Internat. J. Math. & Math.
(1987)69-78
Sci.
69
Vol. l0 No.
ON POLYNOMIAL EQUATIONS IN BANACH SPACE,
PERTURBATION TECHNIQUES AND APPLICATIONS
IOANNIS K. ARGYROS
Department of Mathematic
The University of Iowa
Iowa City, Iowa 522h2
(Received November 21, 1985 and in revised form May 23 1986)
ABSTRACT.
We use perturbation techniques to solve the polynom/al equation in Banach
Our techniques provide more accurate information on the location of solutions
space.
and yield existence and uniqueness in cases not covered before.
An example is given
to justify our method.
KEY WORDS AND PKRASES.
Perturbation, contraction.
46B15,65
1980AMSSUBJECT CLASSIFICATION CODE(S).
i.
INTRODUCTION.
-
In this paper we use perturbation techniques to find solutions of the abstract
polynomial equation of degree
x
in a Banach space
Obviously
X
(i.i)
(x)
k,
x+_
over the field
F
+---
(.)
+MIX+M0
of real or complex numbers.
is a natural generalization of the scalar polynomial equation of
the first kind to the more abstract setting of a Banach space.
The case
case
k a 2.
similar to
k
2
has been examined in
[i], [7], [8].
Here we investigate the
The principal new idea in this paper is the introduction of an equation
(i.i),
z
Fk(Z)
+N2 z2+Nlz+M0.
Nkzk+Nk_Izk-l+
The’results are then obtained under suitable choices of the
Our method is a generalization of the one’s discussed
namely, the method of successive substitutions and
N ’s,
P
[8], [9]
Newton’s method.
(1.2)
p
1,2,---,k.
by L. B. Rall,
It always provides
a more accurate information on the location of the solutions and it also yields exis-
tence and uniqueness results for (i.i) in the cases not covered before.
and
A2
M2-I
our results coincide with theorems in
[8], [9],
For
z
able to provide more accurate information on the location of the solutions.
In order
to justify this, in Part 2 of our paper we compare our results with the results in
[9], [i0]
equation
using as an example a special case of
(i.i), namely
0
but even then we are
[8],
the famous Chandrasekhar’s
70
I.K. ARGYROS
C[0,13
X
for
d
reav
(1.3) have
(1.3) as
k
0
Stronger resets for
0.5.
[3], [6]
been obtained in
exple to justi o method.
DEFINITION i.
L(X,Y)
Denote by
ear operators from a linear space
X
linear operator from
Y
into
M
X
from
k
the linear space over the field
X
into
Y
and
Y,
will be a point of
x
NOTATION i.
operates on
L(X,Y)
Given a
(il,i2,.-.,ik)
the
.
For example, if a k-linear
Xl,X2,..-,xk E X are given, then
X
Y.
i
from
X
1,2,...,k,
into
Y
DEFINITION 2.
A
E
where
k-linear operator
the
(k-l)-
X
into
the notation
Y
(i)
and a permutation
can be used for
such that
i
k-linear operators
k’.
Xl,
The order of operation is impor-
if
x.
x,x2,..-,x
operates on
and so on.
()Sx...
Thus, there are
operator
L(X)
of the integers
M]
e-eX.
(i)
from
associated with a given
X
into
Y
k-linear
is said to be symmetric
if
for all
l, 2,
i
Rk,
denotes the set of all permutations of the integers
The symmetric
,k.
k-linear operator
=..
is called the mean of
NOTATION 2.
p
Y.
E
k,
used.
If
For
p
Mk-
The notation
L(,Y),
for the result of applying
to x E X p-times will be
P
will represent a (k-p)-linear operator from X into
x
p < k,
then
k,
note that
%x
for all
M2,---,
each
M.I
i
X
XlX2---xk
x2,
by
of the lin-
Y.
Y.
into
k-linear operator
k-linear operator from
ll
of
the convention being that
1
Finally, denote
tant.
X
points
k
z
linear operator
L(}-I,Y)
into the space
F
For k
2,3,--. a
(k-1)-linear operators from
into a linear space
k-linear operator from
is called
operator
or
fo= of
In this paper we Just use
and elsewhere.
BASIC CONCEPTS AND THEOREMS.
2.
i
gener
even more
%x (i)
E Rk, x X. It follows from (2.1) that the multilinear operators
in (1.1) may be assumed to be symmetric without loss of generality, since
in (1.1) may be replaced by Mi, i
2,3,.--,k, without changing the value
71
POLYNOMIAL EQUATIONS IN BANACH SPACE
Yk(x’.
of
Oniess tne contrary is exolicitly staked, ne mttlilinear operators
will be assumed o be symmetric.
