413
Internat. J. Math. & Math. Scl.
Vol. 8 No. 2 (1985) 413-415
A NOTE ON PERIODIC SOLUTIONS OF FUNCTIONAL
DIFFERENTIAL EQUATIONS
S. H. CHANG
Department of Mathematics
Cleveland State University
Cleveland, Ohio 44115
(Received September 15, 1983)
ABSTRACT.
The existence of periodic solution for a certain functional differential
equation with quasibounded nonlinearity is established.
Quasibounded nonlinearity, periodic solution.
1980 MATHEMATICS SUBJECT CLASSIFICATION CODE. 34K15.
KEY WORDS AND PHRASES.
I.
INTRODUCTION.
Let
C
denote the Banach space of continuous
with the supremum norm, i.e.
n
R
given continuous
0 _< t
0
b, let
x
t
C
for each
-valued
x
function
C
be the function in
r’
R
n
valued
II + II
defined on
max
-r <0 -<0
[-r,b)
defined by
r
x
functions on
l(O)l
with
Also for a
0
b
x(t+0)
t(0)
[-r,O]
and for
for all
[-r,O].
Consider the following functional differential equation
x’(t)
L
where
and
f
and
f(t+T,)
linear in
L(t,x
t)
+
are continuous mappings from
f(t,)
for all
for fixed
t, and
[0,)
C
r
into
R
n,
L(t+T,)
L(t,)
and for some T > O, L(t,) is
r
maps closed and bounded sets into bounded sets.
[0,)
(t,)
f
(1.1)
f(t,xt)
C
Assume that the equation
x’(t)
has no nontrivial T-periodic solutions.
(1.2)
L(t,x t)
Also, without loss of generality we assume
T>_r.
Fennell
equation
[2]
(I. I)
has established the existence of T-periodic solution for the
by assuming
lim
uniformly in
t.
f(t,)l
0
(I.3)
It is the purpose of this note to generalize Fennell’s result by
We shall see that the limit in (1.3) can be allowed to
relaxing this requirement.
be positive.
Using the mapping
f
in
(1.1),
we give the following definition.
The function
S.H. CHANG
414
f
if the number
is said to be quasibounded with respect to
fl
lf(ll)l
),!
(max
rain
(1.4)
OtT
Ill
In recent years, equations
with quasibounded nonlinearities have been studied extensively. We shall show that if
is finite; in this case,
f
is called the quasinorm of
f.
is quasibounded and has a quasinorm smaller than a certain positive number then
Eq. (1.1)
Our proof uses a technique general-
has at least one T-periodic solution.
izing that used in [2].
2.
THE RESULTS.
(1.2), the functional differential equation
Under the assumption for
x’(t)
L
where
(i.I)
is the same as in
C
with initial value
XT(,O).
U
U
Then
(2.1)
Rn
is continuous and T-periodic,
Rn denote the solution of
h: [0,)
x(@,h): F-r,m)
and
Let
has a unique T-periodic solution.
(2.1)
e(t,x t) + h(t),
U: C
Let
r
C
r
be the operator defined by
r
(2.1)
is completely continuous and the T-periodic solution of
(I-U)
is determined by the initial function
-lxT (O,h).
Let (t)
be the norm of
L(t,),
the operator
9,(s)ds),
exp(
E
O
and
TE
K
THEOREM.
211 (I-U) -III
(2.2), then (I.I)
Let
@
X
+o
<II +II
lh(s) Ids}exp(
is given by
K
where
J(s)ds),
(2.3)
t ->0,
0
(2.1) and Cronwall’s lemma, will be needed.
be the Banach space of continuous T-periodic functions from
with the supremum norm.
For each
X, let
[O, ) -Rn is continuous and T-periodic. Let
r
l/K,
is
The following inequality
which follows from
C
fl
and has a quasinorm
f
has at least one T-periodic solution.
II xt(,h) ll
n
R
(2.2)
If, in addition to the given assumptions for the equation (I.i),
quasibounded with respect to
PROOF.
+ TE.
Now,
define a mapping
P
P: X +X
by
L(t,x t) +
f(t,t).
()(t)
f(t,t).
[-r,)
(I-U)-IxT(O,()).-
@
x(,()),
i.e.,
P
Then
is the unique
T-periodic solution of
x’(t)
Then
P
is a continuous mapping.
Since
Ifl
I/K,
there exists
e
such that
0
definition of quasiboundedness (1.4) there exists
whenever
II II
->
.
Ifl
O(e)
() and O -<
+
0
e <
I/K.
such that
<- T.
into
Then
Then by the
PERIODIC SOLUTIONS OF FUNCTIONAL DIFFERENTIAL EQUATIONS
415
Let
max{KN, O(e)
M
Then let
We claim that (i) P(D)
{
(2.3),
II PII
D
II sJl
p(e)
and
then
II II
e X:
and (ii)
D
Using the inequality
t-<-T}.
0(e), O
is relatively compact.
we obtain that
max IP(t)
K
max
O<s<T
-< T, if
s
M}.
P(D)
O_<t<T
O
<
and
D
Now for
Cr, II II
max{If(t,)l:
N
Klf(S,s)
<
II sll
JJsJi
-<
<
If(S,s) I.
Klf(S,s)
iJ PJJ
O(e) then
<- M. Thus
IIJJ
<- KN
-<
M
and if
whenever
By Schauder’s fixed point theorem ([3], or see [I, p. 131]) there exists
such that
P
,
D
which completes the proof of the theorem.
COROLLARY (FENNELL [2]).
(i.I), f
D.
#
(ii) can be established by using an argument similar to that used in
This proves (i).
[2].
If, in addition to the given assumptions for the equation
(1.3), then (I.i) has at least one T-periodic
satisfies the condition
solution.
PROOF.
The condition (1.3)
implies that
If
O.
REFERENCES
i.
CRONIN, J., "Fixed points and topological degree in nonlinear analysis", Mathe-
2.
FENNELL, R.E., Periodic solutions of functional differential equations, J. Math.
Anal. App !. 39 (1972), 198-201.
SCHAUDER, J., Der Fixpunktsatz in Funktionalraumen, Studia Math. 2 (1930), 171-180.
matical Surveys,
3.
N
!i,
American Mathematical Society, Providence,
1964L.