Internat. J. Math. & Math. Sci.
Vol. 9 No. 2 (1986) 365-372
365
ANALYTIC SOLUTIONS OF NONLINEAR
NEUTRAL AND ADVANCED DIFFERENTIAL EQUATIONS
JOSEPH WIENER
DEPARTMENT OF MATHEMATICS
PAN AMERICAN UNIVERSITY
EDINBURG, TEXAS
78539
LOKENATH OEBNATH
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF CENTRAL FLORIDA
ORLANDO, FLORIDA 32816
and
S.M. SHAH
DEPARTMENT OF MATHEMATICS
UNIVERSITY OF KENTUCKY
LEXINGTON, KENTUCKY 40506
(Received May 15, 1985)
A study is made of local existence and uniqueness theorems for analytic
ABSTRACT.
solutions of nonlinear differential equations of neutral and advanced types.
results are of special interest for advanced eauations whose solutions,
lose
their
of
margin
Furthermore,
smoothness.
existence
of
entire
These
in general,
solutions
is
established for linear advanced differential systems with polynomial coefficients.
KEY WORDS AND PHRASES.
Existence and uniqueness of holomorphic solutions, Nonlinear
neutral and advanced differential equations.
1980 AMS SUBJECT CLASSIFICATION CODES.
I.
34AI0, 34A34.
INTRODUCTION.
The well-known Izumi theorem [I] states that if in the equation
w
(z) + a (z )w
b(z), X(z)
ai(z)
for
(n)
IzlJ
I,
closed disk
are
(n-I)
(4 (z)) +...+ a (z)w( (z))
regular
n
in
the
disk
Izl
there exists a unique solution with
IzlJ I.
and
he
(L.1)
b(z)
n
given
hi(0)
0,
w(i)(0)
IXI(Z)I<
I,
regular in the
In a recent paper, Cooke and Wiener [2] have generalized this
result for linear neutral equations with infinitely many arguments.
Pelyukh [3] has
studied some nonlinear neutral equations of the Izumi type.
This paper is concerned with the study of local existence and uniqueness theorems
for analytic solutions of nonlinear neutral and advanced differential equations.
theorem
provided.
on
entire
solutions
of
linear
advanced
differential
equations
is
A
also
JOSEPH WIENER, LOKENATH DEBNATH and S. M. SHAH
366
2.
HOLOMORPHIC SOLUTIONS OF NONLINEAR NEUTRAL EQUATIONS.
We consider the equation
f(z,w(z), w((z)), w’(X(z))).
w’(z)
If
(z)
z
at
has a fixed point
in
0
assume z
O,
0
Wo, Wo,
for the unknown value
w’(O).
w’(O))
(li)
w
O.
(2.2)
Assuume for (2.1) the following hypotheses:
Equation (2.1) has a solution
i)
We may always
ordinary differential equations.
0 and prescribe for (2.1) an initial value w(O)
(0)
f(O,
THEOREM I.
for
as
(2.1) gives the equation
in
w’(O)
then an initial value problem for (2.1) can be posed
Zo,
manner
that is,
0
z
Putting
same
the
(2.1)
The
2)
f(z ,w,w ,w
function
R:Izl! to, lw Wol! MO’
is
holomorphlc
O’
w2
M
w
wl
0
w.
w’(O)
wl
in
the
region
M I, where
0+
M
wO
and satisfies a Lipschltz condition
If(z,
L
where
2
<
w,
Wl,
w
I.
(iii) The function k(z)
Then in some disk
is holomorphlc in the disk
Izl
r
(2.1) with the initial values
PROOF.
YI’ Y2 )I
f(z, y,
2)
Izl
r
0
and satisfies in
there exists a unique holomorphic solution of equation
Wo,
w.
We replace (2.1) by the integral equation
z
w(z)
f(s,w(s), w(k(s)), w’(k(s))) as
+
w
0
0
and introduce the operator
z
rg(z)
on the space
w
0
G
+
f(s,g(s), g(l(s)), g’(k(s)))
as
(2.3)
0
of all functions
g(z)
holomorphic in the disk
Izl
r
and satisfy-
ing the conditions.
rM
0
l (z)
r
The value of
r
! O.
r
is to be determined
g’(z)
Since
w
0
M
0
+
Izl
r.
later.
