Advances in Pure Mathematics, 2011, 1, 243-244
doi:10.4236/apm.2011.14043 Published Online July 2011 (http://www.SciRP.org/journal/apm)
An Inequality for Second Order Differential Equation with
Retarded Argument
Erdoğan Şen
Department of Mathematics, Namık Kemal University, Tekirdağ, Turkey
Received April 23, 2011; revised May 10, 2011; accepted May 20, 2011
Abstract
Applications of differential equations with retarded argument can be encountered in the theory of automatic
control, in the theory of self-oscillatory systems, in the study of problems connected with combustion in
rocket engines, in a number of problems in economics, biophysics. The problems in this areas can be solved
reducing differential equations with retarded argument. In this work an important inequality for second order
differential equation with retarded argument is obtained.
Keywords: Differential Equation with Retarded Argument, Inequality
1. Introduction
In this study we consider the equation
  
0Lwtwt 
 

Mt 

wt  w
 t (1)
on an interval
I
. Where

is a real parameter;


M
t

t
and are continuous functions on
I
;


0tt 

10t tI
t
and for each .
2. An Inequality for Second Order
Differential Equation with Retarded
Argument
Theorem. Let us denote by every point with 0jwhich is
satisfying the mean-value theorem for a continuous solu-
tion


j
wt of (1) on


,
jj
tttI

 
 j
tIfor each

and , where
jJ
J
is an index set. Also let us as-
sume that

0
sup
M
tM where 0
M
is a real number.
Then for all
j
t in the equation



00
0
ee
jj
jj
j
kt tkt t
wtwt wt
 


0
j
j (2)
where


1/ 2
2
2
00
3
,1 2
j
jj
wtwtw tkM 


 


Proof. From the mean-value theorem we can write the
followings:






0j
jjj
j
wt wtt
wt
t
 






0j



j
jj j
wttwtw tt


and





0j jj j
wttwtw t

 
 
(3)
2
jj
ut wt
Now we let . Thus
uwwww






where
j
j
wt wt. Then
u wwwwwwww

    
From the definition of a derivative it follows that
ww

 


. Also
j
j
wt wt. Therefore






00
22
jj
j
jj
utwtwtwtw t


w
(4)


0Lw Since satisfies

we have



  
jjjjj
wtwtM twtt 
 
and hence applying (3)
 





 

0
00
j
jj jj
jj
wtwtM wtt
wt Mwt wt


  
  
(5)

244 E. ŞEN
Using (5) in (4) we obtain
 
 
 

 
 
 
00
00
00
22
21 2
jj
j
jj
jj
j
utwt wtwt
wt Mwt wt
Mwtwt


 
  
 2
00
jj
Mwt

Now applying the fact




2
2
00
jj
w t

2jj wtwtwt
we get







2
0
2
00
2
0
1
13
3
21 2
j
jj
j
utM wt
Mwt
Mwt



 
  




2
0
j
wt






or

2
j
j
kut
 
2
ut

This is equivalent to

2
j
jj
tku t
20uku
ku tu  (6)
And these inequalities lead directly to (2). Indeed con-
sider the right inequality which can be written as




22
e2e0
jj
kt kt
uku u



. It is equivalent to



0j j
tt0 t0j t
If we integrate from to obtaining



0
2
2
0
ee 0
jj
j
kt
kt
j
ut ut




or




0
2()
0ejj
j
kt t
j
ut ut 
The left inequality in (6) similarly implies



0
()
00
e,
jj
jj
kt t
jj
wtwtt t



and therefore



00
000
ee,
jj
jj
jjj
kt tkt t
jj
wtwtwtt t
 


0
j
j. The case which is just (2) for tt0
j
j
ttmay be
considered analogically.
3. References
[1] E. A. Coddington and N. Levinson, “Theory of Ordinary
Differential Equations,” McGraw-Hill, New York, 1955.
[2] S. B. Norkin, “Differential Equations of the Second Order
with Retarded Argument,” AMS, Providence, 1972.