Internat. J. Math. & Math. Sci.
(1986) 757-766
757
Vol. 9 No. 4
ON THE CARDINALITY OF SOLUTIONS OF
MULTILINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
IOANNIS K. ARGYROS
Department of Mathematics
University of Iowa
Iowa City, Iowa 5222 U.S.A.
(Received October 18, 1985)
ABSTRACT.
We study the existnece and cardinality of solutions of multilinear differ-
ential equations giving upper bounds on the number of solutions.
KEY WOHDS AND PHRASES.
1980
1.
Cardinality, multilinear bound, beams.
AMS SUBJECT CLASSIFICATION COSE.
3A20, 6B15.
INTRODUCTION.
n(i), i
n(i)
Let
and let
L.I
1,2
CijDJ
J=O
cn(1) (I),
fined on
functions
Cij
tinuous on
I.
i
m
be positive integers such that
[a,b]
I
where
1,2
[1]
Using some ideas from
and
[3]
...
n(m)
(but not necessarily). The coefficient
n(i) are never vanishing real and con-
usually
0,1,2
m, j
n(2)
be regular linear differential operators de-
1,2,...,m
i
n(1)
we study the branching of solutions
u
E
cn(1)(I)
to the multilinear equation
Mu
Equation
(i.i)
(LlU)(L2u)...(LmU)
is related with the null set
()
[u
0
N(M)
cn()()
u
0
which can be infinite dimensional.
We give necessary and sufficient conditions for a
(m-l)-tuple
to be a multiple ordinary branching of a solution to (i.i) where
e
(GI,@2
e
,m_l)
I,
1,2,... ,m-l.
We also study the existence and cardinality of solutions to the initial value
problem
Dn(1)u(z)
where
z,z i
zi,
i
1,2
n(1)-i
(1.3)
I, giving upper bounds on the number of solutions with n multiple
branchings.
Multilinear equations have a rather extensive literature
[3], [h], [6]. A
few
758
I. K. ARGYROS
special cases of applications (e.g., pursuit problems and bending of
formulated in the form
beams) may
be
(1,1).
Finally we study the problem
(1.1)
when it assumes the form
2.
%.
for some function
BASIC THEOREMS.
I,B2,...
Let
DEFINITION i.
N(Lm)
B
Bm
and
N(LI),N(L2),...
denote bases for
and
respectively where
B
LUIi,U2i
Un( i)i
(B.0 N cn(1)(1))
Bj_1
i
dim(Bi)
with
n(i), i
1,2
m
and let
Ej
Obviously
N(LI)
dim(Ej) (J)
N(Lm) N(M). We
with
U N(L2) U
U
< n(J), J
2,3,...,m.
will seek solutions
u
N(M)
of the form
[
n(l
f
E
|
hie)
c..(x)
u(x)
(x)
(2.2)
=n(e+l)
jl
x
e-i
e
e3
j=l
e
Ce+ljUe+lj
nim)
E o .u .=u_ (x)
kJ-i m3mJ m
aeN(Le) U N(Le+I). A
,x
e+l
|
/
for
in
aeI,
N(M)
1,2
e
m-i
and
will be said to have a single
[ae_l,ae+l].
branching at
x
am_l
function of the form
at
ordinar branchin5
x
e
(2.1)
on
A function of the form (2.2) will be said to have a multiple ordinary
,m-2
m_l on I [a,b] with e @e+l e 1
(i,2
Denote the Wronskian
We(Uli,U2i,...,Un(i)i, Ul(i+l)(xo)
by
0),
W __(x
e
e
The following theorem shows when
1,2
N(M)
m-l.
will contain functions having a multiple
ordinary branching.
THEOREM i.
Assume that
n(e)
(e)
+ n(e+l)
n(1) + i, e
2
(2.3)
,m-i
and if
()
E has Just one function __ulj(x)’
having a multiple ordinary branching at
We(e
o,
e
i,
m-i
<=>
J
2,...,m,
(l
m-1
(LiUl(i+l))(i)
then there exists
u
N(M)
if and only if
O, i
i,
m-l.