Clearly, the first restriction on
is the derivative of
_<r
in
!r--j"
(g(z)
Wo)
wz,
we have
I ’I0
Taking in hypothesis (il)
M
0
we conclude that the function
f(z,g(z), g(k(z)), g’(k(z)))
is holomorphic (and
r
is
NONLINEAR NEUTRAL AND ADVANCED DIFFERENTIAL EQUATIONS
bounded) in this disk.
M(r)
Let
If(z,g(z),
max
367
g(X(z)),
g’(X(z)))l
Then from (2.3),
We choose
r M(r)
such that
r
<_
r
M0/r0,
that is
<_ M0/r0,
M(r)
Now, we evaluate
which is always possible to do.
In R the function
f
satisfies a Lipschitz condition
If(z,w,w l,w2)
f(z,y,y
with
We next introduce a metric in the space
d(gl,g 2)
(L
0
+ L I) max
G
L
2
<
I.
by the formula
Igl(z) g2(z)
+ L
Then, from (2.3),
ITg l(z) Tg2(z) <_ L0r
J
(L
0
+ L 1)r
max
gl(z)
g2(z)l
(1
g2()l + L2r max
max
and
max
ITg l(z) Tg2(z) <_ r
d(g l,g2 ).
(2.5)
Furthermore,
d
dz
rg (z)
d
z Tg2(z)l If(z’gl(z)’gl (X(z)),
gl’(k(z)))
f(z,g2(z), g2(X(z)), g(X(z))) <
d(g I,
g2
and
max
lz Tgl(z)
Multiplying (2.5) by
d(Tg l,Tg
2) <
(L
d
z Tg2(z) <- d(gl’g2 )"
0
(r(L
+ L I)
0
and (2.6) by
L
(2.6)
2
+ L I) + L 2) d(g l,g2 ).
and adding yields
JOSEPH WIENER, LOKENATH DEBNATH and S. M. SHAH
368
Finally, the condition
<
r
L2)/(L0
(I
+ L
I)
is a contraction of the space
shows that T
G
This proves the theorem.
into itself.
HOLOMORPHIC SOLUTIONS OF NONLINEAR ADVANCED EQUATIONS.
3.
The equation (see Shah and Wiener [4])
a0w(kz)
w’(z)
of considerable
is
z
>
0
lose
alzw’(kz)
+
a2z2w’’(kz)
If the coefficients are real and
interest.
is of advanced
it
general,
+
0
<
k
<
I,
then for
Furthermore, it appears that advanced equations, in
type.
smoothness, and the method of successive integration
their margin of
shows that after several steps to the right from the initial interval the solution may
not even exist.
Nonetheless, (3.1) admits analytic solutions.
w(0)
then the Inltial-value problem
a.I
with complex constants
and
<
141
<
I,
0
has a unique holomorphic solution, and it is an
In fact, substituting the series
entire function of zero order.
w(z)
in
Namely, if 0
for the complex differential equation (3.1)
w
r. w z
n
n=O
(3.1) yields
Z
n=0
(n+l)Wn+iZ
n
ao An wnzn + n=O (n+l)alxn w n+l
E
n=0
+ Z
(n+2) (n+l)
n=O
a2Xnwn+2Z n+2
and
(a0 kn
(n+l)Wn+l
nal n-I
+
From here, it follows that for large
+
n(n-l)a2%
n-2.
w
n
n’
>_
O.
n,
lWn+I/Wnl <-- cqn
with some constant
c
and
q
<
I.
A nonlinear analogue of (3.1) is the equation
w’(z)
w(O)
THEOREM 2.
:
and
(z)
I,:z) _<
n
fCz,w(z),w(,(z)), zw’(,(z))
,z w
(n)
(3.2)
(),(z))),
w
0.
Assume that the function
.,
f(z,w,wk,wl,w2,...,Wn)
-ol <-’o, I,,,., I<_ c.-,.
zl <_ o, Iw- ol-.< o, Iw,is holomorphic in the disk
Izl
<_
r
0
is holomorphic in
..... o.
and satisfies in it the inequality
NONLINEAR NEUTRAL AND ADVANCED DIFFERENTIAL EQUATIONS
zl
Then in some disk
(3.2).
