Internat. J. Math. & Math. Sci.
Vol. 9 No. 3 (1986) 583-587
583
ITERATIONS CONVERGING TO DISTINCT SOLUTIONS
OF SOME NONLINEAR OPERATOR EQUATIONS IN BANACH SPACE
IOANNIS K. ARGYROS
Department of Mathematics
University of Iowa
Iowa City, IA 52242
(Received April 29, 1985 and in revised form April 18, 1986)
ABSTRACT.
We examine the solvability of multilinear equations of the form
x)
(x,x
-k timeswhere
M
k
X
k-linear operator on a Banach space
is a
KEY WORDS AND PHRASES.
1980
,
y, k
y
and
E X
is fixed.
Multilinear operator, contraation.
AMS SUBJECT CLASSIFICATION CODE.
46B15.
INTRODUCTION.
1.
We study the quadratic equation
B(x,x)
X,
in a Banach space
y
CASE i.
2B(x)
ator
B
is a bounded symmetric bilinear operator on
[2], [3], [7], [9], [i0].
X
is fixed in
where
Let
y
0
and set
is invertible then
x
x
(B())-lB,
and
for some
x
such that the linear oper-
(i.i) becomes
(B(x))-lB(x
y
X
We consider two cases.
h
B(h,h)
where
(.)
y
x)
(1.2)
y
h
and
h
E X
is to be determined
We introduce the iteration
((hn))-l(hn
hn+ I
to find a solution
h
of
(1.2)
)
such that
for some
#
h
h
0
X
x.
It turns out under certain assumptions that iteration (1.3) converges to an
h
E X
such that
CASE 2.
Let
h
# x,
therefore
y # 0,
Xn+ I
x
x
h
is a nonzero solution of
(i.i).
we then introduce the iteration
B(xn )-l (y)
for some
x
0
E X
(1.4)
to find solutions of (i.i).
The results obtained here can be generalized to include multilinear equations of
the form
584
I. K. ARGYROS
y(x,x
x)
y
-k times-
Y
where
X
k-linear operator on
is a
We now state the following lemma.
2.
and
y
X
is fixed in
The proof can be found in
[i0].
[i0].
EXISTENCE THEORY.
LEMMA I.
where
L
Let L
and L
1
2
is invertible, and
I
II (L1
LEMMA 2.
Let
where
(0,r0)
r
PROOF.
#
z
(x)
invertible then
0
be bounded linear operators in a Banach space
IILIII
+ L
2
"IIL211
<
Then
i.
r
X.
Assume that the linear operator
Ix
E U(z,r)
x
E
X
l[(z)-lll] -i.
[fill[
X,
exists, 82nd
l-IlL211 llLl-llI
be fixed in
0
L2 )-I
II L2111
)-i II
is also invertible for all
and
(LI +
(z)
llx-zll
is
< r],
We have
llg(z)-ll
llN(-z)ll
ll-zll
l]gll
llg(
<i
for
r
(0,r0).
and
x
U(z,r).
The result now follows from Lemma i for
DEFINITION i.
-
Assume that the linear operator
P,T
Define the operators
p(x)
and the real polynomials
on
g(x,x) +
f(r), g(r)
U(z,r)
on
B(z)
for some
x, (x)
R
L
r
B(z), L2
I
B(x-z)
is invertible.
0
by
(g(x))-(x-)
by
and
THEOREM i.
Let
z
X
be such that
B(z)
is invertible and that the following
are true"
a)
b)
c)
c
b
> 0;
< 0, b 2
there exists
4ac > O,
r > 0
and
such that
f(r) > 0
and
then the iteration
hn+ i
(h n)-l(hn-y),
n
0,i,2
g(r)
0
NONLINEAR OPERATOR EQUATIONS IN BANACH SPACE
is well defined and it converKes to a unique solution
h
of
585
(1.2)
U(z,r)
in
for
6 U(z,r).
0
PROOF. T is well defined by Lemma 2.
any
h
CLAIJM i.