(2.&)
CARDINALITY OF SOLUTIONS OF MULTILINEAR DIFFERENTIAL EQUATIONS AND
APPLICATIONS
(ii)
ae
and
#
dim(Ej)
e
ae+l’
,m-2
(’2
branching at
PROOF
m,
2
i, j
1
C
u
so that
n(l )(I).
Z CljUlj (k)()
j =z
n(-z)
.
CeUe
j=l
where
m-I
g cgju2j
n(3)
Z
c
3ju3j
j =l
(kl(x)
,Cln(1), C21
C2n(2)
Cm/,
(k)(%)
(k)(2),
k
n(1)
0,I
n(m)
I %0%0 (kl(%-Z)
j=l
In this case (2.5) becomes
CASE (i).
n(e)
U
m-lj m-lj
Cll
j =i
Z c2ju2j (k)()
j =i
j=l
with a
int I
e
having a multiple ordinary
Therefore we must have
n(2)
n(1)
C
N(M)
u
m-1 )"
It is enough to find numbers,
Cn(m
(i
then for every
there exists a
759
(k)
c
(e+l)
(we
take
(ae
#
Cln(e+l
0
Uln(e+l)(e)
0, e=1,2
i).
Cln(e+l
nontrivial solution if and only if
(2.4)
0,1,...,n(1)
m-l, k
(2.6)
The homogeneous equation
(2.6)
has a
holds.
Note that it is easy to verify that
W
e
(%)
CASE
e(ule,u2e ,Un(e)e,Ul(e+l)(De)
-i
Un e e(e) LeUl
en e (e)We (Ule U2e
(ii). If (LeUs (e+l))(e)
0 e
1,2
e
W
.
n(e)
j=l
CejUej
(k)
)(e
e=l,2
we let
m-1
c
s
,m-i
1
(e+l)
e
Otherwise we write
We then work as in Case (i).
the rest coefficients zero.
as
e+l
and
(2.5)
n e+l
(e
Z
Cln(e+l)Uln(e+l)(e)
J=l
cj n(e+l)Ujn(e+l) (e)
1,2
e
m-l, k
n(1).
0,I
Note now, that the rank of the coefficients matrix on the left hand side is
(n(1)+l)
and thus we have a unique solution for the coefficients on the left hand
side for any choice of the coefficients on the right hand side and for any
e
e
I,
1,2,...,m-l.
The next theorem characterizes the conditions with the coefficients in
satisfy in order that multiple branching can occur at
e
e+l’
e
,m-2
1
THEOREM 2.
and
e
(l,a2
m_l
(2.2) must
with
I.
The following are equivalent:
u
(Le
k
N(M) on
n(e+l)
j
i
[c,d] c I
Ce+ljUe+lj
and
(e)
is as in
u
0, e
i
(2.2).
(2.8)
m-2.
n(e)
(Le+l[j=
CejUej])(%)
0,
ke
0,i,
n(e+l)
n(e)
(2.9)
760
I. K. ARGYROS
In particular, (2.8) with
Cej #
N(M) to
U
for
Ce+lj # 0 for at
Uej E Be- Ee+l are
for at least on
0
PROOF.
E
.
# 0,
e+l
wise as in Theorem i, we have that
n(e)
-
Z
(e)
c
j =i
am_
Other-
if and only if
n(e+l)
(k)
CejUej
J =i
N(M)
u
with
on [c,d].
1
the result is trivially true.
..,m-2
1,2
e
(2.9)
and
Ej
Ue+lj
(,{z2
have a multiple branching at
Be
If
least one
both necessary and sufficient conditions
u
e +lJ e +lJ
(k
(e)’
n(1),
k
0,i
e
1,2,... ,m-l.
