611
Internat. J. Math. & Math. Scl.
VOL. 13 NO. 3 (1990) 611-616
REMARKS ON QUADRATIC EQUATIONS IN BANACH SPACE
IOANNIS K. ARGYROS
Department of Mathematical Sciences
New Mexico State University
Las Cruces, NM 88003
(Received May 12, 1988 and in revised form May 29, 1989)
ABSTRACT.
We find new sufficient conditions
for
the uniqueness
of
solutions
of
quadratic and in general multillnear equations in a Banach space X, by assuming the
existence of a certain limit of linear operators in a suitable subspace of X.
I.
INTRODUCTION.
Consider the quadratic equation
(1.1)
y + S(x, x)
x
in a
Banach space X, where y e X is fixed and B is a billnear operator on X [I],
[2].
We can assume without loss of generality that B is symmetric, otherwise we can
replace B in (I.I) by
B(x, y)
=
(B(x, y) + B(y, x)) for all x, y e X
which is a symmetric billnear operator on X and bounded if B is.
Using some ideas given in
[3], [4], and [5] for the
linear case, we derive some
new existence and uniqueness results for the solutions x of (I.1) slmilar to the ones
in [4] for the llnear case and different than the results in [6].
Moreover, under the assumption that for all Zl, z2...,
lira
exists, where B(z
i)
B(Zn)(X)
B(Zl)B(z 2)
denotes a llnear operator on X for i
B(z i)(x)
B(zl,
Zn,
x)
B(x, z i),
we give more insight into the behavior of equation (1.1).
x e X
(1.2)
1,2,
,n such that
I. K. ARGYROS
612
Finally,
be obvious from the proofs of the theorems
will
it
obtained here can be easily generalized
that
the
include multil[near equations
to
results
[2], [7],
[8].
Let z., z,..., z be fixed in X. If the limit in (1.2) exists as
(1.2)
defines
a
linear
operator
L, depending on
% then
n
of
Zl,...,z
on
X.
the
choice
given by
L(x)
lim
n
B(Zl)B(z2)...B(z n )(x).
(1.3)
We denote the domain of L by D(L) and the null space of L by N(L).
By x e D(L) or x e N(L) we mean
L(x) exits or L(x)
0
Note that N(L)
where L is given by (1.3).
,
N(L).
since 0 E
MAIN RESULTS.
The following is essentially Rall’s theorem [6] proved in a different way which
allows us to associate it with (1.3).
THEOREM
all z
E
If
the
linear
operator
B(z)
has
no
nonzero
fixed
point
for
X then (I.I) has at most one solution.
PROOF.
of
2.1.
If (I.I) has no solution there is nothing to prove.
Let x be a solution
(I.I), then any other solution x can be given by
x + h, for some h e X.
x
We now have by (I.I),
x + h
y + B(x + h, x + h)
or
B(z)(h)
By hypothesis h
0.
h, z
2x + h.
Therefore, the solution x is unique.
(I.I) has a solution x then the totality of the solutions is
given by x
x + h where h is a fixed point of the linear operator B(z) with
z
2x + h. The solution x is unique if h
O. Every fixed point of the operator
B(z) is a fixed point of the operator
If
the equation
L(x)
lim
(B(z))n(x),
with z
2
0 if X
N(L).
+ h
n/
The equation
L(h)
h shows that h
613
QUADRATIC EQUATIONS IN BANACH SPACE
Thus, the solution x is unique in this case.
L the condition X
Therefore, for this particular choice of
N(L) is a sufficient condition for the uniqueness of x.
The assumptions that X is indeed a Banah space was not used in the derivation of
It will be used, however, in the following results.
the above results.
We now prove Thoerems 2.2-2.5 which are generalizations of the thoerems in [3],
[5] and of the remarks -5 in [4].
THEOREM 2.2.
Let w be a solution of (I.I).
Define the sequence {x
n
}, n
1,2,
by
y +
Xn
Then
{Xn},
n
I, 2
x
0
B(Xn_ I, Xn_ I)
-w
eD(L),
v
n
z
n-I
+ w, n
I, 2, 3,
B(Xn-l’ Xn-l)) (y + B(w,
B(Xn_l, Xn_ I) B(w, w)
B(Xn_ + w, Xn_ w)
B(Xn_ + w)B(Xn_ 2 + w, Xn_2-w)
(Y +
--B(z )S(
n
Xk_l + w,
k
(2. I)
e X.
We have
Xn
where z
0
converges to some point v e X f and only if
where L is given by (1.3) for z
PROOF.
for some x
k
S(z )(x 0 -w)
Zn-I
I, 2,
w))
n.
hand side of the above exists, therefore
Now if x 0 -w e D(L)the limit on the right
converges to some point
{Xn}, n=l,2,
veX.