<__
369
there exists a unique holomorphic solution of problem
r
Replace (3.2) by the integral equation
PROOF.
w(z)
w
i f(s,w(s), w(k(s)), sw’(k(s))
+
0
s
nn (k(s)))
ds
0
and introduce the operator
z
Th(s)
w
0
+
f(s,h(s)) ds
(3.3)
0
whe re
g(z)
H of all functions
on the space
sng(n)(i(s))),
(g(s), g(A(s)), sg’(i(s))
h(s)
zl
holomorphlc in the disk
<
and satisfy-
r
ing the conditions
g(0)
w
0,
w01 < m, m_< Mi/i!
Ig(z)
The first restriction on
g(z)
of the functlon
Iz ig(i) (z)
for
M
zl <-- r.
max
<
i! m
r
Wn)
__<
g(1)(z)
(i
.....
is the derivative of order i
f(z,h(z))
n)
is holomorphlc in this disk.
Let
Then, from (3.3)
Mr,
i
)dz
R.
in
Mr
such that
d
Th<z))
Since
< Mi,
Therefore, the function
and we choose
x
< r 0.
r
is
n).
0
we have
w0,
If(z,w,wi,w
ITh(z) w01
d
r
(i
< M0,
that is
i-I
)zi, f(z,h<z))
<
r
<
M0/M.
Furthermore,
(i-l)’M
r
i>
i-I
and
z
i d
i
-7
dz
Th<z)l-<
The requirement (i-l)!Mr
tion
f
_<
M
i
r
gives
<
M.1
i-I !M
for
<
satisfies a Lipschitz condition
f(’’’wx’wl
Wn)
Yn )1
f("Y’Y)t’Yl
n
i=l
We introduce a metric in the space
d(hl,h 2)
(L
0
+ L x) max
H
by the formula
Igl(z) g2(z)
n
+
Z
i=l
Then, from (3.3)
Li
max
(z
g
(z)-g
i
<
n.
In
R
the func-
JOSEPH WIENER, LOKENATH DEBNATH and S. M. SHAH
370
IThl(z) Th2(z)
max
! rd(hl,h2)"
(3.4)
Furthermore,
i
i d
max
dz
<
r
Th2(z))
i-I
(f(z,h (z))
dzi-I
(i-l)!
i
r
<
i d
Iz
max
(Thl(Z)
i
d(hl,h2)
(i-l)! r
(L
Multiplying (3.4) by
f(z,h
f(z,h I(z))
max
i-
+
0
h2(z)))
f(z
2(z))l
,n).
(i=l
(3.5)
and (3.5) by
Lk)
L
and adding all inequalities yields
i
n
d(rh l,rh
2) <_
[r(e
+
0
Z (i-l)! L i] r
+
El)
i=l
d(hl,h2).
Finally, if
n
(L
then
T
0
+
I (i-I)! L
+
Lk)
i=l
<
i
(3.6)
1,
H
is a contraction of the space
into itself, which proves the existence and
uniqueness for (3.2).
Theorem 2 holds true if on the right of (3.2) the terms
REMARKS.
are changed to
4.
z
kj w( j
(X(z)),
>
with k.
3
zJw (j)
j.
ENTIRE SOLUTIONS OF LINEAR SYSTEMS.
We are concerned with the equation
M
W’(z)
P
in which
N
l
l
i=0 j=0
ij(z)
THEOREM 3.
Pi0(z)
W(z)
and
P
E
k=O
the complex numbers
n
are d x d
Assume that
>
ij
(n+l) E
are nonsingular for all
I, p
>
satisfy
N
M
E
r.
i=O j=l
n
>
O,
W(0)
W
0
matrices.
are polynomials of degree not exceeding
Pij(z)
P
E
Pio kzk Pij (z)
(j
B
Pij(z)W(J)(xijz),
kffij-i
Pijk z
k
I),
N
0
<
Xijl
(n+l)!
(n-j+l)!
where
<
n-j+l
Xij
E
and the matrices
Pij,j-I
is the identity matrix.
