
(
(
))
(
(
() ) 0 ( 0 11−
≤
−−
αα
ds
s
a
p
exp
t
E
K
t
v
t t p p)
(
)
(1
))
(
(
))
(
(
() ) 0 ( 0 11−
≤
−−
αα
ds
s
a
exp
t
E
K
t
v
t t p)
(
))
(
(
))
(
(
() ) 0 ( 1 1 0 −≤
α α As)
(
)
(
))
(
(
)
(
' 't
t
v
t
a
t
D. B. DHAIGUDE & J. V. PATIL*/ BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OF SECOND ORDER INTEGRODIFFERENTIAL EQUATION WITH RETARDED ARGUMENT / IJMA- 2(11), Nov.-2011, Page: 1-8
! ! " $
Taking t = s and integrating from
t
0 to t and change of variable , we haveds
s
s
v
s
a
t
z
t
z
tt
(
(
))
(
)
(
)
)
(
)
(
'0
0
≤
α
α
−
ds
s
s
v
s
a
t
z
t
z
tt
(
(
))
(
)
(
)
)
(
)
(
'0
0
+
α
α
≤
ds
d
a
exp
t
E
K
s
a
K
t
z
st p
t
t
(
)
(
(
))
(
(
)
)
)
(
) 0 ( 1
1 0 )
( ) 0
( α
σ
σ
α
α
−
+
≤
As
u
(
t
)
≤
z
(
t
)
, we get desired result.If
K
≥
0
,we carry out above procedure withK
+
ε
instead of K , whereε
>
0
is arbitrary small constant , and subsequently pass the limit asε
→
0
to obtain desired result.REMARK: Put
α
(
t
)
=
t
,
α
(
t
0)
=
0
, we get the non linear inequality due to Pachpatte, [see,Theorem 2.7.1 in[5] ]In this section we discuss the applications of integral inequality obtain in section 2. The boundedness and asymptotic behaviour of solution of second order integrodifferential equation with retarded argument is studied.
3 APPLICATIONS:
Theorem: 3.1 . Let the function k and f in (1.1) satisfy the conditions,
p
x
s
h
x
s
t
k
(
,
,
)
|
(
)
|
|
|
≤
(3.1)
|)
|
|
)(|
(
|
)
,
,
(
|
f
t
x
z
≤
g
t
x
+
z
(3.2)where g and h are real valued continuous functions defined on
I
and
(
)
<
∞
0
ds
s
g
t
t , 0
h
(
s
)
ds
<
∞
tt and let
)
(
1
1
max
=
''
t
l
M
I
t∈
−
N
MM
'
2=
, let
α
(
t
)
=
t
−
l
(
t
)
,α
(
t
)
≤
t
. Leta
(
t
)
=
Ng
(
t
+
l
(
t
))
,b
(
t
)
=
h
(
t
+
l
(
t
))
. Ifx
1(
t
)
and)
(
2
t
x
are bounded solutions of equation (1.2) then the corresponding solution x(t) of equation (1.1) is bounded on I .Proof : Let
x
1(
t
)
andx
2(
t
)
be solution of (1.2) such that ,1
=
)]
(
)
(
)
(
)
(
)[
(
' 21 ' 2
1
t
x
t
x
t
x
t
x
t
r
−
(3.3)Suppose that x(t) is any solution of (1.1) then on using variation of constants formula x(t) may be expressed as,