The above can be written in the form
n(e
CejUej
j=l
]n(e+l)
[Ue+lJ’J=l
where
if
B e 13
Ue+lj
can ,be written
-
Ee+l
c
(e)’
(k)(e)
eo.Uej
0.
n(e+l)
and
-i
Ce+lj #
(?.0)
n(1),
0,i
1,2,...,m-i
Here
c’
Ce+lJ
Cej
eJ
otherwise.
eJ
u
eCn(e)+l) (x)
O, k
c’ej
e(uli,u2i
Un(e)i,Un(e)+l i)(ge
We (Uli ’u2i
Un (e)i ’Un (e)+li )(e
j=l
n(1),
0,i
Now, (2 ii) has a nontrivial solution for
W
(k)
e
n(e)+l
C
Ce+lj Ue+lj
J=l
and at least one
Ee+l, c’.
e
j=l
n(e+l)
-(ae
k
c’e(n(e)+l)
Now set
(k)
Ce+lj Ue+lJ (k)(x)
1,2
e
(2.10)
and
(.u.)
m-l.
if and only if
O,
but
i.e., if and only if
On the other hand,
if
(2.10)
-1
Un( e )i )(Ce )LeUe (n (e)+i
en( e (e)We Uli ’u2i
(2.8) holds and at least one Cp+lj # 0.
a
has a nontrivial branching at
u
m-1
if and only
has a nontrivial solution for the coefficients on the right hand side.
before we set
c
-i
e (( e+l )+i
n(e)
(2.10)
As
and
Z
Uee+l,+l(X)
and
(l
(Ce)’
J=l
(k)(x)
c’.u
ej ej
can now be written as
n(e+l)+l
j=l
Ce+ljUe+lj
(k)
(e)
0, k
0,1,2
n(1),
e
1
m-i
(2.12)
or
Ad =0
e e
(2.13)
in matrix form,
where A
is the coefficient matrix in (2.12) and
the unknown
e
e
There will exist a nontrivial solution d # 0, e
1,2,...,m-1 if and only
e
if the rank of A
e
1,2
m-1
n(e+l)
But
the
n(e+l)
X n(e+l) principle
e
vector.
submatrix of
Ae
Ae a n(e+l)
Therefore (2
is the Wronskianmatrix evaluated at
13) will have
ae
Hence the rank of
a nontrivial solution if and only if the rank
CARDINALITY OF SOLUTIONS OF MULTILINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
A
of
to
n(e+l).
is
e
A
Now elementary row operations on
e
761
show that this is equivalent
(2.9).
N(M)
We now show that
may contain infinitely many linearly independent functions.
Assume that either Case (i) holds in theorem for infinitely many
THEOREM 3.
or Case (ii) holds.
m-li )’ i 1,2
i’"%- i=l
(li’a2
u (li,2i...m_li)
sequence
branching at
[Uli2i"
the set
PROOF.
<
ei
"m-li ]n,i=l
N(M)
such that
with
ei
ei+l’
has a multiple
%’"%-.
e+li’
e
m-2, i
i
I
1,2
for every
and
n.
We may assume without loss of generality that
1,2,
e
1,2,
u
is linearly independent on
We proceed by induction.
i
<
In either case, there is a
m-2.
Ee+I
Ue+lj’s
Choose
then
n(e+l)
i
L
P
Hence
u
el
(x) #
0
Ce+ljUe+lj (x) #
--[el’’e+l)l]’
on
so
u
# o
ae’" "%-i (x)
[a,b].
I
on
[e]_ ’(e+l) ]"
0, x
Now suppose that
,
u
---%_
Suppose that there exist constants
i
1,2,...,n
i
1,2,...,n+l-
are linearly independent.
n+l
Z .u
=
if
0
dn+I
dependent.
then
If
dn+I #
u
in+la2n+l "m-ln+l
for all
I
x
e
en+l
whereas
(en’aen+l)’
DEFINITION 2.
so
Then since
L.u, i
1,2,...,m
1
i
ae
must
l=l
Define the set
1
S.1
Ee+l
(aen,aen+l)
x
for some
0.
are continuous functions on
Sm
We show that
Se_1
I
the
S.’s,
1
In particular, any point
[ae_l,ae+l]
e
1,2,
together with any limit point of the set of points at
Se+
Se_
a contradiction.
by setting
I.