Conversely, if {xn }, n
x0
I, 2,
converges to some v e X as n
then
w e D(L)
Sometimes this weaker condition sufficies, as indicatted next:
THEOREM 2.3.
that x 0
1,2
Let x 0 be an approximate solution of equation (I.I) in the sense
B(xo, 0) y e N(L), where L is given by (1.3) with z k Xk_ + xk, k
and the x’s are given by (2.1).
n
x
Then every limit point of the sequence {x }, n
n
(1.1).
PROOF.
As in Theorem 2.2,
Xn+
Xn B(Zn)B(Zn-l)’’’B(Zl)(X0 Xl
2,
satisfies equation
I.K. ARGYROS
614
where z
k
Xk_ + Xk,
Now
x for n
if x
n
follows if we let n
I, 2,
k
in
i
n
n.
,
i
then
n
,
i
result
the
and
B(Xn, Xn )"
y +
Xn+
x for n
n+l
Furthermore we can show:
If x e D(L) is a solution of
THEOREM 2.4.
in D(L) where L is given by (1.3) with z
PROOF.
y + B(x, x) and w
Since x
B(x + w) and hence w
x e D(L).
(l.l),
then every other solution w is
x + w, n
0, I, 2,
y + B(w, w) then w
x) + x
(w
Therefore w
E
x is a fixed point of
D(L).
The followlng result complements Theorem 2.2.
Equation (I.I) has a solution x E D(L) where L is as in Theorem 2.2
THEOREM 2.5.
if and only if the sequence {t
s
o
{s o + s
n
+
+ s
n
converges, where
y,
k-!
I
Sk
PROOF.
BCs j’ Sk- ),
3-’-I
0
The given sequence {t
0
O,
2
n
n
I, 2,
corresponds to
{Xn+
with x
0
0.
X then x is a solution of (1.1). By Theorem 2.2.,
0 e D(L), the converse follows from
x).
Now since x 0
If it converges to some x
x E D(L) (for x
k
w
Theorem 2.2.
3.
APPLICATIONS.
We now complete this paper with an example.
Let X =IR 2 and consider the bilinear
operator on X given by
S(x,y)
’(x
x
2)
Yl
0
y
Xl
xlY
xlY
x2Y 2
x2Y 2
x
x
x
2
x
2
2
+ x 22
x
2
2
y
y
]
Note that B is symmetric bilinear operator on X.
0 + B(x, x)
x
or
x
Y2
-x2
+
x
-I
0
We consider the equation
615
QUADRATIC EQUATIONS IN BANACH SPACE
Define a norm on
m2
by
z
Let z
then
+
zl
xlY
z!
xly|
x2Y 2
B(z)B(x) (y)
ZlXlY
+
ZlXlY
+
z
If we apply the above for
g
+
x2Y 2
Z2XlY
+
Z2XlY
ZlX2Y 2
z2x2Y2]
z2x2Y2J
i
i
i-- 1,
zi=
will be of the form
ZlX2Y 2
]
2,
S(Zn)(X)
B(Zl)B(z 2)
n, then
i
i
2
n
k
k k
k.l ClC2
.
V
2n
kj
where
dk’m ckm
o
e
{x,
k
dd 2"
.d
k
2n+l
I, 2,
zl},
us .t=
C2n+l
I, 2,
n, k
1/41 [0, 1/41, hen
(1/4)n+l 2n+l 0 as n
2
n,
m
I, 2,
x o 0,
ilVnll , 2n
;
therefore
L(x)
so X
0 for all x e [0,
]
x
[0,
]
--X
N(L) and equation (3.1) has a unique solution in X, namely x
0
2
n+l.
I.K. ARGYROS
616
REFERENCES
I.
2.
3.
RALL, LB., Quadratic Equations in Banach Space, Rend. Circ. Math. Palermo IO
(1961), 314-332.
RALL, L.B., Solution of Abstract Polynomial Equations by Iterative Methods, MRC
Technical Report #892 Madison, Wisconsin 1968.
BROWDER, F.E. and PETRYSHYN, W.V., The Solution by Iteration of Linear
Functional Equations in Banach Spaces., Bull. Amer. Math. Soc. 72 (1966),
566-570.
4.
KWON, Y.K. and REDHEFFER, R.M., Remarks on Linear Equations in Banach Space,
5.
REDHEFFER, R.M., Remarks on a Paper of Taussky, Journal of Algebra 2 (1965), 42-
6.
7.
RALL, L.B., On the Uniqueness of Solutions of Quadratic Equations, SlAM Reviews
11(3) (1969), 386-388.
RALL, LB., Computational Solution of Nonlinear Operator Equations, John Wiley
8.
RALL, L.B.,
Arch. Rational Mech. Anal. 32 (Iq69), 247-254.
47.
Publ., 1969.
Nonlinear Functional Analysis and Applications, Academic
Press, 1971.