)
))
)
(
(
,
,
(
)),
(
(
,
(
)]
(
)
(
)
(
)
(
[
)
(
)
(
=
)
(
0 2
1 2 1 0 2 2 1
1
x
t
c
x
t
x
s
x
t
x
t
x
s
f
s
x
s
k
s
x
d
ds
c
t
x
st t
t
−
×
α
σ
α
σ
σ
+
+
(3.4)
where
c
1 andc
2 are constant . From (3.1),(3.2) and (3.4) we have ,ds
d
x
h
s
Mg
ds
s
x
s
Mg
C
t
x
s pt t
t t
t
(
)
|
(
(
))
|
(
)
(
(
)
|
(
(
)
|
))
|
)
(
|
0 0
0
σ
σ
α
σ
α
+
+
≤
(3.5)! ! " %
ds
d
x
l
h
s
l
s
g
MM
ds
s
x
s
l
s
g
MM
C
t
x
s pt t
t t
t
'
(
(
))
|
(
)
|
'
(
(
))(
(
(
))
|
(
)
|
)
|
)
(
|
) 0 ( 2
) (
) 0 ( )
(
) 0
( α
σ
σ
σ
σ
α
α α
α
+
+
+
+
+
≤
(3.6)
x
t
C
Ng
s
l
s
x
s
h
l
x
pd
ds
s t t
t
(
(
))[|
(
)
|
(
(
))
|
(
)
|
]
|
)
(
|
) 0 ( )
(
) 0
( α
σ
σ
σ
σ
α
α
+
+
+
+
≤
x
t
C
a
s
x
s
sb
x
pd
ds
t t
t
(
)[|
(
)
|
(
)
|
(
)
|
]
|
)
(
|
) 0 ( )
(
) 0
( α
σ
σ
σ
α
α
+
+
≤
(3.7)Applying Theorem (2.1) in section 2 to (3.7)
ds
s
a
exp
t
E
s
a
C
t
x
st p
t
t
(
)(
(
))
(
(
))]
[1
|
)
(
|
) 0 ( 1
1 0 )
( ) 0 (
−
+
+
≤
α α
α
where
ds
d
a
p
exp
s
b
p
C
t
E
st t
t p
)
)
(
)
(1
(
)
(
)
(1
1
=
)
(
) 0 ( )
(
) 0 ( 1
0 α
σ
σ
α
α
−
−
×
−
+
−for
o
<
p
<
1
,t
∈
I
.Theorem 3.2 : Let the function k and f in (1.1) satisfy the conditions,
p
x
s
h
x
s
t
k
(
,
,
)
|
(
)
|
|
|
≤
(3.8)
|)
|
))
(
(
|
)(|
(
|
)
,
,
(
|
t
x
z
≤
g
t
x
+
exp
−
β
t
+
l
t
z
(3.9)where
β
>
0
, g and h are real valued continuous functions defined onI
=
[
t
o,
T
)
and(
)
<
∞
0
s
h
t
t , 0
g
(
s
)
<
∞
tt
If
x
1(
t
)
andx
2(
t
)
be solution of (1.2) such that|
x
i(
t
)
|
≤
M
iexp
(
−
β
t
)
i
=
1,2
whereM
i>
0
for i = 1,2 areconstants and
α
(
t
)
=
t
−
l
(
t
),
l
(
t
)
≤
ton
I,
satisfy all conditions stated in Theorem 2 .Let))
(
2
(
))
(
(
=
)
(
t
Ng
s
l
s
exp
s
l
s
a
+
−
β
−
β
andb
(
t
)
=
h
(
σ
+
l
(
σ
))
exp
(
−
βσ
)
then the corresponding solutionx(t) of (1.1) approach zero as
t
→
∞
.Proof : Let
x
1(
t
)
andx
2(
t
)
be solution of (1.2) such that,1
=
)]
(
)
(
)
(
)
(
)[
(
t
x
1t
x
2't
x
1't
x
2t
r
−
Suppose that x(t) is any solution of (1.1) then on using variation of constants formula x(t) may be expressed as,
)
))
)
(
(
,
,
(
)),
(
(
,
(
)]
(
)
(
)
(
)
(
[
)
(
)
(
=
)
(
0 2
1 2 1 0 2 2 1
1
x
t
c
x
t
