Internat. J. Math. & Math. Sci.
VOL. 14 NO. 4 (1991) 757-762
757
ON THE STEPANOV ALMOST PERIODIC SOLUTION OF A SECOND-ORDER
INFINITESIMAL GENERATOR DIFFERENTIAL EQUATION
ARIBINDI SATYANARAYAN RAO
Department of Mathematics, Concordia University
Montreal, P. Quebec, Canada, H3G IM8
(Received April 5, 1990 and in revised form January 14, 1991)
The Stepanov almost periodic solution of a certain second-order
ABSTRACT.
differential equation in a reflexive Banach space is shown to be almost
periodic.
KEY
WORDS AND PHRASES:
bounded
(Stepanov) almost periodic function,
infinitesimal
strongly continuous group,
Bochner
operator,
linear
generator.
1980
AMS
PRIMARY 34Gxx,
SUBJECT CLASSIFICATION CODE.
34GI0,
34C27
SECONDARY 47DI0.
1.
INTRODUCTION.
Let X be a Banach space and J the interval
< t <
A
continuous function f:J- X is said to be (Bochner or strongly) almost
there exists a positive real number r
periodic if, given e > 0
such that any interval of the real line of length r contains at least
one point %
for which
sup
f(t + %)
e
f(t)
(i.I)
t6J
f
A function
bounded or
S p-
6
LPoc
(J;X)
with
p <
1
is said to be Stepanov-
bounded on J if
sP-t6J
f e LPoc (J;X)
f
A function
f(s)
with
1
<
F
p <
(1.2)
is said to be Stepanov
S p-
almost periodic or
almost periodic if, given e > 0
there is a
such that any interval of the real line
positive real number r r(6)
of length r contains at least one point % for which
sup
t6J
We denote by
[f+
L(X,X)
f(s + %)- f(s)
ds] /
<
(1.3)
the set of all bounded linear operators on X
into itself, with the uniform operator topology.
An operator-valued
function T:J L(X,X)
is called a strongly continuous group if
A.S. RAO
758
T(t +. tm)
T(t) T(t_)
T(0)
the identity operator on
I
x 6 X, T( t) x, t6J
for each
(1.4)
t,
for all
X
X
(1.5)
is continuous.
(1.6)
A
of a strongly continuous group
infinitesimal generator
dense in
T:J A(X,X)
is a closed linear operator, with domain D(A)
defined by
X
The
lim T(t) x-x
t
Ax
for
xeD(A)
i. 7
t-0
(see Dunford and Schwartz [3]).
T(t)x, t6J- X
is said to be almost periodic if
T
The group
almost periodic for each
is
x6X
NOTE i.
Suppose A and B are two densely-defined closed linear
and a
operators, having their domains and ranges in a Banach space X
function
f:J-X
Then a solution of the differential
is continuous.
equation
ul/(t)
Au/(t)
+ Bu(t) + f(t)
a.e. on J
(1.8)
for
is a twice differentiable function u(t) with u/(t) 6D(A), u(t) 6D(B)
a.e. (almost everywhere) on
t6J and satisfying the equation (1.8)
all
J
Our result i as follows.
X
Suppose
THEOREM.
f’.J
is a reflexive Banach space,
X
is an
S I- almost
periodic continuous function, and A is the infinitesimal
Further, suppose that
generator of an almost periodic group T: J-A (X, X)
is a strong
t6J
for ali
u:J-X, with its derivative uI(t) 6D(A)
solution of the differential equation
u//(t)
L(X,X).
then
2.
A(X,X)
B:J
where
u
Au/(t)
If
and
u
is
+ B(t) u(t) + f(t)
a.e. on J,
S I- almost
periodic and
u
are both almost periodic from
u
is
J
S
to
LEMMAS.
The
LEMMA 1.
representation
u/(t)
PROOF.
derivative
T(t) u/(O)
+
of
any
[r(t-s) u’(s)]
solution
T(t-s) [B(s) u(s)
For an arbitrary but fixed
ds
-
is almost periodic with respect
r(t-s) [u #(s)
t6J
+
(1.9)
to the norm of
bounded on
J
X
of
1.9
f(s)] ds on J.
has
the
(2.1)
we have
Au’(s)]
=r(t-S) [B(s) u(s) + f(s)] a.e. on J,by (1.9).
(2.2)
PERIODIC SOLUTION OF A DIFFERENTIAL EQUATION
Integrating (2.2) from
to
0
t
759
we obtain the representation
(2.1).
g:J X is an almost periodic function, and if
LEMMA 2.
If
G:J- L(X,X)
is an almost periodic group, then G(t)g(t), t6J- X is an
almost periodic function
X a Banach space).
See Zaidman [5].
PROOF.
LEMMA
S 1-
Let
3.
X
reflexive
a
be
space,
Banach
h:J
X
an
almost periodic continuous function, and
=h(s)
H(t)
H
Then
ds
J
on
(2.3)
S I- bounded on
is almost periodic if it is
J
See Notes (ii) of Rao [4].