S U S U...
2
1
at which an ordinary branching occurs on
[ae-l’e+l
Se+1
0
Ix I/(Liu)(x)
belong to
1
1
which ordinary branching occurs since
Se_1
span
d u
i ei
are closed sets and
1,2,3,...,m
THEOREM h
(aen_en+l)
0, x
LeUaen+l(X) #
S.
but
n
-i
(dn+l
x
is linearly in-
n
(e_li,ae+li),
x
(x)
0
I .u
dn+l i=l
i
li2i ..m_li
(x)
in particular for each
L u
when
and
0
(x)
]n+l
[ui2i...m_l.i=l
-
1,2,...,n
0, i
d.1
%i...%_
Se+1
Assume that
Se_1
Se+1
is closed.
is nowhere dense in
u
is nowhere dense in
N(M)
as in
[@e-l’&e+l ]’
(2.2)
e
[e-l’e+l ]’
and
B
e
E
1,2,... ,m-l.
e+l
e
1,2,... ,m-l.
Then
,m-i
762
I. K. ARGYROS
PROOF.
ae-l’ ’ae+l’
interval
N
Se+I, e
le+l’ a’e-1
Se_I
Suppose that
with
1,2,... ,m-I
#
contains a maximal closed
0
e
1 ,2
for
x
[ae_l,ae+I]
,m-1
Then
n e+l
u(x)
e0 e0
(:L)
stants
(2)
Cej
[ae_
I’ ,ae+I’ ].
c
("e-l’ae+l"
Now let
.LI
C
j=l
(i) in Theorem
Then by Case
i there exist con-
such that
Cej
ne)
Uej (x)
c
e3
j=l
ae-i
x
a"e-i
"
x
e+l
x<
e+l
n( e+l
U("e-I
eo
J=l
ne) eo(2)u
c
j=l
N(M)
belongs to
.(x)
c .u
e+lj x
e-I
ej
ej
(x)
ae+l
since
n e+l
at
"
z.
e-1
0
Z CejUej)(x)
e( j=l
0
e
1,2,
j=l
n e+l
L
[e-i ’e+l ]"
e
But
"e+l"
on
]
CejUej )(z)
L
U(’_’e_+/- ’ag+lj x)
Hence
N(Le )"
n e+l
j=l"
Since
n(1)
We now assume that
n(e),
d(Le
n(m)
But this contradicts
N(2)
e
,Zn(1)_l)
]R
n(1)
z
and
I
find
u
if
N(Le
#
N(Le+I),
N(Le ), e
unique solutions belonging to
(the
other cases
given
0
n(1)-l.
0,i
(2.1h)
then we have at least
1,2,...,m-l,
e
i
zi,
is
,m.
such that
Mu (LIU)(L2u)’’" (LmU)
Dn(1)u(z)
1,2
for simplicity
can be dealt analogously) and consider the following problem:
(z0,zI
N(Le
Be U {u(a"e_l,a"e+ljX)]
the proof of Theorem 3 shows that the set
m-i
linearly independent.
CejUej" (x)
m solutions, the
In addition according to
1,2,...,m-1.
Theorems 1 and 2 we may have solutions with one or many multiple ordinary branchings.
In the event that
Sje, J
follows:- let
Le,
1,2,...,n(1),
characteristic equation
containing
1,2,
e
L
and assume
e
then the restriction
z,
have constant coefficients we proceed as
,m
1,2,
e
u
m
u
N(Le
of
u
on
n(1
U(X)
E
c. e
n(l
{eSjeXJ=l
(2.9) we must
spans
N(Le
and
Cje
Le+l(U(e))
O,
I(z)
of
sj eX
are uniquely determined by
have
It follows that
I(z)
can be written
3e
J-1
where
denote the solutions of the
on some subinterval
e
1,2
m-l.