x
s
x
t
x
t
x
s
f
s
x
s
k
s
x
d
ds
c
t
x
st t
t
−
×
α
σ
α
σ
σ
+
+
where
c
1 andc
2 are constant .ds
s
x
s
exp
s
Mg
t
exp
t
Cexp
t
x
tt
(
)
(
)
|
(
(
))
|
)
(
)
(
|
)
(
|
0
α
β
β
β
+
−
−
−
≤
))
(
2
(
)
(
)
(
0
s
l
s
exp
s
Mg
t
exp
tt
−
+
−
+
β
β
sh
x
pd
ds
t0
(
σ
)
|
(
α
(
σ
)
|
σ
)
×
ds
s
x
s
l
s
exp
s
l
s
exp
s
Mg
t
exp
t
Cexp
t
x
tt
(
)
(
2
(
))
(
(
))
|
(
(
))
|
)
(
)
(
|
)
(
|
0
α
β
β
β
β
β
β
+
−
−
+
−
−
≤
))
(
2
(
)
(
)
(
0
s
l
s
exp
s
Mg
t
exp
tt
β
β
β
−
+
−
+
sh
exp
l
exp
l
x
pd
ds
t0
(
σ
)
(
−
βσ
+
β
(
σ
))
(
βσ
−
β
(
σ
))
|
(
α
(
σ
)
|
)
σ
×
ds
s
x
s
l
s
exp
s
l
s
exp
s
Mg
C
t
x
t
exp
tt
(
)
(
2
(
))
(
(
))
|
(
(
))
|
|
)
(
|
)
(
0
α
β
β
β
β
β
≤
+
−
+
−
(
)
(
2
(
))
0
s
l
s
exp
s
Mg
t
t
−
β
+
β
+
sh
exp
l
exp
l
x
pd
ds
t0
(
σ
)
(
−
βσ
+
β
(
σ
))
(
βσ
−
β
(
σ
))
|
(
α
(
σ
))
|
σ
)
D. B. DHAIGUDE & J. V. PATIL*/ BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OF SECOND ORDER INTEGRODIFFERENTIAL EQUATION WITH RETARDED ARGUMENT / IJMA- 2(11), Nov.-2011, Page: 1-8
! ! " &
Making change of variable in above inequality, we have,
ds
s
x
s
exp
s
l
s
exp
s
l
s
g
MM
C
t
x
t
exp
tt
'
(
(
))
(
2
(
(
))
(
)
|
(
)
|
|
)
(
|
)
(
()) 0
(
β
β
β
β
αα
+
−
+
+
+
≤
))
(
)
(
(
2
(
))
(
(
'
) (
) 0
(
MM
g
s
l
s
exp
s
l
s
l
s
t
t
β
β
α
α
+
−
+
+
+
ds
d
x
exp
exp
l
h
ps
t0)
(
(
))
(
)
(
)
|
(
|
)
(
σ
σ
βσ
βσ
σ
σ
α
+
−
×
+
−
−
+
+
≤
C
Ng
s
l
s
exp
s
l
s
exp
s
x
s
ds
t
x
t
exp
tt
(
(
))
(
2
(
))
(
)
|
(
(
))
|
|
)
(
|
)
(
()) 0
(
β
β
β
α
β
αα
ds
d
x
exp
exp
l
h
s
l
s
exp
s
l
s
Ng
s pt t
t
(
(
))
(
2
(
))(
(0)(
(
))
(
)
(
)
|
(
)
|
|
)
)(
) 0
(
β
β
ασ
σ
βσ
βσ
σ
σ
α
α
+
−
−
+
−
))
(
(
|
))
(
(
|
)
(
))[
(
2
(
))
(
(
|
)
(
|
)
(
) 0 ( )
(
) 0
(
β
β
β
φ
σ
σ
β
α α
α
Ng
s
l
s
exp
s
l
s
exp
s
x
s
h
l
C
t
x
t
exp
st t
t
+
−
−
×
+
+
≤
exp
(
−
βσ
)
exp
(
βσ
)
|
x
(
σ
)
|
p|
d
σ
]
ds
(3.10)
ds
d
x
exp
b
t
x
t
exp
s
x
s
exp
s
a
C
t
x
t
exp
s pt t
t
(
)
(
)[|
(
)
|
(
)
|
(
)
|
(
)
(
)
|
(
)
|
]
|
)
(
|
)
(
) 0 ( )
(
) 0
(
β
β
σ
βσ
σ
σ
β
α α
α
+
+
≤
(3.11)Taking
u
(
t
)
=
exp
(
β
t
)
|
x
(
t
)
|
, applying Theorem 1 to inequality (3.11) we have ,ds
d
a
p
exp
s
