OPERATORS, PERTURBATIONS OF NORMALLY SOLVABLE NONLINEAR

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Internat. J. Math. & Math. Sci.
Vol. 8 No. 2 (1985) 241-246
241
PERTURBATIONS OF NORMALLY SOLVABLE
NONLINEAR OPERATORS,
WILLIAM O. RAY and ANITA M. WALKER
Department of Mathematics
The University of Oklahoma
Norman, OK
73019
(Received January 16, 1984)
ABSTRACT.
X
Let
Y
and
F +
F and
be Banach spaces and let
X
differentiable mappings from
Y
to
be Gateaux
In this note we study when the operator
of a surjective
is surjective for sufficiently small perturbations
F
operator
The methods extend previous results in the area of normal solva-
bility for nonlinear operators.
Solvab2llty, normal solvabit2ty, inverse function theorems,
Theorem, contractor detions.
KEY WORDS AND PHRASES.
Ekgand’ s
AMS/MOS Subject classification.
Let
We call
F be a nonlinear mapping from a Banach space X to a Banach space Y
F Gateaux differentiable if for each x X there is an operator F’ (x),
not necessarily linear, mapping
lim
X
h
X
to
=(x)
F(x+th)
This operator
set of adjoint operators
F
F’(x)*
Y
for which
F’(x)(h)
t
t-O
for all
47H15.
is
nory
solvable
if the injectivity of the
implies surjectivity of
F
In this light, the theory of normally solvable operators has been investigated
by, among others, S.I. Phohzhayev [i], F.E. Browder [2], [3], [4] and [5], W.A.
Kirk and J. Caristi
[6], D. Downing
and W.A. Kirk
and W.O. Ray [9] and J. Kolomy [i0].
each of the operators
F’ (x)*
that each
=(x)*
[7], M. Altman [8], W.J. Cramer
Throughout these works the injectivity of
has been implicit assumption.
If it is assumed also
has a bounded inverse, then similar surjectivity results are
obtained by weakening an assumption on the closure (or weak enclosure) of
When
F’ (x)*
equation
has a bounded inverse,
F(x)
0
Newton’s method
F(X)
can be employed to solve the
if the bounded inverse assumption is removed, J. Moser [ii]
Newton’s method
(See also W.O. Ray [12]).
showed that
can be modified to still yield a solution of
F(x)
In this note we avoid the explicit use of adjoint operators; instead we con-
F which we perturb
Under the assumptions we make
sider a uniformly surjective Gateaux differentiable operator
by a "small" Gateaux differentiable operator
on
6’ (x)
we demonstrate, via a transfinite
the operator
F +
Hence
Newton’s
Newton’s method, the
surjectivity of
method is sufficiently regular that small
O.
242
W. O. RAY AND A. M. WALKER
perturbations of a differentiable operator do not hinder the convergence of the
iterates.
As the primary tool in deriving our main result, we make use of an extension
of Caristi’s Theorem formulated in [13]:
(M,d)
Let
THEOREM i.
M
_
Let
g
(0,)
and let
be
be a continuous nonx
M
0
be fixed.
If
g
into itself which satisfies
c(d(x,Xo))d(x,g(x))
then
[0, (R))
c
fc(u)du
increasing function for which
is a mapping of
M[O,(R))
be a complete metric space and let
a lower semicontinuous function.
(g(x))
(x)
()
(x E M)
M
has a fixed point in
Our main result is the following theorem, the proof of which is similar to
those found in the earlier paper [13].
X
Let
and
Y
be Banach spaces and let
differentiable mappings from
X
to
THEOREM 2.
’(x)
Y
Let
c
fc(u)du
non-increasing function for which
F and
[0,(R))
be Gateaux
(0,)
be a continuous
Suppose for each
X
is a bounded linear operator from
to
x
X
that
Y
(i)
and
B(O;c([[x[[))
F’(x)(B(O;I))
Suppose, in addition, for some
(II[I)-II
P
If the mapping
(x)l[
F +
m
..
