m17 x71 hb y87 ff5 fs7 fc0 sc0 ls21">
2
A
tI,
3
A
ttI
 , and

2
f tf



t I
1
Bt
0nn
are n ×
n matrices. Note that,
and I are the zero and the
identity matrices, respectively.
3)
 
T
1
,0 ,,
n
tgtgt

rxx x

1
0
,
n
t
gx
which is a 2m1
vector.
For ,1,2,,ij n
, let is an arbitrary fixed
and let
00t
C
opyright © 2012 SciRes. IJMNTA
M. A. RAMADAN, S. M. EL-KHOLY
126


 
 
11 012010
,,
21 022020
0
,,
10 0
,, ,
,, ,
,
,,
m
ijin j
m
nijnin j
mm
m
zttzttz tt
ZZ
zttzttztt
Ztt ZZ
zttz tt











(3)
be a fundamental matrix solution to the linear system:
A
t
zz (4)
which equal to the identity matrix for .
0
Consider with
tt
00z0
z small enough, and
let us denote by
00t
00
,,ttzz
0
z the unique solution of Equa-
tion (2) which equal to
at . By hypotheses (1)
0
tt
and (2),
0
z
0
,,ttz is defined on a maximal right inter-
val,
0,tl, and satisfies the following integral equation:
 

 

0
1
000 0000000
,,,,,,,, ,,d
t
t
ttZ ttZ ttZstBsstssts
 
zzzzzrzz
This gives us the following integral inequality:


 

0
1
000 0000000
,,,,,e,,,,,d
t
t
ttZ ttZ ttZstBsstssts
 
zzzzzrzz (5)
where .

T
010e
Equations (3) and (4) give us the following differential
equations:

,,ijij nij
zftzz
 ,
,
,
,
(6)
 
,,nijij nij
ztzftz

  (7)

,,injinj ninj
zftzz

 (8)
 
,,nin jin jninj
ztzftz
 
  (9)
for .
,1,2,,ij n

Since is a decreasing function, so Equations (6)
and (7) lead to:
t
 
22 22
,, ,,
1
2ij nijij nij
tz zfttzz



 




0
2d
22
,, 0
e
t
t
f
uu
ij nij
tz zt

 (10)
By the same way we can obtain from Equations (8)
and (9) the following:


0
2d
22
,,
e
t
t
f
uu
in jnin j
tz z

 (11)
For

T
12 12
,,,,,,, m
nn
x
xxyyy Rz, consider
the norm 2
1
m
i
i
z
z, where is the norm defined
in .
m
R
For

T
001 02001020
,,,,,,, m
nn
x
xxyyy Rz we
have:

 
,, ,0,0
0
00
,, ,0,0
0
22
,0,0,0, 0
11 1
,ijin jijin j
nijnin jnijnin j
nn n
ijjinjjnijjnin jj
ij j
ZZZZ
Ztt ZZZ Z
zx zyzx zy

 

 















 
xy
x
zxy
y
satisfies (4), then we get: Using Shwartz inequality, Minkowski inequality [8] and
suitable assumptions lead to:
 

0
2d
00 00
,31e
t
tfu u
Zttn t

z
z (12)
We have also:
00tst 

 
1
00
T
1020 0
,,e
,, ,,,,
m
Ztt Zst
tst tsttst
 





 

1
00
0
1
22
00
1
,,e
,, ,,
i
i
n
ii
n
i
Ztt Zst
ttsttst


Since
2,,
mtst
t
is a decreasing function of , we get:
t
Copyright © 2012 SciRes. IJMNTA
M. A. RAMADAN, S. M. EL-KHOLY 127



 
22 2 2
0000
,,,,,,,,e S

2d
t
f
uu
in
i ini
t tsttsts sstsst
 


 

So,
  

d
1
00 0
,, e
t
S
fu u
Ztt Zstent
(13)
As mentioned the system satisfies the integral inequality (5), then inequalities (12) and (13) give
 

 

0
1
0000000000
,,,,,e,,,,,d
t
t
ttZ ttZ ttZstBsstssts
 
zzzzzrzz
 






0
dd
2
0000 000
,,31ee
tt
ttS
fu utfuu
ttntn tnfs

0
t
,,,df sstss
 
zz z
For all t we can replace g by another function say g
defined as follows. By hypothesis (4) it follows that there
exists a
zzgx (14)
0
such that if
x, then


,tfto
gx x
We defined the function by:
:nn
RRR

g
 

,if
,,if
gt
tgt

ave
xx
gx xx
and we h
x, t0.
ry ,x
It is clear that for eveR

tR

n


,tgx fto
x
is of class and is locally Lipschitzian
g
in

n
CR R

n12
,,,
x
xx
ginal function
. So, we w
g satisfies

ill admit from now that the
all the properties of the ori
g.
Since 1
f
CR
we get from inequality (14) that:
 


0
00 0 0
,,31e
t
tfu
t
ttn t
D

0
00
,, d
t
s
t s
du
z
zz
u
g
zz
0,ttl , with positive constant D [9,10]. This givess
the followin
 