b
p
C
t
E
st t
t p
)
))
(
)
(1
(
)
(
)
(1
1
=
))
(
(
) 0 ( )
(
) 0 ( 1
0 α
σ
σ
α
α
−
−
−
+
−ds
s
a
exp
s
E
C
t
x
t
exp
tt p
t
t
(
(
))
]
(
(
)
[1
|
)
(
|
)
(
()) 0 ( 1
1 0 ) (
) 0 (
−
+
≤
αα α
α
β
(3.12)for
o
<
p
<
1
,t
∈
I
.4 EXAMPLES:
In this section we discuss some simple examples to illustrate our results .
A :- consider the second order integrodifferential equation with retarded argument,
2
1
=
),
4
3
(
sin
)
4
3
(
)
4
3
(
=
2
'
2
'
)
(1
0 2 2
2
p
s
s
x
s
s
x
s
t
t
t
x
t
x
tx
x
t
tt
√
−
−
+
−
+
−
′
−
(4.1)for
0
<
t
<
1
The corresponding second order linear differential equation is,0
=
2
)
'
)
((1
−
t
2x
'+
x
(4.2)where ,
)
4
3
(
sin
)
4
3
(
)
4
3
(
=
))))
(
(
,
,
(
)),
(
(
,
(
0 2 2
0
s
s
x
s
s
x
s
t
t
t
x
t
s
x
s
t
k
t
x
t
f
tt t
t
α
−
+
√
−
−
α
)
,
,
(
t
x
y
f
is continuous on the setI
×
R
×
R
|]
)
4
3
(
sin
)
4
3
(
|
|
)
4
3
(
[|
|=
))))
(
(
,
,
(
)),
(
(
,
(
|
0 2
0
s
s
x
s
s
x
s
t
t
x
t
s
x
s
t
k
t
x
t
f
tt t
t
α
−
+
√
−
−
α
)
4
3
(
sin
)
4
3
(
=
)))
(
(
,
,
(
s
s
x
s
s
x
s
s
x
s
t
k
α
√
−
−
! ! " '
|
)
4
3
(
|
|=
)))
(
(
,
,
(
|
s
s
x
s
s
x
s
t
k
α
√
−
2
1
=
p
t
t
t
4
3
=
)
(
−
α
t
t
)
≤
(
α
,g
(
t
)
=
t
2 ,h
(
s
)
=
s
, clearly g and h are continuous onI
=
(0,1)
. The solutions of equation (4.2) aret
t
x
1(
)
=
1
|
1
1
|
log
2
=
)
(
2
−
−
+
t
t
t
t
x
which are bounded on I ,
I
=
(0,1)
,1
=
))
(
),
(
(
)
(1
−
x
2W
x
1t
x
1t
Observe that all the conditions of Theorem 3.1 are satisfied.Therefore by theorem 3.1 the solution x(t) of equation (1.1) is
bounded on
I
=
(0,1)
B: Consider the second order integrodifferential equation with retarded argument,
]
)
4
3
(
cos
)
4
3
(
)
4
3
(
1
1
1
)
4
3
(
1
1
=
)
(2
2
)
)
(2
(
0 2
' '
ds
s
s
x
s
s
x
t
t
exp
s
t
t
t
x
t
x
t
exp
x
t
exp
tt
√
−
+
√
−
−
+
+
−
+
+
(4.3)for
t
>
1
and the corresponding second order linear differential equation ,0
=
)
(2
2
)
)
(2
(
exp
t
x
' '+
exp
t
x
(4.4)
ds
s
s
x
s
s
x
t
t
exp
s
t
t
t
x
t
s
x
s
t
k
t
x
t
f
tt t
t
4
)
3
(
cos
)
4
3
(
)
4
3
(
1
1
1
)
4
3
(
1
1
=
))))
(
(
,
,
(
))
(
(
,
(
0 2
0
−
−
√
+
−
√
+
+
−
+
α
α
)
,
,
(
t
x
y
f
is continuous on the setI
×