E (0,i)
(2)
and each
has closed graph, then
X
x
P
that
X
is an open mapping from
Y
onto
IIF’(x)-lll
(Note that if F’(x) is linear, invertible and if
c(llxll)
is
then (2)
automatically fulfilled.) As a simple consequence it is enough to
assume that
F and
each have closed graph.
F
Let
THEOREM 3.
differentiable operator
P
The proof that P
graph, then so does
be given as in Theorem 2.
and
P
XY
by
P
in particular, P
F +
If
F
Define the Gateaux
have closed
and
is an open mapping from
X
onto
Y
is open in Theorems 2 and 3 follows readily from a direct
application of Theorem 2.1 of [9].
Instrumental in verifying both the surjectivity
and openness conclusions above are a pair of
"contractor inequalities" (cf. [8]).
These inequalities are given by:
LEMMA.
P
by
a
t
F +
(0,i]
F and
Let
be given as in Theorem 2.
Then there is a
E X
and an
[[P
Px
(,I)
q
Let
such that for each
XY
y
be defined
Y
there is
such that
t(y-Px)[[
qt[ly
Pxl[
(4)
and
[IX
x[[)
c(l[xll)-itlly
pxll
(5)
Similar inequalities can be deduced from injectivity hypotheses
on
(see the survey [14] for a discussion). It is noteworthy that the
uniform surJectivity hypothesis (2) is somewhat weaker than the corresponding
assumptions on the
adjoint P’(x)*
PERTURBATIONS OF NORMALLY SOLVABLE NONLINEAR OPERATORS, I
When specialized to the case that the function
and
Y
be Banach spaces and let
differentiable mappings from
X
to
Let
THEOREM 4.
e’(x)
is a constant, the
I. Rosenholtz and W.O. Ray [15].
above extends a result of
X
c(u)
243
Y
e
F and
x E X
Suppose for each
X
is a bounded and linear operator from
be Gateaux
that
(6)
Y
to
and
F’(x)(B(O;l))
B(O;5)
E (0,i)
Suppose, in addition, for some
(7)
5 > 0
for some
that
6-111G ’(x)
If the operator
particular, if
(8)
P F + e has closed graph, then P is an open surjection. In
F and e each have closed graph, then P is an open surjection.
We now present the proofs of the above theorems; we first verify the Lemma and
then use it to prove our main results.
PROOF OF LEMMA.
then choose
Y
y
Fix
(,i)
q
and
and the conclusions follow for any
x
y
So without loss of generality we may assume
c(llxll)llY-
PxlI-I(y-Px)
F’(x)(w)
c(llxll)llY
F’(x)
geneity of
6
(0,i]
B(O;c(llxll))
F
Set
e
and
Px
y
for some
x
t
P(X)
c(llxll)-llly
h
F’(x)(h)
implies that
P(x)
For each
By (2) there exists a
Pxll-l(y-Px)
By hypothesis both
t 6
y
If
with
Pxllw
llhll
X
x
w E B(O;l)
so that
Then the homo-
c(llxll)-lllY Pxll
are Gateaux differentiable, so we can choose
so small that
llF(x+th)
tF’ (x) (h)II
F(x)
(q-)tIly
Pxll
(9)
and
]Je(x+th)
e(x)
<i(q-)tllY
(10)
and combining (9) and (i0) yields, via the triangle and (3),
x+th
Setting
te’ (x)
P(x)ll
that
+tllF’ (x) (h)
(y-Px)ll + tile’ (x) (h)ll
1/2(q-)tlly
i
Pxll + (q-)tllY
Pxll
+ 0 + tile’ (x) ll-llhll
-< (q-)tllY Pxll + t’c(llxll)c(llxll)-lllY
qtll xl1
-<
xll
giving (4).
II xll
To derive (5) observe that
tllhll
c(llxll)-itllY Pxll
We now prove
Theorem 2.
PROOF.
We begin by demonstrating the surjectivity of
Define a metric
Since
p
X
on
P has closed graph, (x,p)
Fix
y
Y
and
is continuous from
q
(X,p)
(,i)
to
p(x,y)
by
max{(l+q)llx
P
yll
f + 6
c(o)-lll Px
is a complete metric space.
Set
[0, )
@(x)
since
(l+q)(l-q)-lily- Pxll
P has closed graph.