2,3,.-.,k
X,Y
Assume from now on tha%
are Banach spaces.
A linear operaor
DIN!TiON 3.
L
fILl!
is finite.
IILII
Tne quantity
DEFINITION 4.
For
X
from
a
k-linear operator
(k-l)-linear operators from X
of
defined by
L(X,
L-(-’is-l,y)).
ae
NOTATION 3.
(2.2),
L(,Y).
p
and
k
the Banach
norm)
The bound (or
X
k-linear operatoms from
Y
into
x+ _-
operators from
X
DEFINITION 6.
+
on B
by
into
Y.
Let
z
qk (r
i
X
L(,Y)
+ %_ +
Pk
positive solutions
q (r) < i, r
Ix X llx-zll
PROOF.
s
(Sl,S2).
r],
where
Claim I.
Pk
are bounded multilinear
q(r)
of degree
qk
r.
qk(r)-r
has two
0
or none.
& s
2
I
Assume that
THEOREM i.
o’
(r+ llz ll)-
Note that by Descartes rule of signs [5] the equatio=
of degree
is bounded.
r+"
L’ llz II)-llz
Y
into
and define the polynomial
i
)-z
X
from
1,2,-..,k
From now on we assume
be fixed in
Pk
+--- +
Mi,
is said to be bounded if its coefficients
q(r)
Then
r
maps
Pk
has two positive solutions
s
I
< s
2
has a unique fixed point in the ball
such that
(z,r)
(s,s) (Sl,S2).
U(z,r)
into
U(z,r).
l(x-zll l(x)-(z)+(z)-zll l%(-(z)ll+ ll(z)-ll
I% I Ira2 ,(/ ,z )/...+ I% ,( (+ ,z ,)-+ ( llz ,)- ,z II+-- ,z if
.+
+
or
q(r)
(Note
0
Pk
-
Sr
,P(z)-z,
which is true by hypothesis.
that claim i is true even if
Claim 2.
in claim 1,
will be
Note that by Definitions (3) and () if
An abstract polynomial operator
defined by
(x)
k
is said
then
DEFINITION 5.
k
Y
into
L(xk-I,Y),
into
Y.
into
L.
X
from
being considered to be an element of
with
space of bounded
denoted henceforth by
(2.2)
X
to be bounded if it is a bounded linear operator from
space of bounded
is said to be bounded if
(or norm) of
is called the bound
k 2 2,
.
lill
u
Y
into
s
I
s
2
and
is a contraction operator on
r
[Sl,S2]).
U(z,r).
IF
Xl,X2
U(z,r),
then as
72
I.K. ARGYROS
bui
ql (r)
<
by hypothesis.
The result now follows from the contraction mapping principle.
DINITION 7.
%(
Define he polynomial
I1-%_
it;
I- llz IF
+
q}:(r
of degree
k
on
]+
I%-i li. iz !# + %_ (r) + I1. (z)-. It.
+--- +
Note that
z-’
I#o + .." "
(’-)--.
/
IIMo-o " "-5_" "
11.-5. I1"
if
z
*
"-:
I1- I1"
i1"-I1"
I1". "
"
+
(F.
z )-z
" II- I1" I1"
(1.2).
is a fixed point of
The proof of the following theorem follows from Theorem 1 and the above observation.
THEOREM 2. Suppose that there exists a solution z satisfying (1.2) and that
(r)
k(r)-r
point in
has two positive solutions
U(z,r),
where
(s,s)
r
APPLICATIONS.
From now on we assume that
THEOREM 3.
k
s
< s
I
and
1
0.
Then Theorem 2 becomes
has a unique fixed
Consider the equation
M +
z
Suppose that there exists a solution
0
2
N2z
satisfying (3.i) and
z
ilzll < [,/I1!1 (JIl-ll
(I)
Then
(sl,s 2).
2
Pk
2.
+,,/i111):]--.
Then the equation
x
has a unique solution
x
U(z,a),
Mo+M2x
2
(3.2)
where
1
U(z,b)
(II) Moreover, x
b
where
[1-211"2 !1" IIz i1-1:(211"2 I1" IIz I1-- )2_t, i1"2-N2 11{2 Ii" Iiz I12 ]/2:1 (211"2 II)
-.
In practice, an exact solution of the auxiliary equation (3.i) can seldom be
The following theorem, whose proof is similar to that of Theorem 2, guaran-
obtained.
tees that the original equation (3.2) has a solution even when we can only find an
.
approximate solution of
THEOREM
Let z
(3.1).
be fixed in
X
and set
1
_ [_
I1 z2’ o-z I!" IIz
b
IIM-N iiz I1 + IIz
iil
Ii/
POLYNOMIAL EQUATIONS IN BANACH SPACE
73
then
(I)
(II)
equation
(3.2)
U(z,b).
this solution actually lies in
Let us define the operator on
PRO0Y.