Then (4.1) has a unique holomorphic solution
p:
(4.2)
NONLINEAR NEUTRAL AND ADVANCED DIFFERENTIAL EQUATIONS
r.
W(z)
371
W z
n--O
and it is an entire function of zero order.
From (4.3) and (4.2) we obtain
PROOF.
w (j)(z)
(n+j)’
n!
r.
n=O
(n+j)
n!
w(J)(ij n=OZ
Pij (z)W(J)(xij z
z
y.
z
n
Since (4.2) implies
from
n
Xl] Wn+jZ
P
E
k=0
Pij k z
O,
Pijs
P
XIj
(n-s)
for
k
n
m
E
m=O
ij
s
M
Z
(n+l)Wn+l
N
n-j+1
Z
m
sWn-s+j
_<
the index
j-2,
k=s-j+l
(n-k+l)
(n-k-j+l)
Z
k=O
i=0 j=O
Xij Wm+jZ
m’.
Hence, the substitution
n.
to
j-I
n
n-s
Z
s=O
n=O
Wn+jZ
s
in the last sum extends
leads to the equation
Pij ,k+j-I Wn-k+l.
lj
From here,
Wn+1 B-n
Let
llWnll
Cn,
ME NZ
n-j+IE
i=O jr0
k=l
M
>
for k
0
Pijk
p.
N
E
p-j+l
j=0
k=l
z
and for large values of
<
q
Pij ,k+j-IWn-k+l
n >0.
<
I.
kij
n-k-j+l
II PiJ ,k+J-I
Cn-k+l
Furthermore,
< nj
we have
n
(n-k+
(n-k-j+l)!
ijl
ij
(n-k+l),
(n-k-j+l)’
(n-k+l)
(n-k-j+
where
kn-k-j+
then
E
Cn+ --< iiB_-lll if0
since
(n-k+
(n-k-j+l)
Xijl
n-k-J+ <
q
n
Also,
M
N
if0 j=
n!XT;J+l
(n-j+l) Pij
,j-I
)-l’
and for large n,
11"7,’11 _< i.l/(n+l),
with some constant
Therefore,
n+l
-< C
q
p+l
l
k=l
Cn-k+l’
C
COnSt.
(4.4)
JOSEPH WIENER, LOKENATH DEBNATH and S. M. SHAH
372
Denote
0
_< k_<
for large n, it follows
<
Hence, starting with some natural number m,
that
Mn
max
Ck,
n,
then
Cn+l --< C(p+I)qnM"n
Since
C(p+l)q
M
n
n
Cn+
<
M
n
and
M
n+l
M
n
n>m.
=M
m
Successively applying this result to (4.4) yields
Cm+ < C(p+l)q Mm,
Cm+ 2 < C(p+l)qm+IMm+l < C2(p+l)2qmqm+l
Cm+n
< cn(p+l)n q m q m+l q m+2 ...qm+n-I
< cn(p+l)n
q
n(n-l)/2
This estimate for the coefficients
REMARKS.
The
strict
M
M
M
m
m
W
concludes the proof.
IXIjl
inequalities
<
cannot
be
replaced
by
Indeed, the scalar equation
w’(z)
is of type
w(z) + (2z
(4.1),
w(z)
w
I.
with
0
e
has a singularity at
z
2)
w’(z), w(O)
w
0
However, its solution
z/(1-z)
z
I.
REFERENCES
1.
IZUMI, S., On the Theory of the Linear
2.
(1929), 10-18.
COOKE, K. L. and WIENER, J., Distributional and Analytic Solutions of Functional
Differential Equations, J. Math. Anal. Appl. 98 (1984), 111-129.
Functional
Differential Equations, Tohoku
Math. J. 30
3.
PELYUKH, G. P., Existence and Uniqueness of C r+l -Solutlons to Nonlinear Differential-Functional Equations of Neutral Type, Differential-Functlonal and Difference Equations, Kiev, 1981, 57-64 (In Russian).
4.
SHAH, S. M. and WIENER, J., Existence Theorems for Analytic Solutions of Differential and Functional Differential Equations, Indian J. Pure and Appl. Math., to
appear, (1986).