Then
PylI}.
W. O. RAY AND A. M. WALKER
244
We proceed by contradiction, so suppose y
g
X-X by g(x)
it follows that
-
fining
llP(g(x))
Px
t(y-Px)ll
and
qtlly
P(x)
Then by the Lemma, de-
Pxll
(ii)
c(llxll)Ilg(x) xll -< tllY PxII
Note that since y # Px
it follows that g(x) # x for
(12)
each
X
x
Applying
the triangle inequality to (ii) gives
llP(g(x))
Pxll -< (l+q)tlly
Pxll
(13)
A second application of the triangle inequality gives
IIP(g(x))
(l-t)IPx
y
yll -< qtlly
xll
or
Pxll -< (l-q)-l(IIx
tllY
Yll
llP(g(x))
Yl])
(14)
Now, (13) and (14) together imply
llP(g(x))
Pxll -<
(l+q) (l-q) -I (llPx
(x)
yll
llP(g(x))
(g(x))
(15)
Also (12) and (14) imply
(l+q)c (llxll) llg (x)
(l+q) (l-q) -I x (llx(x)
(g(x))
Now let
Then
x
llxll -< (l+q)llxll
llxll
c(0)-lllpx
either case, c(p(x,0)) _<
Now, if
P(0)I
c(llxll
(16)
Consider first if
c((l+q)llxll)
implies that
c(0)-lllpx
_<
]l(g(x))
(l+q)llxll
c(llxll) since c is nonincreasing.
(and thus (l+q)llxll _< c(0)-lllpx
P(0)I
P(0)II)
c(c(0)-lllpx P(0)II) -< c(llxll) Hence in
X
be fixed in
p(x,0)
Likewise, if
then
0
0
Yll-
so
p(x,g(x))= (l+q)llx- g(x)II
p(x,O)
_<
then (16) implies
c(p(x,0))p(x,g(x)) _< c(llxll)(l+q)llx
_< (x)
(g(x))
so (*) holds, while if
p(x,g(x))
c(p(x,0))p(x,g(x))
_<
c(0)-iIIpx- P(g(x))ll
c(llxll)c(0)-lllpx
-< Ii’x
_<
(x)
then (15) implies
(g (x))II
(g(x))
so again (*) holds.
Thus Theorem I implies
y
P(X)
and
P
g
has a fixed point, a contradiction.
Hence
is surjective.
Now, to show P is open as well, fix w X and let 6 > 0
It suffices
B(Pw;) c_ (B(w;6)) for a sufficiently small choice of
So let y
then fly- Pwll <B(Pw;)
Define a mapping
Y
B(w;6)
to show that
by
(x)
Theorem 2.
We will show 0
(B(w;6)) thereby completing the proof of
We accomplish this by applying Theorem 2.1 of [9].
y- Px
By hypothesis, for
q
for which (4) and (5) hold.
(,i)
fixed, there is a
Hence (4) implies
t
(0,I]
and an
X
PERTURBATIONS OF NORMALLY SOLVABLE NONLINEAR OPERATORS, I
[[P- (l-t)xll
245
P-
fly-
-< qtllY
qllII
while (5) implies
xll -< c(llxll) -ltlly
I1
-tllxll
=(llxll)
Now observe that
llxll-<
yields that
c(llxl[)
B(w;6)
x
+ ]lwll
6
(17)
II wll
implies
-<
Applying the function
so
6.
c
IIII -< llx wll
c(6+llwl[) -<
llxll
gives that
So (17) becomes
I1
c(/llwll)-tllxll
xll -<
(IS)
In order to apply Theorem 2.1 of [9] we must verify that
(l-q)
-I Fa
s-iB(s)ds
M
c(6+]lw]])
0 < e <
-I
B(s)
[0, )-[0,)
a,B
for appropriate choices of
and
s
6(l-q)c(6+llwll)e q-I
(19)
_<
and
If we choose
a
c(6+llwll)-lllwlle l-q
a
6
a
and if
then (19) follows:
a
(l-q)-if0 s-iB(s)ds (l-q)-ifoa ds
(l-q) -1 a
(l_q)-lc(6+llwll)-lllxlle l-q
(l-q)-lc(6+llwll) -1 e l-q
(l-q)-ic (6+l]w]])-16 (l-q) c (6+]]wl]) eq-le l-q
_<
_<
=6
x
B(w;6)
0
as required.