T(x)
Claim i.
T
maps
U(z,a)-
has a unique solution zn
U(z,r)
into
M
2:
0
by
+M2 x2.
U(z,r)
for
[b,a).
r
If
x
U(z,r)
then
M2 x2+ M0-z
T(x)- z
(M2-N2)(x-z+z)2+ N2(x-z )2 +2N2z(x-z)+ (N2z 2 +M0-z).
(:11- i!+i1 I1:)
which is true for
T
Claim 2.
r
(iiz IIi1- It/ I1 !1" ii’. I1-) +
+
I1- I!" IIz
-
iiz Ir
0
U(z,r)
then
+
[a,b).
U(z,r).
is a contraction operator on
If
w,v
()-(v)i
% M2 (w-z+v-z+Rz) (w-v)
2(r+llz|)ll!w-vM2
So
T
U(z,r)
is a contraction on
0
for
< r <
a.
Because Theorem h relies on the contraction mapping principle, it actually pro-
(3.2),
vides an iteration procedure for solving
x
0
z
REMARK 1.
converges for any
x
(z,b)
0
progression with quotient
M +
0
(+
z-
-
to the solution
By Theorem 3 we have
COROLLARY i.
M2x
(1 I1" liz II-z)
q
n
1,2,--.
n
i, 2,
x
of
(3.2) at
the rate of a geometric
i1- i111 !1. Iiz 112] /.
IIz !i) I
(111" ii- I1-) -1il-I- II. I1. I1] a/.
Under the hypotheses of Theorem 3, the solution
Theorem 3 satisfies
PROOF.
2
0+M2xn
The iteration
Xn+I
PROOF,
and
M
Xn+I
namely, set
By Theorem 3,
x
obtained in
I.K. ARGYROS
74
so that
coo.u.z
(I)
(II)
IIx
.
or
,.
equation
(3.2)
moreover,
x
Apply
-
"11"011 < z,
11"11
U(MO,a), here
x
-11"1111"o
1111
t
has a ique solution
U(M0,b)
b
PROOF.
uo
X
*c
ii:o i/
eorem
where
I1" I1o I!- -11 I1" "o
11
3 with M2
0
We now state R’s theorem for comparison.
TOR 5.
(I)
If
equation
}M.Mo<I
(3.2)
M
z
d
O.
e
proof c be fod in
[8], [].
then
has a solution
X
x
satising
2 11 !111"o
1111
(II) moreover,
PROPOSITION 1.
x
que
is
in
U(x,R),
here
Asse"
theses of eor 3, 5 e ’satisfied;
(11-12-)) zi l o1 > o.
(I)
(z)
the
+
PROOF.
By Theorem 3,
x-zil
b
so
llx!
b+ll
By Theorem 5,
x
so it is enough to show
or
-
(II.IFII.-. il)11 zll IIz + !i’11 > o
and the result follows from
REMARK 2.
(a)
(II).
If the evaluation of
we can look for a
z
II M2-N2
such that:
in Theorem 3 is difficult, then
POLYNOMIAL EQUATIONS IN BANACH SPACE
75
2J 1t%1 (J IP’-2-21 + 4 It% Ii)
and sar the iteration wiT.,h
z
z"
O
we can apply the theorem in the bali
(b}
z-211%1!.
U(z,a’).
(211%!!. .ll-z)2 1!%1i +1!
211.21
.1i
,
)1i
provided that the quantity under the radical is nonnegative.
b
a’
< a,
EXAMPLE I.
a
and
?/
Also note that sice
c
U(z,a’
c
U(z,a).
For the equation of Chandrasekhar,
l+x(s)01 s x(t )dr,
s
x(s)
X
2
a’ <
we have
U(z,b)
we have
b
where
C[0,1]
Q(x)
x(s)
(s)
X
X
defined by
x(t)dt
M :X X X
2
is quadratic since the symmetric bilinear operator
M2(x,y)
Q
The operator
with the sup-norm.
(3.3)
y(t)dt+y(s)
X
defined by
x(t)dt
satisfies
M2(x,x)
We will prove that the norm
IIM211
Q(x)
for all
in 2.
Now
s- dt
X.
x
in 2
and since always
Q
we obtain
in 2 g
11"2
I1" It-
The proof will be completed if we prove that
I1%11
But by the definition of
in 2.
"2,
ilM2
x
T dt
S
=in2
SO
I1"
in 2.
