CONTRACTORS AND FIXED POINTS
l
inf
~0, AcN(~,B)
and
H(A,B)=
co,
otherwise,
A,BcCL(X).
BcN(E,A}, if the infimum exists,
The function H is called the generalized Hausdorff distance for CL(X)
induced by the norm of X.
D(x,A) = inf{ II x-a II : acA}.
The above theorem holds for nonlinear multivalued operators which are
closed in the following sense:
P:D(P)cX
~
CL(Y), X, Y being Banach spaces, is closed on D(P}, if
xn~x,
YnEPXn and
Yn~Y
imply that XED(P) and yEPX.
9. DIRECTIONAL CONTRACTORS. Let P:D(P}cX be a nonlinear operator from a
Banach space X into a Banach space Y, D(P) being a vector space. Let
Oql be a given constant and let
r(x):v~x
be a bounded linear operator
associated with XED(P). Suppose that for every YEY, there exists a
positive
E=E(x,y}~l
such that
(9.1) IIP(x+Ef(x)y} - Px- EYII
~
qEII Yll.
then r(x) is said to be a directional contractor for P at XED(P).
It follows from this definition that r(x}y=O implies y=O, i.e.,
r(x) is one-to-one.
For operators P=I-F, where Y=X and I is the identity mapping of X,
it is convenient to have contractors r in the form I+r(x). Then the
contractor inequality (9.1) becomes
(9. 2)
II
F(x+Er(x)y) - Fx - Ef(x)y
II~
qEIIY II
for all yEY and some E=E(x,y) with
OE~l.
The following fixed point theorem holds.
THEOREM A[lJ. A closed nonlinear operator F:D(F}cX
~X
which has a bounded
directional contractor r, i.e., satisfying relation (9.2) and
llr(x)
II~
B for all XED(F)
7
and some constant BO, has a fixed point x, i.e. , x=Fx. Moreover, I-F is
a mapping onto X.
This theorem also
ge~eralizes
the well-known Banach fixed point theorem.
In fact, if F:X
~X
is a contraction with Lipschitz constant ql, then
I+r(x) with r(x)=O is obviously a bounded contractor and this notion is
much stronger than a directional contractor. However, since the hypotheses of
the above fixed point theorem are rather weak, we cannot prove the existence
of the inverse mapping of I-F. This theorem results from a more general
Previous Page Next Page