(2.1h).
By
CARDINALITY OF SOLUTIONS OF MULTILINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
.
n(l
j=l
dj ee
where
tj ee
0
763
(2.15)
n(1)
0 Cie+it
c.
d.
je
je
t
i=
s.
ie
s
le
i
e
e
n (1)e
n(1)
1,2
1,2,
(2.16)
,m
(2.17)
Note that each one of the equations in (2.15) can have at most
dje ’s and t j’s are all real [7].
solutions if the
e
Denote by
(2.15),
p
n(1)-i
apl,p2
and assume that
1,2,...,m-i
th
the solutions obtained in the
pn(1)-i
real
equation in
(the other cases can be dealt analogously)
(2.18)
(2.18)
Inequality
branchings, e.g.
THEOREM 5.
a solution
shows that we can have at most
(i’21
If
E N(M)
u
1,2,...,m have constant coefficients, then there exists
(2.2)) to the intial value problem (2.1h) having a
(u as
in
(al,2
multiple ordinary branching
ge
only if
tj’s
e
multiple
We have thus proved.
is one of them.
am-ll
i
Li,
(n(1)-l) m-I ordinary
with
m_l
is a root of the exponential polynomial
are all real and they are given by
(2.16)
Moreover if (2.18) holds there are at most
(u as in (2.2)).
THEOREM 6. Assume that the hypotheses
e
E I,
(2.15),
e
1,2
m-1
where the
(2.17)
(n(1)-l) m-1
dj e
if and
s
and
and
of Theorem
solutions
5 are satisfied.
u
Then there
are at most
(m-i)(n(l )-i )(n(1)-2 )n-i
(u
u
solutions
tiple branchings
e
as in
(2.2)) to
(52
(2.19)
the initial value problem having exactly
I
in
,m_l
where any
m-2
1,2,... ,m-1.
Moreover in this case if there are no solutions with
of the
n+l
e’S
n
mul-
are fixed
multiple branchings
then the total number of solutions to the problem
Mu=O
is bounded by
n-i
(m-1)(n(1)-l)
E
(n(1)-2) j.
(2.20)
J=0
PROOF.
Without loss of generality we can assume that
point at which a branching occurs and
Then
u
N(L2) on
[ii,//+ ],
ble values for
I"
Suppose
of
Then
u
u
occurs.
cJ2(ll )’
j
1,2
,n(1)
N(L2)
w
>
u
N(Ll)
for some
ii
on
such that
l
denote the first
on some subinterval
>
0.
There are at most
I(z)
m-i
[z,all].
possi-
is the next point at which a multiple branching
[ll,W].
Hence there exist uniquely determined
764
I. K. ARGYROS
.
n()
u(x)
dj2 (ail)Uj2(x)
J=l
on
[l,W]
where
]n(1)
uj2oj=l
N(L2).
span
By Theorem 2,
n()
E dj(ll)Uj2)](v)
ILl( j=l
at
v
GII
and
v
w.
Hence there are
m-2
m-1
w’s
possible
argument applies again for the next branching.
any of the
0
w
with
>
This
Ii"
Since this argument can be applied in
rows in (2.18), this proves (2.19).
(2.20) can
(2.19)
easily be proved if we use
for
J
0,1,2
n
and
(a) We can assume in Theorem 6 that any h points h E [1,2
fixed from (al,a
am_I) then proceeding as in Theorem 6 we can prove
2
corresponding relations for (2.19) and (2.20) are respectively
that
Finally
add the results.
REMARK
are
the
m-lS
i.
(m-].- (h-1) )(m(1 )-l )h(n (i)-2 )h-1
and
(m-l- (h-i) (n(l)-l
(b) Up
(m-l)’.
(n (1)-2 )h
hnl
J=0
2.22
till now we obtained the cardinality results in Theorems 5,
above by assuming that
in
2.21
(2.18)
is true and
as in
u
(2.2).