R
×
R
,β
=
1
]
)
4
3
(
cos
)
4
3
(
1
)
4
3
(
)
4
3
(
[
1
1
|
)))
(
(
,
,
(
)),
(
(
,
(
|
0 0
ds
s
s
x
s
s
x
s
t
t
exp
t
t
x
t
s
x
s
t
k
t
x
t
f
tt t
t
α
≤
+
−
+
−
+
√
√
−
−
α
)
4
3
(
cos
)
4
3
(
1
=
)))
(
(
,
,
(
s
s
x
s
s
x
s
t
x
s
t
k
√
−
−
√
α
k is continuous on
I
×
I
×
R
.|
)
4
3
(
|
1
|
)))
(
(
,
,
(
|
s
s
x
s
t
x
s
t
k
√
−
√
≤
α
Note that the functions ,
1
1
=
)
(
+
t
t
g
and
s
t
h
√
1
=
)
(
. g and h are continuous functions on I. The equation (4.4) has the solutions
)
(
cos
)
(
=
)
(
1
t
exp
t
t
x
−
,x
2(
t
)
=
exp
(
−
t
)
sin
(
t
),
t
∈
I
))
(
(
|
)
(
|
x
1t
≤
exp
−
t
D. B. DHAIGUDE & J. V. PATIL*/ BOUNDEDNESS AND ASYMPTOTIC BEHAVIOR OF SECOND ORDER INTEGRODIFFERENTIAL EQUATION WITH RETARDED ARGUMENT / IJMA- 2(11), Nov.-2011, Page: 1-8
! ! " (
))
(
(
|
)
(
|
x
2t
≤
exp
−
t
))
2(
(
=
)
,
(
x
1x
2exp
t
W
−
1
=
)
,
(
))
(2(
t
W
x
1x
2exp
Observe that all the conditions of Theorem 3.2 are satisfied.Hence by applying Theorem 3.2 to solution x(t)of (1.1)
0
|
)
(
|
x
t
→
ast
→
∞
and the conclusion of theorem is true .REFERENCES:
[1] J. S. Beadley,Comparison Theorem for square integrability of solutions of (r(t)y')' +a(t)y=f(t,y), Glasgow Math. J. 13(1), (1972),75-79
[2] R. Bellman, Stability Thoery Of Differential Equaations ,Mc-Graw-Hill, 1953.
[3] P. Hartman,Ordinary Differential Equaations,John Wiley, New York. 1964
[4] B. G. Pachpatte,Boundedness And Asymptotic Behavior Of Second Order Integrodifferential Equations, J. Math. Phy. Sci.92, (1975),171-175
[5] B. G.Pachpatte,Inequalities For Differential and Integral equations,Academic Press, 1998
[6] B. G Pachpatte,On Some Retarded Integral Inequalities And Applications,J.Inequal.Pure And Appl. Math.3(2) Art. 18, 2002
[7] B.G Pachpatte,Integral And Finite Differerce Inequalities And Applications, North Holland, Oxford,2006
[8] P. Waltman,On Asymptotic Behavior Of Solutions Of A Nonlinear Equation, Proc.Amer.Math. Soc.15,6,(1964), 918-923.
[9] P. K. Wong,Bounds For Solutions To A Class Of Non Linear Second Order Differential Equations, J. Differential Equation 7,1, (1970),139-146.