If
U(z,r)
T
U(z,r)
x
U(z,r).
into
then
(x)-(-)-
T()- z
z
[( -(z))(-z)
(x
(z)]
SO
ll(x)
zli
r
if
(using Lemma 1 for
L
B(z)
1
L
and
B(x-z))
2
or
g(r)
0
which is true by
hypothesis.
CLAIM 2.
If
is a contraction operator on
T
U(z,r)
w,v
U(z,r).
then
ll(w)-l[I B(B(v)-l(v-)
,
B(B(v)-l(v-z)) + [([(v)-l(z-))](w-v)l
+
1
z)-lll + ]!II l[(z).-lll2r
i
ll(z)-lll
r
II[III I (z)-lll
ll(w)-l]I
i
So
T
is a contraction on
whSch is true since
THEOR 2.
of
eorem
then if
[ll(z)-l!2z-y--]]"
[I(
U(z,r)
0 < q
if
< i,
where
f(r) > 0.
Asse that there exist
r
> 0, z,
x
X
satisfying the
i d
(a)
0 <
it’ll
< h0
<
& r +
-i +
llzil,
+
PROOF.
x
x
h
4]
the solution
l;ll
Moreover,
llw_ v
r
<
llll
is a nonzero solution of
By Theorem 1
h 6
U(z,r)
h
if
+
(1.2)
;l z l;.
(i.i).
therefore
is such that
hotheses
586
I. K. ARGYROS
II > II
Assume that
0,1,2
k
for
B(hn+l,hn)
h
By iteration (1.3) we have
n.
y
n
or
SO
to show that
llhn+III >
i111
it suffices to show
which is true by
(b).
For consistency we must have
[l-I[
which is true by
(a).
lieu,
<
n --,
The result now follows by taking the limit as
llhll
Finally note that since
>
llII,
h
x
#
0
therefore
x
x
h
in
(2.1).
is a non-
zero solution of (i.i).
Assume that the linear operator
DEFINITION 2.
z
X.
P
Define the operator
and the real polynomials
U(z,r)
on
P(x)
B(x,x)
f(r), g(r)
on
szr2 sr
f(r)
/
/
for some
y,
R
s3,
y
#
B(z)
r >
is invertible for some
0
by
0
by
g (r)
slr
2 +
s2r
+ s
3
where
IIB(.)- II )e
s -i111" IIB(z 11 2
llBll liB(.)-ll
:L IIBli lIB(z)-Zll
(lIBll
-
The proofs of the following theorems are omitted as similar to Theorems i and 2.
Let
THEOREM 3.
be such that the linear operator
X
z
B(z)
is invertible
and that the following are true"
a)
s>
b)
s
.c)
there exists
2
0;
> 0,
s
2
2
4SlS 2 > 0,
r
>
0
and
such that
f(r) > 0
and
g(r)
0
NONLINEAR OPERATOR EQUATIONS IN BANACH SPACE
587
then the iteration
Xn+ 1
for some
x
0
is unique in
THEOR{
Let
p < q
E X is
U(z,r)
4. Let
n)-l(y)
well defined and it converges to a solution
for any
x
x
of
(l.l)
which
6 U(z,r).
0
be such that the hypotheses of Theorem 3 are satisfied.
z,r
be positive numbers such that
a)
then if
B(x
p
llyll
pqlIBIl
[IXol
q
then the solution
x
of
(l.1)
is such that
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i.
2.
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ARGYROS, I.K. On a Contraction Theorem and Applications, (to appear in "Proceedin6s of Symposia in Pure Mathematics" of the American Mathematical
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Quadratic EQuations in Banach Space Perturbation Techniques and
Applications to Chandrasekhar’s and Related EQuations, Ph.D. dissertation,
University of Georgia, Athens, 1984.
3.
ARGYROS, I.K.
4.
5.
6.
BOWDEN, R.L. and ZWEIFEL, P.F.
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7.
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8.
MULLIKIN, T.W.
9.
CHANDRASEK}{AR, S.
Astrophysics J.
(1976), 219-232.
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Solution of the Chandrasekhar
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H
1960.
Equation by Newton’s Method, J.
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i0.
RALL, L.B.
Computational Solution of Nonlinear
Publ., New York,
ii.
Operator Equations,
John Wiley
1968.
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