Hence Theorem 2.1 applies to give the existence of an
Px 0
0
Px 0
y
Thus
and
y 6 P(b(x;6))
for which
Theorem 3 is an easy consequence of the Mean Value Theorem.
PROOF.
{xn}
Let
lim Fx +
n
nN > 0 so that
y
xn
{Xn}
[Ixn
implies
Value Theorem of Mcleod [16"1 to
(Xm)
for which
Xn-X,
> 0
is Cauchy, for every
m,n >_ N
(Xn)
X
be a sequence in
Since
Xmll
E
-ic(0) -I
and
PXn -y
there exists an
Applying the Mean
yields that
E -{’(txn +
(l-t)Xm)(Xn
m)
x
0 < t < i}
from which it follows, via (3), that
llxn xmll
-< llxn
-<
llxn-
xmllsup{ll’ (tXn
xmllsup{c(lltXn
_<
IIxn
xmll.c(O)
+
+
(l-t)xmll
(1-t)xmll)
0 < t < I}
0 < t < i)
< E
if
m,n >_ N
Hence
converges.
{xn
is Cauchy in
By assumption
Y
so the completeness of
has closed graph
and so
lim
n-
Y
xn
implies
x
{Xn}
i.e.
W. O. RAY AND A. M. WALKER
246
Fx,
graph, so
xn y
Fx n +
Since
y
x,
it follows that
Therefore
y
Fx n y
Fx, +
x
x
F has closed
giving P closed
But
Px
graph, as desired.
By Theorem 2, P
is an open surjection.
We conclude by remarking that most of our conclusions are fairly direct consequences of earlier results which have used inequalities (4) and (5) as their main
assumptions.
Theorem 3.2 in
Thus, for example, surjectivity in Theorem 2 is a special case of
[13],
while openness was inferred directly from Theorem 2.1 of [9].
Our main goal here has been to expose a further class of operators to which these
more general results apply.
The inequalities (4) and (5) have come to play a
central role in the theory of normal solvability, and the above results show that
these inequalities and the mapping properties they imply are stable under small
perturbations.
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i.
POHOZHAYEV, S.I.
2.
BROWDER, F.E. Nonlinear operators and nonlinear equations of evolution in
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BROWDER, F.E. Normal solvability and existence theorems for nonlinear mappings
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Roma, Italy, 1971, 19-35.
BROWDER, F.E. Normal solvability for nonlinear mappings and the geometry of
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Italy, 1971, 37-66.
3.
4.
On the normal solvability of nonlinear operators, Do__kl. Akad.
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5.
BROWDER, F.E. Normal solvability for nonlinear mappings in Banach spaces,
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6.
KIRK, W.A., CARISTI, J.
7.
DOWNING, D., KIRK, W.A.
A mapping theorem in metric and Banach spaces,
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A generalization of Caristi’s theorem with applications
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8.
(1977), 339-346.
ALTMAN, M.
Contractors and Contractor Direction: Theory and Applications,
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CRAMER, W. J., RAY, W.O. Solvability of nonlinear operator equations, Pacific
J. Math. 95 (1981), 37-50.
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ii. MOSER, J. A rapidly convergent iteration method and nonlinear differential
equations, Ann. Scuola Norm. Sup.. Pisa__2 (1966), 499-535.
12. RAY, W.O. A rapidly convergent iteration process and Gateaux differentiable
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operators, preprint.
13. RAY, W.O., WALKER, A.M.
Mapping theorems for Gateaux differentiable and
accretive operators, Nonlinear Anal.: TMA 6 (1982), 423-433.
14. RAY, W.O.
Normally solvable nonlinear operators, to appear.
15. ROSENHOLTZ, L., RAY, W.Oo
Mapping theorems for differentiable operators,
Bull. Acad. Polon. Sci. Sr. Sci. Math. 29 (1981), 265-273.
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