6
and in
(a)
But (2.2) can be written
different ways by interchanging the role of the
L.’s,
i
1,2,...,m.
Therefore in general all the cardinality results obtained up till now can be multi-
plied by
(m-i).’
m are nonconstant but continuous (as in the
(c) If the Li, i 1,2,
Introduction) we can restate Theorem 5 and (2.2. However the conclusions and the
proofs are going to be exactly analogous.
We now provide examples for Theorems
and
6
(1.4).
and
APPLICATIONS.
EXAMPLE
i.
Let
m=2
and consider the function
8
1
xl,
x#0
0
x=0
xe
f(x)
Ul(X)
Then
f(x)
e
N(LI), u2
u
I
2f(x)
e
up(x)
N(L2)
u(x)
or
and a
/
That is,
0
[
u
f(x)
ce
de
ce
u(x)
LlU
2F(x)
2f(x),
de fxl
N(M)
f’(x)u,
u’
-
L2u
can be written as
x0
>0
x
0
0
defined by
f
g
x m 0
>0
x
is a limit point of branching points of
u.
u’
2f’(x)u
CARDINALITY OF SOLUTIONS OF MULTILINEAR DIFFERENTIAL EQUATIONS AND APPLICATIONS
In the event that the characteristic equations of
plex roots
(2.15) may
I)
therefore
due to
n
(2.16)
nu, n
and
i
UlX
(2.15)
(2.17)
0,1,2,...
e
ix + i e -ix
e
L
1
(-,).
h, u(O)
O, u’(O)
i.
sin x
1
1
e
1
-2ix
sin 2x
etc
Consider the equation
kM= 07.
dx
Let
1,2,...,m
becomes
u_x,.. y1 e 2ix
EXAMPLE 3.
i
have com-
have infinite solutions to the initial value problem on
even if we have one ordinary multiple branching in
2
2
EXAMPLE 2. Let m 2, L1 D + l, L2 D +
N(L
u
on [-g,] for some
> 0. Then
and
Li,
(D-4)(D-5)(D-6), L 3
(D-I)(D-2)(D-3), L2
(D-7)(D-8)(D-9)
Then
x
l(x)
u2(x)
2e
u3(x)
8e 7x
u
5e
3e
4x
2x + e 3x
6x
9e 5x + &e
8x
15e + 7e 9x
and
For example we can have the solution u
N(M)
given by
and
765
(-,)
Let
766
I. K. ARGYROS
2e
u(x)
5e
x
4x
8e 7x
3e
2x + e 3x
< x
6x
9e 5x + 4e
8x +
15e
7e
in P
9x,
ln(
in 2
X
in()
12
4+llO)
12
x <
+,
etc.
(LI,L2,L3). But we can
(L1,L3,L2), (L2,L1,L3) (L2,L3,L1)
The above are solutions corresponding to the order
tain additional solutions corresponding to
(L 3,LI,L2)
and
ob-
(L3,L2,LI).
REFERENCES
and PRENTER, P.M. On the Branching of Solutions of Quadratic
Differential Equations, Aequationes Mathematicae 10 (197h), 81-96.
l,
ALLGOWER, E.L.
2.
AMES, W.
Ordinary Differential Equations in Transport Processes, Academic Press,
New York, 1968.
CHOW, S.N. and HALE, J.K.
York,
Methods of Bifurcation Theory, Springer Verlag, New
1982.
Introduction to Nonlinear
New York, L962.
DAVIS, H.T.
Differenti.al and. lnte6ral Equatiogs, Dover,
HARTMAN, P. Ordinary Differentialquations, Wiley, New York, 196h.
KAMKE, E. Differential_e_%c_hungen LSs.ungs-__hod___en und Lsungen, Chelsea, New
York, 1959.
LANGER, R.E. On the Zeros of Exponential Sums and Integrals, Bull. Amer. Math.
Soc. 3 (1931), 213-239.