We now apply Theorem 5 and Corollary 2 to (3.3) with
5, equation (3.3) has a unique solution in
R
provided that
i-
4X
in 2
> 0,
i.e.,
U(x,R),
J!-4l
B= XM
2.
According to Theorem
where
in 2
2k in 2
k <
.36067"’’.
According to Corollary 2,
I.K. ARGYROS
76
equation
(3.3,
has a uniaue solution
wner6
l-2X in
2kin 2
(II)
moreover,
U(l,a),
x
l-hl
in 2
> O,
2-9/i-X
l-2k in
a
provided that
where
in 2
2k in 2
i.e.,
k < .36067"".
One can now see by comparing the above that Corollary 2 under the same condition
on
gives a better information on the location of the solution than Theorem
5 [8],
[].
of
Our next goal is to use Theorem h to obtain solutions of (3.3) for a wider range
k. It is not necessary to assume B has any connection with Chandrasekhar’s
equation in Proposition 2 or 3.
PROPOSITION 2.
If
z
E X
is a solution of the equation
+
z
kMe(z,z],..
satisfying
then for
where
c.
[,
I1 IIz II(--x I1 II-IIz II)]-
the conclusions of Theorem 3 for the equation
x
M +
0
klM2(x,x)
hold.
PROOF.
To apply Thoerem 3 we need
I111 <
i:J.. I111 (dl.-hl Iil + Jh I111)]-
since
we have
or by taking the square root of both sides of the above inequality and using
h<
(e
Ie I1"11
II) -x
we get
e
=t =o
If
z
ono
is not
(-) <
oe tm t
i=at zo
IIz II-
exact solution of the quaatic eqtion
z
M
0
+
2z,z),
then we can use the following generalization of Proposition
POLYNOMIAL EQUATIONS IN BANACH SPACE
Le
PROPOSITIOn(
I.
C
z
be flxed in
i
> C
},
and
77
and
k
Tnen for any
satisfying
k < k_ <
M
x
has a unique solution in
U(z,a),
C
the equation
O/XIM2(x,x)
and in fact this solution lies in
U(z,b).
Here
i
I
PROOF.
Similar to Proposition 2.
REMARK 3.
According to Corollary 2 or Theorem
5
and the discussion following
Example i, Chandrasekhar’ s equation
z(s)
has a solution
z
l+kM2(z(s),z(s))
provided that
k <
l+ksz(S)o s+t
.36067376"’’.
the iteration suggested in Remark 1 for a sttitable
range of
l
.424059379"’’.
until
(3
dt
But now using Proposition 3 and
x
0
ZN(I
we can extend the
Here are some characteristic values for
norm of the corresponding approximate solution
ZN(1)
and
C
l
the
I().
llzN()ll
1.44474532
.38h363732
1.53401867
39512252
.399942101
.40524h331
.4
1.5981923
1.6833661
420163281
.k23
1.6961/4492k
.423011429
.42h
1.7005561
1.70073716
.42400047
35
.38
1.558263525
39
.4
.42h059378
.424059379
1.700973721
.424059379
.424059379
Note that the above results coincide at least at six decimal places with the ones
obtained in
[2], [3]
and
[I0].
REFERENCES
i.
Quadratic equations in B..aaach space perturbation techniques and
applications to Chandrasekhar’s and related equations, Ph.D. dissertation,
University of Georgia, Athens, 1984.
ARGYROS, I.K.
I.K. ARGYROS
78
2.
ARGYRO2.. i.}[., On s contraction theorem ana app!icationz, Proc.
Amer. Math. Soc.
3.
CHANDRASEKHAE, S., Radiative transfer, Dover Publ., New York, 1960.
DAVIS, H.T. Introduction to nonlinear differential and intemrai equations, Dover
4.
SL.
Pure Math.,
Publ., New York,
6.
DICKSOE, L.E., New first course in the theory of equations, John Wiley and Sons,
New ork, 1939.
KELLEY, C.., Solution of the Chandrasekhar H-equation by New-ton’ s method,
7.
J. Math. Pvs. 21 (7) (1980), 1625-162E.
MACFARIJLND, J.E., An iterative solution of the quadratic equation in Banach space,
8.
R/ILL, L.B., Quadratic equations in Banach space, Rend. Circ. Mat. Palermo
5.
Trans. Amer. Math. Soc. (1958), 82h-830.
10(1961),
31h-332.
9.
I0.
RALL, L.B., Solution of abstract polynomial equations by iterative methods. Mathematics Research Center, United States Army, The University of Wisconsin
Technical Report 892 (1968).
STIBBS, D.W.N. and WEIR, R.E., On the H-functions for isotropic scattering,
Monthly Not. Ro. Astron. Soc. 119(1959), 512-525.