PROOF OF THEOREM.
PROOF.
3.
From (2.1), we obtain
T(-t) u’(t)
u’(O)
T(-s) [B(s) u(s)
+
f(s)] ds
+
on
J
(3.1)
We write
v(t)
Since
B
B(t) u(t) + f(t)
=<
teJ
since
u
S I- bounded on
J
e
Now, given
0
m( + )
t/l
+]
So
B(t) u(t)
S I- almost
is
-|
uI1
+ /v
-
J
Consider the function on
Since v is
I-
,
on
to
X
it
is
e- almost period of
u(
ds
u()!
(3.4)
a
)
+
a
by (1.2) and (3.3).
S I- almost periodic from
periodic from
vh(t)
()
to
J
(see pp. I0, 77 and 78, Amerio
B(s) u(s)
u(+)
|()
u
from
to be an
z
we may choose
B(s + z)u(s + )
1
we have
(3.3)
B and also an e-S I- almost period of
and Prouse [i]).
Then we have
=/1|
L(X,X),
S I- almost periodic
is
>
.
(3.2)
to
J
is almost periodic from
suIB(t)
Further,
on J.
to
J
Hence v is
X
X
J
v(t + s) ds
for any
h
>
(3.5)
0
S I- almost periodic, it follows easily that
vh(t)
is almost
As shown for scalar-valued functions in
periodic for each fixed h > 0
Besicovitch [2], pp. 80-81, we can prove that vh -v as h- 0+ in the
A.S. RAO
760
S I- sense, that is,
sup[
v(s)
t6Jl
bounded on
ds
0 as h
(3.6)
0+.
T(-s), seJ-L(X,X)
is an almost periodic group.
So,
T(-s)x is almost periodic, and hence is
Thus, by the uniform boundedness principle,
Obviously,
for each
vh(s)
the function
x6X
J
s6J
Now we have
r(-s) v(s)
r(-s) Iv(s)
vh(s)] + r(-s)
vh(s)
(3.8)
and, by (3.7),
sup
t6J
[
r( -s) v( s)
K sup
t6J
fc+1
fly(s)
By LEMMA 2, the functions
X
to
vh s)
j
ds
v(s) ds
T(-t)
Therefore it follows that
vh(t)
0
(3.9)
as h
--
0+.
are almost periodic from
T(-t) v(t)
X
is S
J
to
almost periodic from J
bounded on J
Furthermore, by (3.7), T(-t) u/(t) is S
Hence,
is almost periodic from J to X
by LEMMA 3, T(-t) u/(t)
Therefore,
uI(t) is almost periodic from J to X
by LEMMA 2, T(t) [T(-t) u/(t)]
is bounded on J
u
and hence u is uniformly continuous on J
Consequently, by Theorem VII, p. 78, Amerio and Prouse [i], u is almost
periodic from J to X
completing the proof of the theorem.
So
NOTE 2.
consider
In a reflexive space
X
infinitesimal generator differential equations
u’(t)
u’(t)
[A + B(t)] u(t) + f(t)
Au(t) + f(t)
the
first-order
a.e. on J,
a.e. on J,
(3.10)
(3.11)
f:J- X is an S I- almost periodic continuous function, A is the
and
infinitesimal generator of an almost periodic group T:J- L(X,X)
B:J L(X,X)
is almost periodic with respect to the norm of L (X,X)
Then, from (3.10) and (3.11), we have the representations
where
u(t)
T(t) u(O)
+
T(t-s) [B(s) u(s) + f(s)]ds on J
(3.12)
T(t-s) f(s) ds on J,
(3.13)
and
u(t)
T(t) u(O) +
respectively.
-
From the proof of our THEOREM, it follows that,
S
u:J- D(A)
is an
then
equation (3.10)
almost
ft
is
(a)
if
periodic solution of the differential
J to
X
and
periodic from
almost
S I- bounded solution of the differential
fs an
D (A)
equation (3.11), then it is almost periodic from J to X
(b)
if u:J
PERIODIC SOLUTION OF A DIFFERENTIAL EQUATION
761
REFERENCES
i.
2.
3.
4.
5.
AMERIO, L. and PROUSE, G. Almost Periodic Functions and Functional
Equations, Van Nostrand Reinhold Company (1971).
Almost Periodic Functions, Dover Publications
BESICOVITCH, A.S.
Inc. (1954).
Linear Operators, Part I,
DUNFORD, N. and SCHWARTZ, J.T.
Interscience Publishers Inc., New York (1958).
On Differential Operators with Bohr-Neugebauer Type
RAO, A.S.
Property, J. Diff. Eqs.,13 (1973), 490-494.
ZAIDMAN, S. Sur la Perturbation Presque-Priodique des Groupes et
Semi-groupes de Transformations d’un Espace de Banach, Rend. Matem.
e sue App,.,16 (1957), 197-206.