Internat. J. Math. & Math. Scl.
Vol. 8 No. 2 (1985) 403-406
403
ON THE ORDER OF EXPONENTIAL GROWTH OF THE SOLUTION
OF THE LINEAR DIFFERENCE EQUATION WITH PERIODIC
COEFFICIENT IN BANACH SPACE
H. ATTIA HUSSEIN
Department of Mthematics
Faculty of Science
Alexandria University
Alexandria, Egypt
(Received May 11, 1984)
An equation of the form
ABSTRACT.
Ay
w(t).,
J(t+)
A(t)y
y
f(t)
is
considered, where
and the necessary and sufficient criteria for the exponential
growth of the solution of this equation is obtained.
KEY WORDS AND PHRASES.
Difference equations, solution of exponential growth.
Ig80
MATHEMATICS SUBJECT CLASSIFICATION CODE.
I.
INTRODUCTION.
Let
E
II
II
A(t)
t >-o}
E into itself. We assume that A(t)
linear bounded operators from
Let
Denote by
be a complex Banach space.
strongly continuous in
$9AI0.
a family of
is periodic and
[o, ).
t
E.
be the norm in
Denote by
E
the set of all elements
f(t) E
such that
II f(t)ll
sup
2.
RESULTS.
Let
Ay
exp (-t)
’(t+6)- y(t)
Ay
>
A(t) y
f(t)
0
o
0
<
(R).
y(t)
t
>
be a solution of the difference equation
a
(2.1)
such that
y(t)
where
c)
is the zero of
Let us assume that
f
t
<
(2.2)
E.
E
The solution of equation (2.1) can be written in the
H. A. HUSSEIN
404
form
t-6
t-6
y(t)
A(i) y(i) +
S
6
s f(i)
6
=o
[n6], [a]
t
where
(2.3)
=o
and
a
denotes the greatest positive integer
is a
6
positive integer.
Without loss of generality we suppose that
I, 2
t
Putting
(2.3), one obtains
in
j
n-1
z
j=l
y(t)
n
(I + A(i)) f(j-l)
II
Let w be the period of A(t).
s
y(t)
r=l
I
B(j), j< n
B(n-l) B(n-2)
II B(i)
i=n-i
Substituting
s
(2.4)
f(t-l)
+
i=n-1
is the unit operator.
where
I.
6
t
o
[S w] into equation (2.4), we obtain
11
[I + a(k)]
[ k=w-I
s-r w-1
f (r-l)w+j-l) + f((r-l)w +
11
J
j=l
w-I)}
]
(2.5)
where
A(1) f(w)
a(2) f({w+l)
a(w-l) A(w-2)
A(w-l) A(w-2)
(w)
(w+l)
fl
f2
k=w-
t
t-rw
w-
B w
y(t)
s f.
r=l
((r-1)w + j-l)
+
f((r-l)w+w-1)}.
j=l
The last equation can be written in the fore
y(t)
2i
where
Y
"
t
y
z
y r=l
t-rw
(B-I) -I
w
. fj((r-1)w+j-l)
w-1
(2.6)
+ f((r-l) w + w-l)}
j=1
[1].
is a contour which circumscribes all the specter of the operator B,
It can be seen that if
f
E
(B-
then
I) -1 f
E
for every
y.
From equation (2.6) we obtain a necessary and sufficient criterion for the exponential growth of the solution with an index
o
Set
o
then the solution
y
operator B. Assume that
The following theorem holds-
THEOREM.
that
If
f
Era,
OB
Let
B.
=
I
In
o
B
denote the specter of the
IoI.
of equation (2.1) belongs to
E
B
such
405
DIFFERENCE EQUATION WITH PERIODIC COEFFICIENT IN BANACH SPACE
B
n,
when
B
> n,
when
B
n > n
o
when
o’
o
no.
n <
To prove the sufficiency, we consider the following three, cases:
PROOF.
(1)
If
(2)
If n
n >
1
>
lnlI
then y(t)
Inl
then from
defined by (2.6) belongs to
(2.6) we
E.
obtain
t
II Yll
-<
D z exp (1 In
II
r=l
(t-rw))
w-1
r
j=l
+
D 1 exp
<
D’ exp(nt).t
f((r-1)w + w-l) II}
(t).(w-2) t
t
(nt). + D2exp
<
II
(where D, D I, D 2 and D’ are constants).
E B where B > n.
This means that y
1 In
<
(3) If
II and llfll c exp (t), then from (2.6) we have
IlYll Clexp
(< I Inll).
E1
and y
1
We now prove the necessity"
If
X
Bx 0
is an eigenvector for the operator
B
such that
XoX 0
xo
where
is
llXoll
an element of Banach space such that
I,
by taking
exp (n t).x o equation (2.6) with
f(t)
(B
x
is an eigenvalue and
o
x
Xl,-1 x o
becomes
o
t
y(t)
exp
I
r=l
(&
In
llol
(t-rw))
w-1
I
j=l
Multiplying the last equation by exp (-
y(t) exp (-n o t)
where o
exp(
iot
fj((r-l)w+j-l)
ot),
t
w
where
o
+ f((r-l)w + w-I) }.
w
In
lol,
}1
j=l
fj((r-l)w+j-l)
+ f(r-l)w + w-I)}
t
o t)
exp
+ exp
we have
w-1
exp (- owr)
r=l
arg
y(t) exp (-
(2.7)
w
iet +
T w(1
iet
t
w
T
r-I
n)-l)
exp ((n
r-1
no)Wr).x 0
w-1
z
exp
j-I
(-oWr) fj((r-l)w+
j-I)
H. A. HUSSEIN
406
iot +
(
exp T
exp [’
oI
o
t
w
+ exp
T
[ exp (s-
If
:
r=l
j=1
(-oWr). fj((r-l)w+
exp
Eso
but y
This means that
y
2)
then from (2.8)
y(t) exp (-
(2.8)
cases"
(-Sot)
lim y(t) exp
t
s
j-l)
then by using formula (2.8) we get
> s
If
t- 1] x 0
w-1
I:
Now for the last relation we have the following
I)
So)
o
exp (w(1
Got)
+ exp
)-I +
iBt
T
t
w w-1
z
11
r=1 j=1
t
iot)
w"
1) x 0
(-oWr) fj((r-l)w+
exp
j-l).
Using the last equation we get
lim y(t) exp
(-Sot)
E
This means that
y
3)
then from (2.8) we have y
If s
s
o
s
but Y
E
but y
E
This completes the proof.
ACKNOWLEDGEMENT
The author would like to express his sincerest thanks to Professor Dr. Mahmoud
M. EI-Borai for his advice during the preparation of this paper.
REFERENCES
I.
HUSSEIN, H. A.
On the Bounded Solution of Certain Linear Equations in Partial
Differences Equations, J. Natur. Sci. Math.
2.
2__1, (1981) 165-169.
HUSSEIN, H. A.
Estimation of the Exponential Growth of the Solution of Certain
Linear Partial
Differential
Equations with a Highest Order Term,
Differencial’nye Uravnenija 12 (1976) 2279-2280.
(Russian)