0000 0
,,31 ,,
Dh
ttntett l
zzz (15)
Thus

00
,,tt
z
zas well as

00
,,tt
z
z are bounded
on and so
0,tl

00
,,tt
z
z can be extended to the
right of l. This contradicts the maximality of l. This
means that the solution of Equation (1) is bonded. The
proof of theorem 1 is complete.
Theorem 2
If the hypotheses of theorem 1 are hold and
lowing assumption is satisfied:
1) There exist two constants such that:
u
the fol-
,0hk

2,ftf tkftth

then the zero solution of Equation (1) is uniformly stable
solution.
If in addition
2)

0
dft t
holds, thzero solution en the of
Equation (1) is asymptotically stable.
Proof
n
Our stability question is reduced to the stability of the
zero solutio
0t
z
will be divided into two intervals. In the
to the system (2). The interval of
tfirst interval
we have
0,tth and in the second interval
,th
.
lh
and si
on a ma
Firstly, with nce the hypotheses of theorem
1 are hold, and then the solution
,tz
of Equation
00
,tz
l right interval (1xima) is defined
0,tl. It is
proved in theorem 1 that the solution is bounded and
00
,,tt
z
z cantended to the right of l. Therefore, be ex
00
,,tt
z
z exists on
0,tl with lh.
Assume hl
. We are going to find an estimate
for
00
,,tt
z
z on the interval
,hl . From hypothesis
(1) of theorem 2, we have for thl :
,










,,, d
t
unk fsstgsszz x
dd
000 0
,,31e,e
t
SS
t
fuufu
ttnhhtnh



 zzz z
0
,
h
00






 


dd
0
, ,
t
hS
fu u
nhnkfs fs

z z
00 00
h
00 00
e,,,, d
t
fu ustst s
zzxz ,3 1e,
t
tthtn h

z z
Copyright © 2012 SciRes. IJMNTA
M. A. RAMADAN, S. M. EL-KHOLY
128
with

1
0, nk
nh





(16)










00
d
000
d
00
,,
31e ,,
e
t
h
t
S
fuu
tfuu
h
tt
nh ht
nhnkfs sts



zz
zz
zz
,
,d
(17)
 







d
000
d
00
31e ,,
e,
t
h
t
S
fuu
tfuu
h
vt hht
nhnkfssts



zz
zz
So
,d
 
1,vtnh nkftvtthl



 .
By integration we get:

 
 
1d
00
,,e ,
,
t
h
nhnk fuu
ttt h
thl
vv





zz (18
with
)
 

00
31,,vh nhht
zz
From inequality (18) we can see that

00
,,tt
z
z is
bounded since

00
,,tt
z
z is also bounded, it folls
that For
ow
l .0
we can get:
 



2
e
13013
Dh
nh



From (15) it follows that


00
,,
13
tt nh
zz
for all
0,tth provided that 0
zore,
from (16) and (1t

. Theref
, we ge8)

00
,,ttvh
zz for
all th o, if h, then the solution . S00
t
starting
00
,,ttzz
from any point 0
z
, with 0
z, exists on
0
t
, and satisfies

00
,,tt
zz for al
usly we btain that
l 0
tt. If
and:
0
th, then analogool
 
 
0
1d
0
e ,
t
nhnkfuu




(19
,th
Therefore, with the same
00 0
,,31ttnt
zz z)
t
as before, 0
z im-
plies again

00
,,tt
zHence the
If in
z for all 0
tt.
zero solution is uniformly stable.
addition

0
dft t
is satisfied, then by (16), (18) and (19) it follows tha
zero solution to (1) is asymptotically stable. The proof of
theorem 2 is complete.
3. Examples
We confirm the results of the introduced theorems by
considering two numerical examples for which the func-
tions
t the
f
t,
t
and
tg satisfy theorems assump-
tions.
For the system
 
2,ft t
 
t0
 xxxgx
where
,
12
,
x
xx
.
Example 1

21
t
ft t
,
11tt
,

12
112 2
,, 1
x
xt
gtxxt
and

22
12
212 2
,, 1
x
xt
gtxx t
.
(a)
(b)
Figure 1. Numerical solutions if Example 1, x1 component (a)
and x2 component (b).
Copyright © 2012 SciRes. IJMNTA
M. A. RAMADAN, S. M. EL-KHOLY 129
(a)
(b)
Figure 2. Numerical solutions if example 2, x1 component (a)
and x2 component (b).
Example 2

2
1
ft t
, ,

2
1e
t
t


12
112 2
,,
x
gtxx t
and


22
12
2
212
,,
x
x
t
.
gt
xx
We solved these two examples numerically using
th order Runge-Kutta method. The results of example 1
are drawn as shown in Figure 1 and the results of exam-
own in Figure 2. The curves are
rawn for different initial values of
d 2 demonstrate time increases
all the compents of the solutions tends to zero.
means, that te aptotically stable. W
verify the rightness of our proved theorems.
4. Conclusion
ifo ly stable as well
as
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x.
that, as the
Figures 1 an
on This
he solutions arsymhich
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ear damped vectorial oscillator and the conditions for the
stability of the zero solution to be unrm
asymptotically stable. We verified our theoretical re-
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Copyright © 2012 SciRes. IJMNTA